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Full text of "Physical Chemistry"

TIGHT BINGING 
BOOK 



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Preface to the First Edition 



This book is an account of physical chemistry designed for stude.its in 
the sciences and in engineering. It should also prove useful to chemists in 
industry who desire a review of the subject. 

The treatment is somewhat more precise than is customary in elementary 
books, and most of the important relationships have been given at least a 
heuristic derivation from fundamental principles. A prerequisite knowledge 
of calculus, college physics, and two years of college chemistry is assumed. 

The difficulty in elementary physical chemistry lies not in the mathematics 
itself, but in the application of simple mathematics to complex physical 
situations. This statement is apt to be small comfort to the beginner, who finds 
in physical chemistry his 'ftrst experience with such applied mathematics. 
The familiar x's and /s of the calculus course are replaced by a bewildering 
array of electrons, energy levels, and probability functions. By the time these 
ingredients are mixed well with a few integration signs, it is not difficult to 
become convinced that one is dealing with an extremely abstruse subject. 
Yet the alternative is to avoid the integration signs and to present a series of 
final equations with little indication of their origins, and such a procedure is 
likely to make physical chemistry not only abstruse but also permanently 
mysterious. The derivations are important because the essence of the subject 
is not in the answers we have today, but in the procedure that must be 
followed to obtain these and tomorrow's answers. The student should try not 
only to remember facts but also to learn methods. 

There is more material included in this book than can profitably be dis- 
cussed in the usual two-semester course. There has been a growing tendency 
to extend the course in basic physical chemistry to three semesters. In our 
own course we do not attempt to cover the material on atomic and nuclear 
physics in formal lectures. These subjects are included in the text because 
many students in chemistry, and most in chemical engineering, do not acquire 
sufficient familiarity with them in their physics courses. Since the treatment 
in these sections is fairly descriptive, they may conveniently be used for 
independent reading. 

In writing a book on as broad a subject as this, the author incurs an 
indebtedness to so many previous workers in the field that proper acknow- 
ledgement becomes impossible. Great assistance was obtained from many 
excellent standard reference works and monographs. 

To my colleagues Hugh M. Hulburt, Keith J. Laidler, and Francis O. 
Rice, I am indebted for many helpful suggestions and comments. The 
skillful work of Lorraine Lawrence, R.S.C.J., in reading both galley and 
page proofs, was an invaluable assistance. I wish to thank the staff of 
Prentice-Hall, Inc. for their understanding cooperation in bringing thfc 



vi PREFACE 

book to press. Last, but by no means least, are the thanks due to my 
wife, Patricia Moore, who undertook many difficult tasks in the preparation 
of the manuscript. 

PREFACE TO THE SECOND EDITION 

In preparing the second edition of this book, numerous corrections of 
details and improvements in presentation have been made in every chapter, 
but the general plan of the book has not been altered. My fellow physical 
chemists have contributed generously of their time and experience, suggesting 
many desirable changes. Special thanks in this regard are due to R. M. Noyes, 
R. E. Powell, A. V. Tobolsky, A. A. Frost, and C. O'Briain. A new chapter 
on photochemistry has been added, and recent advances in nuclear, atomic, 
and molecular structure have been described. 

W. J. MOORE 

Btoomington, Indiana 



Contents 



1. The Description of Physicochemical Systems . . 1 

1. The description of our universe, /. 2. Physical chemistry, /. 3. 
Mechanics: force, 2. 4. Work and energy, .?. 5. Equilibrium, 5. 6. 
The thermal properties of matter, 6. 1. Definition of temperature, 8. 

8. The equation of state, 8. 9. Gas thermometry: the ideal gas, JO. 
10. Relationships of pressure, volume, and temperature, 12. 11. Law 
of corresponding states, 14. 12. Equations of state for gases, 75, 13. 
The critical region, 16. 14. The van der Waals equation and lique- 
faction of gases, 18. 15. Other equations of state,' 19. 16. Heat, 19. 

17. Work in thermodynamic systems, 21. 18. Reversible processes, 
22. 19. Mciximum work, 23. 20. Thermodynamics and thermostatics, 
23. 

2. The First Law of Thermodynamics 27 

*1. The history of the First Law, 27. 2. Formulation of the First Law, 
28. 3? The nature of internal energy, 28. 4. Properties of exact differ- 
entials, 29. 5? Adiabatic and isothermal processes, 30. 6? The heat 
content or enthalpy, 30. 1? Heat capacities, 31. 8. The Joule experi- 
ment, 32. 9. The Joule-Thomson experiment, 33. 10. Application of 
the First Law to ideal gases, 34. 1 1 . Examples of ideal-gas calcula- 
tions, 36. 12. Thermochemistry heats of reaction, 38. 13. Heats of 
formation, 39. 14. Experimental measurements of reaction heats, 40. 
15. Heats of solution, 41. 16. Temperature dependence of reaction 
heats, 43. 17. Chemical affinity, 45. 

3. The Second Law of Thermodynamics .... 48 

1. The efficiency of heat engines, 48. 2. The Carnot cycle, 48. 3. The 
Second Law of Thermodynamics, 51. 4. The thermodynamic temper- 
ature scale, 51. 5. Application to ideal gases, 53. 6. Entropy, 53. 1. 
The inequality of Clausius, 55. 8. Entropy changes in an ideal gas, 55. 

9. Entropy changes in isolated systems, 56- 10. Change of entropy in 
changes of state of aggregation, 58. 1 r? Entropy and equilibrium, 58. 
12. The free energy and work functions, 59. 13. Free energy and 
equilibrium, 61. 14. Pressure dependence of the free energy, 61. 15. 
Temperature dependence of free energy, 62. 1 6. Variation of entropy 
with temperature and pressure, 63. 17. The entropy of mixing, 64. 

18. The calculation of thermodynamic relations, 64. 

4. Thermodynamics and Chemical Equilibrium . . 69 

1. Chemical affinity, 69. 2. Free energy and chemical affinity, 71. 3. 
Free-energy and cell reactions, 72. 4. Standard free energies, 74. 5. 
Free energy and equilibrium constant of ideal gas reactions, 75. 6. 
The measurement of homogeneous gas equilibria, 7? 7. The principle 



viii CONTENTS 

of Le Chatelier, 79. 8. Pressure dependence of equilibrium constant, 
80. 9. Effect of an inert gas on equilibrium, 81. 10. Temperature 
dependence of the equilibrium constant, 83. 11. Equilibrium constants 
from thermal data, 85. 12. The approach to absolute zero, 85. 13. 
The Third Law of Thermodynamics, 87. 14. Third-law entropies, 89. 
15. General theory of chemical equilibrium: the chemical potential, 
91. 16. The fugacity, 93. 17. Use of fugacity in equilibrium calcula- 
tions, 95. 

5. Changes of State 99 

1. Phase equilibria, 99. 2. Components, 99. 3. Degrees of freedom, 
100. 4. Conditions for equilibrium between phases, 101. 5. The phase 
rule, 702. 6. Systems of one component water, 104. 1. The Clapey- 
ron-Clausius equation, 105. 8. Vapor pressure and external pressure, 
107. 9. Experimental measurement of vapor pressure, 108. 10. Solid- 
solid transformations the sulfur system, 109. 11. Enantiotropism 
and monotropism, 111. 12. Second-order transitions, 112. 13. High- 
pressure studies, 112. 

6. Solutions and Phase Equilibria 116 

I. The description of solutions, 116. 2. Partial molar quantities: 
partial molar volume, 116. 3. The determination of partial molar 
quantities, 118. 4. The ideal solution Raoult's Law, 120. 5. Equil- 
ibria in ideal solutions, 722. 6. Henry's Law, 722. 7. Two-component 
systems, 123. 8. Pressure-composition diagrams, 723. 9. Temper- 
ature-composition diagrams, 725. 10. Fractional distillation, 725. 

II. Boiling-point elevation, 126. 12. Solid and liquid phases in equil- 
ibrium, 128. 13. The Distribution Law, 130. 14. Osmotic pressure, 
757. 1 5. Measurement of osmotic pressure, 133. 16. Osmotic pressure 
and vapor pressure, 134\+ 17. Deviations from Raoult's Law, 135. 18. 
Boiling-point diagrams, 736. 19. Partial miscibility, 737. 20. Con- 
densed-liquid systems, 739. 21. Thermodynamics of nonideal solu- 
tions: the activity, 747. 22. Chemical equilibria in nonideal solutions, 
143. 23. Gas-solid equilibria, 144. 24. Equilibrium constant in solid- 
gas reactions, 145. 25. Solid-liquid equilibria: simple eutectic dia- 
grams, 145, 26. Cooling curves, 147. 27. Compound formation, 148. 
28. Solid compounds with incongruent melting points, 149. 29. Solid 
solutions, 750. 30. Limited solid-solid solubility, 757. 31. The iron- 
carbon diagram, 752. 32. Three-component systems, 753. 33. System 
with ternary eutectic, 154. 

1. The Kinetic Theory 160 

1 . The beginning of the atom, 160. 2. The renascence of the atom, 
767. 3, Atoms and molecules, 762. 4. The kinetic theory of heat, 763. 
5. The pressure of a gas, 764. 6. Kinetic energy and temperature, 765. 
7. Molecular speeds, 766. 8. Molecular effusion, 766. 9. Imperfect 
gases van \der Waal's equation, 769. 10. Collisions between mole- 
cules, 777. IK Mean free paths, 772. 12. The viscosity of a gas, 773. 
13. Kinetic thec 4 *y of gas viscosity, 775. 14. Thermal conductivity 



CONTENTS ix 

and diffusion, 777, 15. Avogadro's Number and molecular dimen- 
sions, 178. 16. The softening of the atom, 180. 17. The distribution of 
molecular velocities, 181. 18. The barometric formula, 182. 19. The 
distribution of kinetic energies, 183. 20. Consequences of the distribu- 
tion law, 183. 21. Distribution law in three dimensions, 186. 22. The 
average speed, 187. 23. The equipartition of energy, J88. 24. Rota- 
tion and vibration of diatomic molecules, 189. 26. The equipartjtion 
principle and the heat capacity of gases, 792. 27. Brownian motion, 
193. 28. Thermodynamics and Brownian motion, 194. 29. Entropy 
and probability, 795. 

8. The Structure of the Atom 200 

1. Electricity, 200. 2. Faraday's Laws and electrochemical equiva- 
lents, 201. 3. The development of valence theory, 202. 4. The 
Periodic Law, 204. 5. The discharge of electricity through gases, 205. 

6. The electron, 205. 7. The ratio of charge to mass of the cathode 
particles, 206. 8. The charge of the electron, 209. 9 Radioactivity, 
277. 10. The nuclear atom, 272. 1 1. X-rays and atomic number, 213. 
12. The radioactive disintegration series, 27 3. 13. Isotopes, 216. 14. 
Positive-ray analysis, 216. 15. Mass spectra the Dempster method, 
218. 16. Mass spectra Aston's mass spectrograph, 279. 17. Atomic 
weights and isotopes, 227. 18. Separation of isotopes, 223. 19. 
Heavy hydrogen, 225. 

9. Nuclear Chemistry and Physics 228 

1. Mass and energy, 228. 2. Artificial disintegration of atomic nuclei, 
229. 3. Methods for obtaining nuclear projectiles, 237. 4. The 
photon, 232. 5. The neutron, 234. 6. Positron, meson, neutrino, 2J5. 

7. The structure of the nucleus, 236. 8. Neutrons and nuclei, 238. 9. 
Nuclear reactions, 240. 10. Nuclear fission, 241. 11. The trans- 
uranimTLj^meB4^^2<O. 12, Nuclear chain reactions., 243. I IJEnergy 
production by the stars, 244. 14. Tracers, 245. 15. Nuclear spin, 247. 

10. Particles and Waves 251 

1. The dual nature of light, 257. 2. Periodic and wave motion, 257. 
3. Stationary waves, 253. 4. Interference and diffraction, 255. 5. 
Black-body radiation, 257. 6. Plank's distribution law, 259. 7. Atomic 
spectra, 261. 8. The Bohr theory, 252. 9. Spectra of the alkali metals, 
265. 10. Space quantization, 267. 11. Dissociation as series limit, 

268. 12. The origin of X-ray spectra, 268. 13. Particles and waves, 

269. 14. Electron diffraction, 277. 15. The uncertainty principle, 
272. 16. Waves and the uncertainty principle, 274. 17. Zero-point 
energy, 275. 18. Wave mechanics the Schrodinger equation, 275. 
19. Interpretation of the y) functions, 276. 20. Solution of wave 
equation the particle in a box, 277. 21. The tunnel effect, 279. 22. 
The hydrogen atom, 280. 23. The radial wave functions, 282. 
24. The spinning electron, 284. 25. The PauJi Exclusion Principle, 
285. 26, Structure of the periodic table, 285. 27. Atomic energy 
levels, 287. 



x CONTENTS 

11. The Structure of Molecules ....... 295 

1. The development of valence theory, 295. 2. The ionic bond, 295. 
3. The covalent bond, 297. 4. Calculation of the energy in H-H mole- 
cule, 301. 5. Molecular orbitals, 303. 6. Homonuclear diatomic mole- 
cules, 303. 1. Heteronuclear diatomic molecules, 307. 8. Comparison 
of M.O. and V.B. methods, 307. 9. Directed valence, 308. 10. Non- 
localized molecular orbitals, 310. 11. Resonance between valence- 
bond structures, 311. 12. The hydrogen bond, 313. 13. Pipole_ 
momejj*s7 314. 14. Polarization of dielectrics, 314. 15. The induced 
-^polarization, 316. 16. Determination of the dipole moment, 316. 17. 
Dipole moments and molecular structure, 319. 18. Polarization and 
refractivity, 320. 19. Dipole moments by combining dielectric con- 
stant and refractive index measurements, 321. 20. Magnetism and 
molecular structure, 322. 21. Nuclear paramagnetism, 324. 23. 
Application of Wierl equation to experimental data, 329. 24. Mole- 
cular spectra, 331. 25. Rotational levels far-infrared spectra, 333. 
26. Internuclear distances from rotation spectra, 334. 21. Vibrational 
energy levels, 334. 28. Microwave spectroscopy, 336. 29. Electronic 
band spectra, 337. 30. Color and resonance, 339. 31. Raman 
spectra, 340. 32. Molecular data from spectroscopy, 341. 33. Bond 
energies, 342. 

12. Chemical Statistics v ......... 347 

1. The statistical method, 347. 2. Probability of a distribution, 348. 
3. The Boltzmann distribution, 349. 4. Internal energy and heat 
capacity, 552. 5. Entropy and the Third Law, 352. 6. Free energy and 
pressure, 354. 1. Evaluation of molar partition functions, 354. 8. 
Monatomic gases translational partition function, 356. 9. Diatomic 
molecules rotational partition function, 358. 10. Polyatomic mole- 
cules rotational partition 'function, 359. 1 1 . Vibrational partition 
function, 359. 12. Equilibrium constant for ideal gas reactions, 361. 
\ 3. The heat capacity of gases, 361. 14. The electronic partition func- 
tion, 363. 1 5. Internal rotation, 363. 1 6. The hydrogen molecules, 363. 
17. Quantum statistics, 365. 



13. Crystals <2 .......... 369 

1 . The growth and form of crystals, 369. 2. The crystal systems, 370. 
3. Lattices and crystal structures, 371. 4. Symmetry properties, 372. 5. 
Space groups, 374. 6. X-ray crystallography, 375. 7. The Bragg 
treatment, 376. 8. The structures of NaCl and KC1, 377. 9. The 
powder method, 382. 10. Rotating-crystal method, 383. 11. Crystal- 
structure determinations: the structure factor, 384. 12. Fourier 
syntheses, 387. 13. Neutron diffraction, 389. 14. Closest packing of 
spheres, 390. 15. Binding in crystals, 392. 16. The bond model, 392. 
17. The band model, 395. 18. Semiconductors, 398. 19. Brillouin 
zones, 399. 20. Alloy systems electron compounds, 399. 21. Ionic 
crystals, 401. 22. Coordination polyhedra and Pauling's Rule, 403. 
23. Crystal energy the Born-Haber cycle, 405. 24. Statistical thermo- 
dynamics of crystals: the Einstein model, 406. 25. The Debye model, 
408. 



CONTENTS xi 

14. Liquids 413 

1. The liquid state, 413. 2. Approaches to a theory for liquids, 415. 
3. X-ray diffraction of liquids, 4/5. 4. Results of liquid-structure 
investigations, 417. 5. Liquid crystals, 418. 6. Rubbers, 420. 7. 
Glasses, 422. 8. Melting, 422. 9. Cohesion of liquids the internal 
pressure, 422. 10. Intermolecular forces, 424. 11. Equation of state 
and intermolecular forces, 426. 12. The free volume and holes in 
liquids, 428. 13. The flow of liquids, 430. 14. Theory of viscosity, 431. 

15. Electrochemistry 435 



1. Electrochemistry: coulometers, 435. 2. Conductivity measure- 
ments, 435. 3. Equivalent conductivities, 437. 4. The Arrhenius 
ionization theory, 439. 5. Transport numbers and mobilities, 442. 
6. Measurement of transport numbers Hittorf method, 442. 7. 
Transport numbers moving boundary method, 444. 8. Results of 
transference experiments, 445. 9. Mobilities of hydrogen and 
hydroxyl ions, 447. 10. Diffusion and ionic mobility, 447. 1 1 . A solu- 
tion of the diffusion equation, 448. 12. Failures of the Arrhenius 
theory, 450. 1 3. Activities and standard states, 451. 14. Ion activities, 
454. 15. Activity coefficients from freezing points, 455. 16. Activity 
coefficients from solubilities, 456. 17. Results of activity-coefficient 
measurements, 457. IS^^rfie Debye-Htickel theory, 458. 1 9. Poisson's 
equation, 458. 20. Tne Poisson-Boltzmann equation, 460. 21. The 
Debye-Hiickel limiting law, 462. 22. Advances beyond the Debye- 
Hiickel theory, 465. 23. Theory of conductivity, 466. 21. Acids and 
bases, 469. 25. Dissociation constants of acids and bases, 471. 26. 
Electrode processes: reversible cells, 473. 21. Types of half cells, 474. 
28. Electrochemical cells, 475. 29. The standard emf of cells, 476. 30. 
Standard electrode potentials, 478. 31. Standard free energies and 
entropies of aqueous ions, 481. 32. Measurement of solubility pro- 
ducts, 482. 33. Electrolyte-concentration cells, 482. 34. Electrode- 
concentration cells, 483. 

16. Surface Chemistry 498 

1. Surfaces and colloids, 498. 2. Pressure difference across curved 
surfaces, 500. 4. Maximum bubble pressure, 502. 5. The Du Notiy 
tensiometer, 502. 6. Surface-tension data, 502. 1. The Kelvin equa- 
tion, 504. 8. Thermodynamics of surfaces, 506. 9. The Gibbs adsorp- 
tion isotherm, 507. 10. Insoluble surface films the surface balance, 
508. 11. Equations of state of monolayers, 577. 12. Surface films of 
soluble substances, 572. 13. Adsorption of gases on solids, 572. 14. 
The Langmuir adsorption isotherm, 575. 15. Thermodynamics of the 
adsorption isotherm, 576. 16. Adsorption from solution, 577. 17. 
Ion exchange, 518. 18. Electrical phenomena at interfaces, 579. 19. 
Electrokinetic phenomena, 520. 20. The stability of sols, 522. 

17. Chemical Kinetics 528 

1 . The rate of chemical change, 525. 2. Experimental methods ii| 
kinetics, 529. 3. Order of a reaction, 530. 4. Molecularity of a rcac- 



xii CONTENTS 

tion, 531. 5. The reaction-rate constant, 532. 6. First-order rate 
equations, 533. 1. Second-order rate equations, 534. 8. Third-order 
rate equation, 536. 9. Opposing reactions, 537. 10. Consecutive 
reactions, 559. II. Parallel reactions, 541. 12. Determination of the 
reaction order, 541. 13. Reactions in flow systems, 543. 14. Effect of 
temperature on reaction rate, 546. \ 5. Collision theory of gas reac- 
tions, 547. 16. Collision theory and activation energy, 551. 17. First- 
order reactions and collision theory, 557. 18. Activation in many 
degrees of freedom, 554. 19. Chain reactions: formation of hydrogen 
bromide, 555. 20. Free-radical chains, 557. 21. Branching chains 
explosive reactions, 559. 22. Trimolecular reactions, 562. 23. The 
path of a reaction, and the activated complex, 56 3. 24. The transition- 
state theory, 566. 25. Collision theory and transition-state theory, 
568. 26. The entropy of activation, 569. 27. Theory of unimolecular 
reactions, 570. 28. Reactions in solution, 577. 29. Ionic reactions 
salt effects, 572. 30. Ionic reaction mechanisms, 574. 31. Catalysis, 
575. 32. Homogeneous catalysis, 576. 33. Acid-base catalysis, 577. 
34. General acid-base catalysis, 579. 35. Heterogeneous reactions, 
580. 36. Gas reactions at solid surfaces, 582. 37. Inhibition by pro- 
ducts, 583. 38. Two reactants on a surface, 583. 39. Effect of temper- 
ature on surface reactions, 585. 40. Activated adsorption, 586. 41. 
Poisoning of catalysts, 587. 42. The nature of the catalytic surface, 
588. 43. Enzyme reactions, 589. 

18. Photochemistry and Radiation Chemistry ... 595 

1. Radiation and chemical reactions, 595. 2. Light absorption and 
quantum yield, 595. 3. Primary processes in photochemistry, 597. 
4. Secondary processes in photochemistry: fluorescence, 598. 5. 
Luminescence in solids, 601. 6. Thermoluminescence, 60 3. 7. 
Secondary photochemical processes: initiation of chain reactions, 
604. 8. Flash photolysis, 606. 9. Effects of intermittent light, 607. 
10. Photosynthesis in green plants, 609. 11. The photographic pro- 
cess, 6/7. 12. Primary processes with high-energy radiation, 672. 13. 
Secondary processes in radiation chemistry, 614. 14. Chemical 
effects of nuclear recoil, 6/5. 

Physical Constants and Conversion Factors . . . 618 

Name Index 619 

Subject Index 623 



CHAPTER 1 

The Description of Physicochemical Systems 



J/Thc 



The description of our universe. Since man is a rational being, he has 
always tried to increase his understanding of the world in which he lives. 
This endeavor has taken many forms. The fundamental questions of the end 
and purpose of man's life have been illumined by philosophy and religion. 
The form and structure of life have found expression in art. The nature of 
the physical world as perceived through man's senses has been investigated 
by science. 

The essential components of the scientific method are experiment and 
theory. Experiments are planned observations of the physical world. A theory 
seeks to correlate observables with ideals. These ideals have often taken the 
form of simplified models, based again on everyday experience. We have, for 
example, the little billiard balls of the kinetic theory of gases, the miniature 
hooks and springs of chemical bonds, and the microcosmic solar systems of 
atomic theory. 

As man's investigation of the universe progressed to the almost infinitely 
large distances of interstellar space or to the almost infinitesimal magnitudes 
of atomic structures, it began to be realized that these other worlds could not 
be adequately described in terms of the bricks and mortar and plumbing of 
terrestrial architecture. Thus a straight line might be the shortest distance 
between two points on a blackboard, but not between Sirius and Aldebaran. 
We can ask whether John Doe is in Chicago, but we cannot ask whether 
electron A is at point B. 

Intensive research into the ultimate nature of our universe is thus gradu- 
ally changing the meaning we attach to such words as "explanation" or 
"understanding." Originally they signified a representation of the strange in 
terms of the commonplace; nowadays, scientific explanation tends more to 
be a description of the relatively familiar in terms of the unfamiliar, light in 
terms of photons, matter in terms of waves. Yet, in our search for under- 
standing, we still consider it important to "get a physical picture" of the 
process behind the mathematical treatment of a theory. It is because physical 
science is at a transitional stage in its development that there is an inevitable 
question- as to what sorts of concepts provide the clearest picture. 

v^J/Thysical chemistry. There are therefore probably two equally logical 
approaches to the study of a branch of scientific knowledge such as physical 
chemistry. We may adopt a synthetic approach and, beginning with the 
structure and behavior of matter in its finest known states of subdivision, 
gradually progress from electrons to atoms to molecules to states of 

i 



2 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

aggregation and chemical reactions. Alternatively, we may adopt an analyti- 
cal treatment and, starting with matter or chemicals as we find them in the 
laboratory, gradually work our way back to finer states of subdivision as we 
require them to explain our experimental results. This latter method follows 
more closely the historical development, although a strict adherence to his- 
tory is impossible in a broad subject whose different branches have progressed 
at very different rates. 

Two main problems have occupied most of the efforts of physical chem- 
ists: the question of the position of chemical equilibrium, which is the 
principal problem of chemical thermodynamics; and the question of the 
rate of chemical reactions, which is the field of chemical kinetics. Since these 
problems are ultimately concerned with the interaction of molecules, their 
final solution should be implicit in the mechanics of molecules and molecular 
aggregates. Therefore molecular structure is an important part of physical 
chemistry. The discipline that allows us to bring our knowledge of molecular 
structure to bear on the problems of equilibrium and kinetics is found in the 
study of statistical mechanics. 

We shall begin our introduction to physical chemistry with thermo- 
dynamics, which is based on concepts common to the everyday world of 
sticks and stones. Instead of trying to achieve a completely logical presenta- 
tion, we shall follow quite closely the historical development of the subject, 
since more knowledge can be gained by watching the construction of some- 
thing than by inspecting the polished final product. 

j Mechanics: force. The first thing that may be said of thermodynamics 
is that the word itself is evidently derived from "dynamics,'* which is a 
branch of mechanics dealing with matter in motion. 

Mechanics is still founded on the work of Sir Isaac Newton (1642-1727), 
and usually begins with a statement of the well-known equation 



WIth 



/= ma 
dv 



The equation states the proportionality between a vector quantity f, 
called the force applied to a particle of matter, and the acceleration a of the 
particle, a vector in the same direction, with a proportionality factor w, 
called the mass. A vector is a quantity that has a definite direction as well 
as a definite magnitude. Equation (1.1) may also be written 

f. 



where the product of mass and velocity is called the momentum. 

With the mass in grams, time in seconds, and displacement r in centi- 
meters (COS system), the unit force is the dyne. With mass in kilograms, time 



Sec. 4] PHYSICOCHEMICAL SYSTEMS 3 

in seconds, and displacement in meters (MKS system), the unit force is the 
newton. 

Mass might also be introduced by Newton's "Law of Universal Gravi- 
tation," 

f-'~rf 

which states that there is an attractive force between two masses propor- 
tional to their product and inversely proportional to the square of their 
separation. If this gravitational mass is to be the same as the inertia! mass of 
eq. (1.1), the proportionality constant ft 6.66 x 10~ 8 cm 3 sec"" 2 g" 1 . 

The weight of a body, W, is the force with which it is attracted towards 
the earth, and naturally may vary slightly at various points on the earth's 
surface, owing to the slight variation of r 12 with latitude and elevation, and 
of the effective mass of the earth with subterranean density. Thus 



At New York City, g = 980.267 cm per sec 2 ; at Spitzbergen, g = 982.899; 
at Panama, g = 978.243. 

In practice, the mass of a body is measured by comparing its weight by 
meanspf a balance with that of known standards (mjm 2 = W^W^). 
Jltwo 



and energy. The differential element of work dw done by a force 
/ that moves a particle a distance dr in the direction of the force is defined 
as the product of force and displacement, 

dw-^fdr (1.3) 

For a finite displacement from r Q to r t , and a force that depends only on the 
position r, 

*' = P/(r)dr (1.4) 

Jr 

The integral over distance can be transformed to an integral over time: 



f l dr 

M* 

Jt/ at 



t 
Introducing Newton's Law of Force, eq. (1.1), we obtain 

f'i d*rdr , 
w = I m~~~-dt 
Jt dt 2 dt 

Since (d/dt)(dr/dt)* = 2(dr/dt)d*r/dt 2 , the integral becomes 



w = \rnvf - \rnvf (1.5) 

The kinetic energy is defined by 

E K = JlMP 2 



4 PHYSICOCHEMICAL SYSTEMS [Chap, l 

It is evident from eq. (1.5), therefore, that the work expended equals the 
difference in kinetic energy between the initial and the final states, 

\r)dr = E Kl -E KQ (1.6) 

An example of a force that depends only on position r is the force of 
gravity acting on a body falling in a vacuum; as the body falls from a higher 
to a lower level it gains kinetic energy according to eq. (1.6). Since the force 
is a function only of r, the integral in eq. (1.6) defines another function of r, 
which we may write 

J/(r) dr -= - U(r) 

Or /(/)- -dU/dr (1.7) 

This new function U(r) is called the potential energy. It may be noted that, 
whereas the kinetic energy E K is zero for a body at rest, there is no naturally 
defined zero of potential energy; only differences in potential energy can be 
measured. Sometimes, however, a zero of potential energy is chosen by 
convention', an example is the choice U(r) for the gravitational potential 
energy when two bodies are infinitely far apart. 
Equation (1.6) can now be written 

\ dr - Ufa) - Ufa) - E Kl -- E KQ 

The sum of the potential and the kinetic energies, U + E K , is the total 
mechanical energy of the body, and this sum evidently remains constant 
during the motion. Equation (1.8) has the typical form of an equation of 
conservation. It is a statement of the mechanical principle of the conservation 
of energy. For example, the gain in kinetic energy of a body falling in a 
vacuum is exactly balanced by an equal loss in potential energy. A force that 
can be represented by eq. (1.7) is called a conservative force. 

If a force depends on velocity as well as position, the situation is more 
complex. This would be the case if a body is falling, not in a vacuum, but 
in a viscous fluid like air or water. The higher the velocity, the greater is the 
frictional or viscous resistance opposed to the gravitational force. We can no 
longer write /(r) = dU/dr, and we can no longer obtain an equation such 
as (1.8). The mechanical energy is no longer conserved. 

From the dawn of history it has been known that the frictional dissipation 
of energy is attended by the evolution of something called heat. We shall see 
later how the quantitative study of such processes finally led to the inclusion 
of heat as a form of energy, and hence to a new and broader principle of the 
conservation of energy. 

The unit of work and of energy in the COS system is the erg, which is 
the work done by a force of one dyne acting through a distance of one 
centimeter. Since the erg is a very small unit for large-scale processes, it is 



Sec. 5] 



PHYSICOCHEMICAL SYSTEMS 



often convenient to use a larger unit, the joule, which is the unit of work in 
the MKS system. Thus, 

1 joule = 1 newton meter 10 7 ergs 
The joule is related to the absolute practical electrical units since 

1 joule = 1 volt coulomb 
The unit of power is the watt. 

1 watt = 1 joule per sec = 1 volt coulomb per sec = 1 volt ampere 

<-& Equilibrium. The ordinary subjects for chemical experimentation are 
not individual particles of any sort but more complex systems, which may 
contain solids, liquids, and gases. A system is a part of the world isolated 
from the rest of the world by definite boundaries. The experiments that we 
perform on a system are said to measure its properties, these being the attri- 
butes that enable us to describe it with all requisite completeness. This 
complete description is said to define the state of the system. 




A B c 

Fig. l.la. Illustration of equilibrium. 

The idea of predictability enters here; having once measured the prop- 
erties of a system, we expect to be able to predict the behavior of a second 
system with the same set of properties from our knowledge of the behavior 
of the original. This is, in general, possible only when the system has attained 
a state called equilibrium. A system is said to have attained a state of equi- 
librium when it shows no further tendency to change its properties with time. 

A simple mechanical illustration will clarify the concept of equilibrium. 
Fig. l.la shows three different equilibrium positions of a box resting on a 
table. In both positions A and C the center of gravity of the box is lower 
than in any slightly displaced position, and if the box is tilted slightly it will 
tend to return spontaneously to its original equilibrium position. The gravi- 
tational potential energy of the box in positions A or C is at a minimum, and 
both positions represent stable equilibrium states. Yet it is apparent that 
position C is more stable than position A, and a certain large tilt of A will 
suffice to push it over into C. The position A is therefore said to be in meta- 
stable equilibrium. 

Position B is also an equilibrium position, but it is a state of unstable 
equilibrium, as anyone who has tried to balance a chair on two legs will 



PHYSICOCHEMICAL SYSTEMS 



[Chap. 1 



agree. The center of gravity of the box in B is higher than in any slightly dis- 
placed position, and the tiniest tilt will send the box into either position A 
or C. The potential energy at a position of unstable equilibrium is a maximum, 
and such a position could be realized only in the absence of any disturbing 
forces. 

These relations may be presented in more mathematical form by plotting 
in Fig. l.lb the potential energy of the system as a function of height r of 

the center of gravity. Positions of 
stable equilibrium are seen to be 
minima in the curve, and the posi- 
tion of unstable equilibrium is 
represented by a maximum. Posi- 
tions of stable and unstable equi- 
librium thus alternate in any system. 
For an equilibrium position, the 
slope of the U vs. r curve, dU/dr, 
equals zero and one may write the 
equilibrium condition as 

at constant r (= r ), dU 




ABC 
POSITION OF CENTER OF GRAVITY 
Fig. l.lb. Potential energy diagram. 



Although these considerations have been presented in terms of a simple 
mechanical model, the same kind of principles will be found to apply in the 
more complex physicochemical systems that we shall study. In addition to 
purely mechanical changes, such systems may undergo temperature changes, 
changes of state of aggregation, and chemical reactions. The problem of 
thermodynamics is to discover or invent new functions that will play the role 
in these more general systems that the potential energy plays in mechanics. 
^f. The thermal properties of matter. What variables are necessary in order 
to describe the state of a pure substance ? For simplicity, let us assume that 
the substance is at rest in the absence of gravitational and electromagnetic 
forces. These forces are indeed always present, but their effect is most often 
negligible in systems of purely chemical interest. Furthermore let us assume 
that we are dealing with a fluid or an isotropic solid, and that shear forces 
are absent. 

To make the problem more concrete, let us suppose our substance is a 
flask of water. Now to specify the state of this water we have to describe it 
in unequivocal terms so that, for example, we could write to a fellow scientist 
in Pasadena or Cambridge and say, "I have some water with the following 
properties. . . . You can repeat my experiments exactly if you bring a sample 
of water to these same conditions." First of all we might specify how much 
water we have by naming the mass m of our substance; alternatively we 
could measure the volume K, and the density p. 

Another useful property, the pressure, is defined as the force normal to 
unit area of the boundary of a body (e.g., dynes per square centimeter). In 



Sec. 6] PHYSICOCHEMICAL SYSTEMS 7 

a state of equilibrium the pressure exerted by a body is equal to the pressure 
exerted upon the body by its surroundings. If this external pressure is denoted 
by P ex and the pressure of the substance by P, at equilibrium P = P ex . 

We have now enumerated the following properties: mass, volume, den- 
sity, and pressure (m, K, p, P). These properties are all mechanical in nature; 
they do not take us beyond the realm of ordinary dynamics. How many of 
these properties are really necessary for a complete description? We ob- 
viously must state how much water we are dealing with, so let us choose the 
mass m as our first property. Then if we choose the volume F, we do not 
need the density p, since p ml V. We are left with m, V, and P. Then we 
find experimentally that, as far as mechanics is concerned, if any two of 
these properties are fixed in value, the value of the third is always fixed. For 
a given mass of water at a given pressure, the volume is always the same; or 
if the volume and mass are fixed, we can no longer arbitrarily choose the 
pressure. Only two of the three variables of state are independent variables. 

In what follows we shall assume that a definite mass has been taken 
say one kilogram. Then the pressure and the volume are not independently 
variable in mechanics. The value of the volume is determined by the value 
of the pressure, or vice versa. This dependence can be expressed by saying 
that V is a function of P, which is written 

V=-f(P) or F(P 9 K) = (1.9) 

According to this equation, if the pressure is held constant, the volume of 
our kilogram of water should also remain constant. 

Our specification of the properties of the water has so far been restricted 
to mechanical variables. When we try to verify eq. (1.9), we shall find that 
on some days it appears to hold, but on other days it fails badly. The equation 
fails, for example, when somebody opens a window and lets in a blast of 
cold air, or when somebody lights a hot flame near our equipment. A new 
variable, a thermal variable, has been added to the mechanical ones. If the 
pressure is held constant, the volume of our kilogram of water is greater on 
the hot days than on the cold days. 

The earliest devices for measuring "degrees of hotness" were based on 
exactly this sort of observation of the changes in volume of a liquid. 1 In 
1631, the French physician Jean Rey used a glass bulb and stem partly filled 
with water to follow the progress of fevers in his patients. In 1641, Ferdi- 
nand II, Grand Duke of Tuscany, invented an alcohol-in-glass "thermo- 
scope." Scales were added by marking equal divisions between the volumes 
at "coldest winter cold" and "hottest summer heat." A calibration based on 
two fixed points was introduced in 1688 by Dalence, who chose the melting 
point of snow as 10, and the melting point of butter as +10. In 1694 

1 A detailed historical account is given by D. Roller in No. 3 of the Harvard Case 
Histories in Experimental Science, The Early Development of the Concepts of Temperature 
and Heat (Cambridge, Mass.: Harvard Univ. Press, 1950). 



8 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

Rinaldi took the boiling point of water as the upper fixed point. If one adds 
the requirement that both the melting point of ice and the boiling point of 
water are to be taken at a constant pressure of one atmosphere, the fixed 
points^are precisely defined. 

^Definition of temperature. We have seen how our sensory perception 
of relative "degrees of hotness" came to be roughly correlated with volume 
readings on constant-pressure thermometers. We have not yet demonstrated, 
however, that these readings in fact measure one of the variables that define 
the state of a thermodynamic system. 

Let us consider, for example, two blocks of lead with known masses. 
At equilibrium the state of block I can be specified by the independent 
variables P l and F x . Similarly P 2 and K 2 specify the state of block II. If we 
bring the two blocks together and wait until equilibrium is again attained, 
i.e., until P 19 V l9 P& and K 2 have reached constant values, we shall discover 
as an experimental fact that P l9 V 19 P 2 , and K 2 are no longer all independent. 
They are now connected by a relation, the equilibrium condition, which may 
be written 



Furthermore, it is found experimentally that two bodies separately in 
equilibrium with the same third are also in equilibrium with each other. 
That is, if 



and F(P 29 V 29 P 39 K a ) = 

it necessarily follows that 



It is apparent that these equations can be satisfied if the function F has the 
special form 

K 2 ) - o (i. 10) 



Thus F is the difference of two functions each containing properties pertain- 
ing to one body only. The function /(P, V) defined in this way is called the 
empirical temperature t. This definition of / is sometimes called the Zeroth 
Law of Thermodynamics. From eq. (1.10) the condition for thermal equi- 
librium between two systems is therefore 



It may be noted that, strictly speaking, the temperature is defined only 
for a state of equilibrium. The state of our one kilogram of water, or lead, 
is now specified in terms of three thermodynamic variables, P 9 V, and /, of 
which only two are independent. 

8. The equation of state. The properties of a system may be classified as 
extensive or Intensive. Extensive properties are additive; their value for the 
whole system is equal- to the sum of their values for the individual parts. 



Sec. 8] 



PHYSICOCHEMICAL SYSTEMS 



Sometimes they are called capacity factors. Examples are the volume and 
the mass. Intensive properties, or intensity factors, are not additive. Examples 
are temperature and pressure. The temperature of any small part of a system 
in equilibrium is the same as the temperature of the whole. 

If P and V are chosen as independent variables, the temperature is some 
function of P and V. Thus 

) (1.12) 



For any fixed value of t, this equation defines an isotherm of the body under 
consideration. The state of a body in thermal equilibrium can be fixed by 
specifying any two of the three variables, pressure, volume, and temperature. 





2 4 



6 8 10 12 14 16 16 20 
V-LITERS 



400 800 1200 



07 
06 
05 
0.4 
03 
02 
01 
00 




ISOCHORES 



200 400 600 



Fig. 1.2. Isotherms, isobars, and isochores for one gram of hydrogen. 

The third variable can then be found by solving the equation. Thus, by 
analogy with eq. (1.12) we may have: 

V=f(t,P) (1.13) 

P=--f(t,V) (1.14) 

Equations such as (1.12), (1.13), (1.14) are called equations of state. 

Geometrically considered, the state of a body in equilibrium can be 
represented by a point in the PV plane, and its isotherm by a curve in the 
PV plane connecting points at constant temperature. Alternatively, the state 
can be represented by a point in the Vt plane or the Pt plane, the curves 
connecting equilibrium points in these planes being called the isobars (con- 
stant pressure) and isochores or isometrics (constant volume) respectively. 
Examples of these curves for one gram of hydrogen gas are shown in 
Fig. 1.2. 

We have already seen how eq. (1.12) can be the basis for a quantitative 
measure of temperature. For a liquid-in-glass thermometer, P is constant, 
and the change in volume measures the change in temperature. The Celsius 
(centigrade) calibration calls the melting point of ice at 1 atm pressure 0C, 



10 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

and the boiling point of water at I atm pressure 100C. The reading at other 
temperatures depends on the coefficient of thermal expansion a of the thermo- 
metric fluid, 

1 /9K\ 

(1.15) 



where K is the volume at 0C and at the pressure of the measurements. If a 
is a constant over the temperature range in question, the volume increases 
linearly with temperature: 

K t = K + a/K (1.16) 

This is approximately true for mercury, but may be quite far from true for 
other substances. Thus, although many substances could theoretically be 
used as thermometers, the readings of these various thermometers would in 
general agree only at the two fixed points chosen by convention. 

9. Gas thermometry: the ideal gas. Gases such as hydrogen, nitrogen, 
oxygen, and helium, which are rather difficult to condense to liquids, have 
been found to obey approximately certain simple laws which make them 
especially useful as thermometric fluids. 

In his book, On the Spring of the Air, Robert Boyle 2 reported in 1660 
experiments confirming Torricelli's idea that the barometer was supported 
by the pressure of the air. An alternative theory proposed that the mercury 
column was held up by an invisible rigid thread in its interior. In answering 
this, Boyle placed air in the closed arm of a U-tube, compressed it by adding 
mercury to the other arm, and observed that the volume of gas varied in- 
versely as the pressure. He worked under conditions of practically constant 
temperature. 

Thus, at any constant temperature, he found 

PV = constant (1.17) 

If the gas at constant pressure is used as a thermometer, the volume of the 
gas will be a function of the temperature alone. 

By measuring the volume at 0C and at 100C a mean value of a can be 
calculated from eq. (1.16), 

^100 - W + 1005) or a = 



The measurements on gases published by Joseph Gay-Lussac in 1802, 
extending earlier work by Charles (1787), showed that this value of a was a 
constant for "permanent" gases. Gay-Lussac found (1808) the value to 
be 4 ^. By a much better experimental procedure, Regnault (1847) obtained 
2^3. For every one-degree rise in temperature the fractional increase in the 
gas volume is 3 of the volume at 0C. 

2 Robert Boyle's Experiments in Pneumatics, Harvard Case Histories in Experimental 
Science No. 1 (Cambridge, Mass.: Harvard Univ. Press, 1950) is a delightful account of 
this work. 



Sec. 9] 



PHYSICOCHEMICAL SYSTEMS 



11 



Later and more refined experiments revealed that the closeness with 
which the laws of Boyle and Gay-Lussac are obeyed varies from gas to gas. 
Helium obeys most closely, whereas carbon dioxide, for example, is rela- 
tively disobedient. It has been found that the laws are more nearly obeyed 
the lower the pressure of the gas. 

It is very useful to introduce the concept of an ideal gas, one that follows 
the laws perfectly. The properties of such a gas usually can be obtained by 
extrapolation of values measured with real gases to zero pressure. Examples 

36.75i 



36.70 



36.65 



36.60 




200 



1200 



400 600 800 1000 
PRESSURE - mm of Hg 

Fig. 1.3. Extrapolation of thermal expansion coefficients to zero 
pressure. 

are found in some modern redetermi nations of the coefficient a shown 
plotted in Fig. 1.3. The extrapolated value at zero pressure is 

oc () - 36.608 x 10~ 4 , or l/a --- 273.16 

We may use such carefully measured values to define an ideal gas tem- 
perature scale, by introducing a new temperature, 



=t + = t + (273.16 0.01) 
"o 



(1.18) 



The new temperature T is called the absolute temperature (K); the zero on 
this scale represents the limit of the thermal contraction of an ideal gas. 
From eq. (1.16), 

V T V T 

F- ('-I') 



273.16 



where V is now the volume of gas at 0C and standard atmospheric pressure 
P Q9 and V T j> o is the volume at P Q and any other temperature T. The tempera- 
ture of the ice point on the absolute scale is written as T (273.16). 
Boyle's Law eq. (1.17) states that for a gas at temperature T 



12 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

Combining with eq. (1.19), we obtain 

PV^ f ^T=C-T (1.20) 

M) 

The value of the constant C depends on the amount of gas taken, but for a 
given volume of gas, it is the same for ail ideal gases. Thus for 1 cc of gas 
at 1 atm pressure, PV = 7)273. 

For chemical purposes, the most significant volume is that of a mole of 
gas, a molecular weight in grams. In conformity with the hypothesis of 
Avogadro, this volume is the same for all ideal gases, being 22,414 cc at 0C 
and 1 atm. Per mole, therefore, 

PV=RT (1.21) 

where R = 22,414/273.16 - 82.057 cc atm per C. 
For n moles, 

PY=nRT^^-RT (1.22) 

M 

where m is the mass of gas of molecular weight M. In all future discussions 
the volume V will be taken as the molar volume unless otherwise specified. 

It is often useful to have the gas constant in other units. A pressure of 
1 atm corresponds to 76.00 cm of mercury. A pressure of 1 atm in units of 
dynes cm~ 2 is 76.00 /3 H go where /> Hg is the density of mercury at 0C and 
1 atm, and g Q is the standard gravitational acceleration. Thus 1 atm = 
76.00 x 13.595 x 980.665 =1.0130 X 10 6 dyne cm- 2 . The gas constant 
R - 82.057 x 1.0130 x 10 6 - 8.3144 x 10 7 ergs deg~ l mole- 1 == 8.3144 
joules deg" 1 mole* 1 . 

10. Relationships of pressure, volume, and temperature. The pressure, 
volume, temperature (PVT) relationships for gases, liquids, and solids would 
preferably all be succinctly summarized in the form of equations of state of 
the general form of eqs. (1.12), (1.13), and (1.14). Only in the case of gases 
has there been much progress in the development of these state equations. 
They are obtained not only by correlation of empirical PVT data, but also 
from theoretical considerations based on atomic and molecular structure. 
These theories are farthest advanced for gases, but more recent developments 
in the theory of liquids and solids give promise that suitable state equations 
may eventually be available in these fields also. 

The ideal gas equation PV = RT describes the PVT behavior of real 
gases only to a first approximation. A convenient way of showing the devia- 
tions from ideality is to write for the real gas : 

PV=zRT (1.23) 

The factor z is called the compressibility factor. It is equal to PV/RT. For an 
ideal gas z = 1, and departure from ideality will be measured by the deviation 
of the compressibility factor from unity. The extent of deviations from 



Sec. 10] 



PHYSICOCHEMICAL SYSTEMS 



13 



ideality depends on the temperature and pressure, so z is a function of T 
and P. Some compressibility factor curves are shown in Fig. 1.4; these are 
determined from experimental measurements of the volumes of the gases at 
different pressures. 

Useful PVT data for many substances are contained in the tabulated 
values at different pressures and temperatures of thermal expansion co- 
efficients a [eq. (1.15)] and compressibilities /?. 3 The compressibility* is 
defined by 

1 IAV\ 

(1.24) 



The minus sign is introduced because (3V/dP) T is itself negative, the volume 
decreasing with increasing pressure. 

Z2{ C 2 H4/-N2 

^>CH4 




200 400 600 800 1000 
PRESSURE -ATM 

Fig. 1.4. Compressibility factors at 0C. 



1200 



Since V /(P, T), a differential change in volume can be written 5 : 



For a condition of constant volume, V = constant, dV =- 0, and 

- 



,,,6, 



, v (3K/aP) r - 

3 See, for example, International Critical Tables (New York: McGraw-Hill, 1933); also 
J. H. Perry, ed., Chemical Engineers 9 Handbook (New York: McGraw-Hill, 1950), pp. 200, 
205. 

4 Be careful not to confuse compressibility with compressibility factor. They are two 
distinctly different quantities. 

6 Granville, Smith, Longley, Calculus (Boston: Ginn, 1934), p. 412. 



14 



PHYSICOCHEMICAL SYSTEMS 



[Chap. 1 



Or, from eqs. (1.15) and (1.24), (3Pl3T) r =--- a/0. The variation of P with T 
can therefore readily be calculated if we know a and ft. 

An interesting example is suggested by a common laboratory accident, 
the breaking of a mercury-in-glass thermometer by overheating. If a thermo- 
meter is exactly filled with mercury at 50C, what pressure will be developed 
within the thermometer if it is heated to 52C? For mercury, a 1.8 X 
10~ 4 deg- 1 , p - 3.9 x 10- 6 atm- 1 . Therefore (2P/dT) v -- <x/ft =-- 46 atm per 
deg. For AT 2, A/> =-- 92 atm. It is apparent why even a little overheating 
will break the usual thermometer. 

11. Law of corresponding states. If a gas is cooled to a low enough tem- 
perature and then compressed, it can be liquefied. For each gas there is a 
characteristic temperature above which it cannot be liquefied, no matter how 
great the applied pressure. This temperature is called the critical temperature 
T fy and the pressure that just suffices to liquefy the gas at T c is called the 
critical pressure P c . The volume occupied at T c and P c is the critical volume 
V c . A gas below the critical temperature is often called a vapor. The critical 
constants for various gases are collected in Table 1.1. 

TABLE 1.1 
CRITICAL POINT DATA AND VAN DER WAALS CONSTANTS 



Formula 


T c (K) 


P c (atm) 


V t (cc/mole) 


a (I 2 atm/mole 2 ) 


b (cc/mole) 


He 


5.3 


2.26 


57.6 


0.0341 


23.7 


H 2 | 


i 33.3 


12.8 


65.0 


0.244 


26.6 


N 2 


126.1 


33.5 


90.0 


1.39 


39.1 


CO 


134.0 


35.0 


90.0 


1.49 


39.9 


2 


153.4 


49.7 


74.4 


1.36 


31.8 


C 2 H 4 


282.9 


50.9 


127.5 


4.47 


57.1 


CO 2 


304.2 


73.0 


95.7 


3.59 


42.7 


NH 3 


405.6 


111.5 


72.4 


4.17 


37.1 


H 2 O 


647.2 


217.7 


45.0 


5.46 


30.5 


Hg 


1823.0 


200.0 


45.0 


8.09 


17.0 



The ratios of P, K, and T to the critical values P c , K c , and T c are called 
the reduced pressure, volume, and temperature. These reduced variables may 
be written 

P V T 

p r V T (\ ">7\ 

r n ~ p ' ' it ~ is 9 1 R ~ T \ 1 '^') 

* c Y c 2 c 

To a fairly good approximation, especially at moderate pressures, all 
gases obey the same equation of state when described in terms of the reduced 
variables, P n , V w T R , instead of P, K, T. If two different gases have identical 
values for two reduced variables, they therefore have approximately identical 
values for the third: They are then said to be in corresponding states, and 



Sec. 12] 



PHYSICOCHEMICAL SYSTEMS 



15 



LEGEND 

NITROGEN a N-BUTANE 
METHANE a ISOPENTANE 
ETHANE 
ETHYLENE 



N-HEPTANE 

A CARBON DIOXIDE 

WATER 




23456 

REDUCED PRESSURE, P R 

Fig. 1.5. Compressibility factor as function of reduced state variables. 
[From Gouq-Jen Su, Ind. Eng. Chem., 38, 803 (1946).] 

this approximation is called the Law of Corresponding States. This is equiva- 
lent to Sciying that the compressibility facror z is the same function of the 
reduced variables for all gases. This rule is illustrated in Fig. 1.5 for a number 
of different gases, where z PV/RT is plotted at various reduced tempera- 
tures, against the reduced pressure. 

12. Equations of state for gases. If the equation of state is written in terms 
of reduced variables as F(P& V E } ^= T R , it is evident that it contains at least 
two independent constants, characteristic of the gas in question, for example 
P c and K r . Many equations of state, proposed on semi-empirical grounds, 
serve to represent the PVT data more accurately than does the ideal gas 
equation. Several of the best known of these also contain two added con- 
stants. For example: 



16 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

Equation of van der Waals: 



Equation of Berthelot: 

( P 

Equation of Dieterici: 

P(V - b')e a ' IRTV = RT (1.30) 

Van der Waals' equation provides a reasonably good representation of 
the PVT data of gases in the range of moderate deviations from ideality. 
For example, consider the following values in liter atm of the PV product 
for carbon dioxide at 40C, as observed experimentally and as calculated 
from the van der Waals equation: 

P, atm 1 10 50 100 200 500 1100 

PF, obs. 25.57 24.49 19.00 6.93 10.50 22.00 40.00 

PK,calc. 25.60 24.71 19.75 8.89 14.10 29.70 54.20 

The constants a and b are evaluated by fitting the equation to experimental 
PVT measurements, or more usually from the critical constants of the gas. 
Some values for van der Waals' a and b are included in Table 1.1. Berthelot's 
equation is somewhat better than van der Waals' at pressures not much 
above one atmosphere, and is preferred for general use in this range. 

Equations (1.28), (1.29), and (1.30) are all written for one mole of gas. 
For n moles they become: 



f )(V-nb) = nRT 
n*A' 



P(V-nb')e na ' IRTV ^ nRT 

The way in which the constants in these equations are evaluated from 
critical data will now be described, using the van der Waals equation as an 
example. 

13. The critical region. The behavior of a gas in the neighborhood of its 
critical region was first studied by Thomas Andrews in 1869, in a classic 
series of measurements on carbon dioxide. Results of recent determinations 
of these PV isotherms around the critical temperature of 31.01C are shown 
in Fig. 1.6. 

Consider the isotherm at 30.4, which is below T c . As the vapor is com- 
pressed the PV curve first follows AB, which is approximately a Boyle's law 
isotherm. When the point B is reached, liquid is observed to form by the 
appearance of a meniscus between vapor and liquid. Further compression 



Sec. 13] 
75 



74 



<73 



CO 

UJ 



72 



71 



PHYSICOCHEMICAL SYSTEMS 



17 





31.523' 



31.013' 



VAN OER WAALS 




B 



\ 



\ 



30.409 



32 36 40 44 48 52 56 60 

VOLUME 

Fig. 1.6. Isotherms of carbon dioxide near the critical point. 



64 



then occurs at constant pressure until the point C is reached, at which all 
the vapor has been converted into liquid. The curve CD is the isotherm of 
liquid carbon dioxide, its steepness indicating the low compressibility of the 
liquid. 

As isotherms are taken at successively higher temperatures the points of 
discontinuity B and C are observed to approach each other gradually, until 
at 31.0lC they coalesce, and no gradual formation of a liquid is observable. 
This isotherm corresponds to the critical temperature of carbon dioxide. 
Isotherms above this temperature exhibit no formation of a liquid no matter 
how great the applied pressure. 

Above the critical temperature there is no reason to draw any distinction 



18 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

between liquid and vapor, since there is a complete continuity of states. This 
may be demonstrated by following the path EFGH. The vapor at point E, 
at a temperature below T c , is warmed at constant volume to point /% above 
T c . It is then compressed along the isotherm FG, and finally cooled at constant 
volume along GH. At the point //, below T c , the carbon dioxide exists as a 
liquid, but at no point along this path are two phases, liquid and vapor, 
simultaneously present. One must conclude that the transformation from 
vapor to liquid occurs smoothly and continuously. 

14. The van der Waals equation and liquefaction of gases. The van der 
Waals equation provides a reasonably accurate representation of the PVT 
data of gases under conditions that deviate only moderately from ideality. 
When an attempt is made to apply the equation to gases in states departing 
greatly from ideality, it is found that, although a quantitative representation 
of the data is not obtained, an interesting qualitative picture is still provided. 
Typical of such applications is the example shown in Fig. 1.6, where the 
van der Waals isotherms, drawn as dashed lines, are compared with the 
experimental isotherms for carbon dioxide in the neighborhood of the critical 
point. The van der Waals equation provides an adequate representation of 
the isotherms for the homogeneous vapor and even for the homogeneous 
liquid. 

As might be expected, the equation cannot represent the discontinuities 
arising during liquefaction. Instead of the experimental straight line, it 
exhibits a maximum and a minimum within the two-phase region. We note 
that as the temperature gradually approaches the critical temperature, the 
maximum and the minimum gradually approach each other. At the critical 
point itself they have merged to become a point of inflection in the PKcurve. 
The analytical condition for a maximum is that (OP/OK) and (d 2 P/dV 2 ) 
< 0; for a minimum, (ZPfiV) = and (D 2 />/3K 2 ) > 0. At the point of in- 
flection, both the first and the second derivatives vanish, (DP/3K) 



According to van der Waals' equation, therefore, the following three 
equations must be satisfied simultaneously at the critical point (T = T c , 
V= V n P=-.P t ): 

RT r 

*r 



' V, - b V* 

RT f la 



PP\ 

w) - =-- 



(v t -bp y* 

When these equations are solved for the critical constants we find 



(L31) 



Sec. 15] PHYSICOCHEMICAL SYSTEMS 19 

The values for the van der Waals constants are usually calculated from these 
equations. 

In terms of the reduced variables of state, P Jf , V R , and T ]{ , one obtains 
from eq. (1.31): 



The van der Waals equation then reduces to 



As was pointed out previously, it is evident that a reduced equation of 
state similar to (1.32) can be obtained from any equation of state containing 
no more than two arbitrary constants, such as a and b. The Berthelot equa- 
tion is usually used in the following form, applicable at pressures of the order 
of one atmosphere: 

+ 15 (-)] 

15. Other equations of state. In order to represent the behavior of gases 
with greater accuracy, especially at high pressures or near their condensation 
temperatures, it is necessary to use expressions having more than two adjust- 
able parameters. Typical of such expressions is the very general virial equation 
of Kammerlingh-Onnes: 



4 I f 

t ^ 2 i 3 -t- . . 

The factors B(T) 9 C(T) 9 D(T), etc., are functions of the temperature, called 
the second, third, fourth, etc., virial coefficients. An equation like this, 
though difficult to use, can be extended to as many terms as are needed to 
reproduce the experimental PKTdata with any desired accuracy. 

One of the best of the empirical equations is that proposed by Beattie 
and Bridgeman in 1928. 6 This equation contains five constants in addition 
to R, and fits the PKTdata over a wide range of pressures and temperatures, 
even near the critical point, to within 0.5 per cent. 

16. Heat. The experimental observations that led to the concept of tem- 
perature led also to the concept of heat. Temperature, we recall, has been 
defined only in terms of the equilibrium condition that is reached when two 
bodies are placed in contact. A typical experiment might be the introduction 
of a piece of metal at temperature T 2 into a vessel of water at temperature 7\. 
To simplify the problem, let us assume that: (1) the system is isolated com- 
pletely from its surroundings; (2) the change in temperature of the container 
itself may be neglected; (3) there is no change in the state of aggregation of 
either body, i.e., no melting, vaporization, or the like. The end result is that 

' J. A. Beattie and O. C. Bridgeman, Proc. Am. Acad. Arts Sci., 63, 229-308 (1928). 
J. A. Beattie, Chem. Rev., 44, 141-192 (1949). 



20 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

the entire system finally reaches a new temperature T, somewhere between 
7^ and T 2 . This final temperature depends on certain properties of the water 
and of the metal. It is found experimentally that the temperatures can always 
be related by an equation having the form 

C 2 (T 2 ~ 7')=C 1 (r-r 1 ) (1.34) 

Here C\ and C 2 are functions of the mass and constitution of the metal and 
of the water respectively. Thus, a gram of lead would cause a smaller tem- 
perature change than a gram of copper; 10 grams of lead would produce 
10 times the temperature change caused by one gram. 

Equation (1.34) has the form of an equation of conservation, such as 
eq. (1.8). Very early in the development of the subject it was postulated that 
when two bodies at different temperatures are placed in contact, something 
flows from the hotter to the colder. This was originally supposed to be a 
weightless material substance, called caloric. Lavoisier, for example, in his 
Traite elementaire de Chimle (1789), included both caloric and light among 
the chemical elements. 

We now speak of a flow of heat q, given by 

q - C 2 (T 2 - r) - CAT - T,) (1.35) 

The coefficients C are called the heat capacities of the bodies. If the heat 
capacity is reckoned for one gram of material, it is called the specific heat; 
for one mole of material, the molar heat capacity. 

The unit of heat was originally defined in terms of just such an experiment 
in calorimetry as has been described. The gram calorie was the heat that must 
be absorbed by one gram of water to raise its temperature 1C. It followed 
that the specific heat of water was 1 cal per C. 

More careful experiments showed that the specific heat was itself a func- 
tion of the temperature. It therefore became necessary to redefine the calorie 
by specifying the range over which it was measured. The standard was taken 
to be the 75 calorie, probably because of the lack of central heating in 
European laboratories. This is the heat required to raise the temperature of 
a gram of water from 14.5 to 15.5C. Finally another change in the definition 
of the calorie was found to be desirable. Electrical measurements are capable 
of greater precision than calorimetric measurements. The Ninth International 
Conference on Weights and Measures (1948) therefore recommended that 
the joule (volt coulomb) be used as the unit of heat. The calorie, however, is 
still popular among chemists, and the National Bureau of Standards uses a 
defined calorie equal to exactly 4.1840 joules. 

The specific heat, being a function of temperature, should be defined 
precisely only in terms of a differential heat flow dq and temperature change 
dT. Thus, in the limit, eq. (1.35) becomes 

or C = (1.36) 



Sec. 17] 



PHYSICOCHEMICAL SYSTEMS 



21 



The heat added to a body in raising its temperature from 7\ to T 2 is 
therefore 

(1.37) 



*=\ T \ CdT 



Since C depends on the exact process by which the heat is transferred, this 
integral can be evaluated only when the process is specified. 

If our calorimeter had contained ice at 0C instead of water, the heat 
added to it would not have raised its temperature until all the ice had melted. 
Such heat absorption or evolution accompanying a change in state of aggre- 
gation was first studied quantitatively by Joseph Black (1761), who called it 
latent heat. It may be thought of as somewhat analogous to potential energy. 
Thus we have latent heat of fusion, latent heat of vaporization, or latent 
heat accompanying a change of one crystalline form to another, for example 
rhombic to monoclinic sulfur. 

17. Work in thermodynamic systems. In our discussion of the transfer of 
heat we have so far carefully restricted our attention to the simple case in 
which the system is completely isolated and 
is not allowed to interact mechanically with 
its surroundings. If this restriction does not 
apply, the system may either do work on 
its surroundings or have work done on itself. 
Thus, in certain cases, only a part of the 
heat added to a substance causes its tem- 
perature to rise, the remainder being used 
in the work of expanding the substance. The 
amount of heat that must be added to 
produce a certain temperature change depends on the exact process by 
which the change is effected. 

A differential element of work may be defined by reference to eq. (1.3) 
as dw / 'dr, the product of a displacement and the component of force in 
the same direction. In the case of a simple thermodynamic system, a fluid 
confined in a cylinder with a movable piston (assumed frictionless), the work 
done by the fluid against the external force on the piston (see Fig. 1.7) in a 
differential expansion dV would be 




Fig. 1.7. Work in expansion. 



dw - J - A dr - /> ex dV 



Note that the work is done against the external pressure P ex . 
If the pressure is kept constant during a finite expansion from 



w 






dV = 



AK 



(1.38) 

to V* 
(1.39) 



If a finite expansion is carried out in such a way that each successive state 
is an equilibrium state, it can be represented by a curve on the PV diagram, 



22 



PHYSICOCHEMICAL SYSTEMS 



[Chap. 1 



since then we always have P ex = P. This is shown in (a), Fig. 1.8. In this 
case, 

dw = P dV (1.40) 

On integration, 

\v^j*PdY (1.41) 

The value of the integral is given by the area under the PV curve. Only when 
equilibrium is always maintained can the work be evaluated from functions 
of the state of the substance itself, P and Y 9 for only in this case does P -=- P ex . 
It is evident that the work done in going from point I to point 2 in the 
PV diagram, or from one state to another, depends upon the particular path 
that is traversed. Consider, for example, two alternate paths from A to B in 
(b), Fig. 1.8. More work will be done in going by the path ADB than by the 
path ACB, as'is evident from the greater area under curve ADB. If we proceed 



(a) 





Fig. 1.8. Indicator diagrams for work. 

from state A to state B by path ADB and return to A along BCA, we shall 
have completed a cyclic process. The net work done by the system during this 
cycle is seen to be equal to the difference between the areas under the two 
paths, which is the shaded area in (b), Fig. 1.8. 

It is evident, therefore, that in going from one state to another both the 
work done by a system and the heat added to a system depend on the par- 
ticular path that is followed. The reason why alternate paths are possible in 
(b), Fig. 1.8 is that for any given volume, the fluid may exert different pres- 
sures depending on the temperature that is chosen. 

18. Reversible processes. The paths followed in the PV diagrams of 
Fig. 1.8 belong to a special class, of great importance in thermodynamic 
arguments. They are called reversible paths. A reversible path is one connect- 
ing intermediate states all of which are equilibrium states. A process carried 
out along such an equilibrium path will be called a reversible process. 

In order, for example, to expand a gas reversibly, the pressure on the 
piston must be released so slowly, in the limit infinitely slowly, that at every 
instant the pressure everywhere within the gas volume is exactly the same 
and is just equal to the opposing pressure on the piston. Only in this case 
can the state of the gas- be represented by the variables of state, P and V 



Sec. 19] PHYSICOCHEMICAL SYSTEMS 23 

Geometrically speaking the state is represented by a point in the PV plane. 
The line joining such points is a line joining points of equilibrium. 

Consider the situation if the piston were drawn back suddenly. Gas would 
rush in to fill the space, pressure differences would be set up throughout the 
gas volume, and even a condition of turbulence might ensue. The state of the 
gas under such conditions could no longer be represented by the two variables, 
P and V. Indeed a tremendous number of variables would be required, corre- 
sponding to the many different pressures at different points throughout the 
gas volume. Such a rapid expansion is a typical irreversible process; the inter- 
mediate states are no longer equilibrium states. 

It will be recognized immediately that reversible processes are never 
realizable in actuality since they must be carried out infinitely slowly. All 
naturally occurring processes are therefore irreversible. The reversible path 
is the limiting path that is reached as we carry out an irreversible process 
under conditions that approach more and more closely to equilibrium con- 
ditions. We can define a reversible path exactly and calculate the work done 
in moving along it, even though we can never carry out an actual change 
reversibly. It will be seen later that the conditions for reversibility can be 
closely approximated in certain experiments. 

19. Maximum work. In (b), Fig. 1.8, the change from A to B can be 
carried out along different reversible paths, of which two (ACB and ADB) 
are drawn. These different paths are possible because the volume Kis a func- 
tion of the temperature 7, as well as of the pressure P. If one particular tem- 
perature is chosen and held constant throughout the process, only one rever- 
sible path is possible. Under such an isothermal condition the work obtained 
in going from A to B via a path that is reversible is the maximum work possible 
for the particular temperature in question. This is true because in the rever- 
sible case the expansion takes place against the maximum possible opposing 
force, which is one exactly in equilibrium with the driving force. If the 
opposing force, e.g., pressure on a piston, were any greater, the process 
would occur in the reverse direction ; instead of expanding and doing work 
the gas in the cylinder would have work done upon it and would be com- 
pressed. 

20. Thermodynamics and thermostatics. From the way in which the 
variables of state have been defined, it would appear that thermodynamics 
might justly be called the study of equilibrium conditions. The very nature 
of the concepts and operations that have been outlined requires this restric- 
tion. Nowhere does time enter as a variable, and therefore the question of 
the rate of physicochemical processes is completely outside the scope of this 
kind of thermodynamic discussion. It would seem to be an unfortunate 
accident of language that this equilibrium study is called thermodynamics', 
a better term would be thermostatics. This would leave the term thermo- 
dynamics to cover the problems in which time occurs as a variable, e.g., 
thermal conductivity, chemical reaction rates, and the like. The analogy with 



24 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

dynamics and statics as the two subdivisions of mechanics would then be 
preserved. 

Although the thermodynamics we shall employ will be really a thermo- 
statics, i.e., a thermodynamics of reversible (equilibrium) processes, it should 
be possible to develop a much broader study that would include irreversible 
processes as well. Some progress along these lines has been made and the 
field should be a fruitful one for future investigation. 7 

PROBLEMS 

1. The coefficient of thermal expansion of ethanol is given by a 1.0414 
x 10~ 3 t- 1.5672 x 10~ 6 / + 5.148 x 10~ 8 / 2 , where t is the centigrade tem- 
perature. If and 50 are taken as fixed points on a centigrade scale, what 
will be the reading of the alcohol thermometer when an ideal gas thermo- 
meter reads 30C? 

2. In a series of measurements by J. A. Beattie, the following values were 
found for a of nitrogen : 

P (cm) . . . 99.828 74.966 59.959 44.942 33.311 

axlOVK- 1 . . 3.6740 3.6707 3.6686 3.6667 3.6652 

Calculate from these data the melting point of ice on the absolute ideal gas 
scale. 

3. An evacuated glass bulb weighs 37.9365 g. Filled with dry air at 1 atm 
pressure and 25C, it weighs 38.0739 g. Filled with a mixture of methane and 
ethane it weighs 38.0347 g. Calculate the percentage of methane in the gas 
mixture. 

4. An oil bath maintained at 50C loses heat to its surroundings at the 
rate of 1000 calories per minute. Its temperature is maintained by an electri- 
cally heated coil with a resistance of 50 ohms operated on a 110-volt line, 
A thermoregulator switches the current on and off. What percentage of the 
time will the current be turned on? 

5. Calculate the work done in accelerating a 2000 kg car from rest to a 
speed of 50 km per hr, neglecting friction. 

6. A lead bullet is fired at a wooden plank. At what speed must it be 
traveling to melt on impact, if its initial temperature is 25 and heating of 
the plank is neglected? The melting point of lead is 327 and its specific heat 
is 0.030 cal deg~ ' l g- 1 . 

7. What is the average power production in watts of a man who burns 
2500 kcal of food in a day? 

8. Show that 

7 See, for example, P. W. Bridgman, The Nature of Thermodynamics (New Haven: 
Yale Univ. Press, 1941); K. G. Denbigh, The Thermodynamics of the Steady State (London: 
Methuen, 1951). 



Chap. 1] PHYSICOCHEMICAL SYSTEMS 25 

9. Calculate the pressure exerted by 10 g of nitrogen in a closed 1-liter 
vessel at 25C using (a) the ideal gas equation, (b) van der Waals' equation. 

10. Use Berthelot's equation to calculate the pressure exerted by 0.1 g of 
ammonia, NH 3 , in a volume of 1 liter at 20C 

11. Evaluate the constants a and b' in Dieterici's equation in terms of 
the critical constants P c , V c , T c of a gas. 

12. Derive an expression for the coefficient of thermal expansion a for 
a gas that follows (a) the ideal gas law, (b) the van der Waals equation. 

13. The gas densities (g per liter) at 0C and 1 atm of (a) CO 2 and (b) 
SO 2 are (a) 1.9769 and (b) 2.9269. Calculate the molar volumes of the gases 
and compare with the values given by Berthelot's equation. 

14. The density of solid aluminum at 20C is 2.70 g per cc; of the liquid 
at 660C, 2.38 g per cc. Calculate the work done on the surroundings when 
10 kg of Al are heated under atmospheric pressure from 20 to 660C. 

15. One mole of an ideal gas at 25C is held in a cylinder by a piston at 
a pressure of 100 atm. The piston pressure is released in three stages: first to 
50 atm, then to 20 atm, and finally to 10 atm. Calculate the work done by 
the gas during these irreversible isothermal expansions and compare it with 
that done in an isothermal reversible expansion from 100 to 10 atm at 25C. 

16. Two identical calorimeters are prepared, containing equal volumes 
of water at 20.0. A 5.00-g piece of Al is dropped into calorimeter A, and a 
5.00-g piece of alloy into calorimeter B. The equilibrium temperature in A 
is 22.0, that in B is 21.5. Take the specific heat of water to be independent 
of temperature and equal to 4.18 joule deg~ l . If the specific heat of Al is 
0.887 joule deg" 1 , estimate the specific heat of the alloy. 

17. According to Hooke's Law the restoring force/ on a stretched spring 
is proportional to the displacement r (/-=- /cr). How much work must be 
expended to stretch a 10.0-cm-long spring by 10 per cent, if its force constant 
AC 10 5 dynes cm" 1 ? 

18. A kilogram of ammonia is compressed from 1000 liters to 100 liters 
at 50. Calculate the minimum work that must be expended assuming (a) 
ideal gas, (b) van der Waals' equation. 

REFERENCES 

BOOKS 

1. Berry, A. J., Modern Chemistry (Historical Development) (London: 
Cambridge, 1948). 

2. Epstein, P. S., Textbook of Thermodynamics (New York: Wiley, 1937). 

3. Guggenheim, E, A., Modern Thermodynamics by the Methods of Willard 
Gibbs (London: Methuen, 1933). 

4. Keenan, J. G., Thermodynamics (New York: Wiley, 1941). 



26 PHYSICOCHEMICAL SYSTEMS [Chap. 1 

5. Klotz, I. M., Chemical Thermodynamics (New York: Prentice-Hall, 
1950). 

6. Lewis, G. N., and M. Randall, Thermodynamics and the Free Energy of 
Chemical Substances (New York: McGraw-Hill, 1923). 

7. MacDougall, F. H., Thermodynamics and Chemistry (New York: Wiley, 
1939). 

8. Planck, M., Treatise on Thermodynamics (New York: Dover, 1945). 

9. Roberts, J. K., Heat and Thermodynamics (London: Blackie, 1951). 

10. Rossini, F. D., Chemical Thermodynamics (New York: Wiley, 1950). 

11. Sears, F. W., An Introduction to Thermodynamics, The Kinetic Theory of 
Gases, and Statistical Mechanics (Boston: Addison- Wesley, 1950). 

12. Zemansky, M. W., Heat and Thermodynamics (New York: McGraw- 
Hill, 1951). 

ARTICLES 

1. Birkhoff, G. D., Science in Progress, vol. IV, 120-149 (New Haven: 
Yale Univ. Press, 1945), "The Mathematical Nature of Physical Theories." 

2. Brescia, F.,/. Chem. Ed., 24, 123-128 (1947), 'The Critical Temperature." 

3. Reilly, D., /. Chem. Ed., 28, 178-183 (1951), "Robert Boyle and His 
Background." 

4. Roseman, R., and S. KatzofT, J. Chem. Ed., 11, 350-354 (1934), "The 
Equation of State of a Perfect Gas." 

5. Woolsey, G., J. Chem. Ed., 16, 60-66 (1939), "Equations of State." 



CHAPTER 2 

The First Law of Thermodynamics 



1. The history of the First Law. The First Law of Thermodynamics is an 
extension of the principle of the conservation of mechanical energy. This 
extension became natural when it was realized that work could be converted 
into heat, the expenditure of a fixed amount of work always giving rise to 
the production of the same amount of heat. To give the law an analytical 
formulation, it was only necessary to define a new energy function that 
included the heat. 

The first quantitative experiments on this subject were carried out by 
Benjamin Thompson, a native of Woburn, Massachusetts, who became 
Count Rumford of The Holy Roman Empire. Commissioned by the King 
of Bavaria to supervise the boring of cannon at the Munich Arsenal, he 
became impressed by the tremendous generation of heat during this opera- 
tion. He suggested (1798) that the heat arose from the mechanical energy 
expended, and was able to estimate the amount of heat produced by a horse 
working for an hour; in modern units his value would be 0.183 calorie per 
joule. The reaction at the time to these experiments was that the heat was 
produced owing to a lower specific heat of the metal in the form of fine 
turnings. Thus when bulk metal was reduced to turnings it had to release 
heat. Rumford then substituted a blunt borer, producing just as much heat 
with very few turnings. The adherents of the caloric hypothesis thereupon 
shifted their ground and claimed that the heat arose from the action of air 
on the metallic surfaces. Then, in 1799, Sir Humphry Davy provided further 
support for Rumford's theory by rubbing together two pieces of ice by clock- 
work in a vacuum and noting their rapid melting, showing that, even in the 
absence of air, this latent heat could be provided by mechanical work. 

Nevertheless, the time did not become scientifically ripe for a mechanical 
theory of heat until the work of Dalton and others provided an atomic 
theory of matter, and gradually an understanding of heat in terms of 
molecular motion. This development will be considered in some detail in 
Chapter 7. 

James Joule, at the age of twenty, began his studies in 1840 in a labora- 
tory provided by his father in a Manchester brewery. In 1843, he published 
his results on the heating effect of the electric current. In 1849, he carefully 
determined the mechanical equivalent of heat by measuring the work input 
and the temperature rise in a vessel of water vigorously stirred with paddle 
wheels. His value, converted into our units, was 0.241 calorie per joule; the 
accepted modern figure is 0.239. Joule converted electric energy and mechanical 



28 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 

energy into heat in a variety of ways: electric heating, mechanical stirring, 
compression of gases. By every method he found very nearly the same value 
for the conversion factor, thus clearly demonstrating that a given amount of 
work always produced the same amount of heat, to within the experimental 
error of his measurements. 

2. Formulation of the First Law. The interconversion of heat and work 
having been demonstrated, it is possible to define a new function called the 
internal energy E. In any process the change in internal energy A, in passing 
from one state A to another 5, is equal to the sum of the heat added to the 
system q and the work done on the system w. (Note that by convention 
work done by the system is called positive, 4 H>.) Thus, A - q vv. Now 
the first law of thermodynamics states that this difference in energy A 
depends only on the final state B and the initial state A 9 and not on the path 
between A and B. 

&E=E B -E A = q-w (2.1) 

Both q and w depend upon the path, but their difference^ w is independent 
of the path. Equation (2.1) therefore defines a new state function E. Robert 
Mayer (1842) was probably the first to generalize the energy in this way. 
For a differential change eq. (2.1) becomes 

dE = dq - dw (2.2) 

The energy function is undetermined to the extent of an arbitrary addi- 
tive constant; it has been defined only in terms of the difference in energy 
between one state and another. Sometimes, as a matter of convenience, we 
may adopt a conventional standard state for a system, and set its energy in 
this state equal to zero. For example, we might choose the state of the system 
at 0K and 1 atm pressure as our standard. Then the energy E in any other 
state would be the change in energy in going from the standard state to the 
state in question. 

The First Law has often been stated in terms of the universal human 
experience that it is impossible to construct a perpetual motion machine, 
that is, a machine that will continuously produce useful work or energy from 
nothing. To see how this experience is embodied in the First Law, consider 
a cyclic process from state A to B and back to A again. If perpetual motion 
were ever possible, it would sometimes be possible to obtain a net increase 
in energy A > by such a cycle. That this is impossible can be ascertained 
from eq. (2.1), which indicates that for any such cycle &E = (E n E A ) 
+ (E A E B ) = 0. A more general way of expressing this fact is to say that 
for any cyclic process the integral of dE vanishes: 

dE=Q (2.3) 

3. The nature of internal energy. On page 6 we restricted the systems 
under consideration to those in a state of rest in the absence of gravitational 
or electromagnetic fields. With these restrictions, changes in the internal 



Sec. 4] THE FIRST LAW OF THERMODYNAMICS 29 

energy E include changes in the potential energy of the system, and energy 
associated with the addition or subtraction of heat. The potential energy 
changes may be considered in a broad sense to include also the energy 
changes caused by the rearrangements of molecular configurations that take 
place during changes in state of aggregation, or in chemical reactions. 

If the system were moving, the kinetic energy would have to be added to 
E. If the restriction on electromagnetic fields were removed, the definition of 
E would have to be expanded to include the electromagnetic energy. Simi- 
larly, if gravitational effects were of interest, as in centrifugal operations, the 
energy of the gravitational field would have to be included in or added to E 
before applying the First Law. 

In view of these facts, it has been remarked that even if somebody did 
invent a perpetual motion machine, we should simply invent a new variety 
of energy to explain it, and so preserve the validity of the First Law. From 
this point of view, the First Law is essentially a definition of a function called 
the energy. What gives the Law real meaning and usefulness is the practical 
fact that a very small number of different kinds of energy suffice to describe 
the physical world. 

In anticipation of future discussions, it may be mentioned that experi- 
mental proof of the interconversion of mass and energy has been provided 
by the nuclear physicists. The First Law should therefore become a law of 
the conservation of mass-energy, and the extension of thermodynamics along 
these lines is beginning to be studied. The changes in mass theoretically 
associated with the energy changes in chemical reactions are so small that 
they lie just outside the range of our present methods of measurement. Thus 
they need not be considered in ordinary chemical thermodynamics. 

4. Properties of exact differentials. We have seen in Section 1-17 that the 
work done by a system in going from one state to another is a function of 
the path between the states, and that dw is not in general equal to zero. 
The reason was readily apparent when the reversible process was considered. 

Then, dw \ P dV. The differential expression P dV cannot be inte- 

J A J A 

grated when only the initial and final states are known, since P is a function 
not only of the volume Kbut also of the temperature 7, and this temperature 

C B 

may also change along the path of integration. On the other hand, I dE 

can always be carried out, giving E n E A , since is a function of the state 
of the system alone, and is not dependent on the path by which that state is 
reached or on the previous history of the system. 

Mathematically, therefore, we distinguish two classes of differential ex- 
pressions. Those such as dE are called exact differentials since they are 
obtained by differentiation of some state function such as E. Those such as 
dq or dw are inexact differentials, since they cannot be obtained by differen- 
tiation of a function of the state of the system alone. Conversely, dq or dw 
cannot be integrated to yield a q or w. The First Law states that although 



30 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 

dq and dw are not exact differentials, their difference dE = dq dw is an 
exact differential. 

The following statements are mathematically completely equivalent : 

(1) The function E is a function of the state of a system. 

(2) The differential dE is an exact differential. 

(3) The integral of dE about a closed path dE is equal to zero. 

As an important corollary of the fact that it is an exact differential, dE 
may be written 1 

dE - ( ) dx + ( } dy (2.4) 

\dx/ v \cy' x 

where x and y are any other variables of state of the system, for instance 
any two of P 9 T, V. Thus, for example, 

/XF\ 

IT (2.5) 

A further useful property of exact differential expressions is the Euler 
reciprocity relation. If an exact differential is written dE = M dV + TV dT, 
then 



ar/ r \*VJ T (2 ' 6) 

This can be seen immediately from the typical case of eq. (2.5), whence 
eq. (2.6) becomes (3 2 /8FOr) -- (d 2 E/dTdV) since the order of differentiation 
is immaterial. 

5. Adiabatic and isothermal processes. Two kinds of processes occur fre- 
quently both in laboratory experiments and in thermodynamic arguments. 
An isothermal process is one that occurs at constant temperature, T 
constant, dT 0. To approach isothermal conditions, reactions are often 
carried out in thermostats. In an adiabatic process, heat is neither added to 
nor taken from the system; i.e., q = 0. For a differential adiabatic process, 
dq 0, and therefore from eq. (2.2) dE dw. For an adiabatic reversible 
change in volume, dE ==- P dV. Adiabatic conditions can be approached 
by careful thermal insulation of the system. High vacuum is the best insulator 
against heat conduction. Highly polished walls minimize radiation. These 
principles are combined in Dewar vessels of various types. 

6. The heat content or enthalpy. No mechanical work is done during a 
process carried out at constant volume, since V = constant, dV 0, w 0. 
It follows that the increase in energy % equals the heat absorbed at constant 
volume. 

A-? r (2.7) 

If pressure is held constant, as for example in experiments carried out 
under atmospheric pressure, A = E 2 E l = q w q P(V 2 K x ) or 

1 See, e.g., Granviller, Smith, Longley, Calculus (Boston: Ginn, 1934), p. 412. 



Sec. 7] THE FIRST LAW OF THERMODYNAMICS 31 

( 2 + PV*) (Ei + PV\) = <!P> where q p is the heat absorbed at constant 
pressure. We now define a new function, called the enthalpy or heat content 2 
by 

H - E -\ PV (2.8) 

Then A// = H 2 - H - q p (2.9) 

The increase in enthalpy equals the heat absorbed at constant pressure. 

It will be noted that the enthalpy H 9 like the energy *, is a function of the 
state of the system alone, and is independent of the path by which that state 
is reached. This fact follows immediately from the definition in eq. (2.8), 
since ", P, and V are all state functions. 

7. Heat capacities. Heat capacities may be measured either at constant 
volume or at constant pressure. From the definitions in eqs. (1.36), (2.7), and 
(2.9): 

heat capacity at constant volume: C v -= ~ - I I (2.10) 

aT \oTfy 

heat capacity at constant pressure: C P = -~ \ ) (2.11) 

aT \dT/p 

The capital letters C v and C P are used to represent the heat capacities 
per mole. Unless otherwise specified, all thermodynamic quantities that are 
extensive in character will be referred to the molar basis. 

The heat capacity at constant pressure C P is always larger than that at 
constant volume C F , because at constant pressure part of the heat added to 
a substance is used in the work of expanding it, whereas at constant volume 
aH of the added heat produces a rise in temperature. An important equation 
for the difference C P C v can be obtained as follows: 

C " - C * = (I'),- (D K = (fH' + P ()- (D F (2 ' 12) 

Since, dE -- 

/3\ /3\ /3K\ p\ 

Substituting this value in eq. (2.12), we find 

v l/dy\ 

(2.13) 



The term P(dV/dT) P may be seen to represent the contribution to the 
specific heat C P caused by the expansion of the system against the external 

2 Note carefully that heat content H and heat capacity dqjdT are two entirely different 
functions. The similarity in nomenclature is unfortunate, and the term enthalpy is therefore 
to be preferred to heat content. 



32 



THE FIRST LAW OF THERMODYNAMICS 



[Chap. 2 



pressure P. The other term (dE/dV) T (dy/3T)j> is the contribution from the 
work done in expansion against the internal cohesive or repulsive forces of 
the substance, represented by a change of the energy with volume at constant 
temperature. The term (dE/3V) T is called the internal pressure? In the case 
of liquids and solids, which have strong cohesive forces, this term is large. 
In the case of gases, on the other hand, the term (dE/dV) T is usually small 
compared with P. 

In fact, the first attempts to measure (d/dY) T for gases failed to detect 
it at all. These experiments were carried out by Joule in 1843. 

8. The Joule experiment. Joule's drawing of his apparatus is reproduced 
in Fig. 2.1, and he described the experiment as follows. 4 

I provided another copper receiver () which had a capacity of 134 cubic 
inches. ... I had a piece D attached, in the center of which there was a bore J of 

an inch diameter, which could be closed per- 
fectly by means of a proper stopcock. . . . 

Having filled the receiver R with about 22 
atmospheres of dry air and having exhausted the 
receiver E by means of an air pump, I screwed 
them together and put them into a tin can con- 
taining 161 Ib. of water. The water was first 
thoroughly stirred, and its temperature taken by 
the same delicate thermometer which was made 
use of in the former experiments on mechanical 
equivalent of heat. The stopcock was then 
opened by means of a proper key, and the air 
allowed to pass from the full into the empty re- 
ceiver until equilibrium was established between 
the two. Lastly, the water was again stirred and 
Fig. 2.1. The Joule experiment. its temperature carefully noted. 

Joule then presented a table of experimental data, showing that there was 
no measurable temperature change, and arrived at the conclusion that "no 
change of temperature occurs when air is allowed to expand in such a manner 
as not to develop mechanical power" (i.e., so as to do no external work). 

The expansion in Joule's experiment, with the air rushing from R into 
the evacuated vessel , is a typical irreversible process. Inequalities of tem- 
perature and pressure arise throughout the system, but eventually an equi- 
librium state is reached. There has been no change in the internal energy of 
the gas since no work was done by or on it, and it has exchanged no heat 
with the surrounding water (otherwise the temperature of the water would 
have changed). Therefore AE 0. Experimentally it is found that Ar 0. 
It may therefore be concluded that the internal energy must depend only on 
the temperature and not on the volume. More mathematically expressed: 

3 Note that just as d/<V, the derivative of the energy with respect to a displacement, is a 
force, the derivative with respect to volume, 5E/DK, is a force per unit area or a pressure. 

4 Phil. A%., 1843, p.- 263. 



Sec. 9] 



THE FIRST LAW OF THERMODYNAMICS 



33 



while </K>0 



Since 



and 



it follows that 



Joule's experiment, however, was not capable of detecting small effects, 
since the heat capacity of his water calorimeter was extremely large compared 
with that of the gas used. 

9. The Joule-Thomson experiment. William Thomson (Lord Kelvin) 
suggested a better procedure, and working with Joule, carried out a series of 
experiments between 1852 and 1862. Their apparatus is shown schematically 
in Fig. 2.2. The principle involved 
throttling the gas flow from the high 
pressure A to the low pressure C side 
by interposing a porous plug B. In 
their first trials, this plug consisted of 
a silk handkerchief; in later work, 
porous meerschaum was used. In this 
way, by the time the gas emerges into 
C it has already reached equilibrium 
and its temperature can be measured 

directly. The entire system is thermally insulated, so that the process is an 
adiabatic one, and q 0. 

Suppose that the fore pressure in A is /\, the back pressure in C is P 2 > 
and the volumes per mole of gas at these pressures are V and K 2 , respec- 
tively. The work per mole done on the gas in forcing it through the plug is 
then P^, and the work done by the gas in expanding on the other side is 
P 2 V 2 . The net work done by the gas is therefore w P 2 V 2 P^V^ 

It follows that a Joule-Thomson expansion occurs at constant enthalpy, 
since 

A E 2 E l q w --= w 

E 2 EI PI V\ ^2 ^2 

E 2 + P 2 V 2 ^E,+ P.V, 




Fig. 2.2. The Joule-Thomson experi- 
ment. 



The Joule-Thomson coefficient, /i JmTf9 is defined as the change of temperature 
with pressure at constant enthalpy: 

/^r\ 

(2.14) 

This quantity is measured directly from the temperature change A7 of the 
gas as it undergoes a pressure drop A/> through the porous plug. Some 



34 



THE FIRST LAW OF THERMODYNAMICS 



[Chap. 2 



experimental values of the J.-T. coefficients, which are functions of tem- 
perature and pressure, are collected in Table 2.1. 

TABLE 2.1 

JOULE-THOMSON COEFFICIENTS FOR CARBON DIOXIDE* 
/* (C per atm) 



Tempera- 
ture (K) 


Pressure (atm) 





















1 


10 


40 


60 


80 


100 


i 














220 


2.2855 


2.3035 












250 


1.6885 


1.6954 


1.7570 










275 


1.3455 


1.3455 


1.3470 










300 


\ 1.1070 


1.1045 


1.0840 


1.0175 


0.9675 






325 


0.9425 


0.9375 


0.9075 


0.8025 


0.7230 


0.6165 


0.5220 


350 


0.8195 


0.8150 


0.7850 


0.6780 


0.6020 


0.5210 


0.4340 


380 


0.7080 


0.7045 


0.6780 


0.5835 


0.5165 


0.4505 


0.3855 


400 


0.6475 


0.6440 


0.6210 


0.5375 


0.4790 


0.4225 


0.3635 



* From John H. Perry, Chemical Engineers' Handbook (New York: McGraw-Hill, 
1941). Rearranged from Int. Crit. Tables, vol. 5, where further data may be found. 

A positive //< corresponds to cooling on expansion, a negative \i to warm- 
ing. Most gases at room temperatures are cooled by a J.-T. expansion. 
Hydrogen, however, is warmed if its initial temperature is above -80C, 
but if it is first cooled below 80C it can then be cooled further by a J.-T. 
effect. The temperature 80C at which jn ^ is called the Joule-Thomson 
inversion temperature for hydrogen. Inversion temperatures for other gases, 
except helium, lie considerably higher. 

10. Application of the First Law to ideal gases. An analysis of the theory 
of the Joule-Thomson experiment must be postponed until the Second Law 
of Thermodynamics has been studied in the next chapter. It may be said, 
however, that the porous-plug experiments showed that Joule's original con- 
clusion that (9/OK) T ^ for all gases was too broad. Real gases may have 
a considerable internal pressure and work must be done against the cohesive 
forces when they expand. 

An ideal gas may now be defined in thermodynamic terms as follows: 

(1) The internal pressure (9/3K) T = 0. 

(2) The gas obeys Boyle's Law, PV = constant at constant T. 

It follows from eq. (2.5) that the energy of an ideal gas is a function of 
its temperature alone. Thus dE -~- (3E/dT) y dT = C v dT and C v = dE/dT. 
The heat capacity of an ideal gas also depends only on its temperature. 
These conclusions greatly simplify the thermodynamics of ideal gases, so 
that many thermodynamic discussions are carried on in terms of the ideal 
gas model. Some examples follow: 



Sec. 10] THE FIRST LAW OF THERMODYNAMICS 35 

Difference in heat capacities. When eq. (2.13) is applied to an ideal gas, 
it becomes 



Then, since PV = RT 

/9F\ _R 

and Cp C v = R (2.15) 

Heat capacities are usually given in units of calories per degree per mole, 
and, in these units, 

R - 8.3144/4.1840 

= 1. 9872 caldeg- 1 mole- 1 
Temperature changes. Since dE C v dT 

Likewise for an ideal gas: 

dH = C P dT 

and A// = H 2 - H =j*' C v dT (2.17) 

Isothermal volume or pressure change. For an isothermal change in an 
ideal gas, the internal energy remains constant. Since dT - and 
-0, 



and dq = dw = P dV 

Since p = 



f 2 P 
\ dq = \ 

Ji Ji 



dv 

RT- 

i V 



or = w- *nn-^ = RTln (2.18) 

Since the volume change is carried out reversibly, P always having its equi- 
librium value RT/V, the work in eq. (2.18) is the maximum work done in an 
expansion, or the minimum work needed to effect a compression. The equa- 
tion tells us that the work required to compress a gas from 10 atm to 100 atm 
is just the same as that required to compress it from 1 atm to 10 atm. 



36 



THE FIRST LAW OF THERMODYNAMICS 



[Chap. 2 



Reversible adiabatic expansion. In this case, dq = 0, and dE = dw ~ 
~PdV. 



From eq. (2.16) dw = C v dT 

For a finite change w \C V dT 

J i 

We may write eq. (2.19) as C v dT + P dV - 

_ dT dV 



Whence 






(2.19) 
(2.20) 

(2.21) 



Integrating between 7\ and 7^, and ^ and K 2 , the initial and final tempera- 
tures and volumes, we have 

C v In J + R In =-- (2.22) 

' 1 ^1 

This integration assumes that C v is a constant, not a function of T. 

We may substitute for R from eq. (2.15), and using the conventional 
symbol y for the heat capacity ratio C^/Cy we find 




ISOTHERMAL 

^v 

ADIABATIC 



(y- 



7\ 
n 



Therefore, 

Since, for an ideal gas, 



(2.23) 



Fig. 2.3. Isothermal and 
adiabatic expansions. 



(2.24) 



It has been shown, therefore, that for a reversible adiabatic expansion of an 
ideal gas 

PV Y ^ constant (2.25) 

We recall that for an isothermal expansion PV constant. 

These equations are plotted in Fig. 2.3. A given pressure fall produces a 
lesser volume increase in the adiabatic case, owing to the attendant fall in 
temperature during the adiabatic expansion. 

11. Examples of ideal-gas calculations. Let us take 10 liters of gas at 
and 10 atm. We therefore have 100/22.414 4.457 moles. We shall calculate 
the final volume and the work done in three different expansions to a final 
pressure of 1 atm. The heat capacity is assumed to be C v = $R, independent 
of temperature. 

Isothermal reversible expansion. In this case the final volume 

V 2 = P^Pi - (10)(10)/(1) - 100 liters 



Sec. 11] THE FIRST LAW OF THERMODYNAMICS 37 

The work done by the gas in expanding equals the heat absorbed by the gas 
from its environment. From eq. (2.18), for n moles, 

y 

q ^ w nRTln - 
V\ 

- (4.457)(8.314)(273.2)(2.303) log (10) 
-23,3 10 joules 

Adiabatlc reversible expansion. The final volume is calculated from 
eq. (2.24), with 

C P ($R f R) 5 






Thus ^2 H ~ I K! (10) 3/5 - 10 - 39.8 liters 

\ * 2 

The final temperature is obtained from P 2 V 2 nRT 2 : 

P,V, (1)(39.8) 

Tz ~ n'R (4.457X0.08205) m * K 

For an adiabatic process, q =- 0, and A q u ^- - iv. Also, since C r is 
constant, eq. (2.16) gives 

A - rtCjAr n%R(T 2 7\) 9125 joules 

The work done by the gas on expansion is therefore 9125 joules. 

Irreversible adiabatic expansion. Suppose the pressure is suddenly released 
to 1 atm and the gas expands adiabatically against this constant pressure. 
Since this is not a reversible expansion, eq. (2.24) cannot be applied. Since 
q = 0, A ~w. The value of A depends only on initial and final states: 

A- - w =-/iC r (7* 2 - 7\) 
Also, for a constant pressure expansion, we have from eq. (1.39), 



Equating the two expressions for vv, we obtain 



The only unknown is T 2 : 

-^(r-2732)=l^ 2 

2 f \ 1 10 

> 

T z - 174.8K 

Then A = vv =- f Rn(\14.S 273.2) 

=- 5470 joules 



38 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 

Note that there is considerably less cooling of the gas and less work done 
in the irreversible adiabatic expansion than in the reversible expansion. 

12. Thermochemistry heats of reaction. Thermochemistry is the study of 
the heat effects accompanying chemical reactions, the formation of solutions, 
and changes in state of aggregation such as melting or vaporization. Physico- 
chemical changes can be classified as endothermic, accompanied by the 
absorption of heat, or exothermic, accompanied by the evolution of heat. 

A typical example of an exothermic reaction is the burning of hydrogen: 

H 2 f- i- O 2 - H 2 O (gas) f 57,780 cal at 18C 

A typical endothermic reaction would be the reverse of this, the decom- 
position of water vapor: 

H 2 O - H 2 -f I O 2 - 57,780 cal at 18C 

Heats of reaction may be measured at constant pressure or at constant 
volume. An example of the first type of experiment is the determination of 
the heat evolved when the reaction takes place at atmospheric pressure in an 
open vessel. If the reaction is carried out in a closed autoclave or bomb, the 
constant-volume condition holds. 

By convention, reaction heats are considered positive when heat is ab- 
sorbed by the system. Thus an exothermic reaction has a negative "heat of 
reaction." From eq. (2.7) the heat of reaction at constant volume, 

Q v - A F (2.26) 

From eq. (2.9) the heat of reaction at constant pressure, 

Q P - A//,> - A + P AK (2.27) 

The heat of reaction at constant volume is greater than that at constant 
pressure by an amount equal to the external work done by the system in 
the latter case. In reactions involving only liquids or solids AFis so small 
that usually P AK is negligible and Q v & Q P . For gas reactions, however, 
the P A V terms may be appreciable. 

The heat change in a chemical reaction can best be represented by 
writing the chemical equation for the reaction, specifying the states of all 
the reactants and products, and then appending the heat change, noting the 
temperature at which it is measured. Since most reactions are carried out 
under essentially constant pressure conditions, A// is usually chosen to 
represent the heat of reaction. Some examples follow: 

(1) SO 2 (1 atm) + | O 2 (1 atm) - SO 3 (1 atm) 

A// 298 - -10,300 cal 

(2) CO 2 (1 atm) + H 2 (1 atm) - CO (I atm) + H 2 O (1 atm) 

A// 29 8 - 9860 cal 

(3) AgBr (cryst>+ \ C1 2 (1 atm) - AgCl (cryst) + \ Br 2 (liq) 

A//008- -6490 cal 



Sec. 13] 



THE FIRST LAW OF THERMODYNAMICS 



39 



As an immediate consequence of the First Law, A or A// for any 
chemical reaction is independent of the path; that is, independent of any 
intermediate reactions that may occur. This principle was first established 
experimentally by G. H. Hess (1840), and is called The Law of Constant Heat 
Summation. It is often possible, therefore, to calculate the heat of a reaction 
from measurements on quite different reactions. For example: : 

(1) COC1 2 -! H 2 S =-- 2 HCl + COS A// 298 - -42,950 cai 

(2) COS + H 2 S - H 2 O (g) + CS 2 (1) A// 298 - +3980 cai 

(3) COC1 2 | 2 H 2 S - 2 HCl + H 2 O (g) + CS 2 (1) 

A #298 = "38,970 cai 

13. Heats of formation. A convenient standard state for a substance may 
be taken to be the state in which it is stable at 25C and 1 atm pressure; thus, 
oxygen as O 2 (g), sulfur as S (rhombic crystal), mercury as Hg (1), and so on. 
By convention, the enthalpies of the chemical elements in this standard state 
are set equal to zero. The standard enthalpy of any compound is then the 
heat of the reaction by which it is formed from its elements, reactants and 
products all being in the standard state of 25C and 1 atm. 
For example: 

(1) S + O 2 - SO 2 A// 298 - -70,960 cai 



(2) 



2 Al + | O 2 - A1 2 O 



23 



A// 298 - -380,000 cai 



The superscript zero indicates we are writing a standard heat of formation 
with reactants and products at 1 atm; the absolute temperature is written as 
a subscript. Thermochemical data are conveniently tabulated as heats of 
formation. A few examples, selected from a recent compilation of the 
National Bureau of Standards, 5 are given in Table 2.2. The standard heat of 
any reaction at 25C is then readily found as 'the difference between the 
standard heats of formation of the products and of the reactants. 

TABLE 2.2 
STANDARD HEATS OF FORMATION AT 25C 



Compound 


State 


A/^298.1 

(kcal/mole) 


Compound 


State 


A#298.16 

(kcal/mole) 


H 2 


g 


-57.7979 


H 2 S 


g 


-4.815 


H 2 


1 


-68.3174 


H 2 SO 4 




-193.91 


H 2 2 


g 


-31.83 


SO, 


g 


-70.96 


HF 


g 


-64.2 


S0 3 


g 


-94.45 


HCl 


g 


-22.063 


CO 


g 


-26.415? 


HBr 


g 


-8.66 


CO 2 


g 


-94.0518 


HI 


g 


+ 6.20 


SOC1 2 


1 


-49.2 


HI0 3 


c 


-57.03 


S 2 C1 2 


g 


-5.70 


5 The Bureau is publishing a comprehensive collection of thermodynamic data, copies 
of which are to be deposited in every scientific library ("Selected Values of Chemical 



Thermodynamic Properties'*). 



40 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 

Many of our thermochemical data have been obtained from measure- 
ments of heats of combustion. If the heats of formation of all its combustion 
products are known, the heat of formation of a compound can be calculated 
from its heat of combustion. For example 

(1) C 2 H 6 j I O 2 - 2 CO 2 ~| 3 H 2 O (1) A// 298 - -373.8 kcal 

(2) C (graphite) f O 2 -- CO 2 A// 298 = -94.5 kcai 

(3) H 2 + I O 2 - H 2 O (1) A// 298 - -68.3 kcai 

(4) 2 C f 3 H 2 -- C 2 H 6 A// 298 --= -20.1 kcal 

The data in Table 2.3 were obtained from combustion heats by F. D. 
Rossini and his co-workers at the National Bureau of Standards. The 
standard state of carbon has been taken to be graphite. 

When changes in state of aggregation occur, the appropriate latent heat 
must be added. For example: 

S (rh) { O 2 SO 2 A// 298 -70.96 kcal 

S (rh) S (mono) A// 298 ^ -0.08 kcal 

S (mono) + O 2 - SO 2 A// 298 . 70.88 kcal 

14. Experimental measurements of reaction heats. 6 The measurement of 
the heat of a reaction consists essentially of a careful determination of the 
amount of the chemical reaction that produces a definite measured change 
in the calorimeter, and then the measurement of the amount of electrical 
energy required to effect exactly the same change. The change in question is 
usually a temperature change. A notable exception is in the ice calorimeter, 
in which one measures the volume change produced by the melting of ice, 
and thereby calculates the heat evolution from the known latent heat of 
fusion of ice. 

The A// values in Table 2.3 were obtained by means of a combustion- 
bomb calorimeter. It is estimated that the limit of accuracy with the present 
apparatus and technique is 2 parts in 10,000. Measurements with a bomb 
calorimeter naturally yield A values, which are converted to A//'s via 
eq. (2.27). 

A thermochemical problem of great interest in recent years has been the 
difference in the energies of various organic compounds, especially the hydro- 
carbons. It is evident that extremely precise work will be necessary to evaluate 
such differences from combustion data. For example, the heat contents of 
the five isomers of hexane differ by 1 to 5 kcal per mole, while the heats of 
combustion of the hexanes are around 1000 kcal per mole; even a 0.1 per cent 
uncertainty in the combustion heats would lead to about a 50 per cent un- 
certainty in the energy differences. Important information about such small 

a Clear detailed descriptions of the experimental equipment and procedures can be 
found in the publications of F. D. Rossini and his group at the National Bureau of Stan- 
dards, J. Res. ofN.B.S., rf,. 1 (1930); 13, 469 (1934); 27, 289 (1941). 



Sec. 15] 



THE FIRST LAW OF THERMODYNAMICS 



41 



TABLE 2.3 
HEATS OF FORMATION OF GASEOUS HYDROCARBONS 



Substance 
Paraffins : 
Methane 
Ethane 
Propane 
rt-Butane 
Isobutane 
w-Pentane 
2-Methylbutane 
Tetramethylmethane 

Monolefines: 
Ethylene 
Propylene 
1-Butene 
cis-2-Butene 
trans-2-Butene 
2-Methylpropene 
1-Pentene 

Diolefines: 
Allene 

1,3-Butadiene 
1,3-Pentadiene 
1,4-Pentadiene 

Acetylenes : 
Acetylene 
Methylacetylene 
Dimethylacetylene 



Formula 



CH 4 
C 2 H 

C 4 H 10 
C 4 H, 
C,H 12 

QH!! 



C 2 H 4 
C,H. 
C 4 H 8 
C 4 H fl 
C,H 8 



j (cal/mole) 

17,8651 74 
-20,191 L 108 
24,750 f 124 
29,715 153 
31,350 I 132 
34,7394, 213 
36,671 I 153 
39,410 L 227 

12,556 1 67 
4956 t 110 

383 JL 180 
-1388 J 180 

- 2338 180 

- 3205 J. 165 
-4644 .{ 300 

46,046 260 
26,865 -> 240 
18,885 -f- 300 
25,565 j : 300 

54,228 J. 235 
44,309 }- 240 
35,221 -{- 355 



energy differences can be obtained for unsaturated hydrocarbons by measure- 
ment of their heats of hydrogenation. This method has been developed to a 
high precision by G. B. Kistiakowsky and his co-workers at Harvard. 7 

It is evident that in calorimetric experiments for example, in a deter- 
mination of a heat of combustion the chemical reaction studied may 
actually occur at a very elevated temperature. One measures, however, the 
net temperature rise after equilibrium has been reached, and this usually 
amounts to only a few degrees, owing to the high heat capacity of the calori- 
meter. Since AE or A// depends only on the initial and final states, one 
actually measures the A or A//, therefore, at around 25C, even though 
temperatures of over 2000C may have been attained during the actual 
combustion process. 

15. Heats of solution. In many chemical reactions, one or more of the 
reactants are in solution, and the investigation of heats of solution is an 
important branch of thermochemistry. It is necessary to distinguish the 
integral heat of solution and the differential heat of solution. The distinction 

7 Kistiakowsky, et al., /. Am. Cfiem. Soc., 57, 876 (1935). 



42 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 

between these two terms can best be understood by means of a practical 
example. 

If one mole of alcohol (C 2 H 5 OH) is dissolved in nine moles of water, the 
final solution contains 10 moles per cent of alcohol. The heat absorbed is the 
integral heat of solution per mole of alcohol to form a solution of this final 
composition. If the mole of alcohol is dissolved in four moles of water, the 
integral heat of solution has a different value, corresponding to the formation 



3000 



m 2000 



o 

2 



I 

"? 1000 



V 10 20 30 40 50 60 70 80 
MOLES WATER/MOLES ALCOHOL 

Fig. 2.4. Heat of solution of ethyl alcohol in water at 0C. 

of a 20 mole per cent solution. The difference between any two integral heats 
of solution yields a value for the integral heat of dilution. The example can 
be written in the form of thermochemical equations as follows: 

(1) C 2 H 5 OH + 9 H 2 O - C 2 H 5 OH (10 mole % solution) 

A// 273 = -2300 



(2) C 2 H 5 OH -f- 4 H 2 O - C 2 H 5 OH (20% solution) 

(3) C 2 H 5 OH (20% solution) + 5 H 2 O -= C 2 H 5 OH (10% solution; 



A// 273 - -1500 



A// 273 = -800 

The heat of dilution from 20 to 10 per cent amounts to 800 cal per mole. 
It is evident that the heat evolved ( A//) when a mole of alcohol is 
dissolved in water depends upon the final concentration of the solution. If 
one plots the measured integral heat of solution against the ratio moles 
water per mole alcohol (njn a \ the curve in Fig. 2.4 is obtained. As the 
solution becomes more and more dilute, njn a approaches infinity. The 
asymptotic value of the heat of solution is called the heat of solution at 
infinite dilution, A//^, For alcohol in water at 0C, this amounts to 3350 



Sec. 16] THE FIRST LAW OF THERMODYNAMICS 43 

calories. The values of A// 80lution generally become quite constant with in- 
creasing dilution, so that measured values in dilute solutions are usually 
close to A//^. Often one finds literature values for which the dilution is not 
specified. These are written, for aqueous solution, simply as in the following 
example: 

NaCl + x H 2 O - NaCl (aq) h 1260 cal : 

In the absence of more detailed information, such values may be taken to 
give approximately the A// at infinite dilution. 

The integral solution heats provide an average A// over a range of con- 
centrations. For example, if alcohol is added to water to make a 50 mole 
per cent solution, the first alcohol added gives a heat essentially that for the 
solute dissolving in pure water, whereas the last alcohol is added to a solution 
of about 50 per cent concentration. For theoretical purposes, it is often 
necessary to know what the A// would be for the solution of solute in a 
solution of definite fixed concentration. Let us imagine a tremendous volume 
of solution of definite composition and add one more mole of solute to it. 
We can then suppose that this addition causes no detectable change in the 
concentration. The heat absorbed in this kind of solution process is the 
differential heat of solution. The same quantity can be defined in terms of a 
very small addition of dn moles of solute to a solution, the heat absorbed 
per mole being dq/dn and the composition of the solution remaining un- 
changed. Methods of evaluating the differential heat will be considered in 
Chapter 6. 

16. Temperature dependence of reaction heats. Reaction heats depend on 
the temperature and pressure at which they are measured. We may write the 
energy change in a chemical reaction as 

~ ^reactants 



3 pr0(1 



From eq. (2.10), - = C rprod - C Freact = AC r (2.28) 






Similarly, f , = C 'i* ' Q'react - AQ, (2.29) 

Integrating, at a constant pressure of 1 atm, so that A// is the standard A//, 
we obtain 



A// T> - A// Ti - ACp dT (2.30) 

JT, 

These equations were first ^et forth by G. R. Kirchhoff in 1858. They 
state that the difference between the heats of reaction at 7\ and at T 2 is equal 
to the difference in the amounts of heat that must be added to reactants and 



44 



THE FIRST LAW OF THERMODYNAMICS 



[Chap. 2 



products at constant pressure to raise them from 7\ to T 2 . This conclusion 
is an immediate consequence of the First Law of Thermodynamics. 

In order to apply eq. (2.30), expressions are required for the heat capaci- 
ties of reactants and products over the temperature range of interest. Over 
a short range, these may often be taken as practically constant, and we 
obtain: 

A// r A// TI ---= ACV(7 2 Tj) 

More generally, the experimental heat-capacity data will be represented by 
a power series: 

CV -a + bT-\- cT 2 f . . . (2.31) 

Examples of such heat-capacity equations are given in Table 2.4. These 
three-term equations fit the experimental data to within about 0.5 per cent 

TABLE 2.4 
HEAT CAPACITY OF GASES (273-1 500K)* 

C P = a 4- bT i cT 2 (C/. in calories per deg per mole) 



Gas 





b x 10 3 


c x 10 7 


H, 


6.9469 


-0.1999 


4.808 


o; 


6.148 


3.102 


-9.23 


CI 2 


7.5755 


2.4244 


-0.650 


Br 2 


8.4228 


0.9739 


-3.555 


N., 


6.524 


1.250 


-0.01 


CO 


6.420 


1.665 


-1.96 


HCl 


6.7319 


0.4352 


3.697 


HBr 


6.5776 


0.9549 


1.581 


H,O 


7.256 


2.298 


2.83 


C0 2 


6.214 


10.396 


-35.45 


Benzene 


0.283 


77.936 


-262.96 


n-Hexane 


7.313 


104.906 


-323.97 


CH 4 




3.381 


18.044 


-43.00 



* H. M. Spencer, /. Am. Chem. Soc., 67, 1858 (1945). Spencer and Justice, ibid., 56, 
2311 (1934). 

over a temperature range from 0C to 1250C. When the series expression 
for AC/> is substituted 8 in eq. (2.30), the integration can be carried out 
analytically. Thus at constant pressure, for the standard enthalpy change, 

rf(A//) - AQ, dT - (A + BT + CT 2 + . . .)dT 

A// - A// -f -AT + i BT* + J Cr 3 + . . . (2.32) 

Here A// is the constant of integration. 9 Any one measurement of A// at 

8 For a typical reaction, N 8 -f- ; J H 2 - NH 3 ; AC P = C PNHi - i C^ t - J PH . 

9 If the heat-capacity equations are valid to 0K, we may note that at T = 0, A/f 
= A// , so that the integration constant can be interpreted as the enthalpy change in the 
reaction at 0K. 



Sec. 16] THE FIRST LAW OF THERMODYNAMICS 45 

a known temperature T makes it possible to evaluate the constant A// in 
eq. (2.32). Then the A// at any other temperature can be calculated from 
the equation. If the heat capacities are given in the form of a C P vs. T curve, 
a graphical integration is often convenient. 

Recently rather extensive enthalpy tables have become available, which 
give H as a function of T over a wide range of temperatures. The use of 
these tables makes direct reference to the heat capacities unnecessary. 

16. Chemical affinity. Much of the earlier work on reaction heats was 
done by Julius Thomsen and Marcellin Berthelot, in the latter part of the 
nineteenth century. They were inspired to carry out a vast program of 
thermochemical measurements by the conviction that the heat of reaction 
was the quantitative measure of the chemical affinity of the reactants. In the 
words of Berthelot, in his Essai de Mecanique chimique (1878): 

Every chemical change accomplished without the intervention of an external 
energy tends toward the production of the body or the system of bodies that sets 
free the most heat. 

This principle is incorrect. It would imply that no endothermic reaction 
could occur spontaneously, and it fails to consider the reversibility of chemi- 
cal reactions. In order to understand the true nature of chemical affinity and 
of .he driving force in chemical reactions, it is necessary to go beyond the 
Firs. Law of Thermodynamics, and to investigate the consequences of the 
second fundamental law that governs the interrelations of work and heat. 

PROBLEMS 

1. Calculate A and A// when 100 liters of helium at STP are heated to 
100C in a closed container. Assume gas is ideal with C v ^R. 

2. One mole of ideal gas at 25C is expanded adiabatically and reversibly 
from 20 atm to 1 atm. What is the final temperature of the gas, assuming 

Cy = \R1 

3. 100 g of nitrogen at 25C arc held by a piston under 30 atm pressure. 
The pressure is suddenly released to 10 atm and the gas adiabatically 
expands. If C v for nitrogen - 4.95 cal per deg, calculate the final tempera- 
ture of the gas. What are A and A// for the change? Assume gas is 
ideal. 

4. At its boiling point (100C) the density of liquid water is 0.9584 g per 
cc; of water vapor, 0.5977 g per liter. Calculate the maximum work done 
when a mole of water is vaporized at the boiling point. How does this compare 
with the latent heat of vaporization of water? 

5. If the Joule-Thomson coefficient is /< t/ T ^ 1.084 deg per atm and the 
heat capacity C P = 8.75 cal per mole deg, calculate the change in enthalpy 
A// when 50 g of CO 2 at 25C and 1 atm pressure are isothermally com- 
pressed to 10 atm pressure. What would the value be for an ideal gas? 



46 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 

6. Using the heat-capacity equation in Table 2.4, calculate the heat 
required to raise the temperature of one mole of HBr from to 500C. 

7. In a laboratory experiment in calorimetry 100 cc of 0.500 TV acetic acid 
are mixed with 100 cc of 0.500 N sodium hydroxide in a calorimeter. The 
temperature rises from 25.00 to 27.55C. The effective heat capacity of the 
calorimeter is 36 cal per deg. The specific heat of 0.250 N sodium acetate 
solution is 0.963 cal deg- 1 g- 1 and its density, 1.034g per cc. Calculate the 
heat of neutralization of acetic acid per mole. 

8. Assuming ideal gas behavior, calculate the values of &E 29Q for SO 3 
(g), H 2 O (g), and HCl (g) from the A// 298 values in Table 2 2. 

9. From the heats of formation in Table 2.3, calculate A// 298 for the 
following cracking reactions: 

C 2 H 6 + H 2 - 2 CH 4 

/i-C 4 H 10 | 3 H 2 -- 4 CH 4 
iso-C 4 H 10 + 3 H 2 - 4 CH 4 

10. The heat of sublimation of graphite to carbon atoms has been esti- 
mated as 170 kcal per mole. The dissociation of molecular hydrogen into 
atoms, H 2 - 2 H, has A// - 103.2 kcal per mole. From these data and the 
value for the heat of formation of methane, calculate the A// for C (g) + 
4 H (g) CH 4 (g). One fourth of this value is a measure of the "energy of 
the C H bond" in methane. 

11. Assuming that the energy of the C H bond in ethane, C 2 H 6 , is the 
same as in methane (Problem 10) estimate the energy of the C C bond in 
ethane from the heat of formation in Table 2.3. 

12. When w-hexane is passed over a chromia catalyst at 500C, benzene 
is formed: C 6 H 14 (g) - C 6 H 6 (g) f 4 H 2 , A// 298 ^ 59.78 kcal per mole. 
Calculate A// for the reaction at 500C (Table 2.4). 

13. Derive a general expression for A// of the water gas reaction 
(H 2 + CO 2 -= H 2 O f CO) as a function of temperature. Use it to calculate 
A// at 500K and 1000K. 

14. From the curve in Fig. 2.4, estimate the heat evolved when 1 kg of 
a 10 per cent (by weight) solution of ethanol in water is blended with 1 kg of 
a 75 per cent solution of ethanol in water. 

15. If a compound is burned under adiabatic conditions so that all the 
heat evolved is utilized in heating the product gases, the maximum tempera- 
ture attained is called the adiabatic flame temperature. Calculate this tem- 
perature for the burning of ethane with twice the amount of air (80 per cent 
N 2 , 20 per cent O 2 ) needed for complete combustion to CO 2 and H 2 O. Use 
heat capacities in Table 2.4, but neglect the terms cT 2 . 

16. Show that for a van der Waals gas, (3E/3K) r a/Y 2 . 

17. Show that (dEpP) v - 0C r /a. 



Chap. 2] THE FIRST LAW OF THERMODYNAMICS 47 

REFERENCES 

BOOKS 
See Chapter 1, p. 25. 

ARTICLES 

1. Parks, G. S., J. Chem. Ed., 26, 262-66 (1949), "Remarks on the History 
of Thermochemistry." 

2. Menger, K., Am. J. Phys. 18, 89 (1950), "The Mathematics of Elementary 
Thermodynamics." 

3. Sturtevant, J. M., Article on "Calorimetry" in Physical Methods of 
Organic Chemistry, vol. 1, 311-435, edited by A. Weissberger (New York: 
Interscience, 1945). 



CHAPTER 3 

The Second Law of Thermodynamics 



1. The efficiency of heat engines. The experiments of Joule helped to 
disprove the theory of "caloric" by demonstrating that heat was not a 
"substance" conserved in physical processes, since it could be generated by 
mechanical work. The reverse transformation, the conversion of heat into 
useful work, had been of greater interest to the practical engineer ever since 
the development of the steam engine by James Watt in 1769. Such an engine 
operates essentially as follows: A source of heat (e.g., a coal or wood fire) 
is used to heat a "working substance" (e.g., steam), causing it to expand 
through an appropriate valve into a cylinder fitted with a piston. The ex- 
pansion drives the piston forward, and by suitable coupling mechanical 
work can be obtained from the engine. The working substance is cooled by 
the expansion, and this cooled working substance is withdrawn from the 
cylinder through a valve. A flywheel arrangement returns the piston to its 
original position, in readiness for another expansion stroke. In simplest 
terms, therefore, any such heat engine withdraws heat from a heat source, 
or hot reservoir, converts some of this heat into work, and discards the 
remainder to a heat sink or cold reservoir. In practice there are necessarily 
frictional losses of work in the various moving components of the engine. 

The first theoretical discussions of these engines were expressed in terms 
of the caloric hypothesis. The principal problem was to understand the 
factors governing the efficiency e of the engine, which was measured by the 
ratio of useful work output w to the heat input q 2 . 

w 
e = - (3.1) 

?2 

A remarkable advance towards the solution of this problem was made in 
1824 by a young French engineer, Sadi Carnot, in a monograph, Reflexions 
sur la Puissance motrice du Feu. 

2. The Carnot cycle. The Carnot cycle represents the operation of an 
idealized engine in which heat is transferred from a hot reservoir at tem- 
perature / 2 i s Partly converted into work, and partly discarded to a cold 
reservoir at temperature t v (Fig. 3. la.) The working substance through 
which these operations are carried out is returned at the end to the same 
state that it initially occupied, so that the entire process constitutes a com- 
plete cycle. We have written the temperatures as t l and t 2 to indicate that 
they are empirical temperatures, measured on any convenient scale what- 
soever. The various steps in the cycle are carried out reversibly. 



Sec. 2] 



THE SECOND LAW OF THERMODYNAMICS 



49 



To make the operation more definite, we may consider the working sub- 
stance to be a gas, and the cyclic process may be represented by the indicator 



HOT 
RESERVOIR 
t2 


(SOURCE) 




<*2 




WORK 



1 2 




V, V 4 



(0) 



(b) 



Fig. 3.1. The essential features of the heat engine (a) and the Carnot 
cycle for its operation shown on an indicator diagram (b). 

diagram of Fig. 3.1b. The steps in the working of the engine for one complete 
cycle are then : 

(1) Withdrawal of heat --^- q 2 from a hot reservoir at temperature t 2 by 
the isothermal reversible expansion of the gas from V v to V 2 . Work 
done by gas H^. 

(2) Adiabatic reversible expansion from V 2 to K 3 , during which q = 0, 
gas does work w 2 and cools from / 2 to t r 

(3) Isothermal reversible compression at t l from K 3 to F 4 . Work done 
by the gas w 3 . Heat q l absorbed by the cold reservoir at t v 

(4) Adiabatic reversible compression from K 4 to V 19 gas warming from 
t l to t 2 . Work done by gas ^ n> 4 , q = 0. 

The First Law of Thermodynamics requires that for the cyclic process 
A = 0. Now A is the sum of all the heat added to the gas, q = q 2 q ly 
less the sum of all the work done by the gas, w = \v l + w 2 vv 3 ~ vv 4 . 

A* ^ q w ^ q 2 ~ q l w ^= 

The net work done by the engine is equal, therefore, to the heat taken 
from the hot reservoir less the heat that is returned to the cold reservoir: 
w = #2 ~~ 9i- The efficiency of the engine is: 

- 



<tl 92 

Since every step in this cycle is carried out reversibly, the maximum 



50 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

possible work is obtained for the particular working substance and tem- 
peratures considered. 1 

Consider now another engine operating, for example, with a different 
working substance. Let us assume that this second engine, working between 
the same two empirical temperatures / 2 and t l9 is more efficient than engine 1 ; 
that is, it can deliver a greater amount of work, w' > w, from the same 
amount of heat q 2 taken from the hot reservoir. (See Fig. 3. la.) It 
could accomplish this only by discarding less heat, q < q l9 to the cold 
reservoir. 

Let us now imagine that, after the completion of a cycle by this sup- 
posedly more efficient engine, the original engine is run in reverse. It therefore 
acts as a heat pump. Since the original Carnot cycle is reversible, all the heat 
and work terms are changed in sign but not in magnitude. The heat pump 
takes in q l of heat from the cold reservoir, and by the expenditure of work 
- w delivers q 2 of heat to the hot reservoir. 

For the first process (engine 2) w' q 2 q^ 

For the second process (engine 1) - w - q 2 + q l 

Therefore, the net result is: w' w = q l q^ 

Since w' > w, and q l > <//, the net result of the combined operation of these 
two engines is that an amount of heat, q q l <?/, has been abstracted 
from a heat reservoir at constant temperature t l and an amount of work 
w" = w' H' has been obtained from it, without any other change what- 
soever taking place. 

In this result there is nothing contrary to the First Law of Thermo- 
dynamics, for energy has been neither created nor destroyed. The work done 
would be equivalent to the heat extracted from the reservoir. Nevertheless, 
in all of human history, nobody has ever observed the isothermal conversion 
of heat into work without any concomitant change in the system. Think 
what it would imply. It would not be necessary for a ship to carry fuel: this 
wonderful device would enable it to use a small fraction of the immense 
thermal energy of the ocean to turn its propellers and run its dynamos. Such 
a continuous extraction of useful work from the heat of our environment has 
been called "perpetual motion of the second kind," whereas the production 
of work from nothing at all was called "perpetual motion of the first 
kind." The impossibility of the latter is postulated by the First Law of 
Thermodynamics; the impossibility of the former is postulated by the 
Second Law. 

If the supposedly more efficient Carnot engine delivered the same amount 
of work w as the original engine, it would need to withdraw less heat q 2 < q 2 

1 In the isothermal steps, the maximum work is obtained on expansion and the mini- 
mum work done in compression of the gas (cf. p. 23). In the adiabatic steps A" = w, and 
the work terms are constant once the initial and final states are fixed. 



Sec. 3] THE SECOND LAW OF THERMODYNAMICS 51 

from the hot reservoir. Then the result of running engine 2 forward and 
engine 1 in reverse, as a heat pump, would be 

(2) w^ fc'-fc' 

(1) _ W = -ft+ft 



?2 - ? 2 = ft ~ ft ^ <7 

This amounts to the transfer of heat </ from the cold reservoir at t l to the 
hot reservoir at t 2 without any other change in the system. 

There is nothing in this conclusion contrary to the First Law, but it is 
even more obviously contrary to human experience than is perpetual motion 
of the second kind. We know that heat always flows from the hotter to the 
colder region. If we place a hot body and a cold body together, the hot one 
never grows hotter while the cold one becomes colder. We know in fact that 
considerable work must be expended to refrigerate something, to pump heat 
out of it. Heat never flows uphill, i.e., against a temperature gradient, of its 
own accord. 

3. Th^Second Law of Thermodynamics- This Second Law may be ex- 
pressed precisely in various equivalent forms. For example: 

The principle of Thomson. It is impossible by a cyclic process to take heat 
from a reservoir and convert it into work without, in the same operation. 
transferring heat from a hot to a cold reservoir. 

The principle of Clausius. It is impossible to transfer heat from a cold to 
a warm reservoir without, in the same process, converting a certain amount 
of work into heat. 

Returning to Carnot's cycle, we have seen that the supposition that one 
reversible cycle may exist that is more efficient than another has led to results 
contradicting human experience as embodied in the Second Law of Thermo- 
dynamics. We therefore conclude that all reversible Carnot cycles operating 
between the same initial and final temperatures must have the same efficiency. 
Since the cycles are reversible, this efficiency is the maximum possible. It is 
completely independent of the working substance and is a function only of 
the working temperatures : 

*--f(ti,*J 

4. The thermodynamic temperature scale. The principle of Clausius may 
be rephrased as "heat never flows spontaneously, i.e., without the expenditure 
of work, from a colder to a hotter body." This statement contains essentially 
a definition of temperature, and we may recall that the temperature concept 
was first introduced as a result of the observation that all bodies gradually 
reach a state of thermal equilibrium. 

Lord Kelvin was the first to use the Second Law to define a thermo- 
dynamic temperature scale, which is completely independent of any thermo- 
metric substance. The Carnot theorem on the efficiency of a reversible cycle 



52 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

may be written: Efficiency (independent of working substance) = (q 2 qi)/q 2 
= /'('i. '2). or 1 - ft/ft =/'(^i, > 2 )- Therefore 

-=/('i, /a) (3-3) 

ft 

We have written /'('i, / 2 ) an d/('i> '2) 1 ~-/'('i '2) to indicate unspecified 
functions of / t and / 2 - 

Consider two Carnot cycles such that: qjq 2 =/(^i ^)J ft/?3 "/(^ '3)- 
They must be equivalent to a third cycle, operating between / t and / 3 , with 
^ A'i '3)- Therefore 

^^^Wa) 

^ * 2 



But, if this condition is satisfied, we can write: J(t l9 t 3 ) = F(t^)IF(t^\f(t^ t 3 ) 
= F(t 2 )/F(t 3 ). That is, the efficiency function, f(t l9 / 2 ), is the quotient of a 
function of t l alone and a function of t 2 alone. It follows that 

* = (3.4) 



Lord Kelvin decided to use eq. (3.4) as the basis of a thermodynamic 
temperature scale. He took the functions F(t^) and F(/ 2 ) to have the simplest 
possible form, namely, 7\ and To. Thus a temperature ratio on the Kelvin 
scale was defined as equal to the ratio of the heat absorbed to the heat 
rejected in the working of a reversible Carnot cycle. 

** = P (3-5) 

<7i 7*i 

The efficiency of the cycle, eq. (3.2), then becomes 



The zero point of the thermodynamic scale is physically fixed as the 
temperature of the cold reservoir at which the efficiency becomes equal to 
unity, i.e., the heat engine is perfectly efficient. From eq. (3.6), in the limit 
as 7\->0, <?-> 1. 

The efficiency calculated from eq. (3.6) is the maximum thermal efficiency 
that can be approached by a heat engine. Since it is calculated for a reversible 
Carnot cycle, it represents an ideal that actual irreversible cycles can never 
achieve. Thus with a heat source at 120C and a sink at 20C, the maximum 
thermal efficiency is 100/393 =- 25.4 per cent. If the heat source is at 220 
and the sink still at 20, the efficiency is raised to 200/493 = 40.6 per cent. 
It is easy to see why the trend in power plant design has been to higher tem- 
peratures for the heat source. In practice, the efficiency of steam engines 
seldom exceeds 80 per cent of the theoretical value. Steam turbines generally 



Sec. 5] 



THE SECOND LAW OF THERMODYNAMICS 



53 



can operate somewhat closer to their maximum thermal efficiencies, since 
they have fewer moving parts and consequently lower frictional losses. 

5. Application to ideal gases. Temperature on the Kelvin, or thermo- 
dynamic, scale has been denoted by the symbol T, which is the same symbol 
used previously for the absolute ideal gas scale. It can be shown that these 
scales are indeed numerically the same by running a Carnot cycle with an 
ideal gas as the working substance. 

Applying eqs. (2.18) and (2.20) to the four steps: 

(1) Isothermal expansion: \\\ ~ q 2 RT 2 In K 2 /K t 

C T 

(2) Adiabatic expansion: w 2 ~- * C v dT; q 

J TI 

(3) Isothermal compression: u 3 q i RT In VJV% 

CT 

(4) Adiabatic compression: \v 4 = * C v dT; q -- 

j TI 

By summation of these terms, the net work obtained is w --= 



l + w 2 



RT 2 ln V 2 /y i + RT\\n 



Since, from eq. (2.22), 



K,/^ - K 3 /K 4 , 
w - R(T 2 - T,) In ^ 



- T, 






92 7 2 

Comparison with eq. (3.6) completes the proof of the identity of the ideal 
gas and thermodynamic temperature scales. 

6. Entropy. Equation (3.6) for a reversible Carnot cycle operating be- 
tween T 2 and 7\ irrespective of the working substance may be rewritten 

?2 7\ 

Now it can be shown that any cyclic process can be broken down into a 
number of Carnot cycles. Consider the perfectly general ABA of Fig. 3.2. 
The area of the figure has been divided 
into a number of Carnot cycles by the 
crosshatched system of isothermals 
and adiabatics. The outside bound- 
aries of these little cycles form the 
heavy zigzag curve which follows quite 
closely the path of the general cycle 
ABA. The inside portions of the little 
Carnot cycles cancel out, since each 
section is traversed once in the for- 
ward direction and once in the reverse 
direction. For example, consider the Fig. 3.2. General cycle broken down 
isothermal xy which belongs to an into Carnot cycles. 




ISOTHERMALS 



ADIABATICS 



54 



THE SECOND LAW OF THERMODYNAMICS 



[Chap. 3 



expansion in the small cycle /?, and to a compression in the small cycle a, 
all the work and heat terms arising from it thereby being canceled. 

If eq. (3.7) is now applied to all these little Carnot cycles, we have for 
the zigzag segments V q\T = 0. As the Carnot cycles are made smaller and 
smaller, the boundary curve approaches more and more closely to that for 
the general cyclic process ABA. In the limit, for differential Carnot cycles, 
the area enclosed by the crooked boundary becomes identical with the area 
of the cycle ABA. We can then replace the summation of finite terms by the 
integration of differentials and obtain 2 

(3.8) 



r- 

This equation holds true for any reversible cyclic process whatsoever. 



Fig. 3-3. Carnot cycle on a TS diagram. 

It may be recalled (p. 30) that the vanishing of the cyclic integral means 
that the integrand is a perfect differential of some function of the state of 
the system. This new function is defined by 



*3. 

T 



(for a reversible process) 



(3.9) 



Thus, 



. , c c_i_c c o 

- *-J/? A ~> 'i I'i 

The function 5 was first introduced by Clausius in 1850, and is called the 
entropy. Equation (3.9) indicates that when the inexact differential expression 

dq is multiplied by 1/r, it becomes an exact differential; the factor \JT is 

CB 
called an integrating multiplier. The integral dq KV is dependent on the 

f ft * A. 

path, whereas I dq rev /T is independent of the path. This, in itself, is an 
alternative statement of the Second Law of Thermodynamics. 

It is interesting to consider the TS diagram in Fig. 3.3, which is analogous 
to the PV diagram of Fig. 1.8. In the PV case, the area under the curve is a 

2 See P. S. Epstein, Textbook of Thermodynamics (New York: Wiley, 1938), p. 57. 



Sec. 7] THE SECOND LAW OF THERMODYNAMICS 55 

measure of the work done in traversing the indicated path. In the TS diagram, 
the area under the curve is a measure of the heat added to the system. Tem- 
perature and pressure are intensity factors ; entropy and volume are capacity 
factors. The products P dV and T dS both have the dimensions of energy. 

7. The inequality of Clausius. Equation (3.8) was obtained for a reversible 
cycle. Clausius showed that for a cycle into which irreversibility enters at 
any stage, the integral of dq\T is always less than zero. 

?<0 (3.10) 

The proof is evident from the fact that the efficiency of an irreversible 
Carnot cycle is always less than that of a reversible cycle operating between 
the same two temperatures. For the irreversible case, we therefore conclude 
from eq. (3.6) that 



</2 " T <2 

Then, instead of eq. (3.7), we find that 

*_*<(> 

T, 7', 

This relation is extended to the general cycle, by following the argument 
based on Fig. (3.2). Instead of eq. (3.8), which applies to the reversible case, 
we obtain the inequality of Clausius, given by eq. (3.10). 

8. Entropy changes in an ideal gas. The calculation of entropy changes in 
an ideal gas is particularly simple because in this case (3/<)K) T 0, and 
heat or work terms due to cohesive forces need not be considered at any 
point. For a reversible process in an ideal gas, the First Law requires that 

RT dV 
dq - dE + PdV~ Cy dT -\ y- 

Therefore, ^ = ^ + ^ (3.,,) 

On integration, AS 1 = S 2 - S l =- J 2 C v d\n T f J 2 RdlnV 
If C r is independent of temperature, 

AS- C F lnp+ Rln^ (3.12) 

7\ V l 

For the special case of a temperature change at constant volume, the 
increase in entropy with increase in temperature is therefore 



AS- C F ln^ (3.13) 

If the temperature of one mole of ideal gas with C y ^ 3 is^ doubled, the 



56 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

entropy is increased by 3 In 2 --- 2.08 calories per degree, or 2.08 entropy 
units (eu). 

For the case of an isothermal expansion, the entropy increase becomes 

AS- /?ln~- R\n Pl (3.14) 

YI PZ 

If one mole of ideal gas is expanded to twice its original volume, its entropy 
is increased by R In 2 1.38 eu. 

9. Entropy changes in isolated systems. The change in entropy in going 
from a state A to a state B is always the same, irrespective of the path between 
A and B, since the entropy is a function of the state of the system alone. It 
makes no difference whether the path is reversible or irreversible. Only in 
case the path is reversible, however, is the entropy change given by 



AS S tt -S A --j ^ (3.15) 

In order to evaluate the entropy change for an irreversible process, it is 
necessary to devise a reversible method for going from the same initial to 
the same final state, and then to apply eq. (3.15). 

In any completely isolated system we are restricted to adiabatic processes, 
since no heat can either enter or leave such a system. 3 For a reversible process 
in an isolated system, therefore, dq and dS dq/T 0, or S --- constant. 
If one part of the system increases in entropy, the remaining part must 
decrease by an exactly equal amount. 

A fundamental example of an irreversible process is the transfer of heat 
from a hot to a colder body. We can make use of an ideal gas to carry out 
the transfer reversibly, and thereby calculate the entropy change. The gas is 
placed in thermal contact with the hot body at T 2 and expanded reversibly 
and isothermally until it takes up heat equal to q. To simplify the argument, 
it is assumed that the bodies have heat capacities so large that changes in 
their temperatures on adding or withdrawing heat q are negligible. The gas 
is then removed from contact with the hot reservoir and allowed to expand 
reversibly and adiabatically until its temperature falls to T v Next it is placed 
in contact with the colder body at 7\ and compressed isothermally until it 
gives up heat equal to q. 

The hot reservoir has now lost entropy = q/T 2 , whereas the cold reservoir 
has gained entropy ^ q/T r The net entropy change of the reservoirs has 
therefore been AS - <//7\ - q/T 2 . Since T 2 > 7\, AS > 0, and the entropy 
has increased. The entropy of the ideal gas, however, has decreased by an 
exactly equal amount, so that for the entire isolated system of ideal gas plus 
heat reservoirs, AS for the reversible process. If the heat transfer had 

3 The completely isolated system is, of course, a figment of imagination. Perhaps our 
whole universe might be considered as an isolated system, but no small section of it can be 
rigorously isolated. As usual, -the precision and sensitivity of our experiments must be 
allowed to determine how the system is to be defined. 



Sec. 9] THE SECOND LAW OF THERMODYNAMICS 57 

been carried out irreversibly, for example by placing the two bodies in direct 
thermal contact and allowing heat cj to flow along the finite temperature 
gradient thus established, there would have been no compensating entropy 
decrease. The entropy of the isolated system would have increased during 
the irreversible process, by the amount AS <//7\ qlT 2 . 

We shall now prove that the entropv of an isolated system always increases 
during an irreversible process. The proof of 
this theorem is based on the inequality of 
Clausius. Consider in Fig. (3.4) a perfectly 
general irreversible process in an isolated 
system, leading from state A to state B. It is 
represented by the dashed line. Next consider 
that the system is returned to its initial state 
A by a reversible path represented by the 
solid line from B to A. During this reversible Fig- 3.4. A cyclic process. 
process, the system need not be isolated, and 

can exchange heat and work with its environment. Since the entire cycle is 
in part irreversible, cq. (3.10) applies, and 




Writing the cycle in terms of its two sections, we obtain 

^<0 (3.16) 



The first integral is equal to zero, since during the process A >- B the system 
is by hypothesis isolated and therefore no transfer of heat is possible. The 
second integral, from eq. (3.15), is equal to S t S H . Therefore eq. (3.16) 
becomes 

S A -- S H < 0, or SH S A > 

We have therefore proved that the entropy of the final state B is always 
greater than that of the initial state A, if A passes to B by an irreversible 
process in an isolated system. 

Since all naturally occurring processes are irreversible, any change that 
actually occurs spontaneously in nature is accompanied by a net increase in 
entropy. This conclusion led Clausius to his famous concise statement of the 
laws of thermodynamics. "The energy of the universe is a constant; the 
entropy of the universe tends always towards a maximum." 

This increasing tendency of the entropy has also been expressed as a 
principle of the degradation of energy, by which it becomes less available 
for useful work. Thus temperature differences tend to become leveled out, 
mountains tend to become plains, fuel supplies become exhausted, and work 
is frittered away into heat by frictional losses. Interesting philosophical 



58 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

discussions have arisen from the entropy concept, notably the suggestion of 
Sir Arthur Eddington that, because of its continuously increasing character, 
"entropy is time's arrow"; that is, the constantly increasing entropy of the 
universe is the physical basis of our concept of time. The "meaning" of 
entropy will be displayed in another aspect when we discuss its statistical 
interpretation. 

10. Change of entropy in changes of state of aggregation. As an example 
of a change in state of aggregation we may take the melting of a solid. At a 
fixed pressure, the melting point is a definite temperature T m at which solid 
and liquid are in equilibrium. In order to change some of the solid to liquid, 
heat must be added to the system. As long as both solid and liquid are 
present, this added heat does not change the temperature of the system, but 
is absorbed by the system as the latent heat of fusion X f of the solid. Since 
the change occurs at constant pressure, the latent heat, by eq. (2.9), equals 
the difference in enthalpy between liquid and solid. Per mole of substance, 

A, -- A// 7 -- //i, (|U j d //solid 

At the melting point, liquid and solid exist together in equilibrium. The 
addition of a little heat would melt some of the solid, the removal of a little 
heat would solidify some of the liquid, but the equilibrium between solid 
and liquid would be maintained. The Litent heat is necessarily a reversible 
heat, because the process of melting follows a path consisting of successive 
equilibrium states. We can therefore evaluate the entropy of fusion AS/ by 
a direct application of the relation A5 - </ r ev/^ which applies to any rever- 
sible isothermal process. 

A//, 
T f 



^liquid Ssolid ~ AS, - ^ (3.17) 



For example, 4 A//, for ice is 1430 cal per mole, so that AS, = 1430/273.2 
= 5.25 cal deg" 1 mole" 1 . 

By an exactly similar argument the entropy of vaporization AS y , the 
latent heat of vaporization A// v , and the boiling point T b are related by 

-A _ A// * 
^vapor ~~ ^liquid ~" AS V ,= (3.18) 

A similar equation holds for a change from one form of a polymorphic 
solid to another, if the change occurs at a T and P at which the two forms 
are in equilibrium, and if there is a latent heat A associated with the trans- 
formation. For example, grey tin and white tin are in equilibrium at 13C 
and 1 atm, and A = 500 cal. Then AS, = 500/286 = 1.75 cal deg- 1 mole- 1 . 

11. Entropy and equilibrium. Now that the entropy function has been 
defined and a method outlined for the evaluation of entropy changes, we 
have gained a powerful tool for our attack on the fundamental problem of 

4 Further typical data are 'shown in Table 14.1 in sec. 14.8. 



Sec. 12] THE SECOND LAW OF THERMODYNAMICS 59 

physicochemical equilibrium. In our introductory chapter, the position of 
equilibrium in purely mechanical systems was shown to be the position of 
minimum potential energy. What is the criterion for equilibrium in a thermo- 
dynamic system? 

Any spontaneously occurring change in an isolated system is accom- 
panied by an increase in entropy. From the First Law of Thermodynamics 
we know that energy can be neither created nor destroyed, so that the 
internal energy of an isolated system must be constant. The only way such 
a system could gain or lose energy would be by some interaction with its 
surroundings, but the absence of any such interaction is just what we mean 
when we say that the system is "isolated" no work is done on it; no heat 
flows across its boundaries. If we restrict work to PV work (expansion or 
compression), and exclude linear or surface effects, it follows also that 
the volume of an isolated system must remain constant. An isolated 
system may be defined, therefore, as a system of constant energy and constant 
volume. The first sentence of this paragraph can thus be rephrased: In a 
system at constant E and K, any spontaneous change is accompanied by an 
increase in entropy. 

Now a system is said to be at equilibrium when it has no further tendency 
to change its properties. The entropy of an isolated system will increase until 
no further spontaneous changes can occur. When the entropy reaches its 
maximum, the system no longer changes: the equilibrium has been attained. 
A criterion for thermodynamic equilibrium is therefore the following: In a 
system at constant energy and volume, the entropy is a maximum. At constant 
E and K, the S is a maximum. 

If instead of a system at constant E and K, a system at constant 5 and 
V is considered, the equilibrium criterion takes the following form: At 
constant S and V, the E is a minimum. This is just the condition applicable in 
ordinary mechanics, in which thermal effects are excluded. 

The drive, or perhaps better the drift, of physicochemical systems toward 
equilibrium is therefore compounded of two factors. One is the tendency 
toward minimum energy, the bottom of the potential energy curve. The 
other is the tendency toward maximum entropy. Only if E is held con- 
stant can S achieve its maximum; only if S is held constant can E 
achieve its minimum. What happens when E and 5 are forced to strike a 
compromise? 

12. The free energy and work functions. Chemical reactions are rarely 
studied under constant entropy or constant energy conditions. Usually the 
physical chemist places his systems in thermostats and investigates them 
under conditions of approximately constant temperature and pressure. 
Sometimes changes at constant volume and temperature are followed, for 
example, in bomb calorimeters. It is most desirable, therefore, to obtain 
criteria for thermodynamic equilibrium that will be applicable under these 
practical conditions. 



60 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

To this end, two new functions have been invented, defined by the 
following equations: 

A - E - TS (3.19) 

F-- H TS (3.20) 

A is called the work function', F is called the free energy.^ Both A and F, by 
their definitions in terms of state functions, are themselves functions of the 
state of the system alone. 

For a change at constant temperature, 

A/I - A 7AS (3.21) 

If this change is carried out reversibly, T AS q, and A/J A - q or 

- A/* - u Wx (3.22) 

The work is the maximum obtainable since the process is reversible. When 
the system isothermally performs maximum work u' mttx , its work function 
decreases by A/f. In any naturally occurring process, which is more or less 
irreversible, the work obtained is always less than the decrease in A. 
From cqs. (3.19) and (3.20), since H E \ PV, 

F---- A \ PV (3.23) 

For a change at constant pressure, 

AF - &A \ P AF (3.24) 

From eqs. (3.22) and (3.24), at constant temperature and pressure, 

-AF ,,' max P&Y (3.25) 

The decrease in free energy equals the maximum work less the work done 
by the expansion of the system at constant pressure. This work of expansion 
is always equal to P(V^ VJ P AK no matter how the change occurs, 
reversibly or irreversibly, provided the external pressure is kept constant. 
The net work over and above this is given by - AF/or a reversible process. 
For an irreversible process the net work is always less than A/ 7 . It may be 
zero as, for example, in a chemical reaction carried out in such a way that 
it yields no net work. Thus the combustion of gasoline in an automobile 
engine yields net work, but burning the same gasoline in a calorimeter yields 
none. The value of AFfor the change is the same in either case, provided the 
initial and final states are the same. 

A helpful interpretation of the entropy can be obtained in terms of the 
new functions A and F. From eqs. (3.19) and (3.20), we can write for a change 
at constant temperature, 

A/4- A-TAS (3.21) 

AF ----- A// - T AS (3.26) 

5 Sometimes A is called the Helmholtz free energy, and F the Gibbs free energy or 
thermodynamic potential. 



Sec. 13] THE SECOND LAW OF THERMODYNAMICS 61 

The change in the work function in an isothermal process equals the change 
in the energy minus a quantity TAS that may be called the unavailable 
energy. Similarly, the change in free energy equals the total change in en- 
thalpy minus the unavailable energy. 

13. Free energy and equilibrium. The free energy function F may be used 
to define a condition for equilibrium in a form that is more directly applicable 
to experimental situations than the criteria in terms of the entropy. We have 
seen that for a reversible process occurring at constant temperature and 
pressure the net work done by the system is equal to the decrease in free 
energy. For a differential change, therefore, under these reversible (i.e., 
equilibrium) conditions at constant temperature and pressure, 

dF^ -</w net (3.27) 

Now most chemical laboratory experiments are carried out under such 
conditions that no work is obtained from the system or added to the system 
except the ordinary PV work, 6 so that dw net ^ 0. In these cases the equili- 
brium criterion becomes simply: 

At constant T and P, dF (3.28) 

This may be stated as follows: Any change in a system at equilibrium at 
constant temperature and pressure is such that the free energy remains constant. 
Thus we have obtained an answer to the question of how the drive to- 
ward maximum entropy and the drive toward minimum energy reach a 
compromise as a system tends toward equilibrium. From eq. (3.26) it is 
evident that an increase in S and a decrease in H both tend to lower the 
free energy. Therefore the third criterion for equilibrium can be written: at 
constant T and P, the F is a minimum. A similar discussion of eq. (3.19) 
provides the equilibrium condition at constant temperature and volume: 
at constant T and K, the A is a minimum. These are the equilibrium conditions 
that are of greatest use in most chemical applications. 

14. Pressure dependence of the free energy. From eq. (3.20), F = H TS 
= E + PV TS. Differentiating, we obtain 

dF = dE+PdY + VdP - TdS - S dT 
Since dE = TdS - P dV 

cJF = VdP - SdT (3.29) 



Therefore, I I - V (3.30) 

For an isothermal change from state (1) to state (2): 

F 2 - F, = bF^dF^l VdP (3.31) 

6 Notable exceptions are experiments with electrochemical cells, in which electric work 
may be exchanged with the system. A detailed discussion is given in Chapter 15. 



62 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

In order to integrate this equation, the variation of V with P must be 
known for the substance being studied. Then if the free energy is known at 
one pressure, it can be calculated for any other pressure. If a suitable equa- 
tion of state is available, it can be solved for V as a function of P, and 
eq. (3.31) can be integrated after substituting this f(P) for V. In the simple 
case of the ideal gas, V - RT/P, and 



F 2 F t - AF RT\n^ (3.32) 

This gives the increase in free energy on compression, or decrease on ex- 
pansion. For example, if one mole of an ideal gas is compressed isother- 
mally at 300K to twice its original pressure, its free energy is increased by 
1.98 x 300 In 2 - 413 calories. 

15. Temperature dependence of free energy. From eq. (3.29), at constant 
pressure, 



377 " S (3 ' 33) 

To integrate this equation, we must know S as a function of temperature. 
This question is considered in the next section. An alternative expression 
can be obtained by combining eq. (3.33) with eq. (3.20): 



/DF\ F H 
W7 7 > ~T~ 



(3.34) 



For isothermal changes in a system, the variation of AF with temperature 7 
is then 

/DAF\ , AF A// 

(IT \r ~ AS =- - T - < 3 - 35 > 

This is called the Gibbs-Helmholtz equation. It permits us to calculate the 
change in enthalpy A// from a knowledge of AF and the temperature co- 
efficient of AF. Since 

</(AF) AF 



d /AF\ d( 
dT \T ) " T " 



dT T 2 
the Gibbs-Helmholtz equation can be written in the alternative forms: 

A// 

T ' ' 



Or, = A// 

L 3(1 IT) \,, 

7 For example the free energy change AF of a chemical reaction might be studied at a 
series of different constant temperatures, always under the same constant pressure. The 
equation predicts how the observed AF depends on the temperature at which the reaction is 
studied. 



Sec. 16] 



THE. SECOND LAW OF THERMODYNAMICS 



63 



Thus the slope of the plot of &F/Tvs. 1/7 is A//, the change in enthalpy. 
Important applications of these equations to chemical reactions will be con- 
sidered in the next chapter. They are especially important because many 
chemical processes are carried out in thermostats under practically constant 
atmospheric pressure. 

16. Variation of entropy with temperature and pressure. Besides its useful- 
ness in the formulation of equilibrium conditions, the free-energy function 
can be used to derive important relations between the other thermodynamic 
variables. Consider, for example, the 
mathematical identity 



\3Tjp D: 

By virtue of eqs. (3.30) and (3.33), this 
identity yields an expression for the Cp 
pressure coefficient of the entropy: 8 T" 



( 



<>> 



Thus at constant temperature, dS 
IP, so that 




f J vaK\ f 1 '- 

AS- L_ <//>-= - a* 

J/>, \dTj r Jl\ 



dP 

(3.38) 



Fig. 3.5, Graphical evaluation of the 
entropy change with temperature. 



To evaluate this integral, the equation of state or other PVT data must be 
available. For an ideal gas, (3Fpr) 7 > R/P. In this case eq. (3.37) becomes 
dS RdlnP, or AS =- RlnPJP^^ Rln VJV 19 as already shown in 
Section 3.8. 

The temperature variation of the entropy can be calculated as follows: 
At constant pressure, 

_ dq __ dH C P dT 

~ T 7 " T ~ f~ 
At constant volume, 

C v dT 



//c _ ^ - _ . 

^ ~ T ~ T ~ 

Thus at constant pressure, 

S=\C P dlnT+ const 

~dT+ const; AS 
8 Alternatively, apply Euler's rule to eq. (3.29). 



(3.39) 
(3.40) 



= r " 

Jr, f 



dT 



(3.41) 



64 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

When C/* is known as a function of 7", the entropy change is evaluated by 
the integration in eq. (3.41). This integration is often conveniently carried 
out graphically, as in Fig. 3.5: if C V \T is plotted against 7, the area under 
the curve is a measure of the entropy change. The entropy change is also the 
area under the curve of C P vs. In T. 

17. The entropy of mixing. Consider two gases at a pressure P. If these 
gases are brought together at constant temperature and pressure, they will 
become mixed spontaneously by interdiffusion. The spontaneous process will 
be associated with an increase in entropy. This entropy of mixing is of interest 
in a number of applications, and it can be calculated as follows. 

In the final mixture of gases the partial pressure of gas (1) is P 1 = A^P, 
of gas (2), P 2 X 2 P, where X l and X 2 are the mole fractions. 9 The AS of 
mixing is equal to the AS required to expand each gas from its initial pressure 
P to its partial pressure in the gas mixture. On the basis of one mole of ideal 
gas mixture, 

AS XT.R In - P - f X 2 R In ~ 

- X^ In 1 X 2 R In 

AS - -R(X l In X l + X 2 In X 2 ) 
This result can be extended to any number of gases in a mixture, yielding 

AS- -R2X t lnX t (3.42) 

The equation is only approximately valid for liquid and solid solutions. 

Let us calculate the entropy of mixing of the elements in air, taking the 
composition to be 79 per cent N 2 , 20 per cent O 2 , and 1 per cent argon. 

AS - -/?(0.79 In 0.79 + 0.20 In 0.20 + 0.01 In 0.01) 
1.10 cal per deg per mole of mixture 

18. The calculation of thermodynamic relations. One great utility of 
thermodynamics is that it enables us by means of a few simple paper-and- 
pencil operations to avoid many tedious and difficult laboratory experiments. 
The general aim is to reduce the body of thermodynamic data to relations 
in terms of readily measurable functions. Thus the coefficients (3K/3r) P , 
(3P/37V, and (3K/3P) r can usually be measured by straightforward experi- 
ments. The results are often expressed implicitly in the equation of state for 
the substance, of the general formf(P, V, T) = 0. 

The heat capacity at constant pressure C P is usually measured directly 
and C v can then be calculated from it and the equation of state. Thermo- 
dynamics itself does not provide any theoretical interpretation of heat 

9 See Chapter 6, Section 1.. 



Sec. 18] THE SECOND LAW OF THERMODYNAMICS 65 

capacities, the magnitudes of which depend on the structures and con- 
stitutions of the substances considered. 

The basic thermodynamic relations may be reduced to a few fundamental 
equations: 

(1) H -^ E + PV 

(2) A -- E TS 

(3) F=E+-Py-TS 

(4) dE^TdS PdV 

(5) dH = TdS+ VdP 

(6) dA - -SdT- PdV 

(7) dF- SdT f- VdP 

Since dA and dF are perfect differentials, they obey the Euler condition 
eq. (2.6), and therefore from (6) and (7) 

(8) (*S 

(9) QS/aP) T - - 
By the definition of the heat capacities, 

(10) C t , - 

(11) ? ~ 

These eleven equations are the starting point for the evaluation of all others. 10 
The relation dE TdS P dV may be considered as a convenient ex- 
pression of the combined First and Second Laws of Thermodynamics. By 
differentiating it with respect to volume at constant temperature, ($EfiV) T 
- T(3S/dV) T - P. Then, since (3S/dV) T -- (dPpT) r , 

This equation has often been called a thermodynamic equation of state, since 
it provides a relationship among P, T, K, and the energy E that is valid for 
all substances. To be sure, all thermodynamic equations are in a sense 
equations of state, since they are relations between state variables, but 
equations like eq. (3.43) are particularly useful because they are closely 
related to the ordinary PVT data. 

It is now possible by means of eq. (3.43) to prove the statement in the 
previous chapter that a gas that obeys the equation PV ~ RT has a zero 
internal pressure, (dE/3V) T . For such a gas T(3P/dT) y =-- RTJV --=-- P, so that 



An equation similar to eq. (3.43) can be obtained in terms of the enthalpy 
instead of the energy: 



(3.44) 
10 A. Tobolsky, /. Chem. Phys., 10, 644 (1942), gives a useful general method. 



66 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

An important application of this equation is the theoretical discussion of 
the Joule-Thomson experiment. Since 



it follows from eq. (3.44) that 

TQVfiT),. - V 



(3.45) 



It is apparent that the Joule-Thomson effect can be either a warming or 
a cooling of the substance, depending on the relative magnitudes of the two 
terms in the numerator of eq. (3.45). In general, a gas will have one or more 
inversion points at which the sign of the coefficient changes as it passes 
through zero. The condition for an inversion point is that 



=- V 
p 

A coefficient of thermal expansion is defined by 

1 



so that the Joule-Thomson coefficient vanishes when K -- <xK o r. For an 
ideal gas this is always true (Law of Gay-Lussac) so that //./.T. is always zero 
in this case. For other equations of state, it is possible to derive /i JmTm from 
eq. (3.45) without direct measurement, if C P data are available. 

These considerations are very important in the design of equipment for 
the liquefaction of gases. Usually, the gas is cooled by doing external work 
in an adiabatic expansion until it is below its inversion point, after which 
further cooling is accomplished by a Joule-Thomson expansion. A further 
discussion of the methods used for attaining very low temperatures will be 
postponed till the next chapter. We shall then see that these low-temperature 
studies have an important bearing on the problem of chemical equilibrium. 

PROBLEMS 

1. A steam engine operates between 120 and 30C. What is the minimum 
amount of heat that must be withdrawn from the hot reservoir to obtain 
1000 joules of work? 

2. Compare the maximum thermal efficiencies of heat engines operating 
with (a) steam between 130C and 40C, (b) mercury vapor between 380C 
and 50C. 

3. A cooling system is designed to maintain a refrigerator at 20C in 
a room at ambient temperature of 25C. The heat transfer into the refrigera- 
tor is estimated as 10 4 joules per min. If the refrigerating unit is assumed to 



Chap. 3] THE SECOND LAW OF THERMODYNAMICS 67 

operate at 50 per cent of its maximum thermal efficiency, estimate the power 
(in watts) required to operate the unit. 

4. Prove that it is impossible for two reversible adiabatics on a P-V 
diagram to intersect. 

5. One mole of an ideal gas is heated at constant pressure from 25 to 
300C. Calculate the entropy change AS if C v -= $R. 

6. Find the increase in , //, 5, A, and Fin expanding 1.0 liter of an ideal 
gas at 25C to 100 liter at the same temperature. 

7. Ten grams of carbon monoxide at 0C are adiabatically and reversibly 
compressed from 1 atm to 20 atm. Calculate A, A//, AS for the change in 
the gas. Assume C v = 4.95 cal per deg mole and ideal gas behavior. Would 
it be possible to calculate AF from the data provided? 

8. At 5C the vapor pressure of ice is 3.012 mm and that of supercooled 
liquid water is 3.163 mm. Calculate the AFper mole for the transition water 
-> ice at -5C. 

9. One mole of an ideal gas, initially at 100C and 10 atm, is adiabatically 
expanded against a constant pressure of 5 atm until equilibrium is reattained. 
If c r -= 4.50 -f 0.0057 calculate A, A//, AS for the change in the gas. 

10. Calculate AS when 10 g of ice at 0C are added to 50 g of water at 
40C in an isolated system. The latent heat of fusion of ice is 79.7 cal per g; 
the specific heat of water, 1 .00 cal per g deg. 

11. The following data are available for water: latent heat of vaporization 
9630 cal per mole; latent heat of fusion 1435 cal per mole. Molar heat 
capacities: solid, C P = -0.50 + 0.030 T\ liquid, C P = 18.0; vapor, C P = 
7.256 + 2.30 x 10~ 3 r+ 2.83 x 10~ 7 r 2 . Calculate AS when one mole of 
water at 100K is heated at constant pressure of 1 atm to 500K. 

12. Derive an expression for the Joule-Thomson coefficient of a van der 
Waals gas. 

13. Calculate the AS per liter of solution when pure N 2 , H 2 , and NH 3 
gases are mixed to form a solution having the final composition 20 per cent 
N 2 , 50 per cent H 2 , and 30 per cent NH 3 (at S.T.P.). 

14. Prove that a gas that obeys Boyle's Law and has zero internal pressure 
follows the equation of state, PV = RT. 

15. For each of the following processes, state which of the quantities A, 
A//, AS, AF, A/* are equal to zero. 

(a) An ideal gas is taken around a Carnot cycle. 

(b) H 2 and O 2 react to form H 2 O in a thermally isolated bomb. 

(c) A nonideal gas is expanded through a throttling valve. 

(d) Liquid water is vaporized at 100C and 1 atm pressure. 

16. Derive the expression (3///3P) r = T(dSfiP) T + V. 
11. Derive: (2C P /dP) T = -T(yv/dT*) M >. 



68 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 

18. Evaluate the following coefficients for (a) an ideal gas; (b) a van der 
Waals gas: (yppT*) y ; (3/aP) T ; (<>PfiV) 8 \ (9 2 K/arV 

19. Derive expressions for: (a) (dA/dP) T in terms of P and V\ (b) 
(dF/dT)^ in terms of A and T. 

20. Bridgman obtained the following volumes for methanol under high 
pressure, relative to a volume 1.0000 at 0C and I kg per cm 2 : 

P, kg/cm 2 1 500 1000 2000 3000 4000 5000 

Vol. at 20 1.0238 0.9823 0.9530 0.9087 0.8792 0.8551 0.8354 
Vol. at 50 1.0610 1.0096 0.9763 0.9271 0.8947 0.8687 0.8476 

Use these data to estimate the AS when 1 mole of methanol at 35C and 1 kg 
per cm 2 pressure is compressed isothermally to 5000 kg per cm 2 . 

REFERENCES 

BOOKS 
See Chapter 1, p. 25. 

ARTICLES 

1. Buchdahl, H. A., Am. J. Phys., 17, 41-46 (1949), "Principle of Cara- 
theodory." 

2. Crawford, F. H., Am. J. Phys., 17, 1-5 (1949), "Jacobian Methods in 
Thermodynamics." 

3. Darrow, K. K., Am. J. Phys., 12, 183-96 (1944), "Concept Of Entropy." 

4. Dyson, F. J., Scientific American, 191, 58-63 (1954), "What is Heat?" 

5. LaMer, V. K., O. Foss, and H. Reiss, Ann. N. Y. Acad. Sci., 51, 605-26 
(1949), "Thermodynamic Theory of J. N. Br0nsted." 



CHAPTER 4 

Thermodynamics and Chemical Equilibrium 



1. Chemical affinity. The problem of chemical affinity may be sum- 
marized in the question, "What are the factors that determine the position 
of equilibrium in chemical reactions?" 

The earliest reflections on this subject were those of the ancient al- 
chemists, who endowed their chemicals with almost human natures, and 
answered simply that reactions occurred when the reactants loved each other. 
Robert Boyle, in The Sceptical Chymyst (1661), commented upon these 
theories without enthusiasm: "I look upon amity and enmity as affections 
of intelligent beings, and I have not yet found it explained by any, how those 
appetites can be placed in bodies inanimate and devoid of knowledge or of 
so much as sense." 

Isaac Newton's interest in gravitational attractions led him to consider 
also the problem of chemical interaction, which he thought might spring 
from the same causes. Thus in 1701, he surveyed some of the existing 
experimental knowledge, as follows: 

When oil of vitriol is mix'd with a little water . . . in the form of spirit of vitriol, 
and this spirit being poured upon iron, copper, or salt of tartar, unites with the 
body and lets go the water, doth not this show that the acid spirit is attracted by the 
water, and more attracted by the fix'd body than by the water, and therefore lets 
go the water to close with the fix'd body? And is it not also from a natural attrac- 
tion that the spirits of soot and sea-salt unite and compose the particles of sal- 
ammoniac . . . and that the particles of mercury uniting with the acid particles of 
spirit of salt compose mercury sublimate, and with particles of sulphur, compose 
cinnaber . . . and that in subliming cinnaber from salt of tartar, or from quick 
lime, the sulphur by a stronger attraction of the salt or lime lets go the mercury, and 
stays with the fix'd body ? 

Such considerations achieved a more systematic form in the early 
"Tables of Affinity," such as that of Etienne Geoffroy in 1718, which re- 
corded the order in which acids would expel weaker acids from combination 
with bases. 

Claude Louis de Berthollet, in 1801, pointed out in his famous book, 
Essai de statique chimique, that these tables were wrong in principle, since the 
quantity of reagent present plays a most important role, and a reaction can 
be reversed by adding a sufficient excess of one of the products. While serving 
as scientific adviser to Napoleon with the expedition to Egypt in 1799, he 
noted the deposition of sodium carbonate along the shores of the salt lakes 
there. The reaction Na 2 CO 3 + CaCl 2 = CaCO 3 + 2 NaCl as carried out in 
the laboratory was known to proceed to completion as the CaCO 3 was 

69 



70 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

precipitated. Berthollet recognized that, under the peculiar conditions of large 
excess of sodium chloride that occurred in the evaporating brines, the 
reaction could be reversed, converting the limestone into sodium carbonate. 

Berthollet, unfortunately, pushed his theorizing too far, and finally main- 
tained that the actual composition of chemical compounds could be changed 
by varying the proportions of the reaction mixture. In the ensuing contro- 
versy with Louis Proust the Law of Definite Proportions was well established, 
but Berthollet's ideas on chemical equilibrium, the good with the bad, were 
discredited, and consequently neglected for some fifty years. 1 

It is curious that the correct form of what we now know as the Law of 
Chemical Equilibrium was arrived at as the result of a series of studies of 
chemical reaction rates, and not of equilibria at all. In 1850, Ludwig Wilhelmy 
investigated the hydrolysis of sugar with acids and found that the rate was 
proportional to the concentration of sugar remaining undecomposed. In 
1862, Marcellin Berthelot and Pean de St. Gilles reported similar results in 
their famous paper 2 on the hydrolysis of esters, data from which are shown 
in Table 4.1. The effect on the products of varying the concentrations of the 
reactants is readily apparent. 

TABLE 4.1 

DATA OF BERTHELOT AND ST. GILLES ON THE REACTION C 2 H 5 OH -}- CH 3 COOH ^ 
CH 3 COOC 2 H 5 ! H 2 

(One mole of acetic acid is mixed with varying amounts of alcohol, and the amount of ester 
present at equilibrium is found) 



Moles of 
Alcohol 


Moles of Ester 
Produced 


Equilibrium Constant 
[EtAc][H 2 0] 
[EtOH][HAc] 


0.05 


0.049 


2.62 


0.18 


0.171 


3.92 


0.50 


0.414 


3.40 


1.00 


0.667 


4.00 


2.00 


0.858 


4.52 


8.00 


0.966 


3.75 



In 1863, the Norwegian chemists C. M. Guldberg and P. Waage expressed 
these relations in a very general form and applied the results to the problem 
of chemical equilibrium. They recognized that chemical equilibrium is a 
dynamic and not a static condition. It is characterized not by the cessation 
of all reaction but by the fact that the rates of the forward and reverse 
reactions have become the same. 

Consider the general reaction, A + B ^ C + D. According to the "law 
of mass action," the rate of the forward reaction is proportional to the 

1 We now recognize many examples of definite departures from stoichiometric com- 
position in various inorganic compounds such as metallic oxides and sulfides, which are 
appropriately called "berthollide compounds." 

2 Ann. chim. phys., [3] 65, 385 (1862). 



Sec. 2] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 71 

concentrations of A and of B. If these are written as (A) and (/?), K forward = 
k\ (A)(B). Similarly, K backward = k*> (Q(D). At equilibrium, therefore, 

^forward ~ ^backward so tnat 



Thus (C}(D} 

ThUS > 



More generally, if the reaction is aA + bB cC + dD, at equilibrium 

< 



Equation (4.1) is a statement of Guldberg and Waage's Law of Chemical 
Equilibrium. The constant K is called the equilibrium constant of the reaction. 
It provides a quantitative expression for the dependence of chemical affinity 
on the concentrations of reactants and products. By convention, the con- 
centration terms for the reaction products are always placed in the numerator 
of the expression for the equilibrium constant. 

Actually, this work of Guldberg and Waage does not constitute a general 
proof of the equilibrium law, since it is based on a very special type of rate 
equation, which is certainly not always obeyed, as we shall see when we take 
up the study of chemical kinetics. Their recognition that chemical affinity is 
influenced by two factors, the "concentration effect" and what might be 
called the "specific affinity," depending on the chemical nature of the reacting 
species, their temperature, and pressure, was nevertheless very important. 
The equilibrium law will subsequently be derived from thermodynamic 
principles. 

2. Free energy and chemical affinity. The free-energy function described 
in Chapter 3 provides the true measure of chemical affinity under conditions 
of constant temperature and pressure. The free-energy change in a chemical 
reaction can be defined as AF ^ F produt . t8 ^reactants- When the free-energy 
change is zero, there is no net work obtainable by any change or reaction 
at constant temperature and pressure. The system is in a state of equilibrium. 
When the free-energy change is positive for a proposed reaction, net work 
must be put into the system to effect the reaction, otherwise it cannot take 
place. When the free-energy change is negative, the reaction can proceed 
spontaneously with the accomplishment of useful net work. The larger the 
amount of this work that can be accomplished, the farther removed is the 
reaction from equilibrium. For this reason, AF has often been called the 
driving force of the reaction. From the statement of the equilibrium law, it 
is evident that this driving force depends on the concentrations of the re- 
actants and products. It also depends on their specific chemical constitution, 
and on the temperature and pressure, which determine the molar free-energy 
values of reactants and products. 

If we consider a reaction at constant temperature, e.g., one conducted in 



72 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

a thermostat, AF = A// -f T AS. The driving force is seen to be made 
up of two parts, a A// term and a 7 AS term. The A// term is the 
reaction heat at constant pressure, and the T AS term is the heat change 
when the process is carried out reversibly. The difference is the amount of 
reaction heat that can be converted into useful net work, i.e., total heat minus 
unavailable heat. 

If a reaction at constant volume and temperature is considered, the 
decrease in the work function, A/l = AF + T AS, should be used as 
the proper measure of the affinity of the reactants, or the driving force of 
the reaction. The constant volume condition is much less usual in laboratory 
practice. 

It is now apparent why the principle of Berthelot and Thomsen (p. 45) 
was wrong. They considered only one of the two factors that make up the 
driving force of a chemical reaction, namely, the heat of reaction. They 
neglected the T AS term. The reason for the apparent validity of their prin- 
ciple was that for many reactions the A// term far outweighs the T AS term. 
This is especially so at low temperatures; at higher temperatures the TAS 
term naturally increases. 

The fact that the driving force for a reaction is large (AF is a large nega- 
tive quantity) does not mean that the reaction will necessarily occur under 
any given conditions. An example is a bulb of hydrogen and oxygen on the 
laboratory shelf. For the reaction, H 2 + \ O 2 == H 2 O (g), AF 298 = 54,638 
cal. Despite the large negative AF, the reaction mixture can be kept for years 
without any detectable formation of water vapor. If, after ten years on the 
shelf, a pinch of platinum-sponge catalyst is added, the reaction takes place 
with explosive violence. The necessary affinity was certainly there, but the 
rate of attainment of equilibrium depended on entirely different factors. 

Another example is the resistance to oxidation of such extremely active 
metals as aluminum and magnesium. 2 Mg + O 2 (l atm) = 2 MgO (c); 
AF 298 136,370 cal. In this case, after the metal is exposed to air it 
becomes covered with a very thin layer of oxide and further reaction occurs 
at an immeasurably slow rate since the reactants must diffuse through the 
oxide film. Thus the equilibrium condition is never attained. The incendiary 
bomb and the thermit reaction, on the other hand, remind us that the 
large AF for this reaction is a valid measure of the great affinity of the 
reactants. 

3. Free-energy and cell reactions. Reactions occurring in electrochemical 
cells with the production of electric energy are of especial interest in the 
discussion of free-energy changes, since they can be carried out under con- 
ditions that are almost ideally reversible. This practical reversibility is 
achieved by balancing the electromotive force of the cell by an opposing 
emf which is imperceptibly less than that of the cell. Such a procedure can 
be accomplished with the laboratory potentiometer, in which an external 
source of emf, such as a battery, is balanced against the standard cell. The 



Sec. 3] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 73 

arrangement for this "compensation method" is shown in Fig. 4.1. When 
the opposing emf 's are balanced by adjustment of the slide wire 5, there is 
no detectable deflection of the galvanometer G. 

An electrochemical cell converts chemical free energy into electric free 
energy. The electric energy is given by the product of the emf of the cell 
times the amount of electricity flowing through it. Michael Faraday showed, 
in 1834, that a given amount of electricity was always produced by or would 
produce the same amount of chemical reaction. For one chemical equivalent 
of reaction the associated amount of 

electricity is called the Faraday, ^", CELL AR 

and is equal to 96,519 coulombs. 
Thus the electric energy available per 
mole of reaction equals zS^ ", where 
z is the number of equivalents per 
mole and S is the emf of the cell. A GALVANOMETER( 
convenient energy unit is therefore 
the volt-coulomb or joule. 

When the reaction is carried out UNKNOWN 

emt 

reversibly, this energy is the maxi- _. . . _ . ,, . r 

/ ., , , p- 7 , . Fig. 4.1. Compensation method for 

mum available, or the net woi k w . mea suring the emf of a cell without drawing 
If the reaction is carried out at a current from it. When there is no deflection 
finite rate, some of the energy is of galvanometer ff x - (SXISS')tf t . 
expended in overcoming the electric 

resistance of the cell, appearing as heat. This Joule heat, 7 2 /?, is the electrical 
analogue of the frictional heat produced in irreversible mechanical pro- 
cesses. We may now write, if ^ is the reversible emf, 

(4.2) 




This equation provides a direct method for evaluating the free-energy change 
in the cell reaction. If we know the temperature coefficient of the emf of the 
cell, we can also calculate A// and AS for the reaction by means of eq. (3.35), 
which on combination with eq. (4.2) yields the relations 



A T-f >r<3r \ JP T \ AC T^ (A. "\\ 

l\rt = 2^ \6 1 -TIL], IAO Z^ (4..J) 

\ #77 dT 

In a later chapter, devoted to electrochemistry, we shall see that it is possible 
to carry out many changes by means of reversible cells, and thereby to 
evaluate AF and A// for the changes from measurements of the emf and its 
temperature coefficient. 

A cell that is occasionally used as a laboratory standard of emf is the 
Clark cell shown in Fig. 4.2. The reaction in this cell is Zn -f Hg 2 SO 4 
ZnSO 4 -f 2 Hg, or more simply, Zn -f 2 Hg+ = Zn++ + 2 Hg. The emf of 
the cell is 1 .4324 volts at 1 5C and the temperature coefficient dSjdT = 
0.00119 volt per degree. It can therefore be calculated that for the cell 



74 



THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 



ZINC 

SULFATE 

SOLUTION 



ZINC 
AMALGAM 




SOLID 
Hg 2 S0 4 

MERCURY 



Fig. 4.2. A typical electrochemical 
cell: the Clark cell. 



reaction AF = (-1.4324 x 2 x 96,519 - -276,510 joule. From eq. (4.3), 
AS- (-0.00119 x 2 x 96,519) - -229.7 joule deg- 1 mole" 1 . Whence, 
A// - AF f T AS = -276,510 - 66,200 - -342,710 joule. The value of 
A// obtained from thermochemical data is 339,500, in good agreement 
with the electrochemical value. 

Since the temperature coefficient is negative, heat is given up to the 

surroundings during the working of this cell, and the net work obtainable, 

A/% is less than the heat of the reaction. There are other cells for which 

the temperature coefficient is positive. 
These cells absorb heat from the environ- 
ment, and their work output, under re- 
versible conditions, is greater than the 
heat of the reaction. 

These relationships, discovered theo- 
retically by Willard Gibbs in 1876, were 
first applied to experimental cases by 
Helmholtz in 1882. Before that time it was 
thought, reasoning from the First Law, 
that the maximum work output that could 
be achieved was the conversion of all of the 
heat of reaction into work. The Gibbs- 
Helmholtz treatment shows clearly that 
the work output is governed by the value 

of AFfor the cell reaction, not by that of A// The working cell can either 
reversibly absorb heat from or furnish heat to its environment. This reversible 
heat change then appears as the T AS term in the free-energy expression. 
4. Standard free energies. In Chapter 2 (p. 39) the definition of standard 
states was introduced in order to simplify calculations with energies and 
enthalpies. Similar conventions are very helpful for use with free-energy data. 
Various choices of the standard state have been made, one that is frequently 
used being the state of the substance under one atmosphere pressure. This is 
a useful definition for gas reactions ; for reactions in solution, other choices 
of standard state may be more convenient and will be introduced as needed. 
A superscript zero will be used to indicate a standard state of 1 atm pressure. 
The absolute temperature will be written as a subscript. 

The most stable form of an element in the standard state (1 atm pressure) 
and at a temperature of 25C will by convention be assigned a free energy 
of zero. 

The standard free energy of formation of a compound is the free energy 
of the reaction by which it is formed from its elements, when all the reactants 
and products are in the standard state. For example: 

H 2 (I atm) + i O 2 (1 atm) - H 2 O (g; 1 atm) AF 298 = -54,638 

S (rhombic crystal) + 3 F 2 (1 atm) - SF 6 (g; 1 atm) AF 298 = -235,000 



Sec. 5] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 



75 



In this way it is possible to make tabulations of standard free energies 
such as that given by Latimer, 3 examples from which are collected in 
Table 4.2. Some of these free-energy values are determined directly from 
reversible cell emf's but most are obtained by other methods to be described 

later. 

TABLE 4.2 

STANDARD FREE ENERGIES OF FORMATION OF CHEMICAL COMPOUNDS AT 25C 



Compound 


State 


AF 298 
(kcaljmole) 


Compound 


State 


AF2,8 

(kcallmole) 


AgCl 


c 


-26.22 


H 2 


g 


-54.638 


Agl 


c 


-15.81 


H 2 2 


g 


-24.73 


CaCl 2 


aq 


-195.36 


H 2 2 


1 


-28.23 


CaC0 3 


c 


-207.22 


H 2 S 


g 


-7.87 


CH 4 


g 


- 12.09 


NaCl 


c 


-91.70 


C 2 H 2 


g 


50.0 


NH 3 


g 


-3.94 


C 2 H 4 


g 


16.28 


N 2 O 


g 


24.93 


C 2 H 6 


g 


-7.79 


NO 


g 


20.66 


CO 


g 


-32.79 


N0 2 


g 


12.27 


C0 2 


g 


-94.24 


N 2 4 


g 


23.44 


CuO 


c 


-30.4 


3 


g 


39.4 


Cu 2 


c 


-35.15 


SO 2 


g 


-71.74 


H 2 


1 


-56.693 


HCOOH 


1 


-86.4 



Free-energy equations can be added and subtracted just as thermo- 
chemical equations are, so that the free energy of any reaction can be cal- 
culated from the sum of the free energies of the products minus the sum of 
the free energies of the reactants. 

\F C V F V F 

^-* r ~~ Z, r products Z, r reactants 

If we adopt the convention that moles of products are positive and moles of 
reactants negative in the summation, this equation can be written concisely as 

AF - 2 ", F> (4.4) 

For example: 

Cu 2 (c) T- NO (g) ^ 2 CuO (c) + i N 2 (g) 

From Table 4.2, 

AF = 2 (-30.4) + i (0) - 20.66 - (-35.15) - -46.31 kcal 

5. Free energy and equilibrium constant of ideal gas reactions. Many im- 
portant applications of equilibrium theory are in the field of homogeneous 
gas reactions, that is, reactions taking place entirely between gaseous pro- 
ducts and reactants. To a good approximation in many such cases, the gases 
may be considered to obey the ideal gas laws. 

The variation at constant temperature of the free energy of an ideal gas 

8 W. M. Latimer, The Oxidation States of the Elements, 2nd ed. (New York: Prentice- 
Hall, 1952). 



76 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

is given from eq. (3.29) as dF = V dP = RTdln P. Integrating from F and 
P, the free energy and pressure in the chosen standard state, to F and P, 
the values in any other state, F - F = RTln (P/P). Since P = 1 atm, this 
becomes 

F-F = RT\nP (4.5) 

Equation (4.5) gives the free energy of one mole of an ideal gas at pressure 
P and temperature 7, minus its free energy in a standard state at P = I atm 
and temperature T. 

If an ideal mixture of ideal gases is considered, Dalton's Law of Partial 
Pressures must be obeyed, and the total pressure is the sum of the pressures 
that the gases would exert if each one occupied the entire volume by itself. 
These pressures are called the partial pressures of the gases in the mixture, 
Pj, 7*2 ... P n . Thus if /? t is the number of moles of gas / in the mixture, 

RT 
P-lP, = -fI", (4.6) 

For each individual gas / in the mixture eq. (4.5) can be written 

F l -F? = RTlnP, (4.7) 

For n t moles, n t (F l F) = RTn t In P t . For a chemical reaction, therefore, 
from eq. (4.4), 

AF - AF = RT 2 n t In P t (4.8) 

If we now consider the pressures P t to be the equilibrium pressures in 
the gas mixture, AF must equal zero for the reaction at equilibrium. Thu 
we obtain the important relation 

- AF = RT 2 n t In P** (4.9) 

AF 
or 2 , ^ P** - - 

Since AF is a function of the temperature alone, the left side of this ex- 
pression is equal to a constant at constant temperature. For a typical reaction, 
aA + bB --= cC 4- dD, the summation can be written out as 



This expression is simply the logarithm of the equilibrium constant in terms 
of partial pressures, K p . Equation (4.9) therefore becomes 

-AF- RTlnK, (4.10) 

The analysis in this section has now established two important results. 
The constancy of the expression 



Sec. 6] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 77 

at equilibrium has been proved by thermodynamic arguments. This con- 
stitutes a thermodynamic proof of the Law of Chemical Equilibrium. Second, 
an explicit expression has been derived, eq. (4.10), which relates the equili- 
brium constant to the standard free-energy change in the chemical reaction. 
We are now able, from thermodynamic data, to calculate the equilibrium 
constant, and thus the concentration of products from any given concentra- 
tion of reactants. This was one of the fundamental problems that chemical 
thermodynamics aimed to answer. 

Sometimes the equilibrium constant is expressed explicitly in terms of 
concentrations c t . For an ideal gas PI n^RT/V) c { RT. Substituting in 
eq. (4.11), we find 



CA^B 
K p = K c (RT)* n (4.12) 

Here K c is the equilibrium constant in terms of concentrations (e.g., moles 
per liter) and A is the number of moles of products less that of reactants in 
the stoichio metric equation for the reaction. 

Another way of expressing the concentrations of the reacting species is 
in terms of mole fractions. The mole fraction of component / in a mixture 
is defined by 

*, - Y n ( 4 - 13 > 

It is the number of moles of a component / in the mixture divided by the 
total number of moles of all the components. It follows that P t = 
Therefore the equilibrium constant in terms of the mole fractions is 



Since K 9 for ideal gases is independent of pressure, it is evident that K x is 
a function of pressure except when A/2 = 0. It is thus a "constant" only with 
respect to variations of the A"s at constant T and P. 

6, The measurement of homogeneous gas equilibria. The experimental 
methods for measuring gaseous equilibria can be classified as either static or 
dynamic. 

In the static method, known amounts of the reactants are introduced 
into suitable reaction vessels, which are closed and kept in a thermostat 
until equilibrium has been attained. The contents of the vessels are then 
analyzed in order to determine the equilibrium concentrations. If the reaction 
proceeds very slowly at temperatures below those chosen for the experiment, 
it is sometimes possible to "freeze the equilibrium" by chilling the reaction 
vessel rapidly. The vessel may then be opened and the contents analyzed 
chemically. 



78 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

This was the procedure used by Max Bodenstein 4 in his classic investiga- 
tion of the hydrogen-iodine equilibrium: H 2 + I 2 = 2 HI. The reaction pro- 
ducts were treated with an excess of standard alkali; iodide and iodine were 
determined by titration, and the hydrogen gas was collected and its volume 
measured. For the formation of hydrogen iodide, A/? = 0; there is no change 
in the number of moles during the reaction. Therefore K v = K c K x . 

If the initial numbers of moles of H 2 and I 2 are a and b, respectively, 
they will be reduced to a x and b x with the formation of 2x moles of 
HI. The total number of moles at equilibrium is therefore a -f b + c, where 
c is the initial number of moles of HI. 

Accordingly the equilibrium constant can be written 

- 2 *) 



The (a + b + c) terms required to convert "number of moles" into "mole 
fraction" have been canceled out between numerator and denominator. In 
a run at 448C, Bodenstein mixed 22.13 cc at STP of H 2 with 16.18 of I 2 , 
and found 25.72 cc of HI at equilibrium. Hence 

K~ - _ 25/72i - - -215 

(22.13 - 12.86)(16.18 - 12.86) 

In the dynamic method for studying equilibria, the reactant gases are 
passed through a thermostated hot tube at a rate slow enough to allow 
complete attainment of equilibrium. This condition can be tested by making 
runs at successively lower flow rates, until there is no longer any change in 
the observed extent of reaction. The effluent gases are rapidly chilled and 
then analyzed. Sometimes a catalyst is included in the hot zone to speed the 
attainment of equilibrium. This is a safer method if a suitable catalyst is 
available, since it minimizes the possibility of any back reaction occurring 
after the gases leave the reaction chamber. The catalyst changes the rate of 
a reaction, not the position of final equilibrium. 

These flow methods were extensively used by W. Nernst and F. Haber 
(around 1900) in their pioneer work on technically important gas reactions. 
An example is the "water-gas equilibrium," which has been studied both 
with and without an iron catalyst. 5 The reaction is 

H 2 + C0 2 - H 2 + CO, and K 9 = ^ HiQ f CQ 



If we consider an original mixture containing a moles of H 2 , b moles of 
CO 2 , c moles of H 2 O, and d moles of CO, the analysis of the data is as 
follows. 

4 Z.physik. Chem., 22, \ (1897); 29, 295 (1899). 

5 Z. anorg. Chem., 38. 5 (1904). 



Sec. 7] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 



79 



Constituent 


Original 
Moles 


Moles 


H 2 


a 


a 


X 


C0 2 


b 


b 


x 


H 2 


c 


c 


+ x 


CO 


d 


d 


+ x 



At Equilibrium 


Mole Fraction 




Partial 
Pressure 


a x/(a -f b + c -f 


d) 


((a - x)ln]P 


b - x/(a + b + c + 


d) 


Kb - x)/n]P 


(c + x)l(a b + c- 


f d) 


[(c + x)ln\P 


(d + x)l(a 4- b + c 


+ d) 


[(d + x)/n]P 



Total Moles at Equilibrium a + b j rc + d = n 

Substituting the partial pressure expressions, we obtain 



^ 



The values for the equilibrium composition, obtained by analysis of the 
product gases, have been used to calculate the constants in Table 4.3. 

TABLE 4.3 
THE WATER GAS EQUILIBRIUM H 2 -f CO 2 = H 2 O -f CO; temperature 986C 



Initial Composition 
(moles per cent) 


Equilibrium Composition 
(moles per cent) 


K. 


C0 2 


H, 


C0 2 


H 2 


CO = H 2 O 


10.1 


89.9 


0.69 


80.52 


9.40 


.59 


30.1 


69.9 7.15 


46.93 


22.96 


.57 


49.1 


51.9 


21.44 


22.85 


27.86 


.58 


60.9 


39.1 


34.43 


12.68 


26.43 


.61 


70.3 


29.7 


47.51 


6.86 


22.82 


.60 



It is often possible to calculate the equilibrium constant for a reaction 
from the known values of the constants of other reactions. This is a principle 
of great practical utility. For example, from the dissociation of water vapor 
and the water-gas equilibrium one can calculate the equilibrium constant for 
the dissociation of carbon dioxide. 



H a O 
CO 2 
CO 2 



= H 



H 2 - H 2 O 



2 
CO 

CO + O 2 



It is apparent that AT/ -= K V 'K V . 

7. The principle of Le Chatelier. The effects of such variables as pressure, 
temperature, and concentration on the position of chemical equilibrium have 
been succinctly summarized by Henry Le Chatelier (1888). "Any change in 



80 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

one of the variables that determine the state of a system in equilibrium 
causes a shift in the position of equilibrium in a direction that tends to 
counteract the change in the variable under consideration." This is a prin- 
ciple of broad and general utility, and it can be applied not only to chemical 
equilibria but to equilibrium states in any physical system. It is indeed possible 
that it can be applied also with good success in the psychological, economic, 
and sociological fields. 

The principle indicates, for example, that if heat is evolved in a chemical 
reaction, increasing the temperature tends to reverse the reaction; if the 
volume decreases in a reaction, increasing the pressure shifts the equilibrium 
position farther toward the product side. Quantitative expressions for the 
effect of variables such as temperature and pressure on the position of 
equilibrium will now be obtained by thermodynamic methods. 

8. Pressure dependence of equilibrium constant. The equilibrium constants 
K p and K c are independent of the pressure for ideal gases; the constant K x 
is pressure-dependent. Since K x = K P P An , In K x -= In K v - A In P. 



dP " P RT ' 

When a reaction occurs without any change in the total number of moles 
of gas in the system, A/? = 0. An example is the previously considered water 
gas reaction. In these instances the constant K p is the same as K x or K c , and 
for ideal gases the position of equilibrium does not depend on the total 
pressure. When AH is not equal to zero, the pressure dependence of K x is 
given by eq. (4.15). When there is a decrease in the mole number (A/z < 0) 
and thus a decrease in the volume, K x increases with increasing pressure. If 
there is an increase in n and V (A/7 > 0), K x decreases with increasing 
pressure. 

An important class of reactions for which A -- is that of dissociation- 
association equilibria. An extensively studied example is the dissociation of 
nitrogen tetroxide into the dioxide, N 2 O 4 2 NO 2 . In this case, K p = 
PxoJP$ t o t ' Jf one m l e f N 2 O 4 is dissociated at equilibrium to a fractional 
extent a, 2a moles of NO 2 are produced. The total number of moles at 
equilibrium is then (1 a) ~\- 2a = 1 + a. It follows that 



(\-a)/(\+a) 1-0 2 
Since for this reaction AA? -^ ~f 1, 



p ~ * 

When a is small compared to unity, this expression predicts that the degree 
of dissociation a shall vary inversely as the square root of the pressure. 
Experimentally it is found that N 2 O 4 is appreciably dissociated even at 



Sec. 9] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 



81 



room temperatures. As a result, the observed pressure is greater than that 
predicted by the ideal gas law for a mole of N 2 ^4 s * nce ea h m l e yields 
1 -\- a moles of gas after dissociation. Thus P (ideal) RTfV, whereas 
P (observed) - (1 + a)RT/V. Hence a - (K/*r)(/> ob8 - /> ldcal ). 

This behavior provides a very simple means for measuring a. For example, 
in an experiment at 318K and 1 atm pressure, a is found to be 0.38. There- 
fore K x = 4(0.38) 2 /(1 - 0.38 2 ) - 0.67. At 10 atm pressure, K x - 0.067 and 
a is 0.128. 

Among the most interesting dissociation reactions are those of the 
elementary gases. The equilibrium constants for a few of these are collected 
in Table 4.4. 

TABLE 4.4 
EQUILIBRIUM CONSTANTS OF DISSOCIATION REACTIONS 



Temp. 


K. 






rig 2 H 


M 
2 ^ M 


C1 2 - 2 Cl 


Br 2 - 2 Br 


600 


1.4 x 10~ 37 


3.6 x 10~ 33 


1.3 x 10- 66 


4.8 x 10~ 16 


6.18 x 10~ 12 


800 


9.2 X 10~ 27 


1.2 x 10~ 23 


5.1 x 10~ 41 


1.04 x 10~ 10 


1.02 x 10- 7 


1000 


3.3 x 10- 20 


7.0 x 10~ 18 


1.3 x 10~ 31 


2.45 x 10~ 7 


3.58 x 10~ 5 


1200 


8.0 x 10~ 16 


5.05 x 10' 14 


2.4 x 10~ 26 


2.48 x 10~ 5 


1.81 x 10- 8 


1400 


1.1 x 10~ 12 


2.96 x 10- 11 


7.5 x 10~ 21 


8.80 x 10~ 4 


3.03 x I0~ a 


1600 


2.5 x 10~ 10 


3.59 x 10~ 9 


1.8 x 10- 17 


1.29 x 10~ 2 


2.55 x 10- 1 


1800 


1.7 x 10- 8 


1.52 x 10~ 7 


7.6 x 10- 16 


0.106 




2000 


5.2 x 10- 7 


3.10 x 10- 


9.8 x 10~ 13 


0.570 





9. Effect of an inert gas on equilibrium. In reactions in which there is no 
change in the total number of moles, AH = 0, and the addition of an inert 
gas cannot affect the composition of the equilibrium mixture. If, however, 
A ^ 0, the inert gas must be included in calculating the mole fractions and 
the total pressure P. Let us consider as an example the technically important 
gas reaction, SO 2 + \ O 2 = SO 3 . In this case A = |, and K p = K X P~ 1/2 . 
Let the initial reactant mixture contain a moles of SO 2 , b moles of O 2 , and 
c moles of inert gas, for example N 2 . If y moles of SO 3 are formed at equi- 
librium, the equilibrium mole fractions are 



b - (y/2) 



y 



Here n is the total number of moles at equilibrium: n a + b + c (y/2). 
The equilibrium constant, 



K = K P 1/2 = 
* 



y/n 



yn 



1/2 



[(a 



[a - 



82 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

It follows that = / 

a y 

/?S 3 __ 

"so, 

where w s()i , w 80a , w 0i , /? are the equilibrium mole numbers. 

Let us now consider three cases. (1) If the pressure is increased by com- 
pressing the system without addition of gas from outside, n is constant, and 
as P increases, n$ Jn$ 0t a l so increases. (2) If an inert gas is added at constant 
volume, both n and P increase in the same ratio, so that the equilibrium 
conversion of SO 2 to SO 3 , w SO| //7 SOi remains unchanged. (3) If an inert gas 
is added at constant pressure, n is increased while P remains constant, and 
this dilution of the mixture with the inert gas decreases the extent of con- 
version /f so > s <v 

This reaction is exothermic, and therefore increasing the temperature 
decreases the formation of products. The practical problem is to run the 
reaction at a temperature high enough to secure a sufficiently rapid velocity, 
without reaching so high a temperature that the equilibrium lies too far to 
the left. In practice, a temperature around 500C is chosen, with a platinum 
or vanadium-pentoxide catalyst to accelerate the reaction. The equilibrium 
constant from 700 to 1200K is represented quite well by the equation 
In K p = (22,6QO/ RT) - (21.36/7?). At 800K, therefore, K v - 33.4. 

Let us now consider two different gas mixtures, the first containing 
20 per cent SO 2 and 80 per cent O 2 at 1 atm pressure, and a second containing 
in addition a considerable admixture of nitrogen, e.g., 2 per cent SO 2 , 8 per 
cent O 2 , 90 per cent N 2 , at 1 atm pressure. Letting y moles SO 3 at equi- 
librium, we obtain: 

I II 

K s - ffpP 1 / 2 = 33.4 K x - K p P l l 2 = 33.4 

y y 



1 ~ (y/2) 1 - (y/2) 

0.2 -y roSj- O/2)] 1 / 1 0.02 - y ["0.08 -(y/2)] 1 / 2 

1 - (y/2) I 1 - '(y/2) J 1 - (y/2) I 1 - (^T J 

/ - 2.000/ -I- 0.681^ - 0.0641 =0 / - 0.1985/ + 6.81 X 10~ 3 j 

y = 0.190 64.06 X 10~ 6 = 

y - 0.0180 

95 % conversion of SO 2 to SO 3 90 % conversion of SO 2 to SO 3 

The cubic equations that arise in problems like these are probably best 
solved by successive approximations. Beginning with a reasonable value 
guessed for the percentage conversion, a sufficiently accurate solution can 
usually be obtained after three or four trials. 



Sec. 10] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 



83 



10. Temperature dependence of the equilibrium constant. An expression 
for the variation of K P with temperature is derived by combining eqs. (4.10) 
and (3.36). Since 

v (4.10) 



and 



therefore 



A// c 

r 2 



dT 



(3.36) 



(4.16) 



It is apparent that if the reaction is endothermic (A// positive) the 
equilibrium constant increases with temperature; if the reaction is exothermic 
(A// negative) the equilibrium con- 
stant decreases as the temperature is 
raised. 

Equation (4.16) can also be written: 

iH (4.17) 



d(\IT) 



R 



Thus if In K p is plotted against \/T 9 
the slope of the curve at any point is 
equal to A// //?. As an example of 
this treatment, the data for the varia- 
tion with temperature of the 2 HI --= 
H 2 + I 2 equilibrium are plotted in 
Fig. 4.3. The curve is almost a straight 
line, indicating that A// is approxi- 
mately constant for the reaction over 
the experimental temperature range. 
The value calculated from the slope at 
400C is A// = 7080 cal. 

It is also possible to measure the 
equilibrium constant at one tempera- 
ture and with a value of A// obtained 

from thermochemical data to calculate the constant at other temperatures. 
Equation (4.16) can be integrated, giving 



In 



3.3U 


\ 










C f\f\ 


\ 












\ 


\ 












\ 














\ 






A on 






\ 


t 






SLOPE 


w* 


- 


V 




* ^n 


^ 


..... 


h- 


h .v 


\ 



V L25 



150 



1.75 



Fig. 4.3. The variation with temper- 
ature of K f = PH 2 Pi.JPm". (Data of 
Bodenstein.) 



Since, over a short temperature range, A/f may often be taken as approxi- 
mately constant, one obtains 



In 



KJTj -A// 



R 



(4.18) 



84 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

If the variations of the heat capacities of the reactants and products are 
known as functions of temperature, an explicit expression for the tempera- 
ture dependence of A// can be derived from Kirchhoff 's equation (2.29). 
This expression for A// as a function of temperature can then be substituted 
into eq. (4.16), whereupon integration yields an explicit equation for K 9 as 
a function of temperature. This has the form 

In K, - - A// //?r + A In T + BT + CT* . . . + I (4.19) 

In this case, as usual, the value of the integration constant / can be deter- 
mined if the value of K p is known at any one temperature, either experiment- 
ally or by calculation from AF. It will be recalled that one value of A// is 
needed to determine A// , the integration constant of the Kirchhoff 
equation. 

To summarize, from a knowledge of the heat capacities of the reactants 
and products, and of one value each for A// and A^,it is possible to calculate 
the equilibrium constant at any temperature. 

As an example, consider the calculation of the constant for the water- 
gas reaction as a function of the temperature. 

CO + H 2 (g) - H 2 + C0 2 ; K v - ^ co> 

-CO r H,0 

From Table 4.2, the standard free-energy change at 25C is: 

A/r 298 = -94,240 - (-54,640 - 32,790) - -6810 

6810 
298^ 

From the enthalpies of formation on page 39, 

A# 298 - -94,050 - (-57,800 - 26,420) = -9830 

The heat capacity table on page 44 yields for this reaction 



Thus In K V298 = - = 1 1.48, or K v298 = 9.55 X 10* 



= -0.515 + 6.23 x 10~ 3 r- 29.9 x 10~ 7 r 2 
From eq. (2.32), 

A// = A// - 0.5157+ 3.12 x 10~ 3 r 2 - 10.0 x 1Q-T 3 

Substituting A// = -9830, T = 298K, and solving for A// , we get 
A// = 9921. Then the temperature dependence of the equilibrium 
constant, eq. (4.19), becomes 



By inserting the value ofln K v at 298K, the integration constant can be 



Sec. 11] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 85 

evaluated as / = 3.97. The final expression for K v as a function of tem- 
perature is, therefore, 

In K - -3.97 + 9 - - 0.259 In T + 1.56 x I0~ 3 r - 2.53 x 10~ 7 r 2 

For example, at 800K, In K 9 = 1.63, K v - 5.10. 

11. Equilibrium constants from thermal data. We have now seen how a 
knowledge of the heat of reaction and of the temperature variation of the 
heat capacities of reactants and products allows us to calculate the equi- 
librium constant at any temperature, provided there is a single experimental 
measurement of either K 9 or AF at some one temperature. If an independent 
method is available for finding the integration constant /in eq. (4.19), it will 
be possible to calculate K 9 without any recourse to experimental measure- 
ments of the equilibrium or of the free-energy change. This calculation would 
be equivalent to the evaluation of the entropy change, AS , from thermal 
data alone, i.e., heats of reaction and heat capacities. If we know AS and 
A//, K p can be found from AF - A// - TA5. 

From eq. (3.41), the entropy per mole of a substance at temperature T 
is given by 

5 = f r C P rflnr+S 



where 5 is the entropy at 0K. 6 If any changes of state occur between the 
temperature limits, the associated entropy changes should be added. For a 
gas at temperature Tthe general expression for the entropy therefore becomes 

o 'Q, cryst din T + ^^ +J^C P ^dln T 

A// r T 

+ -=r SE + Cp^dlnT+S, (4.20) 

* b J T 



/' 

Jo 



All these terms can be measured except the constant S . The evaluation 
of this constant becomes possible by virtue of the third fundamental law of 
thermodynamics. 

12. The approach to absolute zero. The laws of thermodynamics are in- 
ductive in character. They are broad generalizations having an experimental 
basis in certain human frustrations. Our failure to invent a perpetual-motion 
machine has led us to postulate the First Law of Thermodynamics. Our 
failure ever to observe a spontaneous flow of heat from a cold to a hotter 
body or to obtain perpetual motion of the second kind has led to the state- 
ment of the Second Law. The Third Law of Thermodynamics can be based 
on our failure to attain the absolute zero of temperature. A detailed study 
of refrigeration principles indicates that the absolute zero can never be 
reached. 

8 Be careful not to confuse 5, the entropy in the standard state of 1 atm pressure, and 
S" , the entropy at 0K. 



86 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

Most cryogenic systems have depended on the cooling of a gas by an 
adiabatic expansion. This effect was first described by Clement and Desormes 
in 1819. If a container of compressed air is vented to the atmosphere, the 
outrushing gas must do work to push back the gas ahead of it. If the process 
is carried out rapidly enough, it is essentially adiabatic, and the gas is cooled 
by the expansion. 

To obtain continuous refrigeration, some kind of cyclic process must be 
devised; simply opening a valve on a tank of compressed gas is obviously 
unsatisfactory. 7 Two methods of controlled expansion can be utilized: 
(1) a Joule-Thomson expansion through a throttling valve; (2) an expansion 
against a constraining piston. In the latter case, the gas does work against 
the external force and also against its internal cohesive forces. In the Joule- 
Thomson case, only the internal forces are operative, and these change in 
sign as the gas passes through an inversion point. It was shown on page 66 
that in order to obtain cooling // t/ T = (\/C P )[T(dV/dT) P V] must be 
positive. 

In 1860, Sir William Siemens devised a countercurrent heat exchanger, 
which greatly enhanced the utility of the Joule-Thomson method. This was 
applied in the Linde process for the production of liquid air. Chilled com- 
pressed gas is cooled further by passage through a throttling valve. The 
expanded gas passes back over the inlet tube, cooling the unexpanded gas. 
When the cooling is sufficient to cause condensation, the liquid air can be 
drawn off at the bottom of the apparatus. Liquid nitrogen boils at 
77K, liquid oxygen at 90K, and they are easily separated by fractional 
distillation. 

In order to liquefy hydrogen, it is necessary to chill it below its Joule- 
Thomson inversion temperature at 193K; the Linde process can then be 
used to bring it below its critical temperature at 33K. The production 
of liquid hydrogen was first achieved in this way by James- Dewar in 
1898. 

The boiling point of hydrogen is 22K. In 1908, Kammerlingh-Onnes, 
founder of Leiden's famous cryogenic laboratory, used liquid hydrogen to 
cool helium below its inversion point at 100K, and then liquefied it by an 
adaptation of the Joule-Thomson principle. Temperatures as low as 0.84K 
have been obtained with liquid helium boiling under reduced pressures. 
This temperature is about the limit of this method, since enormous pumps 
become necessary to carry off the gaseous helium. 

Let us consider more carefully this cooling produced by evaporating 
liquid from a thermally isolated system. The change in state, liquid -> vapor, 
is a change from the liquid, a state of low entropy and low energy, to the 
vapor, a state of higher entropy and higher energy. The increase in entropy 
on evaporation can be equated to A// vap /r. Since the system is thermally 

7 This method is used, however; in a laboratory device for making small quantities of 
"dry ice,** solid carbon dioxide. 



Sec. 13] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 87 

isolated, the necessary heat of vaporization can come only from the liquid 
itself. Thus the temperature of the liquid must fall as the adiabatic evaporation 
proceeds. 

In 1926, a new refrigeration principle was proposed independently by 
W. F. Giauque 8 and P. Debye. This is the adiabatic demagnetization method. 
Certain rare earth salts have a high paramagnetic susceptibility? i.e., in a 
magnetic field they tend to become highly magnetized, but when the field is 
removed, they lose their magnetism immediately. In 1933, Giauque per- 
formed the following experiment. A sample of gadolinium sulfate was cooled 
to 1.5K in a magnetic field of 8000 oersteds, and then thermally isolated. 
The field was suddenly shut off. The salt lost its magnetism spontaneously. 
Since this was a spontaneous process, it was accompanied by an increase in 
the entropy of the salt. The magnetized state is a state of lower energy and 
lower entropy than the demagnetized state. The change, magnetized -> de- 
magnetized, is therefore analogous to the change, liquid -> vapor, discussed 
in the preceding paragraph. If the demagnetization occurs in a thermally 
isolated system, the temperature of the salt must fall. 

When the field was turned off in Giauque's experiment, the temperature 
fell to 0.25K. In 1950, workers at Leiden 10 reached a temperature of 
0.0014K by this method. Even the measurement of these low temperatures 
is a problem of some magnitude. The helium vapor-pressure thermometer is 
satisfactory down to about 1K. Below this, the Curie- Weiss expression for 
the paramagnetic susceptibility, # ^= const/r, can be used to define a 
temperature scale. 

The fact that we have approached to within a few thousandths of a 
degree of absolute zero does not mean that the remaining step will soon be 
taken. On the contrary, it is the detailed analysis of these low-temperature 
experiments that indicates most definitely that zero degrees Kelvin is 
absolutely unattainable. 

The Third Law of Thermodynamics will, therefore, be postulated as 
follows: "It is impossible by any procedure, no matter how idealized, to 
reduce the temperature of any system to the absolute zero in a finite number 
of operations." 11 

13. The Third Law of Thermodynamics. How does the Third Law 
answer the question of the value of the entropy of a substance at 
T = 0K, the integration constant S Q in eq. (4.20)? Since absolute zero is 
unattainable, it would be more precise to ask what is the limit of S as T 
approaches 0. 

Consider a completely general process, written as a -> b. This may be a 
chemical reaction, a change in temperature, a change in the magnetization, 

8 /. Am. Chem. Soc., 49, 1870 (1927). 

9 Cf. Sec. 1 1-20. 

10 D. de Klerk, M. J. Steenland, and C. J. Gorter, Physica, 16, 571 (1950). 

11 R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics (London: Cam- 
bridge, 1940), p. 224. 



88 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

or the like. The entropies of the system in the two different states a and b 
can be written as : 



(4.21) 

S aQ and S bQ are the limiting entropy values as T approaches zero. 

Let us start with the system a at a temperature T f and allow the process 
a -> b to take place adiabatically and reversibly, the final temperature being 
T" . The entropy must remain constant, so that S a = S b , or 



+J7 Q d In T = 5 &0 +/J' C b dlnT 



In order for the temperature T" in the final state to equal zero, it would be 
necessary to have 

S bQ -S a0 ^*' C a d\nT (4.22) 



As T -> 0, C n -> 0. Now if 5 50 > 5 a0 it is possible to choose an initial T' 
that satisfies this equation, since the integral is a positive quantity. In this 
way the process a -> b could be used to reach the absolute zero starting from 
this T 1 . This conclusion, however, would be a direct contradiction of the 
Third Law, the principle of the unattainability of absolute zero. The only 
escape is to declare that S b0 cannot be greater than 5 a0 . Then there can be 
no 7" that satisfies the condition (4.22). The same reasoning, based upon 
the reverse process b -> a, can be used to show that 5* a0 cannot be greater 
than S bQ . 

Since S aQ can be neither greater than nor less than S 60 , it must be equal 
to S bQ . In order to conform with the principle of the unattainability of 
absolute zero, therefore, it is necessary to have 

5 rt0 -5 60 or AS = (4.23) 

This equation indicates that for any change in a thermodynamic system 
the limiting value of AS as one approaches absolute zero is equal to zero. 
The change in question may be a chemical reaction, a change in physical 
state such as magnetized ^ demagnetized, or in general any change that can 
in principle be carried out reversibly. This requirement of a possible reversible 
process is necessary, since otherwise there would be no way of evaluating 
the AS for the change being considered. 12 The statement in eq. (4.23) is the 

12 This restriction may be a little too severe. In one-component systems, changes of one 
polymorphic crystal to another may also have A5 = 0. Examples are white tin -> grey tin, 
diamond -> graphite, monochnic sulfur -> rhombic sulfur, zinc blende -> wurtzite. The heat 
capacity of the metastable form can be measured at low temperatures, and by extrapolation 
to 0K and assuming 5 0, it is possible to obtain a "Third-Law entropy,*' as defined in 
the next section. 



Sec. 14] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 89 

famous heat theorem first proposed by Walther Nernst in 1906. It has served 
as a useful statement of the Third Law of Thermodynamics. 

Certain types of systems therefore do not fall within the scope of eq. (4.23). 
For example, any reaction that changed the identity of the chemical elements, 
i.e., nuclear transmutation, would not be included, since there is no thermo- 
dynamic method of calculating AS for such a change. This restriction, of 
course, does not affect chemical thermodynamics in any way, since the nuclei 
of the elements retain their identities in any chemical change. 

Another class of changes that must be excluded from eq. (4.23) comprises 
those in which the system passes from a metastable to a more stable state. 
Such changes are essentially irreversible and can proceed in one direction 
only, namely, toward the more stable states. Certain systems can become 
"frozen" in nonequilibrium states at low temperatures. Examples are glasses, 
which can be regarded as supercooled liquids, and solid solutions and alloys, 
in which there is a residual entropy of mixing. At sufficiently low tempera- 
tures, the glass is metastable with respect to the crystalline silicates of which 
it is composed, and the solid solutions are less stable than a mixture of 
pure crystalline metals. Yet the rate of attainment of equilibrium becomes 
so slow in the very cold solids that transformations to the more stable 
states do not occur. Such systems have an extra entropy, which can be 
considered as an entropy of mixing, and this may persist at the lowest tem- 
peratures attainable experimentally. This fact does not contradict eq. (4.23) 
because a change such as "metastable glass ~> crystalline silicates" cannot 
be carried out by a reversible isothermal path. Hence these metastable states 
are said to be "nonaccessible," and the changes do not lie within the scope 
of eq. (4.23). These cases will be discussed later from a statistical point of 
view in Chapter 12. 

14. Third-law entropies. Only changes or differences in entropy have any 
physical meaning in thermodynamics. When we speak of the entropy of a 
substance at a certain temperature, we really mean the difference between its 
entropy at that temperature and its entropy at some other temperature, 
usually 0K. Since the chemical elements are unchanged in any physico- 
chemical process, we can assign any arbitrary values to their entropies at 
0K without affecting in any way the values of AS for any chemical change. 
It is most convenient, therefore, to take the value of 5 for all the chemical 
elements as equal to zero. This is a convention first proposed by Max Planck 
in 1912. 

It then follows, from eq. (4.23), that the entropies of all pure chemical 
compounds in their stable states at 0K are also zero, because for their 
formation from the elements, AS = 0. This formulation is equivalent to 
setting the constant 5 in eq. (4.20) equal to zero. 

It is now possible to use heat-capacity data extrapolated to 0K to deter- 
mine so-called third-law entropies, which can be used in equilibrium calcula- 
tions. As an example, the determination of the standard entropy, S Q 29B , for 



90 



THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 



TABLE 4.5 

EVALUATION OF ENTROPY OF HYDROGEN CHLORIDE FROM HEAT-CAPACITY 
MEASUREMENTS 



Contribution 



1. Extrapolation from 0-1 6K (Debye Theory, Sec. 13-23) 

2. lC,,d\r\ Tfor Solid I from 16 J ~98.36 

3. Transition, Solid I Solid II, 2843/98.36 

4. JO</ln 7 for Solid II from 98.36 r -l 58.91 

5. Fusion, 476.0/158.91 

6. JCV/ln Tfor Liquid from 158.91-! 88.07 

7. Vaporization, 3860/188.07 

8. JCW In Tfor Gas from 188.07-298.15K 



caljdeg mole 

0.30 
7.06 
2.89 
5.05 
3.00 
2.36 
20.52 
3.22 
S ~~~ 5 -44.40 0. 10 



hydrogen chloride gas is shown in Table 4.5. The value S 2 98 44.4 eu 
is that for HC1 at 25C and 1 atm pressure. A small correction due to non- 
ideality of the gas raises the figure to 44.7. A number of third-law-entropies 
are collected in Table 4.6. 

TABLE 4.6 

THIRD-LAW ENTROPIES 

(Substances in the Standard State at 25C) 



Substance 



H 2 

D 2 

He 

N 2 

2 

C1 2 

HC1 

CO 



Mercury 

Bromine 

Water 

Methanol 

Ethanol 



C (diamond) 
C (graphite) 
S (rhombic) 
S (monoclinic) 

Ag 
Cu 
Fe 

Na 



(calldeg mole) 



31.2 
34.4 
29.8 
45.8 
49.0 
53.2 
44.7 
47.3 



17.8 
18.4 
16.8 
30.3 
38.4 



0.6 
1.4 
7.6 
7.8 

10.2 
8.0 
6.7 

12.3 



Gases 



Liquids 



Solids 



Substance 



C0 2 

H 2 

NH 3 

S0 2 

CH 4 

C 2 H 2 

C 2 H 4 

C 2 H, 



Benzene 

Toluene 

Diethylether 

rt-Hexane 

Cyclphexane 



K 

I 2 

NaCl 

KCl 

KBr 

KI 

AgCl 

Hg 2 Cl 2 



298 

(calldeg mole) 



51.1 
45.2 
46.4 
59.2 
44.5 
48.0 
52.5 
55.0 



41.9 
52.4 
60.5 
70.6 
49.2 



16.5 
14.0 
17.2 
19.9 
22.5 
23.4 
23.4 
46.4 



Sec. i5] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 



91 



The standard entropy change AS in a chemical reaction can be calculated 
immediately, if the standard entropies of products and reactants are known. 

AS - 2 ", S? 

One of the most satisfactory experimental checks of the Third Law is pro- 
vided by the comparison of AS values obtained in this way from low- 
temperature heat capacity measurements, with AS values derived either 
from measured equilibrium constants and reaction heats or from the tem- 
perature coefficients of celt emf's eq. (4.3). Examples of such comparisons 
are shown in Table 4.7. The Third Law is now considered to be on a firm 
experimental basis. Its full meaning will become clearer when its statistical 
interpretation is considered in a later chapter. 

The great utility of Third Law measurements in the calculation of 
chemical equilibria has led to an intensive development of low-temperature 
heat-capacity techniques, using liquid hydrogen as a refrigerant. The ex- 
perimental procedure consists essentially in a careful measurement of the 
temperature rise that is caused in an insulated sample by a carefully measured 
energy input. 

We have now seen how thermodynamics has been able to answer the 
old question of chemical affinity by providing a quantitative method for 
calculating (from thermal data alone) the position of equilibrium in chemical 
reactions. 

TABLE 4.7 
CHECKS OF THE THIRD LAW OF THERMODYNAMICS 



Reaction 



Ag (c) f i Br 2 (1) - AgBr (c) 
Ag (c) + * C1 2 (g) - AgCl (c) 
Zn (c) + J 2 (g) = ZnO (c) 
C M 2 (g) = CO (g) 
CaC0 3 (c) - CaO (c) + CO 2 (g) 



Temp. 
(K) 


Third Law AS 
(cal/deg mole) 


Experimental 

AS 


Method 


265.9 


-3.01 0.40 


-3.02 0.10 


emf 


298.16 


-13.85 0.25 


-13.73 0.10 


emf 


298.16 


-24.07 0.25 


-24.24 0.05 


K and Atf 


298.16 


-20.01 0.40 


-2 1.38 0.05 


#andA# 


298.16 


38.40 0.20 


38.03 0.20 


K and Atf 



15. General theory of chemical equilibrium: the chemical potential. We 

have so far confined our attention to equilibria involving ideal gases. The 
relations discovered are of great utility, and are accurate enough for the 
discussion of most homogeneous gas equilibria. Some gas reactions, how- 
ever, are carried out under such conditions that the ideal gas laws are no 
longer a good approximation. Examples include the high-pressure syntheses 
of ammonia and methanol. In addition, there are the great number of 
chemical reactions that occur in condensed phases such as liquid or even 
solid solutions. In order to treat these reactions especially, a more general 
equilibrium theory will be needed. 



92 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

The composition of a system in which a chemical reaction is taking place 
is continually changing, and the state of the system is not defined by specifying 
merely the pressure, volume, and temperature. In order to discuss the changes 
of composition it is necessary to introduce, in addition to P 9 V, and T, new 
variables that are a measure of the amount of each chemical constituent of 
the system. As usual, the mole will be chosen as the chemical measure, with 
the symbols n l9 w 2 n 3 . . . n l representing the number of moles of constituent 
1, 2, 3, or i. 

It then follows that each thermodynamic function depends on these /i/s 
as well as on P, V, and T. Thus, E - E(P, V, T, n t ); F = F(P, V, T, n,\ etc. 
Consequently, a perfect differential, for example of the free energy, becomes 



By eq. (3.29) dF ~ ~S dT + V dP for any system of constant composition, 
i.e., when all dn % 0. Therefore 



= -SdT + VdP + - dn, (4.25) 

T,P,n, 

The coefficient (dF/dn t ) T P n> , first introduced by Gibbs, has been given a 
special name because of its great importance in chemical thermodynamics. 
It is called the chemical potential, and is written as 

(4 ' 26) 

It is the change of the free energy with change in number of moles t of 
component /, the temperature, the pressure, and the number of moles of all 
other components in the system being kept constant. Using the new symbol, 
eq. (4.25) becomes 

dF - -5 dT + VdP + 2 Hi dn, (4.27) 

i 

At constant temperature and pressure, 

</F=2^<**. (4-28) 

The condition for equilibrium, dF 0, then becomes 

I to **< = <> (4.29) 

i 

For an ideal gas, the chemical potential is simply the free energy per mole 
at pressure P t . Therefore from eq. (4.7), 

(4.30) 



The value of //, for the ideal gas is the same whether the ideal gas is pure 
gas at a pressure P t or is in an ideal gas mixture 13 at partial pressure P t . If, 

13 This statement is a definition of an ideal gas mixture. To be precise, one must distin- 
guish an ideal gas mixture from a mixture of ideal gases. There might be specific interactions 
between two ideal gases that would cause their mixture to deviate from ideality. 



Sec. 16] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 93 

however, the gas mixture is not ideal, this identity no longer holds true. 
Various interaction forces come into operation, and the evaluation of ^ { 
becomes a separate experimental problem in each case. 

16. The fugacity. Because relations such as eq. (4.30) lead to equations 
of such simple form in the development of the theory of chemical equilibrium, 
it is convenient to introduce a new function, called the fugacity of the sub- 
stance, that preserves the form of eq. (4.30) even for nonideal systems. 
Therefore we write 

dfi = VdP = RTdlnf, and /i, - - RTln^ 

J i 

where/- is the fugacity of the substance, and/? is its fugacity in the standard 
state. It now becomes desirable to change the definition of the standard state 
so that instead of the state of unit pressure, it becomes the state of unit 
fugacity,/? = 1. Then 

Vi-tf-RTlnfi (4.31) 

Now the treatment of equilibrium in Section 4-5 can be carried through 
in terms of the fugacity and chemical potential. This leads to an expression 
for the equilibrium constant which is true in general, not only for real 
(nonideal) gases but also for substances in any state of aggregation what- 
soever: 

f cf d 



f ~~ f af b 
JA JB 

-A// - RT \nK f (4.32) 

The fugacity of a pure gas or of a gas in a mixture can be evaluated if 
sufficiently detailed PVT data, are available. This discussion will be limited 
to an illustration of the method for determining the fugacity of a pure gas. 
In this case, 

dF=dp=VdP (4.33) 

If the gas is ideal, V = RT/P. For a nonideal gas, this is no longer true. We 
may write a - F ideal - K real - (RT/P) - K, whence V = (RT/P) ~ a. Sub- 
stituting this expression into eq. (4.33), we find 

RTdlnf - dF=dfA = RTctlnP - adP 
The equation is integrated from P = to P. 

RT\ f dlnf=RTf P d\nP-( P adP 
J/,p=o J JP-O Jo 

As its pressure approaches zero, a gas approaches ideality, and for an ideal 
gas the fugacity equals the pressure, /= P [cf. eqs. (4.30 and (4.31)]. The 
lower limits of the first two integrals must therefore be equal, so that we 
obtain 

RT\nf= RT In P -J P <2 dP (4.34) 



94 



THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 



This equation enables us to evaluate the fugacity at any pressure and 
temperature, provided PVT data for the gas are available. If the deviation 
from ideality of the gas volume is plotted against P, the integral in eq. (4.34) 
can be evaluated graphically. Alternatively, an equation of state can be used 
to calculate an expression for a as a function of P, making it possible to 
evaluate the integral by analytical methods. 

The fugacity may be thought of as a sort of idealized pressure, which 
measures the true escaping tendency of a gas. In Chapter 1, it was pointed 




V 2 



6 8 10 12 14 16 18 20 22 24 
REDUCED PRESSURF-Ffe 



Fig. 4.4. Variation of activity coefficient with reduced pressure at 
various reduced temperatures. 

out that the deviations of gases from ideality are approximately determined 
by their closeness to the critical point. This behavior is confirmed by the 
fact that at the same reduced pressures all gases have approximately the 
same ratio of fugacity to pressure. The ratio of fugacity to pressure is called 
the activity coefficient, y =f/P. Figure 4.4 shows a family of curves 14 relating 
the activity coefficient of a gas to its reduced pressure P K at various values 
of the reduced temperature T lf . To the approximation that the law of corre- 
sponding states is valid, all gases have the same value of y when they are in 
corresponding states, i.e., at equal P R and T I{ . This is a very useful principle, 

14 Newton, Ind. Eng. Chem., 27, 302 (1935). Graphs for other ranges of P R and T R are 
included in this paper. 



Sec. 17] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 



95 



for it allows us to estimate the fugacity of a gas solely from a knowledge of 
its critical constants. 

17. Use of fugacity in equilibrium calculations. Among the industrially 
important gas reactions that are carried out under high pressures is the 
synthesis of ammonia: | N 2 + $ H 2 = NH 3 . This reaction has been carefully 
investigated up to 1000 atm by Larson and Dodge. 15 The per cent of NH 3 
in equilibrium with a three-to-one H 2 -N 2 mixture at 450C and various total 
pressures is shown in Table 4.8. In the third column of the table are the 
values of K p = PyuJPy^Pa?' 2 calculated from these data. 

Since K p for ideal gases should be independent of the pressure, these 
results indicate considerable deviations from ideality at the higher pressures. 
Let us therefore calculate the equilibrium constant K f using Newton's graphs 
to obtain the fugacities. We are therefore adopting the approximation that 
the fugacity of a gas in a mixture is determined only by the temperature and 
by the total pressure of the gas mixture. 

Consider the calculation of the activity coefficients at 450C (723K) and 
600 atm. 

PC T C P R T R r 

N 2 '33.5 126 17.9 5.74 1.35 

H 2 . 12.8 33.3 46.8 21.7 1.19 

NH 3 . . . 111.5 405.6 5.38 1.78 0.85 

The activity coefficients y are read from the graphs, at the proper values of 
reduced pressure P H and reduced temperature T R . (Only the NH 3 values are 
found in Fig. 4.4; the complete graphs must be consulted for the other gases.) 

TABLE 4.8 
EQUILIBRIUM IN THE AMMONIA SYNTHESIS AT 450C WITH 3 : 1 RATIO OF H 2 TO N 2 



Total 


Per cent 








Pressure 


NH 3 at 


K v 


K Y 




(atm) 


Equilibrium 








10 


2.04 


0.00659 


0.995 


0.( 


30 


5.80 


0.00676 


0.975 


O.C 


50 


9.17 


0.00690 


0.945 


O.C 


100 


16.36 


0.00725 


0.880 


O.C 


300 


35.5 


0.00884 


0.688 


O.C 


600 


,53.6 


0.01294 


0.497 


O.C 


1000 


69.4 


0.02328 


0.434 


O.C 



Since the fugacity/^ yP, we can write in general K f = K Y K^ where 
in this case K v = ^H^N^H, 372 - The values of K v and K f are shown in 
Table 4.8. There is a marked improvement in the constancy of K f as com- 
pared with K v . Only at 1000 atm does the approximate treatment of the 
fugacities appear to fail. To carry out an exact thermodynamic treatment, it 

15 /. Am. Chem. Soc., 45, 2918 (1923); 46, 367 (1924). 



96 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

would be necessary to calculate the fugacity of each gas in the particular 
mixture under study. This would require very extensive PVT data on the 
mixture. 

Often, knowing AF for the reaction, we wish to calculate the equilibrium 
concentrations in a reaction mixture. The procedure is to obtain K f from 
AF = RTln K f , to estimate K y from the graphs, and then to calculate 
the partial pressures from K p K f /K y . 

PROBLEMS 

1. The emf of the cadmium-calomel cell in which the reaction is Cd + 
Hg 2 ++ = Cd++ + 2 Hg, can be represented by: ? = 0.6708 - 1.02 x 
10~ 4 (/ 25) 2.4 x 10~ 6 (r - 25) 2 , where t is the centigrade temperature. 
Calculate AF, AS, and A// for the cell reaction at 45C. 

2. From the standard free energies in Table 4.2 calculate A/ 70 and K v at 
25C for the following reactions : 

(a) N 2 O + 4 H 2 - 2 NH 3 + H 2 O (g) 

(b) H 2 2 (g)-H 2 0(g) + |0 2 
(c) CO + H 2 O (1) = HCOOH (1) 

3. At 900K the reaction C 2 H 6 = C 2 H 4 + H 2 has A// - 34.42, AF 
= 5.35 kcal. Calculate the per cent H 2 present at equilibrium if pure C 2 H 6 
is passed over a dehydrogenation catalyst at this temperature and 1 atm 
pressure. Estimate the per cent H 2 at equilibrium at 1000K. 

4. If an initial mixture of 10 per cent C 2 H 4 , 10 per cent C 2 H 6 , and 80 per 
cent N 2 is passed over the catalyst at 900K and 1 atm, what is the per cent 
composition of effluent gas at equilibrium? What if the same mixture is used 
at 100 atm? (Cf. data in Problem 3.) 

5. The equilibrium LaCl 3 (s) + H 2 O (g) - LaOCl (s) + 2 HCl (g). [/. 
Am. Chem. Soc. 9 74, 2349 (1952)] was found to have K p - 0.63 at 804K, 
and 0.125 at 733K. Estimate A// for the reaction. If the equilibrium HCI 
vapor pressure at 900K is 2.0mm estimate the equilibrium H 2 O vapor 
pressure. 

6. From the data in Table 4.4, calculate the heat of dissociation of O 2 
into 2 O at 1000K. Similarly, calculate A// 1000 for H 2 = 2 H. Assuming 
atomic H and O are ideal gases with C P = f /*, and using the Q>'s for H 2 
and O 2 in Table 2.4, calculate A// 298 for 2 H + O = H 2 O (g). The heat of 
formation of H 2 O(g) is 57.80 kcal. One-half the heat calculated in this 
problem is a measure of the "strength of the O H bond" in water. 

7. For the reaction N 2 O 4 2 NO 2 , calculate K P , K x , K c at 25C. and 
1 atm from the free energies of formation of the compounds (Table 4.2). 

8. PC1 5 vapor decomposes on heating according to PC1 5 = PC1 3 + C1 2 . 
The density of a sample t>f partially dissociated PC1 5 at 1 atm and 230C was 



15 
0.89 


20 
2.07 


30 
4.59 


40 
6.90 


50 
8.53 


60 
9.76 


70 
10.70 


80 
11.47 


90 
12.10 


100 
12.62 


120 
13.56 


140 
14.45 


160 
15.31 


180 
16.19 


200 
17.08 


220 
17.98 


240 
18.88 


260 
25.01 


280 

25.17 


300 

25.35 



Chap. 4] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 97 

found to be 4.80 g per liter. Calculate the degree of dissociation a and AF 
for the dissociation at 230C. 

9. The following results were obtained for the degree of dissociation of 
CO 2 (CO 2 = CO + i O 2 ) at 1 atm: 

K . . . 1000 1400 2000 

a . . . 2.0 x 10~ 7 1.27 x 10~ 4 1.55 x 10~ 2 

What is AS for the reaction at 1400K? 

10. The free energy of formation of H 2 S is given by AF = 19,200 
+ 0.94rin T - 0.001 65T 2 - 0.00000037r 3 + 1-657. H 2 + J S 2 (g) - H 2 S 
(g). If H 2 S at 1 atm is passed through a tube heated to 1200K, what is per 
cent H 2 in the gas at equilibrium? 

11. Jones and Giauque obtained the following values for C P of nitro- 
methane. 16 

K 
C P 

K 
C P 

The melting point is 244.7K, heat of fusion 2319cal per mole. The 
vapor pressure of the liquid at 298. 1K is 3.666 cm. The heat of vaporization 
at 298. 1K is 9147 cat per mole. Calculate the Third-Law entropy of CH 3 NO 2 
gas at 298. 1K and 1 atm pressure (assuming ideal gas behavior). 

12. Using the Third-Law entropies in Table 4.6 and the standard heats 
of formation calculate the equilibrium constants at 25C of the following 
reactions : 

H 2 + C1 2 - 2 HC1 
CH 4 + 2 2 = C0 2 + 2 H 2 (g) 
2Ag(s) + Cl 2 -2AgCi(s) 

13. For the reaction CO + 2 H 2 - CH 3 OH (g), AF - -3220 cal at 
700K. Calculate the per cent CH 3 OH at equilibrium with a 2 : 1 mixture of 
H 2 + CO at a pressure of 600 atm using (a) ideal gas law, (b) Newton's 
fugacity charts. 

14. At high temperature and pressure, a quite good equation of state for 
gases is P(V b) = RT. Calculate the fugacity of N 2 at 1000 atm and 
1000C according to this equation, if b = 39.1 cc per mole. 

15. Show that 



T,F,n, \s,P f n, W t ' S,V,n, 

16. Amagat measured the molar volume of CO 2 at 60C. 

Pressure, atm . . 13.01 35.42 53.65 74.68 85.35 

Volume, cc . . 2000.0 666.7 400.0 250.0 200.0 

16 /. Am. Chem. Soc., 69, 983 (1947). 



98 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 

Calculate the activity coefficient y = f/P for CO 2 at 60C and pressures of 
10, 20, 40, and 80 atm. 

17. When rt-pentane is passed over an isomerization catalyst at 600 K, 
the following reactions occur : 

(A) CH 3 CH 2 CH 2 CH 2 CH 3 - CH 3 CH(CH 3 )CH 2 CH 3 (B) 

- C(CH 3 ) 4 (C) 

The free energies of formation at 600K are: (A) 33.79, (B) 32.66, (C) 35.08 
kcal per mole. Calculate the composition of the mixture when complete 
equilibrium is attained. 

18. For the reaction 3 CuCl (g) - Cu 3 C) 3 (g), Brewer and Lofgren [J. 
Am. Chem. Soc., 72, 3038 (1950)] found AF - -126,400 - 12.5iriog T 
+ 104.7 T. What are the A// and AS of reaction at 2000K? What is the 
equilibrium mole fraction of trimer in the gas at 1 atm and 2000K? 

REFERENCES 

BOOKS 

1. Kubaschewski, O., and E. L. Evans, Metallurgical Thermochemistry 
(London: Butterworth, 1951). 

2. Putnam, P. C., Energy in the Future (New York: Van Nostrand, 1953). 

3. Squire, C. F., Low Temperature Physics (New York: McGraw-Hill, 1953). 

4. Wenner, R. R., Thermochemical Calculations (New York: McGraw-Hill, 
1941). 

Also see Chapter 1, p. 25. 

ARTICLES 

1. Chem. Revs., 39, 357-481 (1946), "Symposium on Low Temperature 
Research." 

2. Daniels, F., Scientific American, 191, 58-63 (1954), "High Temperature 
Chemistry," 

3. Huffman, H. M.,' Chem. Revs., 40, 1-14 (1947), "Low Temperature 
Calorimetry." 

4. Lemay, P., and R. Oesper, /. Chem. Ed., 23, 158-65, 230-36 (1946), 
"Claude Berthollet " 

5. Oesper, R., /. Chem. Ed., 21, 263-64 (1944), "H. Kammerlingh-Onnes." 

6. Urey, H. C., /. Chem. Soc., 562-81 (1947), "Thermodynamic Properties 
of Isotopic Substances." 

7. Walden, P., /. Chem. Ed., 31, 27-33 (1954), "Beginnings of the Doctrine 
of Chemical Affinity." 

8. Watson, R. G., Research, 7, 34-40 (1954), "Electrochemical Generation 
of Electricity." 



CHAPTER 5 

Changes of State 



1. Phase equilibria. Among the applications of thermodynamics is the 
study of the equilibrium conditions for changes such as the melting of ice, 
the solution of sugar, the vaporization of benzene, or the transformation of 
monoclinic to rhombic sulfur. Certain fundamental principles are applicable 
to all such phenomena, which are examples of "changes in state of aggrega- 
tion" or "phase changes." 

The word phase is derived from the Greek (pa.ai<t, meaning "appearance." 
If a system is "uniform throughout, not only in chemical composition, but 
also in physical state," 1 it is said to be homogeneous, or to consist of only 
one phase. Examples are a volume of air, a noggin of rum, or a cake of ice. 
Mere difference in shape or in degree of subdivision is not enough to deter- 
mine a new phase. Thus a mass of cracked ice is still only one phase. 2 

A system consisting of more than one phase is called heterogeneous. 
Each physically or chemically different, homogeneous, and mechanically 
separable part of a system constitutes a distinct phase. Thus a glassful of 
water with cracked ice in it is a two-phase system. The contents of a flask 
of liquid benzene in contact with benzene vapor and air is a two-phase 
system; if we add a spoonful of sugar (practically insoluble in benzene) we 
obtain a three-phase system: a solid, a liquid, and a vapor phase. 

In systems consisting entirely of gases, only one phase can exist at equi- 
librium, since all gases are miscible in all proportions (unless, of course, a 
chemical reaction intervenes, e.g., NH 3 + HC1). With liquids, depending on 
their mutual miscibility, one, two, or more phases can arise. Many different 
solid phases can coexist. 

2. Components. The composition of a system may be completely de- 
scribed in terms of the "components" that are present in it. The ordinary 
meaning of the word "component" is somewhat restricted in this technical 
usage. We wish to impose a requirement of economy on our description of 
the system. This is done by using the minimum number of chemically distinct 
constituents necessary to describe the composition of each phase in the 
system. The constituents so chosen are the components. If the concentrations 
of the components are stated for each phase, then the concentrations in each 
phase of any and all substances present in the system are uniquely fixed. 
This definition may be expressed more elegantly by saying that the com- 

1 J. Willard Gibbs. 

2 This is because we are assuming, at this stage in our analysis, that a variable surface 
area has no appreciable effect on the properties of a substance. 

QO 



100 CHANGES OF STATE [Chap. 5 

ponents are those constituents whose concentrations may be independently 
varied in the various phases. 

Consider, for example, a system consisting of liquid water in contact 
with its vapor. We know that water is composed of hydrogen and oxygen, 
but these elements are always present in fixed and definite proportions. The 
system therefore contains one component only. 

Another example is the system consisting of calcium carbonate, calcium 
oxide, and carbon dioxide. A chemical reaction between these compounds is 
possible, CaCO 3 CaO + CO 2 . In this case, three phases are present, 
gaseous CO 2 , solid CaCO 3 and CaO. Two components are required in order" 
to describe the composition of all of these phases, the most suitable choice 
being CaO and CO 2 . 

A less obvious example is the system formed by water and two salts 
without a common ion, e.g., H 2 O, NaCl, KBr. As a result of interaction 
between ions in solution four different salts, or their hydrates, may occur in 
solid phases, namely NaCl, KBr, NaBr, KC1. In order to specify the com- 
position of all possible phases, four components are necessary, consisting of 
water and three of the possible salts. This fixes the concentrations of three 
of the four ions in any phase, and the fourth is fixed by the requirement of 
over-all electrical neutrality. 

Careful examination of each individual system is necessary in order to 
decide the best choice of components. It is generally wise to choose as com- 
ponents those constituents that cannot be converted into one another by 
reactions occurring within the system. Thus CaCO 3 and CaO would be a 
possible choice for the CaCO 3 CaO + CO 2 system, but a poor choice 
because the concentrations of CO 2 would have to be expressed by negative 
quantities. While the identity of the components is subject to some degree 
of choice, the number of components is always definitely fixed for any 
given case. 

Even the last statement should perhaps be modified, because the actual 
choice of the number of components depends on how precisely one wishes 
to describe a system. In the water system, there is always some dissociation 
of water vapor into hydrogen and oxygen. At moderate temperatures, this 
dissociation is of no consequence in any experimental measurements, and to 
consider it in deciding the number of components would be unduly scrupu- 
lous. 3 The precision with which experimental data on the system can be 
obtained should be allowed to decide borderline cases. 

3. Degrees of freedom. For the complete description of a system, the 
numerical values of certain variables' must be reported. These variables are 

3 It is worth noting that the mere dissociation of water into hydrogen and oxygen does 
not create new components, because the proportion of H 2 to O 2 is always fixed at 2:1, 
since we exclude the possibility that additional H 2 or O 2 can be added to the system. 
The reason why an extra component, either H 2 or O 2 , might conceivably be required is 
that H 2 and O 2 dissolve to different extents in the water, so that their ratio is no longer 
fixed at 2:1 in each phase. 



Sec. 4] CHANGES OF STATE 101 

chosen from among the "state functions" of the system, such as pressure, 
temperature, volume, energy, entropy, and the concentrations of the various 
components in the different phases. Values for all of the possible variables 
need not be explicitly stated, for a knowledge of some of them definitely 
determines the values of the others. For any complete description, however, 
at least one capacity factor is required, since otherwise the mass of the system 
is undetermined, and one is not able, for example, to distinguish between a 
system containing a ton of water and one containing a few drops. 

An important feature of equilibria between phases is that they are in- 
dependent of the actual amounts of the phases that may be present. 4 Thus 
the vapor pressure of water above liquid water in no way depends on the 
volume of the vessel or on whether a few milliliters or many gallons of water 
are in equilibrium with the vapor phase. Similarly, the concentration of a 
saturated solution of salt in water is a fixed and definite quantity, regardless 
of whether a large or a small excess of undissolved salt is present. 

In discussing phase equilibria, we therefore need not consider the capacity 
factors, which express the absolute bulk of any phase. We consider only the 
intensity factors, such as temperature, pressure, and concentrations. Of these 
variables a certain number may be independently varied, but the rest are 
fixed by the values chosen for the independent variables and by the thermo- 
dynamic requirements for equilibrium. The number of the intensive state 
variables that can be independently varied without changing the number of 
phases is called the number of degrees of freedom of the system, or sometimes 
the variance. 

For example, the state of a certain amount of a pure gas may be specified 
completely by any two of the variables, pressure, temperature, and density. 
If any two of these are known, the third can be calculated. This is therefore 
a system with iwo degrees of freedom, or a bi variant system. 

In the system "water water vapor," only one variable need be specified 
to determine the state. At any given temperature, the pressure of vapor in 
equilibrium with liquid water is fixed in value. This system has one degree 
of freedom, or is said to be univariant. 

4. Conditions for equilibrium between phases. In a system containing 
several phases, certain thermodynamic requirements for the existence of 
equilibrium may be derived. 

For thermal equilibrium it is necessary that the temperatures of all the 
phases be the same. Otherwise, heat would flow from one phase to another. 
This intuitively recognized condition may be proved by considering two 
phases a and /? at temperatures r a , T ft . The condition for equilibrium at 
constant volume and composition is given on p. 59 as dS 0. Let 5 a and 
S ft be the entropies of the two phases, and suppose there were a transfer of 
heat dq from a to /? at equilibrium. 

4 This statement is proved in the next Section. It is true as long as surface area variations 
are left out of consideration. (See Chapter 16.) 



102 CHANGES OF STATE [Chap. 5 

Then dS = dS* + dS ft = or - -| + -| - 

whence 7 a - 7* (5.1) 

For mechanical equilibrium it is necessary that the pressures of all the 
phases be the same. Otherwise, one phase would increase in volume at the 
expense of another. This condition may be derived from the equilibrium 
condition at constant over-all volume and temperature, dA 0. Suppose 
one phase expanded into another by 6V. Then 



or P - Pft (5.2) 

In addition to the conditions given by eqs. (5.1) and (5.2), a condition is 
needed that expresses the requirements of chemical equilibrium. Let us con- 
sider the system with phases a and ft maintained at constant temperature 
and pressure, and denote by n*, /?/*, the numbers of moles of some particular 
component / in the two phases. From eq. (3.28) the equilibrium condition 
becomes dF 0, or 

dF | - tf*-=0 (5.3) 



Suppose that a process occurred by which dn t moles of component / were 
taken from phase a and added to phase ft. (This process might be a chemical 
reaction or a change in aggregation-state.) Then, by virtue of eq. (4.28), 
eq. (5.3) becomes 

-^ /i f to, + /*//!, = 

or ^ - p* (5.4) 

This is the general condition for equilibrium with respect to transport of 
matter between phases, including chemical equilibrium between phases. For 
any component / in the system, the value of the chemical potential /^ must 
be the same in every phase. 

An important symmetry between the various equilibrium conditions is 
apparent in the following summary: 

Capacity Intensity Equilibrium 

factor factor condition 

S T T* = TP 

V P P = Pft 



5. The phase rule. Between 1875 and 1878, Josiah Willard Gibbs, Pro- 
fessor of Mathematical Physics at Yale University, published in the Trans- 
actions of the Connecticut Academy of Sciences a series of papers entitled 
"On the Equilibrium of Heterogeneous Substances." In these papers Gibbs 
disclosed the entire science, of heterogeneous equilibrium with a beauty and 
preciseness never before and seldom since seen in thermodynamic studies. 



Sec. 5] CHANGES OF STATE 103 

Subsequent investigators have had little to do save to provide experimental 
illustrations for Gibbs's equations. 

The Gibbs phase rule provides a general relationship among the degrees 
of freedom of a system/, the number of phases /?, and the number of com- 
ponents c. This relationship always is 

f^c-p + 2 (5.5) 

The derivation proceeds as follows: 

The number of degrees of freedom is equal to the number of intensive 
variables required to describe a system, minus the number that cannot be 
independently varied. The state of a system containing p phases and c com- 
ponents is specified at equilibrium if we specify the temperature, the pressure, 
and the amounts of each component in each phase. The total variables 
required in order to do this are therefore pc -\- 2. 

Let n* denote the number of moles of a component / in a phase a. Since 
the size of the system, or the actual amount of material in any phase, does 
not affect the equilibrium, we are really incerested in the relative amounts of 
the components in the different phases and not in their absolute amounts. 
Therefore, instead of the mole numbers n* 9 the mole fractions X? should be 
used. These are given by 



For each phase, the sum of the mole fractions equals unity. 

Xf + X,' + AV [-... -+ X? = 1 
or 2 X>" =- 1 (5-6) 

I 

If all but one mole fraction are specified, that one can be calculated from 
eq. (5.6). If there are/? phases, there are/? equations similar to eq. (5.6), and 
therefore p mole fractions that need not be specified since they can be cal- 
culated. The total number of independent variables to be specified is thus 
pc + 2 p or p(c 1) + 2. 

At equilibrium, the eqs. (5.4) impose a set of further restraints on the 
system by requiring that the chemical potentials of each component be the 
same in every phase. These conditions are expressed by a set of equations 
such as : 

V* = fit = /*!" = ... 

^ - ^ = ^ - - ' - (5.7) 



Each equality sign in this set of equations signifies a condition imposed on 
the system, decreasing its variance by one. Inspection shows that there are 
therefore c(p 1) of these conditions. 



104 



CHANGES OF STATE 



[Chap. 5 



The degrees of freedom equal the total required variables minus the 
restraining conditions. Therefore 

f-=p(c- l) + 2-c(/>- 1) 

f=c-p + 2 (5.8) 

6. Systems of one component water. In the remainder of this chapter, 
systems of one component will be considered. These systems comprise the 
study of the conditions of equilibrium in changes in the state of aggregation 
of pure substances. 

From the phase rule, when c* !,/= 3 /?, and three different cases 
are possible: 

p ^ l,/ r - 2 bi variant system 
p --= 2,f^= 1 univariant system 
p _z_- 3,/ invariant system 

These situations may be illustrated by the water system, with its three 
familiar phases, ice, water, and steam. Since the maximum number of degrees 
of freedom is two, any one-component system can be represented by a two- 
dimensional diagram. The most convenient variables are the pressure and 
the temperature. The water system is shown in Fig. 5.1. 




.0075 100 
TEMPERATURE - *C 



374 



Fig. 5.1. The water system schematic. (Not drawn to scale.) 

The diagram is divided into three areas, the fields of existence of ice, 
water, and steam. Within these single-phase areas, the system is bivariant, 
and pressure and temperature may be independently varied. 



Sec. 7] CHANGES OF STATE 105 

Separating the areas are lines connecting the points at which two phases 
may coexist at equilibrium. Thus the curve AC dividing the liquid from the 
vapor region is the familiar vapor-pressure curve of liquid water. At any 
given temperature there is one and only one pressure at which water vapor 
is in equilibrium with liquid water. The system is univariant, having one 
degree of freedom. The curve AC has a natural upper limit at the point C, 
which is the critical point, beyond which the liquid phase is no longer 
distinguishable from the vapor phase. 

Similarly, the curve AB is the sublimation-pressure curve of ice, giving 
the pressure of water vapor in equilibrium with solid ice, and dividing the 
ice region from the vapor region. 

The curve AD divides the solid-ice region from the liquid-water region. 
It shows how the melting temperature of ice or the freezing temperature of 
water varies with the pressure. It is still an open question whether such 
curves, at sufficiently high pressures, ever have a natural upper limit beyond 
which solid and liquid are indistinguishable. 

These three curves intersect at a point A, at which solid, liquid, and vapor 
are simultaneously at equilibrium. This point, which occurs at 0.0075C and 
4.579 mm pressure, is called a triple point. Since three phases coexist, the 
system is invariant. There are no degrees of freedom and neither pressure 
nor temperature can be altered even slightly without causing the disappear- 
ance of one of the phases. 

It should be noted that this triple point is not the same as the ordinary 
melting point of ice, which by definition is the temperature at which ice and 
water are in equilibrium under an applied pressure of 1 atm or 760 mm. 
This temperature is, by definition, 0C. 

Liquid water may be cooled below its freezing point without solidifying. 
In AE we have drawn the vapor-pressure curve of this supercooled water, 
which is a continuous extension of curve AC. It is shown as a dotted line 
on the diagram since it represents a metastable system. Note that the meta- 
stable vapor pressure of supercooled water is higher than the vapor pressure 
of ice. 

The slope of the curve A D, the melting-point curve, is worth remarking. 
It shows that the melting point of ice is decreased by increasing pressure. 
This is a rather unusual behavior; only bismuth and antimony among 
common substances behave similarly. These substances expand on freezing. 
Therefore the Le Chatelier principle demands that increasing the pressure 
should lower the melting point. The popularity of ice skating and the flow 
of glaciers are among the consequences of the peculiar slope of the melting 
point curve for ice. For most substances, the density of the solid is greater 
than that of the liquid, and by Le Chatelier's principle, increase in pressure 
raises the melting point. 

7. The Clapeyron-CIausius equation. There are two fundamental theoreti- 
cal equations that govern much of the field of phase equilibrium. The first 



106 CHANGES OF STATE [Chap. 5 

is the Gibbs phase rule, which determines the general pattern of the phase 
diagram. The second is the Clapeyron-Clausius equation, which determines 
the slopes of the lines in the diagram. It is a quantitative expression for the 
Le Chatelier principle as it applies to heterogeneous systems. First proposed 
by the French engineer Clapeyron in 1834, it was placed on a firm thermo- 
dynamic foundation by Clausius, some thirty years later. 

From eq. (5.4) the condition for equilibrium of a component / between 
two phases, a and /7, is ju^ -= ///. For a system of one component, the 
chemical potentials // are identical with the free energies per mole F, so that 
F* F ft at equilibrium. Consider two different equilibrium states, at slightly 
separated temperatures and pressures : 

(1) T,P, F* - F ft . 

(2) T + dT, P + dP, F* + dF* - F ft \- dF?. 

It follows that dF* - dF fl . The change in F with T and P is given by 
eq. (3.29), dF = V dP - S dT. Therefore, V dP - 5 a dT - V & dP - S? dT 9 or 

dP_S'-S^AS 

dT VP - K a AK ^ ' ' 

If the heat of the phase transformation is /I, AS is simply A/T where T 
is the temperature at which the phase change is occurring. The Clapeyron- 
Clausius equation is now obtained as 

^ - -A (5 10) 

dT (5 ' 10) 



This equation is applicable to any change of state: fusion, vaporization, 
sublimation, and changes between crystalline forms, provided the appro- 
priate latent heat is employed. 

In order to integrate the equation exactly, it would be necessary to know 
both X and AK as functions of temperature and pressure. 5 The latter corre- 
sponds to a knowledge of the densities of the two phases over the desired 
temperature range. In most calculations over short temperature ranges, 
however, both X and AKmay be taken as constants. 

In the case of the change "liquid ^ vapor," several approximations are 
possible, leading to a simpler equation than eq. (5.10), 



dT 

Neglecting the volume of the liquid compared with that of the vapor, and 
assuming ideal gas behavior for the latter, one obtains 

d In P _ A vap 

~W = Kf* (5 ' 12) 

6 A good discussion of the temperature variation of A is given by Guggenheim, Modern 
Thermodynamics y p. 57. The variation with pressure of A and A Kis much less than that with 
temperature. 



Sec. 8] CHANGES OF STATE 107 

A similar equation would be a good approximation for the sublimation 
curve. 

Just as was shown for eq. (3.36), this may also be written 

wTn = ~R (5 - 13) 

If the logarithm of the vapor pressure is plotted against 1/r, the slope of 
the curve at any point multiplied by R yields a value for the heat of vapori- 
zation. In many cases, since X is effectively constant over short temperature 
ranges, a straight-line plot is obtained. This fact is useful to remember in 
extrapolating vapor pressure data. 

When A is taken as constant, the integrated form of eq. (5.12) is 

ln -h -*(?-) (5 - 14) 

An approximate value for A vap can often be obtained from Troutorfs 
Rule (1884): 

^ & 22 cal deg" 1 mole" 1 

The rule is followed fairly well by many nonpolar liquids (Sec. 14-8). It is 
equivalent to the statement that the entropy of vaporization is approximately 
the same for all such liquids. 

8. Vapor pressure and external pressure. It is of interest to consider the 
effect of an increased hydrostatic pressure on the vapor pressure of a liquid. 
Let us suppose that an external hydrostatic pressure P p is imposed on a 
liquid of molar volume V v Let the vapor pressure be P, and the molar 
volume of the vapor V g . Then at equilibrium at constant temperature: 

</F vap = rfF llq or V g dP - V, dP e 
or ~ Vl (5.15) 

dl f V fl 

This is sometimes called the Gibbs equation. If the vapor is an ideal gas, this 
equation becomes 

"(-.SrH 

Since the molar volume of the liquid does not vary greatly with pressure, 
this equation may be integrated approximately, assuming constant V t : 

n \r f n n '\ 



In theory, one can measure the vapor pressure of a liquid under an 
applied hydrostatic pressure in only two ways: (1) with an atmosphere of 
"inert" gas; (2) with an ideal membrane semipermeable to the vapor. In 



108 



CHANGES OF STATE 



[Chap. 5 



practice, the inert gas will dissolve in the liquid, so that the application of 
the Gibbs equation to the problem is dubious. The second case is treated in 
the theory of osmotic pressure. 

As an example of the use of eq. (5.16), let us calculate the vapor pressure of 
mercury under an external pressure of 1000 atm at 100C. The density is 
13.352 gem- 3 ; hence V, = M/p = 200.61/13.352 - 15.025 cm 3 , and 

P l 15.025(1000-1) 
In >l = lilQrx 373.2 

Therefore, Pi/P 2 = 1.633. The vapor pressure at 1 atm is 0.273 mm, so that 
the calculated vapor pressure at 1000 atm is 0.455 mm. 

9. Experimental measurement of vapor pressure. Many different experi- 
mental arrangements have been employed in vapor-pressure measurements. 
One of the most convenient static methods is the Smith- Menzies isoteniscope 
shown in Fig. 5.2. The bulb and short attached U-tube are filled with the 



TO AIR 



THERMOMETER 



ISOTENISCOPE 





TO VACUUM 



BALLAST 
VOLUME 



THERMOSTAT MANOMETER 
Fig. 5.2. Vapor pressure measurement with isoteniscope. 

liquid to be studied, which is allowed to boil vigorously until all air is re- 
moved from the sample side of the U-tube. At each temperature the external 
pressure is adjusted until the arms of the differential U-tube manometer are 
level, and the pressure and temperature are then recorded. 

The gas-saturation method was used extensively by Ramsay and Young. 
An inert gas is passed through the liquid maintained in a thermostat. The 
volume of gas used is measured, and its final vapor content or the loss in 
weight of the substance being studied is determined. If care is taken to ensure 
saturation of the flowing gas, the vapor pressure of the liquid may readily 
be calculated. 

Some experimentally measured vapor pressures are collected 6 in 
Table 5.1. 



6 A very complete compilation is given by D. R. Stull, Ind. Eng. Chem., 39, 517-550 
(1947). 



Sec. 10] 



CHANGES OF STATE 

TABLE 5.1 
TYPICAL VAPOR PRESSURE DATA 



109 





Vapor Pressure in Millimeters of Mercury 


Temp. 




CO 


CC1 4 


CH 3 COOH 


C 2 H 5 OH 


(C 2 H 5 ) 2 


C 7 H 16 


QH 5 -CH 8 


:H 2 o 









12.2 


185.3 


11.45 




4.579 


10 





-- 


23.6 


291.7 


20.5 





9.209 


20 


91 


11.7 


43.9 


442.2 


35.5 





17.535 


30 


143.0 


20.6 


78.8 


647.3 


58.35 


36.7 


31.824 


40 


215.8 


34.8 


135.3 


921.3 


92.05 


59.1 


55.324 


50 


317.1 


56.6 


222.2 


1277 


140.9 


92.6 


92.51 


60 


450.8 


88.9 


352.7 





208.9 


139.5 


149.38 


70 


622.3 


136.0 


542.5 





302.3 


202.4 


233.7 


80 


843 


202.3 


812.6 


426.6 


289.7 


355.1 


90 


1122 


293.7 


1187 





588.8 


404.6 


525.76 


100 


1463 


417.1 








795.2 


557.2 


760.00 




! 













10. Solid-solid transformations the sulfur system. Sulfur provides the 
classical example of a one-component system displaying a solid-solid trans- 
formation. The phenomenon of polymorphism, discovered by Mitscherlich 
in 1821, is the occurrence of the same chemical substance in two or 
more different crystalline forms. In the case of elements, it is called 
a I lot ropy. 

Sulfur occurs in a low-temperature rhombic form and a high-temperature 
monoclinic form. The phase diagram for the system is shown in Fig. 5.3. 
The pressure scale in this diagram has been made logarithmic in order to 
bring the interesting low-pressure regions into prominence. 

The curve AB is the vapor-pressure curve of solid rhombic sulfur. At 
point B it intersects the vapor-pressure curve of monoclinic sulfur BE, and 
also the transformation curve for rhombic-monoclinic sulfur, BD. This inter- 
section determines the triple point B, at which rhombic and monoclinic sulfur 
and sulfur vapor coexist. Since there are three phases and one component, 
f= c p-^-2 3 3 0, and point B is an invariant point. It occurs at 
0.01 mm pressure and 95.5C. 

The density of monoclinic sulfur is less than that of rhombic sulfur, and 
therefore the transition temperature (S r ^ S m ) increases with increasing 
pressure. 

Monoclinic sulfur melts under its own vapor pressure of 0.025 mm at 
120C, the point E on the diagram. From E to the critical point F there 
extends the vapor-pressure curve of liquid sulfur EF. Also from , there 
extends the curve ED, the melting-point curve of monoclinic sulfur. The 
density of liquid sulfur is less than that of the monoclinic solid, the usual 
situation in a solid-liquid transformation, and hence ED slopes to the right 
as shown. The point E is a triple point, S m -S liq -S vap . 

The slope of BD is greater than that of ED, so that these curves intersect 



110 



CHANGES OF STATE 



[Chap. 5 



at Z), forming a third triple point on the diagram, S f -S m -S llq . This occurs at 
155 and 1290atm. At pressures higher than this, rhombic sulfur is again 
the stable solid form, and DG is the melting-point curve of rhombic sulfur 
in this high-pressure region. The range of stable existence of monoclinic 
sulfur is confined to the totally enclosed area BED. 

Besides the stable equilibria represented by the solid lines, a number of 
metastable equilibria are easily observed. If rhombic sulfur is heated quite 
rapidly, it will pass by the transition point B without change and finally melt 



10' 



IO 



10 



10* 



RHOMBIC 




80 90 100 110 120 130 140 150 160 
TEMPERATURE 

Fig. 5.3. The sulfur system. 



to liquid sulfur at 1 14C (point //). The curve BH is the metastable vapor- 
pressure curve of rhombic sulfur, and the curve EH is the metastable vapor 
pressure curve of supercooled liquid sulfur. Extending from H to D is the 
metastable rhombic melting-point curve. Point H is a metastable triple point, 
S r -S liq -S vap . 

All these metastable equilibria are quite easily studied because of the 
extreme sluggishness that characterizes the rate of attainment of equilibrium 
between solid phases. 

In this discussion of the sulfur system, the well-known equilibrium 
between SA and S /4 in liquid sulfur has not been taken into consideration. If 
this occurrence of two distinct forms of liquid sulfur is considered, the sulfur 



Sec. 11] 



CHANGES OF STATE 



111 



system can no longer be treated as a simple one-component system, but 
becomes a "pseudobinary" system. 7 

11. Enantiotropism and monotropism. The transformation of monoclinic 
to rhombic sulfur under equilibrium conditions of temperature and pressure 
is perfectly reversible. This fact is, of course, familiar, since the transforma- 
tion curve represents a set of stable equilibrium conditions. Such a change 
between two solid forms, occurring in a region of the phase diagram where 
both are stable, is called an enantiotropic change. 

On the other hand, there are cases in which the transformation of one 
solid form to another is irreversible. The classical example occurs in the 




VAPOR 



ENANTIOTROPISM 




VAPOR 



MONOTROPISM 



T 
Fig. 5.4. 



Enantiotropic and monotropic changes. 



phosphorus system, in the relations between white (cubic) phosphorus and 
violet (hexagonal) phosphorus. When white phosphorus is heated, trans- 
formation into violet phosphorus occurs at an appreciable rate at tem- 
peratures above 260; but solid violet phosphorus is never observed to 
change into solid white phosphorus under any conditions. In order to obtain 
white phosphorus, it is necessary to vaporize the violet variety, whereupon 
the vapor condenses to white phosphorus. 

Such an irreversible solid-state transformation is called a monotropic 
change. It may be characterized by saying that one form is metastable with 
respect to the other at all temperatures up to its melting point. The situa- 
tion is shown schematically in Fig. 5.4. The transition point (metastable) 
between the two solid forms in this case lies above the melting point of 
either form. 

7 If SA and S^ came to equilibrium quickly when the T or P of the liquid was changed, 
the sulfur system would still have only one component (unary system) as explained in foot- 
note 3. If SA and S^ were present in fixed proportions, which did not change with rand P, 
because the time of transformation was very long compared with the time of the experiment, 
the sulfur system would have two components (binary system). In fact it appears that the 
time of transformation is roughly comparable with the time of most experiments, so that 
the observed behavior is partly unary and partly binary, being called "pseudobinary." 



112 CHANGES OF STATE [Chap. 5 

Actually, the phosphorus case is complicated by the occurrence of several 
molecular species, P 2 , P 4 , P % , and so on, so that considerations based on a 
one-component system must be applied with caution. 

12. Second-order transitions. The usual change of state (solid to liquid, 
liquid to vapor, etc.) is called & first-order transition. At the transition tem- 
perature T t at constant pressure, the free energies of the two forms are equal, 
but there is a discontinuous change in the slope of the F vs. T curve for the 
substance at T t . Since (3F/3T) S, there is therefore a break in the 
S vs. T curve, the value of AS at T t being related to the observed latent heat 
for the transition by AS = XjT t . There is also a discontinuous change in 
volume AF, since the densities of the two forms are not the same. 

A number of transitions have been studied in which no latent heat or 
density change can be detected. Examples are the transformation of certain 
metals from ferromagnetic to paramagnetic solids at their Curie points, the 
transition of some metals at low temperatures to a condition of electric 
superconductivity, and the transition observed in helium from one liquid 
form to another. 8 In these cases, there is a change in slope, but no dis- 
continuity, in the S vs. T curve at T t . As a result, there is a break AC P in 
the heat capacity curve, since C p = T(dS/dT) r . Such a change is called a 
second-order transition. 

13. High-pressure studies. It is only a truism that our attitude toward 
the physical world is conditioned by the scale of magnitudes provided in 
our terrestrial environment. We tend, for example, to classify pressures or 
temperatures as high or low by comparing them with the fifteen pounds per 
square inch and 70F of a spring day in the laboratory, despite the fact that 
almost all the matter in the universe exists under conditions very different 
from these. Thus, even at the center of the earth, by no means a large astro- 
nomical body, the pressure is around 1,200,000 atm, and substances at this 
pressure would have properties quite unlike those to which we are accus- 
tomed. At the center of a comparatively small star, like our sun, the pressure 
would be around ten billion atmospheres. 

The pioneer work of Gustav Tammann on high-pressure measurements 
has been greatly extended over the past twenty years by P. W. Bridgman 
and his associates at Harvard. Pressures up to 400,000 atm have been 
achieved and methods have been developed for measuring the properties of 
substances at 100,000 atm. 9 

The attainment of such pressures has been made possible by the con- 
struction of pressure vessels of alloys such as Carboloy, and by the use of a 
multiple-chamber technique. The container for the substance to be studied 
is enclosed in another vessel, and pressure is applied both inside and outside 
the inner container, usually by means of hydraulic presses. Thus although 

8 W. H. Keesom, Helium (Amsterdam: Elsevier, 1942). 

* For details see P. W. Bridgman, The Physics of High Pressures (London: Bell & Co., 
1949), and his review article, Rev. Mod. Phys., 18, 1 (1946). 



Sec. 13] 



CHANGES OF STATE 



113 



the absolute pressure in the inner vessel may be 100,000 atm, the pressure 
differential that its walls must sustain is only 50,000 atm. 

High-pressure measurements on water yielded some of the most inter- 
esting results, which are shown in the phase diagram of Fig. 5.5. The melting 
point of ordinary ice (ice I) falls on compression, until a value of 22.0C 
is reached at 2040 atm. Further increase in pressure results in the transforma- 
tion of ice I into a new modification, ice III, whose melting point increases 



9000 



8000- 



7000 - 



6000 - 



5000 - 



1 4000 - 



3000- 



2000 - 



1000 - 



-20 20 
TEMPERATURE -*C 

Fig. 5.5. Water system at high pressures. 

with pressure. Altogether six different polymorphic forms of ice have been 
found. There are six triple points shown on the water diagram. Ice VII is an 
extreme high-pressure form not shown on the diagram; at a pressure of 
around 20,000 atm, liquid water freezes to ice VII at about 100C. Ice IV is 
not shown. Its existence was indicated by the work of Tammann, but it was 
not confirmed by Bridgman. 

PROBLEMS 

1. From the following data, roughly sketch the phase diagram for carbon 
dioxide: critical point at 31C and 73 atm; triple point (solid-liquid-vapor) 
at 57 and 5.3 atm; solid is denser than liquid at the triple point. Label 
all regions on the diagram. 

2. Roughly sketch the phase diagram of acetic acid, from the data: 

(a) The low-pressure a form melts at 16.6C under its own vapor pressure 
of 9. 1 mm. 




114 CHANGES OF STATE [Chap. 5 

(b) There is a high-pressure /? form that is denser than the a, but both 
a and /? are denser than the liquid. 

(c) The normal boiling point of liquid is 1 18C. 

(d) Phases a, /?, liquid are in equilibrium at 55C and 2000 atm. 

3. Sketch the liquid-solid regions of the phase diagram of urethane. 
There are three solid forms, a, /?, y. The triple points and the volume changes 
AF in cc per kg at the triple points are as follows: 

(a) liq, a, /? P - 2270 atm / - 66C AK (I - a) = 25.3 

(I - ft - 35.5 
( a _ ft ^ 10.2 

(b) liq, ft y P - 4090 atm / - 77C A V: (I - ft) = 18.4 

(1 - y) - 64.0 
0? - y) - 45.6 

(c) a, /?, y - P = 3290 atm / - 25.5C A K: (a - ft - 9.2 

(/? - y) - 48.2 
(a - y) - 57.4 

4. The density /> of ice at 1 atm and 0C is 0.917 g per cc. Water under 
the same conditions has p -=-- l.OOOg per cc. Estimate the melting point of 
ice under a pressure of 400 atm assuming that p for both ice and water is 
practically constant over the temperature and pressure range. 

5. Bridgman found the following melting points / (C) and volume 
changes on melting AK(cc per g) for Na: 

P, kg/cm 2 . 1 2000 4000 6000 

/ . 97.6 114.2 129.8 142.5 

AK . . 0.0279 0.0236 0.0207 0.0187 

Estimate the heat of fusion of sodium at 3000 atm. 

6. Estimate the vapor pressure of mercury at 25C assuming that the 
liquid obeys Trouton's rule. The normal boiling point is 356.9C. 

7. The vapor pressure of solid iodine is 0.25 mm and its density 4.93 at 
20C. Assuming the Gibbs equation to hold, calculate the vapor pressure of 
iodine under a 1000-atm argon pressure. 

8. In a determination of the vapor pressure of ethyl acetate by the gas 
saturation method 100 liters of nitrogen (STP) were passed through a 
saturator containing ethyl acetate at 0C, which lost a weight of 12.8g. 
Calculate vapor pressure at 0C. 

9. The vapor pressures of liquid gallium are as follows: 

/, C . 1029 1154 1350 

P, mm . . 0.01 0.1 1.0 

Calculate A//, AF, and AS for the vaporization of gallium at 1154C. 

10. At 25C, the heat of combustion of diamond is 94.484 kcal per mole 
and that of graphite is 94.030. The molar entropies are 0.5829 and 1.3609 cal 
per deg mole, respectively. Find the AFfor the transition graphite -> diamond 
at 25C and 1 atm. The densities are 3.513 g per cc for diamond and 2.260 



Chap. 5] CHANGES OF STATE 115 

for graphite. Estimate the pressure at which the two forms would be in 
equilibrium at 25C. You may assume the densities to be independent of 
pressure. 

11. Sketch graphs of F, S, V, Q> against T at constant P, and P at 
constant T, for typical first- and second-order phase transitions. 

12. From the data in Table 5.1, plot log P vs. T~ l for water and calculate 
the latent heats of vaporization of water at 20 and at 80C. 

REFERENCES 

BOOKS 

1. Bridgman, P. W., The Physics of High Pressures (London: Bell, 1949). 

2. Findlay, A., The Phase Rule (New York: Dover, 1945). 

3. Marsh, J. S., Principles of Phase Diagrams (New York: McGraw-Hill, 
1935). 

4. Ricci, J. E., The Phase Rule and Heterogeneous Equilibrium (New York: 
Van Nostrand, 1951). 

5. Tammann, G., The States of Aggregation (New York: Van Nostrand, 
1925). 

6. Wheeler, L. P.,Josiah WillardGibbs(New Haven: Yale Univ. Press, 1953). 

ARTICLES 

1. Bridgman, P. W., Science in Progress, vol. Ill, 108-46 (New Haven: Yale 
Univ. Press, 1942), "Recent Work in the Field of High Pressures." 

2. Garner, W. E.,/. Chem. Soc., 1961-1973 (1952), "The Tammann Memor- 
ial Lecture." 

3. Staveley, L. A. K., Quart. Rev., 3, 65-81 (1949), "Transitions in Solids 
and Liquids." 

4. Swietoslawski, W., J. Chem. Ed., 23, 183-85 (1946), "Phase Rule and the 
Action of Gravity." 

5. Ubbelohde, A. R., Quart. Rev., 4, 356-81 (1950), "Melting and Crystal 
Structure." 



CHAPTER 6 

Solutions and Phase Equilibria 



1. The description of solutions. As soon as systems of two or more com- 
ponents are studied, the properties of solutions must be considered, for a 
solution is by definition any phase containing more than one component. 
This phase may be gaseous, liquid, or solid. Gases are in general miscible 
in all proportions, so that all mixtures of gases, at equilibrium, are solutions. 
Liquids often dissolve a wide variety of gases, solids, or other liquids, and 
the composition of these liquid solutions can be varied over a wide or narrow 
range depending on the particular solubility relationships in the individual 
system. Solid solutions are formed when a gas, a liquid, or another solid 
dissolves in a solid. They are often characterized by very limited concentra- 
tion ranges, although pairs of solids are known, for example copper and 
nickel, that are mutually soluble in all proportions. 

It is often convenient in discussing solutions to call some components 
the solvents and others the solutes. It should be recognized, however, that 
the only distinction between solute and solvent is a verbal one, although the 
solvent is usually taken to be the constituent present in excess. 

The concentration relations in solutions are expressed in a variety of 
units. The more important of these are summarized in Table 6.1. 

TABLE 6.1 
CONCENTRATION OF SOLUTIONS 



Name 


Symbol 


Definition 


Molar 
Molai 
Volume molal 
Weight per cent 
Mole fraction 


c 
m 
m f 

% 
X 


Moles of solute in 1 liter solution 
Moles of solute in 1000 g solvent 
Moles of solute in 1 liter solvent 
Grams of solute in 100 g solution 
Moles of solute divided by total 
number of moles of all components 



2. Partial molar quantities: partial molar volume. The equilibrium prop- 
erties of solutions are described in terms of the thermodynamic state func- 
tions, such as P, T, K, ", 5, F, //. One of the most important problems in 
the theory of solutions is how these properties depend on the concentrations 
of the various components. In discussing this question, it will be assumed 
that the solution is kept at constant over-all pressure and temperature. 

Consider a solution containing n A moles of A and n B moles of B. Let the 
volume of the solution be K, and assume that this volume is so large that 



Sec. 2] SOLUTIONS AND PHASE EQUILIBRIA 117 

the addition of one extra mole of A or of B does not change the concentration 
of the solution to an appreciable extent. The change in volume caused by 
adding one mole of A to this large amount of solution is then called the 
partial molar volume of A in the solution at the specified pressure, tempera- 
ture, and concentration, and is denoted by the symbol V A . It is the change 
of volume K, with moles of A, n A , at constant temperature, pressure; and 
moles of B, and is therefore written as 



One reason for introducing such a function is that the volume of a 
solution is not, in general, simply the sum of the volumes of the individual 
components. For example, if 100ml of alcohol are mixed at 25C with 
100 ml of water, the volume of the solution is not 200 ml, but about 190 ml. 
The volume change on mixing depends on the relative amount of each 
component in the solution. 

\fdn A moles of A and dn B moles of B are added to a solution, the increase 
in volume at constant temperature and pressure is given by the complete 
differential, 

-(")*, + () *. 

A'* M WB'* A 

or dV --= V A dn A + P yy dn tt (6.2) 

This expression can be integrated, which corresponds physically to increasing 
the volume of the solution without changing its composition, V A and V n 
hence being held constant. 1 The result is 

V = V A n A + V B n B (6.3) 

This equation tells us that the volume of the solution equals the number of 
moles of A times the partial molar volume of A, plus the number of moles 
of B times the partial molar volume of B. 
On differentiation, eq. (6.3) yields 

<*V= VA dn A + n A d? A + V B dn B + n B dV B 
By comparison with eq. (6.2), it follows that 



or dV A - - dV B (6.4) 

n A 

Equation (6.4) is one example of the Gibbs-Duhem equation. This par- 
ticular application is in terms of the partial molar volumes, but any other 

1 Mathematically, the integration is equivalent to the application of Euler's theorem to 
the homogeneous differential expression. See D. V. Widder, Advanced Calculus (New York: 
Prentice-Hall, 1947), p. 15. 



118 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

partial molar quantity may be substituted for the volume. These partial 
molar quantities can be defined for any extensive state function. For example: 



Sl . a . 

WA'T,I',H. 

' ' n 

The partial molar quantities are themselves intensity factors, since they 
are capacity factors per mole. The partial molar free energy is the chemical 
potential /i. 

All the thermodynamic relations derived in earlier chapters can be 
applied to the partial molar quantities. For example: 

/v ' (6 ' 5) 



The general thermodynamic theory of solutions is expressed in terms of 
these partial molar functions and their derivatives just as the theory for 
pure substances is based on the ordinary thermodynamic functions. 

3. The determination of partial molar quantities. The evaluation of the 
partial quantities will now be described, using the partial molar volume as 
an example. The methods for ff A , S A , F A9 and so on, are exactly similar. 

The partial molar volume V A , defined by eq. (6.1), is equal to the slope 
of the curve obtained when the volume of the solution is plotted against the 
molal concentration of A. This follows since the molal concentration m A is 
the number of moles of A in a constant quantity, namely 1000 grams, of 
solvent B. 

The determination of partial molar volumes by this slope method is 
rather inaccurate; the method of intercepts is therefore usually preferred. To 
employ this method, a quantity is defined, called the molar volume of the 
solution v, which is the volume of the solution divided by the total number 
of moles of the various constituents. For a two-component solution: 



"A + "B 
Then, Y = v (n A + n B ) 

and P A = 

Now the derivative with respect to mole number of A, n A , is transformed 
into a derivative with respect to mole fraction of B, X B . 

dv 



sinc e XK = ^> 1^1 =- JL 

("A + ' 



Sec. 3] 



SOLUTIONS AND PHASE EQUILIBRIA 



119 



Thus eq. (6.6) becomes : V A = v 



V 



n B dv 
n A + n s dX B 

x dv 

A B Ti7~ 



(6.7) 



The application of this equation is illustrated in Fig. 6.1, where v for a 
solution is plotted against the mole fraction. The slope S^ is drawn tangent 
to the curve at point P, corresponding to a definite mole fraction X B ' . The 
line ^iA 2 is drawn through P parallel to O^O 2 . Therefore the distance 



Si 




2 



XB i 

MOLE FRACTION OF B-X B 
Fig. 6.1. Determination of partial molar volumes intercept method. 

O l A l v, the molar volume corresponding to X B . The distance S l A l is 
equal to the slope at X B multiplied by X B , i.e., to the term in eq. (6.7), 
X n (dv/dX B ). It follows that O^ = O l A l S l A l equals V A , the partial 
molar volume of A in the solution. It can readily be shown that the intercept 
on the other axis, O 2 5 2 , is the partial molar volume of B, P B . This con- 
venient intercept method is the one usually used to determine partial molar 
quantities. It is not restricted to volumes, but can be applied to any extensive 
state function, 5, H, E, F 9 and so on, given the necessary data. It can also 
be applied to heats of solution, and the partial molar heats of solution so 
obtained are the same as the differential heats described in Chapter 2. 

If the variation with concentration of a partial molar quantity is known 
for one component in a binary solution, the Gibbs-Duhem equation (6.4) 
permits the calculation of the variation for the other component. This cal- 
culation can be accomplished by graphical integration of eq. (6.4). For 
example: 



120 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

where X is the mole fraction. If X B \X A is plotted against V n , the area under 
the curve gives the change in V A between the upper and lower limits of 
integration. The V A of pure A is simply the molar volume of pure A, and 
this can be used as the starting point for the evaluation of V A at any other 
concentration. 

4. The ideal solution Raoult's Law. The concept of the "ideal gas" has 
played a most important role in discussions of the thermodynamics of gases 
and vapors. Many cases of practical interest are treated adequately by means 
of the ideal gas approximations, and even systems deviating largely from 
ideality are conveniently referred to the norm of behavior set by the ideal 
case. It would be most helpful to find some similar concept to act as a guide 
in the theory of solutions, and fortunately this is indeed possible. Because 
they are very much more condensed than gases, liquid or solid solutions 
cannot be expected to behave ideally in the sense of obeying an equation of 
state such as the ideal gas law. Ideality in a gas implies a complete absence 
of cohesive forces; the internal pressure, (3E/c>V) T 0. Ideality in a solution 
is defined by complete uniformity of cohesive forces. If there are two com- 
ponents A and B, the forces between A and A, B and /?, and A and B are 
all the same. 

A property of great importance in the discussion of solutions is the vapor 
pressure of a component above the solution. This partial vapor pressure 
may be taken as a good measure of the escaping tendency of the given species 
from the solution. The exact measure of this escaping tendency is the fugacity, 
which becomes equal to the partial pressure when the vapor behaves as an 
ideal gas. The tendency of a component to escape from solution into the 
vapor phase is a very direct reflection of the physical state of affairs within 
the solution, 2 so that by studying the escaping tendencies, or partial vapor 
pressures, as functions of temperature, pressure, and concentration, we 
obtain a description of the properties of the solution. 

This method is a direct consequence of the relation between chemical 
potential and fugacity. If we have a solution, say of A and J9, the chemical 
potential of A in the solution must be equal to the chemical potential of A 
in the vapor phase. This is related to the fugacity by eq. (4.31), since 



If we know the pressure, the temperature, and the chemical potentials of 
the various components, we then have a complete thermodynamic descrip- 
tion of a system, except for the absolute amounts of the various phases. The 
partial vapor pressures are important because they are an approximate 
indication of the chemical potentials. 

A solution is said to be ideal if the escaping tendency of each component 

2 One may think of an analogy in which a nation represents a solution and its citizens 
the molecules. If life in the nation is a good one, the tendency to emigrate will be low. This 
presupposes, of course, the absence of artificial barriers. 



Sec. 4] 



SOLUTIONS AND PHASE EQUILIBRIA 



121 



is proportional to the mole fraction of that component in the solution. It is 
helpful to look at this concept from a molecular point of view. Consider an 
ideal solution of A and B. The definition of ideality then implies that a 
molecule of A in the solution will have the same tendency to escape into the 
vapor whether it is surrounded entirely by other A molecules, entirely by 
B molecules, or partly by A and partly by B molecules. This means that the 
intermolecular forces between A and A, A and B, and B and B, are all 
essentially the same. It is immaterial to the behavior of a molecule what 
sort of neighbors it has. The escaping tendency of component A from such 



200 



180 



160 

0140 

E 

6120 





a: 60 


<40 

20 



TOTAL 

VAPOR 
*-- ^PRESSURE 




. 

A 
C 2 H 4 Br 2 



.2 3 .4 .5 .6 .7 .8 .9 1.0 
MOLE FRACTION OF C 3 H 6 Br 2 B 



C 3 H 6 Br 2 



Fig. 6.2. Pressures of vapors above solutions of ethylene bromide and 
propylene bromide at 85C. The solutions follow Raoult's Law. 

an ideal solution, as measured by its partial vapor pressure, is accordingly 
the same as that from pure liquid A, except that it is proportionately reduced 
on account of the lowered fraction of A molecules in the solution. 

This law of behavior for the ideal solution was first given by Francois 
Marie Raoult in 1886, being based on experimental vapor-pressure data. It 
can be expressed as 

PA = *A PA (6-9) 

Here P A is the partial vapor pressure of A above a solution in which its 
mole fraction is X A , and P A is the vapor pressure of pure liquid A at the 
same temperature. 

If the component B added to pure A lowers the vapor pressure, eq. (6.9) 
can be written in terms of a relative vapor pressure lowering, 



(6.10) 



1 22 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

This form of the equation is especially useful for solutions of a relatively 
involatile solute in a volatile solvent. 

The vapor pressures of the system ethylene bromide propylene bromide 
are plotted in Fig. 6.2. The experimental results almost coincide with the 
theoretical curves predicted by eq. (6.9). The agreement with Raoult's Law 
in this instance is very close. 

Only in exceptional cases are solutions found that follow Raoult's Law 
closely over an extended range of concentrations. This is because ideality 
in solutions implies a complete similarity of interaction between the com- 
ponents, which can rarely be achieved. 

This equality of interaction leads to two thermodynamic conditions: 
(1) there can be no heat of solution; (2) there can be no volume change on 
mixing. Hence, AF 80lution - and A// 80hltion - 0. 

5. Equilibria in ideal solutions. If we wish to avoid the assumption that 
the saturated vapor above a solution behaves as an ideal gas, Raoult's Law 
may be written 

fA~X A ft (6.11) 

where / 4 and/jj' are the fugacities of A in the solution, and in pure A. It is 
evident from eq. (6.8) that 

dp -- RTd\nf A RTd\nX A (6.12) 

Then, following the sort of development given in Section 4-5, one obtains 
for the equilibrium constant in an ideal solution 

-AF --= RT\r\K x 

with K <'-'r* (6 ' 13) 

A A A B 

for the typical case. 

6. Henry's Law. Consider a solution of component B, which may be 
called the solute, in A 9 the solvent. If the solution is sufficiently diluted, a 
condition ultimately is attained in which each molecule of B is effectively 
completely surrounded by component A. The solute B is then in a uniform 
environment irrespective of the fact that A and B may form solutions that 
are far from ideal at higher concentrations. 

In such a very dilute solution, the escaping tendency of B from its uniform 
environment is proportional to its mole fraction, but the proportionality 
constant k no longer is P#. We may write 

PB - kX B (6.14) 

This equation was established and extensively tested by William Henry 
in 1803 in a series of measurements of the pressure dependence of the solu- 
bility of gases in liquids. Some results of this type are collected in Table 6.2. 
The fc's are almost constajit, so that Henry's Law is nearly but not exactly 
obeyed. 



Sec. 7] 



SOLUTIONS AND PHASE EQUILIBRIA 



123 



TABLE 6.2 

THE SOLUBILITY OF GASES IN WATER (ILLUSTRATING HENRY'S LAW P B = kX B , THE GAS 
PRESSURE BEING IN ATM AND X B BEING THE MOLE FRACTION) 



Partial Pressure 


Henry's Law Constant (k x 10~ 4 ) 

* 


(atm) 










N 2 at 19.4 


O 2 at 23 


H 2 at 23 






1 


1.18 


8.24 


4.58 




2.63 


8.32 


4.59 


7.76 


3.95 


8.41 


4.60 


7.77 


5.26 


8.49 


4.68 


7.81 


6.58 


8.59 


4.73 


7.89 


7.90 


8.74 


4.80 


8.00 


9.20 


8.86 


4.88 


8.16 



As an example, let us calculate the volume of oxygen (at STP) dissolved 
in 1 liter of water in equilibrium with air at 23. From eq. (6.14) the mole 
fraction of O 2 is X n = P#/k. Since P B = 0.20, and from the table k = 
4.58 x 10 4 , X n - 4.36 x 10~ 6 . In 1 liter of H 2 O there are 1000/18 = 
55.6 moles. Thus X R n n l(n u + 55.6), or n B = 2 A3 x 10~ 4 . This number 
of moles of oxygen equals 5.45 cc at STP. 

Henry's Law is not restricted to gas-liquid systems, but is followed by a 
wide variety of fairly dilute solutions and by all solutions in the limit of 
extreme dilution. 

7. Two-component systems. For systems of two components the phase 
rule, /= c p + 2, becomes /= 4 p. The following cases are possible: 

p ^= [ , / ^ 3 trivariant system 

p 2, / 2 bivariant system 

p 3, f= 1 univariant system 

p = 4, f ~ invariant system 

The maximum number of degrees of freedom is three. A complete 
graphical representation of a two-component system therefore requires a 
three-dimensional diagram, with coordinates corresponding to pressure, 
temperature, and composition. Since a three-dimensional representation is 
usually inconvenient, one variable is held constant while the behavior of 
the other two is plotted. In this way, plane graphs are obtained showing 
pressure vs. composition at constant temperature, temperature vs. com- 
position at constant pressure, or pressure vs. temperature at constant 
composition. 

8. Pressure-composition diagrams. The example of a (P-X) diagram in 
Fig. 6.3a shows the system ethylene bromide-propylene bromide, which obeys 
Raoiilt's Law quite closely over the entire range of compositions. The 
straight upper line represents the dependence of the total vapor pressure 



124 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



above the solution on the mole fraction in the liquid. The curved lower line 
represents the dependence of the pressure on the composition of the vapor. 

Consider a liquid of composition X 2 at a pressure P 2 (point C on the 
diagram). This point lies in a one-phase region, so there are three degrees of 
freedom. One of these degrees is used by the requirement of constant tem- 
perature for the diagram. Thus for any arbitrary composition X 2 , the liquid 
solution at constant T can exist over a range of different pressures. 

As the pressure is decreased along the dotted line of constant com- 
position, nothing happens until the liquidus curve is reached at B. At this 
point liquid begins to vaporize. The vapor that is formed is richer than the 




140 
138 

P. 136 
134 
132 



VAPOR 




.1 .2 3 .4 .5 .6 .7 .8 .9 1.0 

A *B~" B 

C 2 H 4 Br 2 

Fig. 6.3a. Pressure-composition (mole 



01 2345. 6 789 10 



Fig. 6.3b. Temperature-composition 



fraction) diagram for system obeying diagram for system obeying Raoult's 



Raoult's Law. 



Law. 



liquid in the more volatile component, ethylene bromide. The composition 
of the first vapor to appear is given by the point A on the vapor curve. 

As the pressure is further reduced below B, a two-phase region on the 
diagram is entered. This represents the region of stable coexistence of liquid 
and vapor. The dotted line passing horizontally through a typical point D 
in the two-phase region is called a tie line', it connects the liquid and vapor 
compositions that are in equilibrium. 

The over-all composition of the system is X 2 . This is made up of liquid 
having a composition X { and vapor having a composition X 3 . The relative 
amounts of the liquid and vapor phases required to yield the over-all 
composition are given by the lever rule: if (/) is the fraction of liquid 
and (v) the fraction 3 of vapor, (/)/(r) - (Jjf 3 - X Z )/(X 2 - A\). This rule 

3 Since a mole fraction diagram is being used, (v) is the fraction of the total number of 
moles that is vapor. On a weight fraction diagram, (v} would be the weight fraction that is 
vapor. 



Sec. 9] SOLUTIONS AND PHASE EQUILIBRIA 125 



is readily proved: It is evident that X 2 -= (l)X l I [1 ~ (I)]X& or (7)^ 
(X 2 - *s)/(*i- *s)- Similarly (v) - 1 - (/) (X,- X 2 )/(X l - JIT,). Hence 
(/)/() =(*3 - *,)/(* a - A\), the lever rule. 

As the pressure is still further decreased along BF more and more liquid 
is vaporized till finally, at F, no liquid remains. Further decrease in pressure 
then proceeds in the one-phase, all-vapor region. 

In the two-phase region, the system is bi variant. One of the degrees of 
freedom is used by the requirement of constant temperature; only one 
remains. When the pressure is fixed in this region, therefore, the compositions 
of both the liquid and the vapor phases are also definitely fixed. They are 
given, as has been seen, by the end points of the tie line. 

9. Temperature-composition diagrams. The temperature-composition dia- 
gram of the liquid-vapor equilibrium is the boiling-point diagram of the 
solutions at the constant pressure chosen. If the pressure is one atmosphere, 
the boiling points are the normal ones. 

The boiling-point diagram for a solution in which the solvent obeys 
Raoult's Law can be calculated if the vapor pressures of the pure com- 
ponents are known as functions of temperature (Fig. 6.3b). The two end 
points of the boiling-point diagram shown in Fig. 6.3b are the temperatures 
at which the pure components have a vapor pressure of 760 mm, viz., 
131. 5C and 141. 6C. The composition of the solution that boils anywhere 
between these two temperatures, say at 135C, is found as follows: 

According to Raoult's Law, letting X A be the mole fraction of C 2 H 4 Br 2 , 
760 -= P A X A + P K (\ - X A ). At 135, the vapor pressure of C 2 H 4 Br 2 is 
835mm, of C 3 H 6 Br 2 , 652mm. Thus, 760 - 835 X A -f 652(1 - X A \ or 
X A 0.590, X n ^ 0.410. This gives one intermediate point on the liquidus 
curve; the others are calculated in the same way. 

The composition of the vapor is given by Dalton's Law: 



The vapor-composition curve is therefore readily constructed from the 
liquidus curve. 

10. Fractional distillation. The application of the boiling-point diagram 
to a simplified representation of distillation is shown in Fig. 6.3b* The 
solution of composition X begins to boil at temperature T v The first vapor 
that is formed has a composition Y, richer in the more volatile component. 
If this is condensed and reboiled, vapor of composition Z is obtained. This 
process is repeated until the distillate is composed of pure component A. In 
practical cases, the successive fractions will each cover a range of com- 
positions, but the vertical lines in Fig. 6.3b, may be considered to represent 
average compositions within these ranges. 



126 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



_ 
CONDENSER 



LIQUID 
FEED 



A fractionating column is a device that carries out automatically the 
successive condensations and vaporizations required for fractional distilla- 
tion. An especially clear example of this is the "bubble-cap" type of column 
in Fig. 6.4. As the vapor ascends from the boiler, it bubbles through a film 

of liquid on the first plate. This liquid is 
somewhat cooler than that in the boiler, so 
that a partial condensation takes place. The 
vapor that leaves the first plate is therefore 
richer than the vapor from the boiler in the 
more volatile component. A similar enrich- 
ment takes place on each succeeding plate. 
Each attainment of equilibrium between 
liquid and vapor corresponds to one of the 
steps in Fig. 6.3b. 

The efficiency of a distilling column is 
measured by the number of such equilibrium 
stages that it achieves. Each such stage is 
called a theoretical plate. In a well designed 
bubble-cap column, each unit acts very nearly 
as one theoretical plate. The performance of 
various types of packed columns is also 
described in terms of theoretical plates. The 
separation of liquids whose boiling points lie 
close together requires a column with a con- 
siderable number of theoretical plates. The number actually required de- 
pends on the cut that is taken from the head of the column, the ratio of 
distillate taken off to that returned to the column. 4 

11. Boiling-point elevation. If a small amount of a nonvolatile solute is 
dissolved in a volatile solvent, the solution being sufficiently dilute to behave 
ideally, the lowering of the vapor pressure can be calculated from eq. (6.10). 
As a consequence of the lowered vapor pressure, the boiling point of the 
solution is higher than that of the pure solvent. This fact is evident on 
inspection of the vapor pressure curves in Fig. 6.5. 

The condition for equilibrium of a component A, the volatile solvent, 
between the liquid and vapor phases is simply ju, A v = [i A l . From eq. (6.12), 
t*A l PA + RTln X A , where fi A is the chemical potential of pure liquid 
A, i.e., fJL A when X A 1. At the boiling point the pressure is 1 atm, so that 
/// = iff* th e chemical potential of pure A vapor at 1 atm. Therefore 
(l*A 9 = PA I ) becomes JU A = p% + RT\n X A . For the pure component A 9 the 
chemical potentials // are identical with the molar free energies F. Hence, 




BOILER 



Fig. 6.4. Schematic draw- 
ing of bubble-cap column. 



4 For details of methods for determining the number of theoretical plates in a column, 
see C. S. Robinson and E. R. Gilliland, Fractional Distillation (New York: McGraw-Hill, 



Sec. 11] 



SOLUTIONS AND PHASE EQUILIBRIA 



127 



From eq. (3.36), 5(F/T)/3T = ~H/T\ so that differentiating the above 
yields 

H? - H? = - 



Since H^ H% is the molar heat of vaporization, 



RT 2 



I ATM 



SOLUTION - 
PURE SOLID 



PURE LIQUID - 
PURE SOLID 



rwrt <JWL.II/ i rwr\u s>v*>i_iu 

EQUILIBRIUM^/ f EQUILIBRIUM 

/ 

L f_ + _____ / ___ / ^ 




,^/, 
^<?/ 



'</:*^- 



^:<y 



|m wo 



FR DER 



B.P EL. 



Fig. 6.5. Diagram showing the elevation of the boiling point caused by 
addition of a nonvolatile solute to a pure liquid. 

Taking A as constant over the temperature range, this equation is integrated 
between the limits set by the pure solvent (X A = 1, T = 7^) and the solution 



R r 



TT Q 



When the boiling-point elevation is not large, TT Q can be replaced by T 2 . 
If X B is the mole fraction of solute, the term on the left can be written 
In (I X B \ and then expanded in a power series. Writing A7^ for the 
boiling-point elevation, T jT , we obtain 



RTJ 



i X 



128 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

When the solution is dilute, X B is a small fraction whose higher powers may 
be neglected. Then, 

RT 2 
AT^y-*-** (6.15) 

^vap 

In the dilute solutions for which eq. (6.15) is valid, it is also a good 
approximation to replace X B by (W B M A )/(W A M S ); W B , M B> and W Ay M A 
being the masses and molecular weights of solute and solvent. Then, 



- _ 

B *vap W A M B ' / vap W A M B 

where / vap is the latent heat of vaporization per gram. Finally W B \W A M B 
is set equal to w/1000, m being the weight molal concentration, moles of 
solute per 1000 grams of solvent. Thus, 






and K B is called the molal boiling-point elevation constant. 

For example, for water T Q = 373.2, / vap 538 cal per g. Hence 

(1.986)(373.2)* 
KB = 1538X1000T = ' 5 4 

For benzene, K B - 2.67; for acetone, 1.67, etc. 

The expression (6.16) is used frequently for molecular-weight determination 
from the boiling-point elevation. From K B and the measured T B , we calculate 
m, and then the molecular weight from M B 1000 W B jmW A . For many 
combinations of solute and solvent, perfectly normal molecular weights are 
obtained. In certain instances, however, there is apparently an association or 
dissociation of the solute molecules in the solution. For example, the molec- 
ular weight of benzoic acid in acetone solution is found to be equal to the 
formula weight of 122.1. In 1 per cent solution in benzene, benzoic acid has 
an apparent molecular weight of 242. This indicates that the acid is to a 
considerable extent dimerized into double molecules. The extent of associa- 
tion is greater in more concentrated solutions, as is required by the Le Chatelier 
principle. From molecular-weight determinations at different concentra- 
tions, it is possible to calculate the equilibrium constant of the reaction 
(C 6 H 5 COOH) 2 = 2 C 6 H 5 COOH. 

12. Solid and liquid phases in equilibrium. The properties of solutions 
related to the vapor pressure are called colligatwe from the Latin, colligatus, 
collected together. They are properties which depend on the collection of 
particles present, that is, on the number of particles, rather than on the kind. 
A colligative property amenable to the same sort of treatment as the boiling- 
point elevation is the depression of the freezing point. That this also has its 
origin in the lowering of the vapor pressure in solutions can be seen by 



Sec. 12] SOLUTIONS AND PHASE EQUILIBRIA 129 

inspection of Fig. 6.5. The freezing point of pure solvent, T mo is lowered to 
T m in the solution. 

It should be understood that "freezing-point depression curve" and 
"solubility curve" are merely two different names for the same thing that 
is, a temperature vs. composition curve for a solid-liquid equilibrium at 
some constant pressure, usually chosen as one atmosphere. Such a diagram 
is shown in Fig. 6.13 (p. 147) for the system benzene-chloroform. The curve 
CE may be considered to illustrate either (1) the depression of the freezing 
point of benzene by the addition of chloroform, or (2) the solubility of solid 
benzene in the solution. Both interpretations are fundamentally equivalent: 
in one case, we consider Tas a function of c; in the other, c as a function of 
T. The lowest point E on the solid-liquid diagram is called the eutectic point 
(evT'^KTO?, "easily melted"). 

In this diagram, the solid phases that separate out are shown as pure 
benzene (A) on one side and pure chloroform (B) on the other. It becomes 
evident in the next section that this is not exactly correct, since there is 
usually at least a slight solid solution of the second component B in the solid 
component A. Nevertheless the absence of any solid solution is in many 
cases a good enough approximation. 

The equation for the freezing-point depression, or the solubility equation 
for ideal solutions, is derived by essentially the same method used for the 
boiling-point elevation. In order for a pure solid A to be in equilibrium with 
a solution containing A, it is necessary that the chemical potentials of A be 
the same in the two phases, JU A * = fi A l . From eq. (6.12) the chemical potential 
of component A in an ideal solution is JU A I = fi A -f RT\t\ X A , where p, A 
is the chemical potential of pure liquid A. Thus the equilibrium condition 
can be written H A * = fi A f RTln X A . Now jti A 8 and fj, A are simply the 
molar free energies of pure solid and pure liquid, hence 

Z^/--=l*XA (6.17) 



Since we have d(F/T)/3T = H/T* from eq. (3.36), differentiation of 
eq. (6.17) with respect to T yields 

1 2 ^/l V 

(6.18) 



RT* RT* dT 

Integrating this expression from T Q9 the freezing point of pure A, mole 
fraction unity, to T, the temperature at which pure solid A is in equilibrium 
with solution of mole fraction X A , we obtain 5 

^fus 



5 It is a good approximation to take A fu8 independent of T over moderate ranges of 
temperature. 



130 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

This is the equation for the temperature variation of the solubility X A of a 
pure solid in an ideal solution. 

As an example, let us calculate the solubility of naphthalene in an ideal 
solution at 25C. Naphthalene melts at 80C, and its heat of fusion at the 
melting point is 4610 cal per mole. Thus, from eq. (6.19), 

4610 (353.2- 1 - 298.2- 1 ) - 2.303 log X A 



1.986 



X A - 0.298 



This is the mole fraction of naphthalene in any ideal solution, whatever the 
solvent may be. Actually, the solution will approach ideality only if the 
solvent is rather similar in chemical and physical properties to the solute. 
Typical experimental values for the solubility X A of naphthalene in various 
solvents at 25C are as follows: chlorobenzene, 0.317; benzene, 0.296; 
toluene, 0.286; acetone, 0.224; hexane, 0.125. 

The simplification of eq. (6.19) for dilute solutions follows from the same 
approximations used in the boiling-point elevation case. The final expression 
for the depression of the freezing point &T F -^ T T is 



RT 

K - 



For example: water, K F = 1.855; benzene, 5.12; camphor, 40.0, and so on. 
Because of its exceptionally large K F , camphor is used in a micro method for 
molecular-weight determination by freezing-point depression. 

13. The Distribution Law. The equilibrium condition for a component A 
distributed between two phases a and ft is p A * /^/, From eq. (6.8), 
f A = //. If the solutions are ideal, Raoult's Law is followed, and f A = 
XA/A* where f A is the fugacity of pure A (equal, if the vapor is an ideal 
gas/ to the vapor pressure P A . Thus X A *f A = JT//J, or X A * - Xj, and 
as long as the solutions are ideal, the solute A must be distributed equally 
between them. 

If the solutions do not follow Raoult's Law, but are sufficiently dilute to 
follow Henry's Law,/^ k A X A , and it follows that 

y a k a 

JT, = F? - * < 6 - 21 ) 

A A K A 

The ratio of the Henry's Law constants, K D , is called the distribution constant 
(or distribution coefficient). Thus K D is a function of temperature and pressure. 
Equation (6.21) is one form of the Nernst Distribution Law* 

In a dilute solution, X A = n A j(n A + n B ) ^ n A jn u & c A M B /\QQQp Ii , 
where C A is the ordinary molar concentration and M B and p B are the mole- 
cular weight and density of the solvent. With this approximation, the ratio 

W. Nernst, Z. physik. Chem., 8, 110 (1891). 



Sec. 14] SOLUTIONS AND PHASE EQUILIBRIA 131 

of mole fractions is proportional to the ratio of molar concentrations, and 
eq. (6.21) becomes 

^4 = K D (6-22) 

CA P 

A test of the Law in this form, for the distribution of iodine between water 
and carbon bisulfide may be seen in Table 6.3. 

TABLE 6.3 
DISTRIBUTION OF I 2 BETWEEN CS 2 AND H 2 O AT 1 8 



c a g I 2 per liter CS 2 


. 174 


129 


66 


41 


7.6 


cP g I 2 per liter H 2 O 


0.41 


0.32 


0.16 


0.10 


0.017 


K D ' - c*/cP 


. 420 


400 


410 


410 


440 



If association, dissociation, or chemical reaction of the distributed com- 
ponent takes place in either phase, modification of the Distribution Law is 
required. For example, if a solute S is partly dimerized to S 2 molecules in 
both phases, there will be two distribution equations, one for monomer and 
one for dimer, but the two distribution constants will not be independent, 
being related through the dissociation constants of the dimers. 

Solvent extraction is an important method for the isolation of pure 
organic compounds. Apparatus has been developed by L. C. Craig 7 at the 
Rockefeller Institute to carry out continuously hundreds of successive stages 
of extraction by the so-called "countercurrent distribution method." 

14. Osmotic pressure. The classical trio of colligative properties, of which 
boiling-point elevation and freezing-point depression are the first two 
members, is completed by the phenomenon of osmotic pressure. 

In 1748, the Abbe Nollet described an experiment in which a solution of 
"spirits of wine" was placed in a cylinder, the mouth of which was closed 
with an animal bladder and immersed in pure water. The bladder was 
observed to swell greatly and sometimes even to burst. The animal membrane 
is semipermeable; water can pass through it, but alcohol cannot. The in- 
creased pressure in the tube, caused by diffusion of water into the solution, 
was called the osmotic pressure (from the Greek, coer/jos "impulse"). 

The first detailed quantitative study of osmotic pressure is found in a 
series of researches by W. Pfeffer, published in 1877. Ten years earlier, 
Moritz Traube had observed that colloidal films of cupric ferrocyanide acted 
as semipermeable membranes. PfefTer deposited this colloidal precipitate 
within the pores of earthenware pots, by soaking them first in copper sulfate 
and then in potassium ferrocyanide solution. Some typical results of measure- 
ments using such artificial membranes are summarized in Table 6.4. 

7 L. C. Craig and D. Craig, "Extraction and Distribution," in Techniques of Organic 
Chemistry, ed. by A. Weissberger (New York: Interscience, 1950). 



132 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



In 1885 J. H. van't Hoff pointed out that in dilute solutions the osmotic 
pressure FI obeyed the relationship II V = nRT, or 

II = cRT (6.23) 

where c = w/Kis the concentration of solute in moles per liter. The validity 
of the equation can be judged by comparison of the calculated and experi- 
mental values of II in Table 6.4. 

TABLE 6.4 
OSMOTIC PRESSURES OF SOLUTIONS OF SUCROSE IN WATER AT 20 



Molal 


Molar 


Observed 


Calculated Osmotic Pressure 


Concen- 


Concen- 


Osmotic 




tration 


tration 


Pressure 








(m) 


(c) 


(atm) 


Eq. (6.23) 


Eq. (6.27) 


Eq. (6.25) 


0.1 


0.098 


2.59 


2.36 


2.40 


2.44 


0.2 


0.192 


5.06 


4.63 


4.81 


5.46 


0.3 


0.282 


7.61 


6.80 


7.21 


7.82 


0.4 


0.370 


10.14 


8.90 


9.62 


10.22 


0.5 


0.453 


12.75 


10.9 


12.0 


12.62 


0.6 


0.533 


15.39 


12.8 


14.4 


15.00 


0.7 


0.610 


18.13 


14.7 


16.8 


17.40 


0.8 


0.685 


20.91 


16.5 


19.2 


19.77 


0.9 


0.757 


23.72 


18.2 


21.6 


22.15 


1.0 


0.825 


26.64 


19.8 


24.0 


24.48 



The essential requirements for the existence of an osmotic pressure are 
two. There must be two solutions of different concentrations (or a pure 
solvent and a solution) and there must be a semipermeable membrane 
separating these solutions. A simple illustration can be found in the case of 
a gaseous solution of hydrogen and nitrogen. Thin palladium foil is appre- 
ciably permeable to hydrogen, but practically impermeable to nitrogen. If 
pure nitrogen is put on one side of a palladium barrier and a solution of 
nitrogen and hydrogen on the other side, the requirements for osmosis are 
satisfied. Hydrogen flows through the palladium from the hydrogen-rich to 
the hydrogen-poor side of the membrane. This flow continues until the 
chemical potential of the H 2 , /%, * s ^ e same on both sides of the barrier. 

In this example, the nature of the semipermeable membrane is rather 
clear. Hydrogen molecules are catalytically dissociated into hydrogen atoms 
at the palladium surface, and these atoms, perhaps in the form of protons 
and electrons, diffuse through the barrier. A solution mechanism of some 
kind probably is responsible for many cases of semipermeability. For 
example, it seems reasonable that protein membranes, like those employed 
by Nollet, can dissolve water but not alcohol. 

In other cases, the membrane may act as a sieve, or as a bundle of capil- 
laries. The cross sections of these capillaries may be very small, so that they 



Sec. 15] 



SOLUTIONS AND PHASE EQUILIBRIA 



133 



can be permeated by small molecules like water, but not by large molecules 
like carbohydrates or proteins. 

Irrespective of the mechanism by which the semipermeable membrane 
operates, the final result is the same. Osmotic flow continues until the 
chemical potential of the diffusing component is the same on both sides, of 
the barrier. If the flow takes place into a closed volume, the pressure therein 
necessarily increases. The final equilibrium osmotic pressure can be cal- 
culated by thermodynamic methods. It is the pressure that must be applied 
to the solution in order to prevent flow of solvent across the semipermeable 
membrane from the pure solvent into the solution. The same effect can be 
produced by applying a negative pressure or tension to the pure solvent. 

15. Measurement of osmotic pressure. We are principally indebted to two 
groups of workers for precise measurements of osmotic pressure: H. N. 
Morse, J. C. W. Frazer, and their colleagues at Johns Hopkins, and the 
Earl of Berkeley and E. G. J. Hartley at Oxford. 8 



CAPILLARY 

MANOMETER 
FOR PRESSURE 
MEASUREMENT 



SOLUTION =-_.- - 



POROUS CELL- 
IMPREGNATED 

WITH 
Cu 2 Fe(CN) 6 



(o) 




APPLIED 
PRESSURE 



CAPILLARY FOR 
MEASURING FLOW 
THROUGH CELL* 



PRESSURE 
GAUGE 




SOLUTION' L WATER ^IMPREGNATED 
CELL 

(b) 



Fig. 6.6. Osmotic pressure measurements: (a) method of Frazer; 
(b) method of Berkeley and Hartley. 

The method used by the Hopkins group is shown in (a), Fig. 6.6. The 
porous cell impregnated with copper ferrocyanide is filled with water and 
immersed in a vessel containing the aqueous solution. The pressure is 
measured by means of an attached manometer. The system is allowed to 
stand until there is no further increase in pressure. Then the osmotic pres- 
sure is just balanced by the hydrostatic pressure in the column of solution. 
The pressures studied extended up to several hundred atmospheres, and a 

8 An excellent detailed discussion of this work is to be found in J. C. W. Frazer's 
article, 'The Laws of Dilute Solutions' 1 in A Treatise on Physical Chemistry, 2nd ed., 
edited by H. S. Taylor (New York: Van Nostrand, 1931), pp. 353-414. 



134 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

number of ingenious methods of measurement were developed. These 
included the calculation of the pressure from the change in the refractive 
index of water on compression, and the application of piezoelectric gauges. 

The English workers used the apparatus shown schematically in (b), 
Fig. 6.6. Instead of waiting for equilibrium to be established and then 
reading the pressure, they applied an external pressure to the solution just 
sufficient to balance the osmotic pressure. This balance could be made very 
precisely by observing the level of liquid in the capillary tube, which would 
fall rapidly if there was any flow of solvent into the solution. 

16. Osmotic pressure and vapor pressure. Consider a pure solvent A that 
is separated from a solution of B in A by a membrane permeable to A alone. 
At equilibrium an osmotic pressure FI has developed. The condition for 
equilibrium is that the chemical potential of A is the same on both sides of 
the membrane, /if // /. Thus the }t A in the solution must equal that of 
the pure A. There are two factors tending to make the value of p A in the 
solution different from that in pure A. These factors must therefore have 
exactly equal and opposite effects on fi A . The first is the change in p A pro- 
duced by dilution of A in the solution. This change causes a lowering of p A 
equal to A/* =- RT\nP 4 /P A [eq. (6.8) with /=/>]. Exactly counteracting 
this is the increase in p A in the solution due to the imposed pressure II. 

From eq. (6.5) dp PdP, so that A/ J n V A dP. 

At equilibrium, therefore, in order that p A in solution should equal p A 
in the pure liquid, J* 1 V A dP -= -RT\n(P A /P A ). If it is assumed that the 

partial molar volume V A is independent of pressure, i.e., the solution is 
practically incompressible, 

PJT = firing (6.24) 

The significance of this equation can be stated as follows: the osmotic 
pressure is the external pressure that must be applied to the solution to 
raise the vapor pressure of solvent A to that of pure A. 

In most cases, also, the partial molar volume of solvent in solution V A 
can be well approximated by the molar volume of the pure liquid V A . In the 
special case of an ideal solution, eq. (6.24) becomes 

HV A - -RT\nX A (6.25) 

By replacing X A by (1 X B ) and expanding as in Section 6-11, the dilute 
solution formula is obtained: 

HV A = RTX B (6.26) 

Since the solution is dilute, 

RT n 
II w -S *< RTm' (6.27) 

This is the equation used by Frazer and Morse as a better approximation 



Sec. 17] 



SOLUTIONS AND PHASE EQUILIBRIA 



135 



than the van't HofT equation (6.23). As the solution becomes very dilute, m' 
the volume molal concentration approaches c the molar concentration, and 
we find as the end product of the series of approximations 

Ft = RTc (6.23) 

The adequacy with which eqs. (6.23), (6.25), and (6.27) represent the 
experimental data can be judged from the comparisons in Table 6.4. 9 

17. Deviations from Raoult's Law. Only a very few of the many liquid 
solutions that have been investigated follow Raoult's Law over the complete 



600 



500 



E 
E 
u -400 

( 
ID 
CO 
C/> 



(T 

a 200 



100 







400 



320 



240 



160 



80 








f).0 .2 .4 .6 .8 

CH 2 (OCH 3 ) 2 MO LE FRACTION CSa 

(a) 



1.0 ^0 .2 A .6 .8 

CS 2 (CH 3 ) 2 CO MOLE FRACTION CHC1 3 CH Cl * 

(b) 



Fig. 6.7. (a) Positive deviation from Raoult's Law the PX diagram of carbon 
bisulfide-methylal system, (b) Negative deviation from Raould's Law the PX 
diagram of chloroform-acetone system. 

range of concentrations. It is for this reason that the greatest practical 
application of the ideal equations is made in the treatment of dilute solutions, 
in which the solvent obeys Raoult's Law and the solute obeys Henry's Law. 
Nevertheless, one of the most instructive ways of qualitatively discussing 
the properties of nonideal solutions is in terms of their deviations from 
ideality. The first extensive series of vapor-pressure measurements, per- 
mitting such comparisons, were those made by Jan von Zawidski, around 
1900. 

Two general types of deviation were distinguished. An example exhibiting 
a positive deviation from Raoult's Law is the system carbon bisulfide- 
methylal, whose vapor-pressure-composition diagram is shown in (a), 
Fig. 6.7. An ideal solution would follow the dashed lines. The positive 

9 The osmotic pressures of solutions of high polymers and proteins provide some of the 
best data on their thermodynamic properties. A typical investigation is that of Shick, Doty, 
and Zimm, /. Am. Chetn. Soc., 72, 530 (1950). 



136 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

deviation is characterized by vapor pressures higher than those calculated 
for ideal solutions. 

The escaping tendencies of the components in the solution are accordingly 
higher than the escaping tendencies in the individual pure liquids. This effect 
has been ascribed to cohesive forces between unlike components smaller 
than those within the pure liquids, resulting in a trend away from complete 
miscibility. To put it naively, the components are happier by themselves than 
when they are mixed together; they are unsociable. These are metaphorical 
expressions; a scientific translation is obtained by equating a happy com- 
ponent to one in a state of low free energy. One would expect that this 
incipient immiscibility would be reflected in an increase in volume on mixing 
and also in an absorption of heat on mixing. 

The other general type of departure from Raoult's Law is the negative 
deviation. This type is illustrated by the system chloroform-acetone in (b), 
Fig. 6.7. In this case, the escaping tendency of a component from solution is 
less than it would be from the pure liquid. This fact may be interpreted as 
being the result of greater attractive forces between the unlike molecules in 
solution than between the like molecules in the pure liquids. In some cases, 
actual association or compound formation may occur in the solution. As a 
result, in cases of negative deviation, a contraction in volume and an evolution 
of heat are to be expected on mixing. 

In some cases of deviation from ideality, the simple picture of varying 
cohesive forces may not be adequate. For example, positive deviations are 
often observed in aqueous solutions. Pure water is itself strongly associated 
and addition of a second component may cause partial depolymerization of 
the water. This would lead to an increased partial vapor pressure. 

A sufficiently great positive deviation from ideality may lead to a maxi- 
mum in the PX diagram, and a sufficiently great negative deviation, to a 
minimum. An illustration of this behavior is shown in (a), Fig. 6.8. It is now 
no longer meaningful to say that the vapor is richer than the liquid in the 
"more volatile component." The following more general statement (Kono- 
valov's Rule) is employed: the vapor is richer than the liquid with which it 
is in equilibrium in that component by addition of which to the system the 
vapor pressure is raised. At a maximum or minimum in the vapor-pressure 
curve, the vapor and the liquid must have the same composition. 

18. Boiling-point diagrams. The PX diagram in (a), Fig. 6.8, has its 
counterpart in the boiling-point diagram in (b), Fig. 6.8. A minimum in the 
PX curve necessarily leads to a maximum in the TX curve. A well known 
example is the system HC1-H 2 O, which has a maximum boiling point (at 
760 mm) of 108.58 at a concentration of 20.222 per cent HC1. 

A solution with the composition corresponding to a maximum or 
minimum point on the boiling-point diagram is called an azeotropic 
solution (c>, "to boil"; arppTros, "unchanging"), since there is no change 
in composition on boiling. Such solutions cannot be separated by isobaric 



Sec. 19] 



SOLUTIONS AND PHASE EQUILIBRIA 



137 



distillation. It was, in fact, thought at one time that they were real chemical 
compounds, but changing the pressure changes the composition of the 
azeotropic solution. 

The distillation of a system with a maximum boiling point can be dis- 
cussed by reference to (b), Fig. 6.8. If the temperature of a solution having 
the composition / is raised, it begins to boil at the temperature t v The first 
vapor that distills has the composition y, richer in component A than is the 
original liquid. The residual solution therefore becomes richer in B; and if 
the vapor is continuously removed the boiling point of the residue rises, as 



LIQUID 




VAPOR 



VAPOR 




LIQUID 



X 
(a) 



A v I 



X 
(b) 



I' B 



Fig. 6.8. Large negative deviation from Raoult's Law. The PX 
curve has a minimum; the TX curve has a maximum. 

its composition moves along the liquidus curve from / toward m. If a frac- 
tional distillation is carried out, a final separation into pure A and the 
azeotropic solution is achieved. Similarly a solution of original composition 
/' can be separated into pure B and azeotrope. 

19. Partial miscibility. If the positive deviations from Raoult's Law 
become sufficiently large, the components may no longer form a continuous 
series of solutions. As successive portions of one component are added to 
the other, a limiting solubility is finally reached, beyond which two distinct 
liquid phases are formed. Usually, but not always, increasing temperature 
tends to promote solubility, as the thermal kinetic energy conquers the 
reluctance of the components to mix freely. In otjier words, the T AS term 
in AF = A// T AS* becomes more important. A solution that displays a 
large positive deviation from ideality at elevated temperatures therefore 
frequently splits into two phases when it is cooled. 

A PC diagram for a partially miscible liquid system, such as aniline and 
water, is shown in (a), Fig. 6.9. The point x lies in the two-phase region and 
corresponds to a system of two liquid solutions, one a dilute solution of 
aniline in water having the composition y, and the other a dilute solution of 



138 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



water in aniline having the composition z. These are called conjugate solutions. 
The relative amounts of the two phases are given by the ratios of the distances 
along the tie line, xy/xz. Applying the phase rule to this two-phase region: 
since p 2 and c -- 2, the system is bivariant. Because of the requirement 
of constant temperature imposed on the PC diagram, only one degree of 
freedom remains. Once the pressure is fixed, the compositions of both phases 
are fixed, which is indeed what the diagram indicates. The over-all com- 
position x is of course not fixed, since this depends on the relative amounts 
of the two conjugate solutions, with which the phase rule is not concerned. 



P 760mm 



VAPOR 




VAPOR 




A B A 

WATER COMPOSITION ANILINE WATER 



B 
COMPOSITION ANILINE 



(a) (b) 

Fig. 6.9. Schematic diagrams for aniline-water system, showing limited 
solubility of liquids, (a) PC diagram, (b) TC diagram. 

Let us follow the sequence of events as the pressure is gradually reduced 
along the line of constant composition, or isopleth, xx', 

At the point P, vapor having a composition corresponding to point Q 
begins to appear. There are now three phases coexisting in equilibrium, so 
that the system is invariant. If the volume available to the vapor is increased, 
the amount of the vapor phase will increase, at constant pressure, until all 
the aniline-rich solution, of composition /?, has vaporized. When this 
process is complete, there will remain a vapor of composition Q and a 
solution of composition N 9 so that the system becomes univariant again as 
the pressure falls below that at P. 

Since the vapor that is formed is richer in aniline, the composition of the 
residual solution becomes rjcher in water. The liquid composition moves 
along the line NL, and the vapor composition moves along QL until all the 



Sec. 20] 



SOLUTIONS AND PHASE EQUILIBRIA 



139 




liquid has been transformed into vapor, at the point M. After this, further 
decrease in pressure proceeds at constant vapor composition along MX' . 

It may be noted that the two conjugate solutions N and R have the same 
total vapor pressure and the same vapor composition. It follows that the 
partial" vapor pressure of component A above a dilute solution of A in B is 
the same as the vapor pressure of A above the dilute solution of B in A. For 
example, if benzene and water are mixed at 25C, two immiscible layers are 
formed, one containing 0.09 per cent C 6 H 6 and 99.91 per cent H 2 O, the other 
99.81 per cent C 6 H 6 and 0.19 per cent H 2 O. The partial pressure of benzene 
above either of these solutions is the 
same, namely 85 mm. 

In (a), Fig. 6.9, the lines NN' and 
RR' are almost vertical, since the 
solubility limits are only slightly de- 
pendent on pressure. Change in tem- 
perature, on the other hand, may 
greatly affect the mutual solubility of 
two liquids. In (b), Fig. 6.9, the TC 
diagram for the water-aniline system is 
drawn for the constant pressure of 
one atmosphere (normal-boiling-point 
diagram). Increasing the temperature 
tends to close the solubilitv gap, the 
difference between the concentrations 
of the two conjugate solutions. 

The interpretation of the solubility gap can be given in terms of the free 
energy of the system. At some constant temperature, let us plot the molar 
free energy of the system, defined as F =-- F/(n A -f- n B ), against the mole 
fraction of B, X B , for both the a and ft phases. In Fig. 6.9b, for example, 
these phases would be the two immiscible liquid solutions. The diagram 
obtained, Fig. 6.10, is an exact analog of Fig. 6.1, which was used for the 
determination of partial molar volumes. In this case, the intercept of the 
common tangent to the two F vs. X curves gives the value of the partial 
molar free energies, or chemical potentials, of the two components. At this 
composition, therefore, /// = /^/, and // yy a /y f /, i.e., the condition for 
equilibrium of components A and B between the two phases is fulfilled. The 
corresponding mole fractions represent the phase-boundary compositions; 
at any composition between X' B and X" B , the system will split into two 
distinct phases, since in this way it can reach its minimum free energy. For 
X B < X' B , however, pure phase a gives the lowest free energy, and for 
X B > X" B , pure phase ft. 

20. Condensed-liquid systems. In (b), Fig. 6.9, the variation of solubility 
with temperature is shown for only one pressure. At high enough tempera- 
tures boiling occurs, and it is therefore not possible to trace the ultimate 



Fig. 6.10. Partial miscibility deter- 
mined by free energy. 



140 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



course of the solubility curves. One might expect that the solubility gap 
would close completely if a high enough temperature could be reached 
before the onset of boiling. This expectation is represented by the dashed line 
in (b), Fig. 6.9. 

A number of condensed systems have been studied, which illustrate com- 
plete liquid-liquid solubility curves. A classical example is the phenol-water 
system of Fig. 6.11 (a). At the temperature and composition indicated by 
the point x, two phases coexist, the conjugate solutions represented by y and 



70 
60 
50 
.40 
30 
20 
10 

c 






X' 


*- 
d 


L 














180 
160 
140 
M20 
IOO 


4 




V 


^ 






Y 


/ 


60 








\ 




-- 




- 




50 

30 
20 

10 

r\ 




J 

TWO 
RE 







































hH 

PH/ 
IGlOf 

N 

-PH 
.GIO 


VSE 




TWO -PHASE 
~ REGION 








' 


X 

X 

*EGIC 


^SE 

)N 




\ 


ii ^H 

ONE 
RE 


(\SE 
SI 




180 
60 
40 






^ 




ONE -PHASE 
REGION 




) 20 40 60 80 100 ^0 20 40 60 80 100 tv t) 20 40 60 8O 10 
PER CENT WATER PER CENT PER CENT NICOTINE 
TRIETHYLAMINE 

(a) (b) (c) 



Fig. 6.11. Partial miscibility of two liquids, (a) phenol-water system, 
(b) tnethylamine-water system, (c) nicotine-water system. 

z. The relative amounts of the two phases are proportional, as usual, to the 
segments of the tie line. 

As the temperature is increased along the isopleth XX \ the amount 
of the phenol-rich phase decreases and the amount of water-rich phase 
increases. 

Finally at Y the compositions of the two phases become identical, 
the phenol-rich phase disappears completely, and at temperatures above Y 
there is only one solution. 

This gradual disappearance of one solution is characteristic of systems 
having all compositions except one. The exception is the composition corre- 
sponding to the maximum in the TC curve. This composition is called the 
critical composition and the temperature at the maximum is the critical 
solution temperature or upper consolute temperature. If a two-phase system 
having the critical composition is gradually heated [line CC in (a), Fig. 6.11] 
there is no gradual disappearance of one phase. Even in the immediate 
neighborhood of the maximum d, the ratio of the segments of the tie line 
remains practically constant. The compositions of the two conjugate solu- 
tions gradually approach each other, until, at the point d, the boundary line 
between the two phases suddenly disappears and a single-phase system 
remains. 



Sec. 21] SOLUTIONS AND PHASE EQUILIBRIA 141 

As the critical temperature is slowly approached from above, a most 
curious phenomenon is observed. Just before the single homogeneous phase 
passes over into two distinct phases, the solution is diffused by a pearly 
opalescence. This critical opalescence is believed to be caused by the scatter- 
ing of light from small regions of slightly differing density, which are formed 
in the liquid in the incipient separation of the two phases. 

Strangely enough, some systems exhibit a lower consolute temperature. 
At high temperatures, two partially miscible solutions are present, which 
become completely intersoluble when sufficiently cooled. An example is the 
triethylamine-water system in (b), Fig. 6.11, with a lower consolute tem- 
perature of 18.5 at 1 atm pressure. It is almost impossible to locate the 
critical composition exactly, since lowering the temperature a fraction of a 
degree greatly increases the solubility. This somewhat weird behavior suggests 
that large negative deviations from Raoult's Law (e.g., compound formation) 
become sufficient at the lower temperatures to counteract the positive 
deviations responsible for the immiscibility. 

Finally, systems have been found with both upper and lower consolute 
temperatures. These are most common at elevated pressures, and indeed 
one would expect all systems with a lower consolute temperature to display 
an upper one at sufficiently high temperature and pressure. An atmospheric- 
pressure example is the nicotine-water system of Fig. 6.11 (c). Having 
come to solutions of this type, we have run the gamut of deviations from 
ideality. 

21. Thermodynamics of nonideal solutions: the activity. A complete 
thermodynamic description of a solution, except for its amount, can be 
expressed in terms of the temperature, the pressure, and the chemical poten- 
tials of the various components. All the other thermodynamic functions can 
be derived from these. 

For a single pure ideal gas, the change in chemical potential is given from 
eq. (4.33) as dp RTdln P. By integration we obtain ^ = // + RT In P, 
where ju is the chemical potential of the gas at one atmosphere pressure. 
For a pure gas, this equation is identical with F F + RTlnP, where F 
is the free energy per mole. 

If the gas is not ideal, the fugacity is defined by the equation ft = JLL + 
RTlnf. Such an equation holds also for any component in a mixture of gases 
(gaseous solution). The constant p is a function of temperature alone. It is 
the chemical potential of the gas in its standard state of unit fugacity, or the 
standard free energy of the gas. 

The same equation is valid for a component in a liquid or solid solution, 
since at equilibrium the chemical potential must be the same in the con- 
densed phase as in the vapor. For a component A, 



(6.8) 
If the vapor above the solution can be considered to behave as an ideal 



142 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

gas, f A P A , and /LI A /t A + RTinP A . For an ideal solution, P A = 
XA/A = X A P A> and therefore 



(6.28) 
nX A 

The two constant terms can be combined, giving 

P.i = 1% + *T\n X A (6.29) 

This is the expression for the chemical potential in an ideal solution, p A 
being the chemical potential of A when X A - \ ; i.e., of pure liquid A. It 
should be clearly understood that p, A is a function of both temperature 
and pressure, in contrast with JU A in eq. (6.8). This is because the vapor 
pressure of the pure liquid, P A in eq. (6.28), is a function of both temperature 
and over-all pressure (p. 107). 

In the discussion of nonideal solutions we can always use the chemical 
potential, obtained from eq. (6.8) in terms of the partial vapor pressure or 
fugacity. Sometimes, however, it is convenient to introduce a new function, 
the activity a, which was invented by G. N. Lewis. It is defined as follows 
so as to preserve the form of eq. (6.29), 

p A - fi A 4 RTlna A (6.30) 

or P^ I- 



where y a/X is called the activity coefficient. 

One advantage of the activity coefficient is that it indicates at a glance 
the magnitude of the deviation from ideality in the solution. In terms of the 
activity, Raoult's Law becomes simply a = X, or y = 1 . 

Comparing eq. (6.30) with eqs. (6.28) and (6.8), we find that 

*A -4 (6.31) 

J A 

The activity is accordingly the ratio of the fugacity to the fugacity in the 
standard state. We have implicitly taken this standard state to be pure A, 
but other definitions might have been used. For a gas/J = 1 and therefore 
a A ~f A , the activity equals the fugacity. 

Equation (6.31) provides the most direct method of determining the 
activity of a component in a solution. It is usually sufficiently accurate to 
ignore gas imperfections and set the fugacity ratio equal to the vapor pressure 
ratio, so that a A = P A /P A . 

Some activities calculated in this way from vapor-pressure data are 
collected in Table 6.5. Once the activity of one component has been obtained 
as a function of concentration, the activity of the other component in 
a binary solution can be calpulated from the Gibbs-Duhem equation. 



Sec. 22] 



SOLUTIONS AND PHASE EQUILIBRIA 



143 



TABLE 6.5 

ACTIVITIES OF WATER AND SUCROSE IN THEIR SOLUTIONS AT 50C OBTAINED FROM 
VAPOR PRESSURE LOWERING AND THE GIBBS-DUHEM EQUATION 



Mole Fraction 
of Water 


Activity of Water 


Mole Fraction 
of Sucrose 


Activity of Sucrose 


X A 


<*A 


X* 


a B 


0.9940 


0.9939 


0.0060 


0.0060T 


0.9864 


0.9934 


0.0136 


0.0136 


0.9826 


0.9799 


0.0174 


0.0197 


0.9762 


0.9697 


0.0238 


0.0302 


0.9665 


0.9617 


0.0335 


0.0481 


0.9559 


0.9477 


0.0441 


0.0716 


0.9439 


0.9299 


0.0561 


0.1037 


0.9323 


0.9043 


0.0677 


0.1390 


0.9098 


0.8758 


0.0902 


0.2190 


0.8911 


0.8140 


0.1089 


0.3045 



Corresponding with eq. (6.4) for the partial molar volumes, we have for the 
partial molar free energies or chemical potentials, 



d / l A ^ - 



n A 



From eq. (6.30), 



d\n a A --= 



d In a,, 



If a B is known as a function of X J}9 a A can be obtained by a graphical 
integration. 

Activities can also be calculated from any of the colligative properties 
related to the vapor pressure. The details of these calculations are to be 
found in various treatises on thermodynamics. 10 

22. Chemical equilibria in nonideal solutions. The activity function defined 
in eq. (6.30) is useful in discussing the equilibrium constants of reactions in 
solution. It is readily proved (cf. p. 76) that for the schematic reaction 

aA + bB^cC + dD 



aS a r 



(6.32) 



and A/* - RTln K a 

In terms of activity coefficients and mole fractions, 



"a A. a A. & y a y b -~r"x 

7 A 7s A A A B 
In an ideal solution, all the activity coefficients become equal to unity, 

10 G. N. Lewis and M. Randall, Thermodynamics and Free Energy of Chemical Sub- 
stances (New York: McGraw-Hill, 1923), p. 278. 



144 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



and the equilibrium constant is simply K x . Extensive data on the activity 
coefficients of components in solutions of nonelectrolytes are not available, 
and the most important applications of eq. (6.32) have been made in electro- 
lytic solutions, which will be discussed in Chapter 15. 

23. Gas-solid equilibria. The varieties of heterogeneous equilibrium that 
have been considered so far have almost all been chosen from systems 
involving liquid and vapor phases only. Some systems of the solid-vapor and 
solid-liquid types will now be described. Most of the examples will be chosen 



200 
160 

160 

140 

: 120 

=JOO 

60 

60 

40 

20 




SATURATED 

SOLUTION * 

CuS0 4 -5H20 

t VAPOR 



CuS0 4 '5l 
CuS0 4 -3H 2 ' 
+ 2H 2 



100 
90 

80 

70 

60 

> 

40 
30 
20 
10 



20 30 40 50 60 70 80 90 KX) 110 120 

t,c 

(0) 



t -50*C 



CuS0 4 3H 2 



GuS0 4 + H 2 
- H 2 



12345 
MOLES H 2 0/MOLE CuS0 4 
IN SOLID 



(b) 



Fig. 6.12. The system CuSO 4 ~H 2 O. 



from two-component systems, with only a brief introduction to three- 
component phase diagrams. 

A two-component gas-solid system in which there is no appreciable 
solid-solution formation is exemplified by: CaCO 3 ^ CaO f CO 2 . Since 
c 2, the degrees of freedom are/^ 4 p. If the two solid phases are 
present, together with the gaseous phase CO 2 , the system is univariant, 
/=4 3=1. At a given temperature, the pressure of CO 2 has a fixed 
value. For example, if CO 2 is admitted to a sample of CaO at 700C, there 
is no absorption of gas until a pressure of 25 mm is reached; then the CaO 
takes up CO 2 at constant pressure until it is completely converted into 
CaCO 3 , whereupon further addition of CO 2 again results in an increase in 
pressure. 

The pressure-temperature diagram for such a system is therefore similar 
to the vapor-pressure curve of a pure liquid or solid. The CO 2 pressure has 
been loosely called the "dissociation pressure of CaCO 3 ." Since the pressure 
has a definite value only when the vapor phase is in equilibrium with both 
solid phases, it is really necessary to speak of the "dissociation pressure in 
the system CaCO 3 -CaO-CO 2 ." 



Sec. 24] SOLUTIONS AND PHASE EQUILIBRIA 145 

The necessity of specifying both the solid phases is to be emphasized in 
systems formed by various salts, their hydrates, and water vapor. The case 
of copper sulfate-water is shown in (a), Fig. 6.12, on a PT diagram, and in 
(b), Fig. 6.12, on a PC diagram. As long as only the two phases are present, 
a salt hydrate can exist in equilibrium with water vapor at any temperature 
if the pressure of water vapor is (1) above the dissociation pressure to lower 
hydrate or anhydrous salt and (2) below the dissociation pressure of the 
next higher hydrate or the vapor pressure of the saturated solution. State- 
ments in the older literature that a given hydrate "loses water at 1 10C" are 
devoid of precise meaning. 

When the pressure of water vapor falls below the dissociation pressure 
for the system, efflorescence occurs, as the hydrate loses water and its surface 
becomes covered with a layer of lower hydrate or anhydrous salt. When the 
vapor pressure exceeds that of the saturated aqueous solution, deliquescence 
occurs, and the surface of hydrate becomes covered with a layer of saturated 
solution. 

24. Equilibrium constant in solid-gas reactions. The equilibrium constant 
for a reaction involving solid phases can be discussed conveniently by con- 
sidering a typical reaction of this kind, the reduction of zinc oxide by carbon 
monoxide, ZnO (s) f CO -> Zn (g) + CO 2 . 

The equilibrium constant in terms of activities can be written as follows: 

K a = gzn * C( \ AF - -RTln K a (6.33) 



The activity is the ratio of the fugacity under the experimental conditions to 
the fugacity in a standard state, f A lf^ The standard state of a pure solid 
component is taken to be its state as a pure solid at one atmosphere pressure. 
The fugacity of the solid varies so slightly with pressure that over a con- 
siderable range of pressure, f A lf% for a solid is effectively a constant equal 
to unity. Making this very good approximation, the expression in eq. (6.33) 
becomes 





a co /co 

If the gases are considered to be ideal the activity ratio equals the partial 
pressure ratio, and K 9 = /WW^co- 

This discussion leads to the following general rule: no terms involving 
pure solid or liquid components need be included in equilibrium constants 
for solid-gas or liquid-gas reactions, unless very high precision is required, 
in which case there may be a small pressure correction to K v or K f . 

Equilibrium data for the reduction of zinc oxide are given in Table 6.6. 

25. Solid-liquid equilibria: simple eutectic diagrams. For two-component 
solid-liquid equilibria in which the liquids are completely intersoluble in all 
proportions and there is no appreciable solids-solid solubility, the simple 



146 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



TABLE 6.6 
THE EQUILIBRIUM ZnO -f CO ^ Zn h CO 2 (TOTAL PRESSURE = 760 MM) 





Equilibrium Concentrations in Vapor 




Temp. (C) 




K = PznPcoJPco 

(atm) 


Per cent CO 


Per cent CO 2 

-= Per cent Zn 


427 


99.98 


0.01 


1.00 x 10~ 8 


627 


99.12 


0.44 


1.95 x 10~ 5 


827 


91.76 


4.12 


1.84 x 10~ 3 


1027 


66.92 


16.54 


4.08 x 10~ 2 


1227 


30.84 


34.58 


3.87 x 10- 1 


1427 


9.8 


45.1 


2.07 



diagram of Fig. 6.13 is obtained. Examples of systems of this type are 
collected in Table 6.7. 

TABLE 6.7 
SYSTEMS WITH SIMPLE EUTECTIC DIAGRAMS SUCH AS FIG. 6.13 











Eutectic 


Component A 


M. pt. A 


Component B 


M. pt. B 





. 




v *"-'/ 




\ ^-v 




Mol 






| 


C 


per cent B 


CHBr 3 


7.5 


C 6 H 8 


5.5 


-26 


50 


CHC1 3 


63 


C 6 H 5 NH 2 


-6 


-71 


24 


Picric acid 


122 


TNT 


80 


60 


64 


Sb 


630 


Pb 


326 


246 


81 


Cd 


321 


Bi 


271 


144 


55 


KC1 


790 


AgCI 


451 


306 


69 


Si 


1412 


Al 


657 


578 


89 


Be 


1282 


Si 


1412 


1090 


32 



Consider the behavior of a solution of composition X on cooling along 
the isopleth XX' . When point P is reached, pure solid A begins to separate 
from the solution. As a result, the residual solution becomes richer in the 
other component B, its composition falling along the line PE. At any point 
Q in the two-phase region, the relative amounts of pure A and residual 
solution are given as usual by the ratio of the tie-line segments. When point 
R is reached, the residual solution has the eutectic composition E. Further 
cooling now results in the simultaneous precipitation of a mixture of A and 
B in relative amounts corresponding to E. 

The eutectic point is an invariant point on a constant pressure diagram; 
since three phases are in equilibrium,/^ c p + 2 = 2 --/? + 2 --= 4 
3=1, and the single degree of freedom is used by the choice of the constant- 
pressure condition. 



Sec. 26] 



SOLUTIONS AND PHASE EQUILIBRIA 



147 



Microscopic examination of alloys often reveals a structure indicating 
that they have been formed from a melt by a cooling process similar to that 
considered along the isopleth XX' of 
Fig. 6.13. Crystallites of pure metal 
are found dispersed in a matrix of 
finely divided eutectic mixture. An 
example taken from the antimony- 
lead system is shown in the photo- 
micrograph of Fig. 6.14. 

26. Cooling curves. The method 
of cooling curves is one of the most 
useful for the experimental study of 
solid-liquid systems. A two-compo- 
nent system is heated until a homo- 
geneous melt is obtained. A thermo- 
couple, or other convenient device 
for temperature measurement, is 




- PER CENT B - 
Fig. 6.13. Simple eutectic diagram for 



two components, A and B, completely inter- 
soluble as liquids but with negligible solid- 
solid solubility. 



immersed in the liquid, which is kept 

in a fairly well insulated container. 

As the system slowly cools, the 

temperature is recorded at regular time intervals. Examples of such curves 

for the system shown in Fig. 6.13 are drawn in Fig. 6.15. 

The curve a for pure A exhibits a gradual decline until the melting point 
of A is reached. It then remains perfectly flat as long as solid and liquid A 




Fig. 6.14. Photomicrograph at 50X of 80 per cent Pb-20 per cent Sb, 
showing crystals of Sb in a eutectic matrix. (Courtesy Professor Arthur 
Phillips, Yale University.) 

are both present, and resumes its decline only after all the liquid has solidified. 
The curve for cooling along the isopleth XX' is shown in b. The decline as 
the homogeneous melt is cooled becomes suddenly less steep when the tem- 
perature is reached corresponding to point P, where the first solid begins to 



148 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



separate from the solution. This change of slope is a consequence of the 
liberation of latent heat of fusion during the solidification of A. The more 
gradual decline continues until the eutectic temperature is reached. Then 
the cooling curve becomes absolutely flat. This is because the eutectic point 
in a two-component system, just as the melting point of one component, is 
an invariant point at constant over-all pressure. If the composition of the 



COMPOSITION 



ALONG \EUTECTIC 

\COMPOSITION 




(0) 



TIME 

Fig. 6.15. Cooling curves for various compositions on the simple eutectic 
diagram of Fig. 6.13. 

system chosen initially happened to be the same as that of the eutectic, the 
cooling curve would be that drawn in c. 

The duration of the constant-temperature period at the eutectic tempera- 
ture is called the eutectic halt. This halt is a maximum for a melt having the 
eutectic composition. 

Each cooling curve determination yields one point on the TC diagram 
(point of initial break in slope) in addition to a value for the eutectic tempera- 
ture. By these methods, the entire diagram can be constructed. 

27. Compound formation. If aniline and phenol are melted together in 
equimolar proportions, a definite compound crystallizes on cooling, 
C 6 H 5 OH-C 6 H 5 NH 2 . Pure phenol melts at 40C, pure aniline at -6.1C, 
and the compound melts at 3lC. The complete TC diagram for this system, 
in Fig. 6.16, is typical of many instances in which stable compounds occur 
as solid phases. The most convenient way of looking at such a diagram 
is to imagine it to be made up of two diagrams of the simple eutectic 
type placed side by side. In this case, one such diagram would be the 
phenol-compound diagram, and the other the aniline-compound diagram. 
The phases corresponding with the various regions of the diagram are 
labeled. 

A maximum such as the point C is said to indicate the formation of a 



Sec. 28] 



SOLUTIONS AND PHASE EQUILIBRIA 



149 



compound with a congruent melting point, since if a solid having the com- 
position C 6 H 5 OH-C 6 H 5 NH 2 is heated to 31C, it melts to a liquid of identical 
composition. Compounds with congruent melting points are readily detected- 



o 

UJ< 

S 1 

liJ 

40 
30 



20 

O 
lr'0 



-10 



-20, 



(b) 



rSOLID PHENOL 
[ + SOLUTION 

SOLID COMPOUND 
SOLUTIONJ 



SOLID PHENOL 

+ 
SOLID COMPOUND 



SOLUTION 




SOLID ANILINE 

+ 
SOLID COMPOUND 



.1 .2 .3 4 .5 .6 .7 .8 .9 1.0 
CgHsOH MOLE FRACTION ANILINE C 6 H 5 NH 2 

(0) 
Fig. 6.16. The system phenol-aniline. 

by the cooling-curve method. A liquid having the composition of the com- 
pound exhibits no eutectic halt, behaving in every respect like a single pure 
component. 

28. Solid compounds with incongruent melting points. In some systems, 
solid compounds are formed that do not melt to a liquid having the same 
composition, but instead decompose before such a melting point is reached. 
An example is the silica-alumina system (Fig. 6.17), which includes a com- 
pound, 3Al 2 O 3 -SiO 2 , called mul/ite. 

If a melt containing 40 per cent A1 2 O 3 is prepared and cooled slowly, 
solid mullite begins to separate at about 1780C. If some of this solid com- 
pound is removed and reheated along the line XX', it decomposes at 1800C 
into solid corundum and a liquid solution (melt) having the composition P. 
Thus: 3Al 2 O 3 -SiO 2 -* A1 2 O 3 + solution. Such a change is called incongruent 
melting, since the composition of the liquid differs from that of the solid. 

The point P is called the incongruent melting point or the peritectic point 
(rrjKTo*, "melting"; TTC/M, "around"). The suitability of this name becomes 
evident if one follows the course of events as a solution with composition 
3Al 2 O 3 -SiO 2 is gradually cooled along XX' . When the point M is reached, 



150 



SOLUTIONS AND PHASE EQUILIBRIA 



[Chap. 6 



solid corundum (A1 2 O 3 ) begins to separate from the melt, whose com- 
position therefore becomes richer in SiO 2 , falling along the line MP. When 
the temperature falls below that of the peritectic at />, the following change 
occurs: liquid + corundum --> mullite. The solid A1 2 O 3 that has separated 



2100 




60 3A1203 80 100 
Si02 A\203 

w PER CENT A1 2 03 

Fig. 6.17. System displaying peritectic. 

reacts with the surrounding melt to form the compound mullite. If a specimen 
taken at a point such as Q is examined, the solid material is found to consist 
of two phases, a core of corundum surrounded by a coating of mullite. It 
was from this characteristic appearance that the term "peritectic" originated. 
29. Solid solutions. Solid solutions are in theory no different from other 
kinds of solution: they are simply solid phases containing more than one 

1500 

1452 




I000 10 20 30 4p 50 60 70 80 90 100 
Cu wt PER CENT NICKEL Ni 

Fig. 6.18. The copper-nickel system a continuous series of solid 
solutions. 

component. The phase rule makes no distinction between the kind of phase 
(gas, liquid, or solid) that occurs, being concerned only with how many 



Sec. 33] 



SOLUTIONS AND PHASE EQUILIBRIA 



155 



pure tin melts at 232C and pure bismuth at 268C, their eutectic being at 
133C and 42 per cent Sn. The Sn-Bi eutectic temperature is lowered by the 
addition of lead to a minimum at 96C and a composition of 32 per cent Pb, 
16 per cent Sn, 52 per cent Bi. This is the ternary eutectic point. 




Pb 



Pb 



Sn 




Bi Sn 





SOLID Pb 
+ SOLUTION 

SOLID Pb ^ L 

+ SOLIDSn Pb /-SOLID Pb 

+ SOLUTION 



SOLUTION 




325 a 315 

(0) (b) 

SOLID Pb + 
SOLID S 
SOLUTION 




182 133 

(0 (d) 

SOLID Pb 4- SOLUTION 
SOLID Pb + SOLID Bi 



SOLUTION 



Bi 



Fig. 6.22. 



The system Pb-Sn-Bi : three-dimensional diagram and iso- 
thermal sections. 



Without using a solid model, the behavior of this system is best illustrated 
by a series of isothermal sections, shown in Fig. 6.22. Above 325C (a), the 
melting point of pure lead, there is a single liquid solution. At around 315C 
(b) the system consists of solid Pb and solution. The section at 182C (c) 
indicates the binary eutectic of Sn and Pb. Below this temperature, solid Pb 
and solid Sn both separate from the solution. At 133C the binary eutectic 
between Sn and Bi is reached (d). Finally, in (e) at 100C there is shown a 
section slightly above the ternary eutectic. 

The subject of ternary diagrams is an extended and very important one, 
and only a few of the introductory aspects have been mentioned. For further 
details some of the special treatises that are available should be consulted. 11 

11 J. S. Marsh, Principles of Phase Diagrams (New York: McGraw-Hill, 1935); G. Mas- 
sing, Introduction to the Theory of Three Component-Systems (New York: Reinhold, 1944). 



156 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

PROBLEMS 

1. Solutions are prepared at 25C containing 1000 g of water and n moles 
of NaCl. The volume is found to vary with n as V =- 1001.38 + 16.6253 + 
1.7738 3/2 + 0.1194fl 2 . Draw a graph showing the partial molar volumes 
of H 2 O and NaCl in the solution as a function of the molality from to 2 
molal. 

2. In the International Critical Tables (vol. Ill, p. 58) there is an extensive 
table of densities of HNO 3 H 2 O solutions. Use these data to calculate, by 
the graphical method of Fig. 6.1, the partial molar volumes of H 2 O and 
HNO 3 in 10, 20, 30, and 40 per cent solutions at 25. 

3. When 2 g of nonvolatile hydrocarbon containing 94.4 per cent C is 
dissolved in 100 g benzene, the vapor pressure of benzene at 20C is lowered 
from 74.66 mm to 74.01 mm. Calculate the empirical formula of the hydro- 
carbon. 

4. Pure water is saturated with a 2 : 1 mixture of hydrogen and oxygen 
at a total pressure of 5 atm. The water is then boiled to remove all the gases. 
Calculate the per cent composition of the gases driven off (after drying). Use 
data from Table 6.2. 

5. Water and nitrobenzene can be considered to be immiscible liquids. 
Their vapor pressures are: H 2 O, 92.5mm at 50C; 760mm at 100C; 
C 6 H 5 NO 2 , 22.4 mm at 10QC; 148 mm at 150C. Estimate the boiling point 
of a mixture of water and nitrobenzene at 1 atm pressure. In a steam dis- 
tillation at 1 atm how many grams of steam would be condensed to obtain 
one gram of nitrobenzene in the distillate? 

6. The following data were obtained for the boiling points at 1 atm of 
solutions of CC1 4 in C 2 C1 4 : 

Mole fraction 
CCI 4 inliq. . 0.000 0.100 0.200 0.400 0.600 0.800 1.000 

Mole fraction 
CCI 4 invap. . 0.000 0.469 0.670 0.861 0.918 0.958 1.000 

Boiling point 
C . . 120.8 108.5 100.8 89.3 83.5 79.9 76.9 

If half of a solution 30 mole per cent in CC1 4 is distilled, what is the com- 
position of the distillate? If a solution 50 mole per cent in CC1 4 is distilled 
until the residue is 20 mole per cent CC1 4 , what is approximate composition 
of the distillate? 

7. A compound insoluble in water is steam distilled at 97.0C, the dis- 
tillate containing 68 wt. per cent H 2 O. The vapor pressure of water is 682 mm 
at 97. What is the molecular weight of the compound? 

8. When hexaphenylethane is dissolved in benzene, the f.pt. depression 
of a 2 per cent solution is 0.219C; the b.pt. elevation is 0.135. Calculate 
the heat of dissociation of hexaphenylethane into triphenylmethyl radicals. 



Chap. 6] SOLUTIONS AND PHASE EQUILIBRIA 1 57 

9. Calculate the weight of (a) methanol, (b) ethylene glycol which, when 
dissolved in 4.0 liters of water, would just prevent the formation of ice at 
-10C. 

10. The solubility of picric acid in benzene is: 

/, C . . 5 10 15 20 25 3t 

g/100gC 6 H 6 . 3.70 5.37 7.29 9.56 12.66 21.38 

The melting points of benzene and picric acid are 5.5 and 121.8C. Calculate 
the heat of fusion of picric acid. 

11. The osmotic pressure at 25C of a solution of /Mactoglobulin con- 
taining 1.346 g protein per 100 cc solution was found to be 9.91 cm of water. 
Estimate the molecular weight of the protein. 

12. For the ideal solutions of ethylene bromide and propylenc bromide 
(p. 124), draw a curve showing how the mole fraction of C 2 H 4 Br 2 in the 
vapor varies with that in the liquid. Use this curve to estimate the number 
of theoretical plates required in a column in order to yield a distillate with 
mole fraction of C 2 H 4 Br 2 0.9 from a solution of mole fraction 0. 1 . Assume 
total reflux. 

13. Calculate the distribution coefficient K for piperidine between water 
and benzene at 20C, given : 

g solute/ 1 00 cc water layer . . 0.635 1.023 1.635 2.694 

g solute/100 cc benzene layer . . 0.550 0.898 1.450 2.325 

14. A solution of 3.795 g sulfur in 100 g carbon bisulfide (b.pt. CS 2 
46.30C; A// vap = 6400 cal per mole) boils at 46.66C. What is the formula 
of the sulfur molecule in the solution? 

15. The melting points and heats of fusion of 0, /?, m dinitrobenzenes are: 
116.9, 173.5, 89.8, and 3905, 3345, 4280 cal per mole [Johnston, J. Phys. 
Chem., 29, 882, 1041 (1925)]. Assuming the ideal solubility law, calculate the 
ternary eutectic temperature and composition for mixtures of o, m, p com- 
pounds. 

16. The following boiling points are obtained for solutions of oxygen and 
nitrogen at 1 atm: 

b.pt., K . . 77.3 78.0 79.0 80.0 82.0 84.0 86.0 88.0 90.1 
Mole % O in liq. . 8.1 21.6 33.4 52.2 66.2 77.8 88.5 100 

Mole%Oinvap. 2.2 6.8 12.0 23.6 36.9 52.2 69.6 100 

Draw the TX diagram. If 90 per cent of a mixture containing 20 per cent O 2 
and 80 per cent N 2 is distilled, what will be the composition of the residual 
liquid and its b.pt. ? Plot an activity a vs. mole fraction X diagram from the 
data. 

17. For a two-component system (A, B) show that: 



158 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 

18. Redder and Barratt [/. Chem. Soc., 537 (1933)] measured the vapor 
pressures of potassium amalgams at 387.5C, at which temperature the vapor 
pressure of K is 3.25 mm, of Hg 1280 mm. 

Mole % K in liq. . 41.1 46.8 50.0 56.1 63.0 72.0 

PofHg, mm . 31.87 17.30 13.00 9.11 6.53 3.70 

PofK, mm . 0.348 0.68 1.07 1.69 2.26 2.95 

Calculate the activity coefficients of K and Hg in the amalgams and plot 
them vs. the composition in the range studied. 

19. The equilibrium pressures for the system CaSO 4 -2 H 2 O = CaSO 4 + 
2 H 2 O, and the vapor pressures of pure water, at various temperatures are: 

/, C 50 55 60 65 

CaSO 4 system, mm . .80 109 149 204 

H 2 O, mm .... 92 118 149 188 

The solubility of CaSO 4 in water is so low that the vapor pressure of the 
saturated solution can be taken to equal that of pure water. 

(a) State what happens on heating the dihydrate in a previously evacu- 
ated sealed tube from 50 to 65C. (b) What solid phase separates when a 
solution of CaSO 4 is evaporated at 65, at 55C? (c) What solid phase 
separates on evaporating at 55 if, when the solution becomes saturated, 
enough CaCl 2 is added to reduce its v.p. by 10 per cent? 

20. Data for the Au-Te system: 

wt. % Te 10 20 30 40 42 50 56.4 60 70 82.5 90 100 

f.pt., C 1063 940 855 710 480 447 458 464 460 448 416 425 453 

Sketch the phase diagram. Label all regions carefully. Describe what happens 
when a melt containing 50 per cent Te is cooled slowly. 

21. The dissociation pressure of galactose monohydrate is given by 
Iog 10 />(mrn) - 7.04 - 1780/7. Calculate AF, A//, AS , at 25C for the 
dissociation. 

22. The solubility of glycine in liquid ammonia was found to be: 

Moles per liter . . . 0.20 0.65 2.52 

/, C -77 -63 -45 

Estimate the heat of solution per mole. 

23. For the free energies of formation of Cu 2 O and CuO the following 
equations are cited : 

Cu 2 0: AF = -40,720 + 1.1771n T - 1.545 x lQr*T* -f 85.77 1 / 2 -f 6.977 
CuO: AF = -37,680 + 1.757 In T - 2.73 X 10- 8 7 2 + 85.77' / 2 + 9.497 

What product is formed when O 2 at 10 mm pressure is passed over copper 
at900C? 

REFERENCES 

BOOKS 

1. Brick, R. M., and A. Phillips, Structure and Properties of Alloys (New 
York: McGraw-Hill, 1949). 



Chap. 6] SOLUTIONS AND PHASE EQUILIBRIA 159 

2. Carney, T. P., Laboratory Fractional Distillation (New York: Macmillan, 
1949). 

3. Guggenheim, E. A., Mixtures (New York: Oxford, 1952). 

4. Hildebrand, J. H., and R. L. Scott, Solubility of Nonelectrolytes (New 
York: Reinhold, 1950). 

5. Hume-Rothery, W., J. W. Christian, and W. B. Pearson, Metallurgical 
Equilibrium Diagrams (London: Inst. of Physics, 1952). 

6. Shand, S. J., Rocks for Chemists (London: Murby, 1952). 

7. Wagner, C, Thermodynamics of Alloys (Cambridge, Mass: Addison- 
Wesley, 1952). 

8. Weissberger, A. (editor), Physical Methods of Organic Chemistry , vol. I 
(New York: Interscience, 1950). Articles on determination of melting 
point, boiling point, solubility, osmotic pressure. 

ARTICLES 

1. Chem. Rev., 44, 1-233 (1949), "Symposium on Thermodynamics of 
Solutions." 

2. Fleer, K. B., /. Chem. Ed., 22, 588-92 (1945), "Azeotropism." 

3. Hildebrand, J. H., J. Chem. Ed., 25, 74-77 (1948), "Ammonia as a 
Solvent." 

4. Teller, A. J., Chem. Eng., 61, 168-88 (1954), "Binary Distillation." 



CHAPTER 7 

The Kinetic Theory 



1. The beginning of the atom. Thermodynamics is a science that takes 
things more or less as it finds them. It deals with pressures, volumes, tem- 
peratures, and energies, and the relations between them, without seeking to 
elucidate further the nature of these entities. For thermodynamics, matter is 
a Continuous substance, and energy behaves in many ways like an incom- 
pressible, weightless fluid. The analysis of nature provided by thermo- 
dynamics is very effective in a rather limited field. Almost from the beginning 
of human thought, however, man has tried to achieve an insight into the 
structure of things, and to find an indestructible reality beneath the ever- 
changing appearances of natural phenomena. 

The best example of this endeavor has been the development of the 
atomic theory. The word atom is derived from the Greek aro/io?, meaning 
"indivisible" ; the atoms were believed to be the ultimate and eternal particles 
of which all material things were made. Our knowledge of Greek atomism 
comes mainly from the long poem of the Roman, Lucretius, De Rerwn 
Natura "Concerning the Nature of Things," written in the first century 
before Christ. Lucretius expounded the theories of Epicurus and of 
Democritus : 

The same letters, variously selected and combined 

Signify heaven, earth, sea, rivers, sun, 

Most having some letters in common. 

But the different subjects are distinguished 

By the arrangement of letters to form the words. 

So likewise in the things themselves, 

When the intervals, passages, connections, weights, 

Impulses, collisions, movements, order, 

And position of the atoms interchange, 

So also must the things formed from them change. 

The properties of substances were determined by the form of their atoms. 
Atoms of iron were hard and strong with spines that locked them together 
into a solid; atoms of water were smooth and slippery like poppy seeds; 
atoms of salt were sharp and pointed and pricked the tongue; whirling atoms 
of air pervaded all matter. 

Later philosophers were inclined to discredit the atomic theory. They 
found it hard to explain the many qualities of materials, color, form, taste, 
and odor, in terms of naked, colorless, tasteless, odorless atoms. Many 
followed the lead of Heraclitus and Aristotle, considering matter to be 
formed from the four "elements," earth, air, fire, and water, in varying 

160 



Sec. 2] THE KINETIC THEORY 161 

proportions. Among the alchemists there came into favor the tria prima of 
Paracelsus (1493-1541), who wrote: 

Know, then, that all the seven metals are born from a threefold matter, namely, 
Mercury, Sulphur, and Salt, but with distinct and peculiar colorings. 

Atoms were almost forgotten till the seventeenth century, as the al- 
chemists sought the philosopher's stone by which the "principles" could be 
blended to make gold. 

2. The renascence of the atom. The writings of Descartes (1596-1650) 
helped to restore the idea of a corpuscular structure of matter. Gassendi 
(1592-1655) introduced many of the concepts of the present atomic theory; 
his atoms were rigid, moved at random in a void, and collided with one 
another. These ideas were extended by Hooke, who first proposed (1678) 
that the "elasticity" of a gas was the result of collisions of its atoms with 
the retaining walls. 

The necessary philosophic background for the rapid development of 
atomism was now provided by John Locke. In his Essay on Human Under- 
standing (1690), he took up the old problem of how the atoms could account 
for all the qualities perceived by the senses in material things. The qualities 
of things were divided into two classes. The primary qualities were those of 
shape, size, motion, and situation. These were the properties inherent in the 
corpuscles or atoms that make up matter. Secondary qualities, such as color, 
odor, and taste, existed only in the mind of the observer. They arose when 
certain arrangements of the atoms of matter interacted with other arrange- 
ments of atoms in the sense organs of the observer. 

Thus a "hot object" might produce a change in the size, motion, or 
situation of the corpuscles of the skin, which then produces in the mind the 
sensations of warmth or of pain. The consequences of Locke's empiricism 
have been admirably summarized by J. C. Gregory. 1 

The doctrine of qualities was a curiously dichotomized version of perception. 
A snowflake, as perceived, was half in the mind and half out of it, for its shape was 
seen but its whiteness was only in the mind. . . . This had quick consequences 
for philosophy. . . . The division between science and philosophy began about the 
time of Locke, as the one turned, with its experimental appliances, to the study of 
the corpuscular mechanism, and the other explored the mind and its ideas. The 
severance had begun between science and philosophy and, although it only gradually 
progressed into the nineteenth century cleft between them, when the seventeenth 
century closed, physical science was taking the physical world for her domain, and 
philosophy was taking the mental world for hers. 

In the early part of the eighteenth century, the idea of the atom became 
widely accepted. Newton wrote in 1718: 

It seems probable to me that God in the beginning formed matter in solid, 
massy, hard, impenetrable, movable particles, of such sizes and figures, and with 
such other properties, and in such proportion, as most conduced to the end for 
which He formed them. 

1 A Short History of Atomism (London: A. & C. Black, Ltd., 1931). 



162 THE KINETIC THEORY [Chap. 7 

Newton suggested, incorrectly, that the pressure of a gas was due to repulsive 
forces between its constituent atoms. In 1738, Daniel Bernoulli correctly 
derived Boyle's Law by considering the collisions of atoms with the container 
wall. 

3. Atoms and molecules. Boyle had discarded the alchemical notion of 
elements and defined them as substances that had not been decomposed in 
the laboratory. Until the work of Antoine Lavoisier from 1772 to 1783, 
however, chemical thought was completely dominated by the phlogiston 
theory of Georg Stahl, which was actually a survival of alchemical concep- 
tions. With Lavoisier's work the elements took on their modern meaning, 
and chemistry became a quantitative science. The Law of Definite Propor- 
tions and The Law of Multiple Proportions had become fairly well established 
by 1808, when John Dalton published his New System of Chemical Philosophy . 
Dalton proposed that the atoms of each element had a characteristic 
atomic weight, and that these atoms were the combining units in chemical 
reactions. This hypothesis provided a clear explanation for the Laws of 
Definite and Multiple Proportions. Dalton had no unequivocal way of 
assigning atomic weights, and he made the unfounded assumption that in 
the most common compound between two elements, one atom of each was 
combined. According to this system, water would be HO, and ammonia NH. 
If the atomic weight of hydrogen was set equal to unity, the analytical data 
would then give O = 8, N - 4.5, in Dalton's system. 

At about this time, Gay-Lussac was studying the chemical combinations 
of gases, and he found that the ratios of the volumes of the reacting gases 
were small whole numbers. This discovery provided a more logical method 
for assigning atomic weights. Gay-Lussac, Berzelius, and others felt that the 
volume occupied by the atoms of a gas must be very small compared with 
the total gas volume, so that equal volumes of gas should contain equal 
numbers of atoms. The weights of such equal volumes would therefore be 
proportional to the atomic weights. This idea was received coldly by Dalton 
and many of his contemporaries, who pointed to reactions such as that 
which they wrote as N + O NO. Experimentally the nitric oxide was 
found to occupy the same volume as the nitrogen and oxygen from which 
it was formed, although it evidently contained only half as many "atoms." 2 
Not till 1860 was the solution to this problem understood by most 
chemists, although half a century earlier it had been given by Amadeo 
Avogadro. In 1811, he published in the Journal de physique an article that 
clearly drew the distinction between the molecule and the atom. The "atoms" 
of hydrogen, oxygen, and nitrogen are in reality "molecules" containing two 
atoms each. Equal volumes of gases should contain the same number of 
molecules (Avogadro's Principle). 

Since a molecular weight in grams-(mole) of any substance contains the 

same number of molecules, 'according to Avogadro's Principle the molar 

2 The elementary corpuscles of compounds were then called "atoms" of the compound. 



Sec. 4] THE KINETIC THEORY 163 

volumes of all gases should be the same. The extent to which real gases 
conform to this rule may be seen from the molar volumes in Table 7.1 cal- 
culated from the measured gas densities. For an ideal gas at 0C and 1 atm, 
the molar volume would be 22,414 cc. The number of molecules in one mole 
is now called Avogadro's Number N. 

TABLE 7.1 
MOLAR VOLUMES OF GASES IN cc AT 0C AND 1 ATM PRESSURE 



Hydrogen . . 22,432 

Helium . . . 22,396 

Methane . . . 22,377 

Nitrogen . . . 22,403 

Oxygen . . . 22,392 

Ammonia . . 22,094 



Argon . . . 22,390 

Chlorine . . . 22,063 

Carbon dioxide . . 22,263 

Ethane . . . 22,172 

Ethylene . . . 22,246 

Acetylene . . . 22,085 



The work of Avogadro was almost completely neglected until it was 
forcefully presented by Cannizzaro at the Karlsruhe Conference in 1860. 
The reason for this neglect was probably the deeply rooted feeling that 
chemical combination occurred by virtue of an affinity between unlike ele- 
ments. After the electrical discoveries of Galvani and Volta, this affinity 
was generally ascribed to the attraction between unlike charges. The idea 
that two identical atoms of hydrogen might combine into the compound 
molecule H 2 was abhorrent to the chemical philosophy of the early nineteenth 
century. 

4. The kinetic theory of heat. Even among the most primitive peoples 
the connection between heat and motion was known through frictional 
phenomena. As the kinetic theory became accepted during the seventeenth 
century, the identification of heat with the mechanical motion of the atoms 
or corpuscles became quite common. 

Francis Bacon (1561-1626) wrote: 

When I say of motion that it is the genus of which heat is a species I would be 
understood to mean, not that heat generates motion or that motion generates heat 
(though both are true in certain cases) but that heat itself, its essence and quiddity, 
is motion and nothing else. . . . Heat is a motion of expansion, not uniformly of 
the whole body together, but in the smaller parts of it ... the body acquires a 
motion alternative, perpetually quivering, striving, and struggling, and initiated by 
repercussion, whence springs the fury of fire and heat. 

Although such ideas were widely discussed during the intervening years, 
the caloric theory, considering heat as a weightless fluid, was the working 
hypothesis of most natural philosophers until the quantitative work of Rum- 
ford and Joule brought about the general adoption of the mechanical theory. 
This theory was rapidly developed by Boltzmann, Maxwell, Clausius, and 
others, from 1860 to 1890. 

According to the tenets of the kinetic theory, both temperature and 
pressure are thus manifestations of molecular motion. Temperature is a 
measure of the average translational kinetic energy of the molecules, and 



164 THE KINETIC THEORY [Chap. 7 

pressure arises from the average force resulting from repeated impacts of 
molecules with the containing walls. 

5. The pressure of a gas. The simplest kinetic-theory model of a gas 
assumes that the volume occupied by the molecules may be neglected com- 
pletely compared to the total volume. It is further assumed that the molecules 
behave like rigid spheres, with no forces of attraction or repulsion between 
them except during actual collisions. 

In order to calculate the pressure in terms of molecular quantities, let us 
consider a volume of gas contained in a cubical box of side /. The velocity 
c of any molecule may be resolved into components u, v, and w, parallel to 
the three mutually perpendicular axes X, Y, and Z, so that its magnitude is 
given by 

C 2^_ U 2 +V 2 + W 2 (7.1) 

Collisions between a molecule and the walls are assumed to be perfectly 
elastic; the angle of incidence equals the angle of reflexion, and the velocity 
changes in direction but not in magnitude. At each collision with a wall 
that is perpendicular to X, the velocity component u changes sign from 
} u to - w, or vice versa; the momentum component of the molecule accord- 
ingly changes from imw to ^mu, where m is the mass of the molecule. The 
magnitude of the change in momentum is therefore 2 mu. 

The number of collisions in unit time with the two walls perpendicular 
to X is equal to w//, and thus the change in the X component of momentum 
in unit time is 2mu - (u/l) -- 2mu 2 /l. 

If there are N molecules in the box, the change in momentum in unit 
time becomes 2(/Ww( 2 )//), where (w 2 ) is the average value of the square of 
velocity component 3 u. This rate of change of momentum is simply the force 
exerted by the molecules colliding against the two container walls normal to 
X, whose area is 2/ 2 . Since pressure is defined as the force normal to unit area, 

_ 2Nm(u 2 ) Nm(rf) 

P ^ 21* -I " ~ V~ 

Now there is nothing to distinguish the magnitude of one particular 
component from another in eq. (7.1) so that on the average (u 2 ) = (v 2 ) = 
(w 2 ). Thus 3(w 2 ) (c 2 ) and the expression for the pressure becomes 

(7.2) 

V ' 



3K 

The quantity (c 2 ) is called the mean square speed of the molecules, and 
may be given the special symbol C 2 . Then C = (c 2 ) 172 is called the root mean 

3 Not to be confused with the square of the average value of the velocity component, 
which would be written (w) 2 . In this derivation we are averaging w 2 , not //. 



Sec. 6] - THE KINETIC THEORY 165 

square speed. The total translational kinetic energy E K of the molecules is 
iNmC*. Therefore from eq. (7.2): 

PV = \NrnC* ^%E K (7.3) 

Since the total kinetic energy is a constant, unchanged by the elastic 
collisions, eq. (7.3) is equivalent to Boyle's Law. 

If several different molecular species are present in a gas mixture, their 
kinetic energies are additive. From eq. (7.3), therefore, the total pressure is 
the sum of the pressures each gas would exert if it occupied the entire volume 
alone. This is Dalton's Law of Partial Pressures. 4 

6. Kinetic energy and temperature. The concept of temperature was first 
introduced in connection with the study of thermal equilibrium. When two 
bodies are-placed in contact, energy flows from one to the other until a state 
of equilibrium is reached. The two bodies are then at the same temperature. 
We have found that the temperature can be measured conveniently by means 
of an ideal-gas thermometer, this empirical scale being identical with the 
thermodynamic scale derived from the Second Law. 

A distinction was drawn in thermodynamics between mechanical work 
and heat. According to the kinetic theory, the transformation of mechanical 
work into heat is simply a degradation of large-scale motion into motion on 
the molecular scale. An increase in the temperature of a body is equivalent 
to an increase in the average translational kinetic energy of its constituent 
molecules. We may express this mathematically by saying that the tempera- 
ture is a function of E K alone, T -^ f(E K ). We know that this function must 
have the special form T %E K /R, or 

E K - $RT (7.4) 

so that eq. (7.3) may be consistent with the ideal-gas relation, PV RT. 

Temperature is thus not only a function of, but in fact proportional to, 
the average translational kinetic energy of the molecules. The kinetic-theory 
interpretation of absolute zero is thus the complete cessation of all molecular 
motion the zero point of kinetic energy. 5 

The average translational kinetic energy may be resolved into components 
in the three degrees of freedom corresponding to velocities parallel to the 
three rectangular coordinates. Thus, for one mole of gas, where TV is 
Avogadro's Number, 



E K = 
For each translational degree of freedom, therefore, from eq. (7.4), 

E' K - \Nm^f) - \RT (7.5) 



* PV = l(E Kl \ E K2 + ...); P,V - \E Kl \ P 2 y =- E 
Therefore, P = /\ + P 2 + . . ., Dalton's Law. 

5 It will be seen later that this picture has been somewhat changed by quantum theory, 
which requires a small residual energy even at the absolute zero. 



166 THE KINETIC THEORY [Chap. 7 

This is a special case of a more general theorem known as the Principle of 
the Equipartition of Energy. 

7. Molecular speeds. Equation (7.3) may be written 

C 2 =--= 3/> (7.6) 

P 

where /> -= NmjV is the density of the gas. From eqs. (7.3) and (7.4) we 
obtain for the root mean square speed C, if M is the molecular weight, 

2 3RT 3RT 

^ ~ ~ ~ 



The average speed c, as we shall see later, differs only slightly from the root 
mean square speed : 



From eq. (7.6), (7.7), or (7.8), we can readily calculate average or root 
mean square speeds of the molecules of any gas at any temperature. Some 
results are shown in Table 7.2. The average molecular speed of hydrogen at 
25C is 1768 m per sec or 3955 mi per hr, about the speed of a rifle bullet. 
The average speed of a mercury vapor atom would be only about 400 mi 

per *hr. 

TABLE 7.2 
AVERAGE SPEEDS OF GAS MOLECULES AT 0C 

Meters/sec 

. 1692.0 

. 1196.0 

. 170.0 

. 600.6 

. 454.2 

. 425.1 

. 566.5 

We note that, in accordance with the principle of equipartition of energy, 
at any constant temperature the lighter molecules have the higher average 
speeds. This principle extends even to the phenomena of Brownian motion, 
where the dancing particles are some thousand times heavier than the 
molecules colliding with them, but nevertheless have the same average 
kinetic energy. 

8. Molecular effusion. A direct experimental illustration of the different 
average speeds of molecules of different gases can be obtained from the 
phenomenon called molecular effusion. Consider the arrangement shown in 
(a), Fig. 7.1. Molecules from a vessel of gas under pressure are permitted to 
escape through a tiny orifice, so small that the distribution of the velocities 



Gas 


Meters/sec 


Gas 


Ammonia 
Argon 
Benzene . 
Carbon dioxide 


. 582.7 
. 380.8 
. 272.2 
. 362.5 


Hydrogen 
Deuterium 
Mercury . 
Methane . 


Carbon monoxide 
Chlorine . 
Helium 


. 454.5 
. 285.6 
. 1204.0 


Nitrogen . 
Oxygen . 
Water 



Sec. 8] 



THE KINETIC THEORY 



167 



of the gas molecules remaining in the vessel is not affected in any way; that 
is, no appreciable mass flow in the direction of the orifice is set up. The 
number of molecules escaping in unit time is then equal to the number that, 
in their random motion, happen to hit the orifice, and this number is pro- 
portional to the average molecular speed. 

In (b), Fig. 7.1 is shown an enlarged view of the orifice, having an area 
ds. If all the molecules were moving directly perpendicular to the opening 
with their mean speed r, in one second all those molecules would hit the 
opening that were contained in an element of volume of base ds and height c, 
or volume c ds, for a molecule at a distance c will just reach the orifice at 




(b) 



Fig. 7.1. Effusion of gases. 

the end of one second. If there are n molecules per cc, the number striking 
would be nc ds. To a first approximation only one-sixth of all the molecules 
are moving toward the opening, since there are six different possible direc- 
tions of translation corresponding to the three rectangular axes. The number 
of molecules streaming through the orifice would therefore be \nc ds, or per 
unit area \nc. 

Actually, the problem is considerably more complicated, since half the 
molecules have a component of motion toward the area, and one must 
average over all the different possible directions of motion. This gives the 
result: number of molecules striking unit area per second = number of 
molecules effusing through unit area per second = \nc. 

It is instructive to consider how this result is obtained, since the averaging 
method is typical of many kinetic-theory calculations. This derivation will be 
the only one in the chapter that makes any pretense of exactitude, and may 
therefore serve also to inculcate a proper suspicion of the cursory methods 
used to obtain subsequent equations. 

If the direction of the molecules is no longer normal to the wall, instead 
of the situation of Fig. 7. 1 , we have that of Fig. 7.2(a). For^any given direction 
the number of molecules hitting ds in unit time will be those contained in a 
cylinder of base ds and slant height c. The volume of this cylinder is c cos ds, 
and the number of molecules in it is nc cos 6 ds. 

The next step is to discover how many molecules out of the total have 



168 



THE KINETIC THEORY 



[Chap. 7 



NORMAL 



velocities in the specified direction. The velocities of the molecules will be 
referred to a system of polar coordinates [Fig. 7.2.(b)] with its origin at the 
wall of the vessel. We call such a representation a plot of the molecular 

velocities in "velocity space." The 
distance from the origin c defines 
the magnitude of the velocity, and 
the angles and <f> represent its 
direction. Any particular direction 
from the origin is specified by the 
differential solid angle doj. The 
fraction of the total number of 
molecules having their velocities 
within this particular spread of 
directions is */a>/477 since 4?r is the 
total solid angle subtended by the 
surface of a sphere. In polar co- 



(c) 




(a) 





(b) 



Fig. 7.2. Calculation of gaseous effusion. 
Element of solid angle is shown in (c). 



ordinates this solid angle is given 6 
by sin 6 dO d<f>. 

The number of molecules hit- 
ting the surface ds in unit time 
from the given direction (6, (/>) be- 
comes ( 1 /47r)nc cos sin 6 dO dc/> ds. 
Or, for unit surface, it is (l/47r)rtccos sin 6 dO d<j>. In order to find the 
total number striking from all directions, dn'/dt, this expression must be 
integrated : 

dn f w/2 f 2 " 1 

-y = --nccosO sin d<f> dO 

at Jo Jo 4-7T 

The limits of integration of <f> are from to 2?r, corresponding to ail the 
directions around the circle at any given 0. Then is integrated from to 
7T/2. The final result for the number of molecules striking unit area in unit 
time is then 

^ - i nc (7.9) 



The steps of the derivation may be reviewed by referring to Fig. 7.2. 
If p is the gas density, the weight of gas that effuses in unit time is 



From eq. (7.8) 



dW^ _ 
~dt " 

dW _ / 
~dt ~~ n 



vl/2 



P \bM) 



(7.10) 
(7.11) 



6 G. P. Harriwell, Principles of Electricity and Electromagnetism (New York: McGraw- 
Hill, 1949), p. 649. 



Sec. 9] 



THE KINETIC THEORY 



169 



For the volume rate of flow, e.g., cc per sec per cm 2 , 

1/2 



dt 



- (--V 

\l-nMJ 



(7.12) 



It follows that at constant temperature the rate of effusion varies in- 
versely as the square root of the molecular weight. Thomas Graham (1848) 
was the first to obtain experimental evidence for this law, which is now 
named in his honor. Some of his data are shown in Table 7.3. 



TABLE 7.3 
THE EFFUSION OF GASES* 



Gas 



Air . 
Nitrogen , 
Oxygen 
Hydrogen 
Carbon dioxide . 





Relative Velocity of Effusion 




Observed 


Calculated from (7.12) 




0) 
1.0160 
0.9503 
3.6070 
0.8354 


(1) 
1.0146 
0.9510 
3.7994 
0.8087 



* Source: Graham, "On the Motion of Gases," Phil. Trans. Roy. Soc. (London), 136, 
573 (1846). 

It appears from Graham's work, and also from that of later experi- 
menters, that eq. (7.12) is not perfectly obeyed. It fails rapidly when one 
goes to higher pressures and larger orifices. Under these conditions the 
molecules can collide many times with one another in passing through the 
orifice, and a hydrodynamic flow towards the orifice is set up throughout 
the container, leading to the formation of a jet of escaping gas. 7 

It is evident from eq. (7.12) that the effusive-flow process provides a 
good method for separating gases of different molecular weights. By using 
permeable barriers with very fine pores, important applications have been 
made in the separation of isotopes. Because the lengths of the pores are 
considerably greater than their diameters, the flow of gases through such 
barriers does not follow the simple orifice-effusion equation. The dependence 
on molecular weight is the same, since each molecule passes through the 
barrier independently of any others. 

9. Imperfect gases van der Waals' equation, The calculated properties 
of the perfect gas of the kinetic theory are the same as the experimental 
properties of the ideal gas of thermodynamics. It might be expected then 
that extension and modification of the simplified model of the perfect gas 
should provide an explanation for observed deviations from ideal-gas 
behavior. 

7 For a discussion of jet flow, see H. W. Liepmann and A. E. Puckett, Introduction to 
Aerodynamics of a Compressible Fluid (New York: Wiley, 1947), pp. 32 et seq. 



170 



THE KINETIC THEORY 



[Chap. 7 



The first improvement of the model is to abandon the assumption that 
the volume of the molecules themselves can be completely neglected in com- 
parison with the total gas volume. The effect of the finite volume of the 
molecules is to decrease the available void space in which the molecules are 
free to move. Instead of the V in the perfect gas equation, we must write 
V b where b is called the excluded volume. This is not just equal to the 
volume occupied by the molecules, but actually to four times that volume. 
This may be seen in a qualitative way by considering the two molecules of 
Fig. 7.3 (a), regarded as impenetrable spheres each with a diameter d. The 



QD o 

o. o 

o 




(o) 







o 



o 



o 

(b) 



Fig. 7.3. Corrections to perfect gas law. (a) Excluded volume, 
(b) Intermolecular forces. 

centers of these two molecules cannot approach each other more closely 
than the distance d\ the excluded volume for the pair is therefore a sphere 
of radius d 2r (where r is the radius of a molecule). This volume is JTrrf 3 -- 
8 . |77r 3 per pair, or 4 . ^vrr 3 which equals 4V m per molecule (where V m is the 
volume of the molecule). 

The consideration of the finite molecular volumes leads therefore to a 
gas equation of the form: P(V b) ---- RT. A second correction to the perfect 
gas formula comes from consideration of the forces of cohesion between the 
molecules. We recall that the thermodynamic definition of the ideal gas 
includes the requirement that (dE/dV) T 0. If this condition is not fulfilled, 
when the gas is expanded work must be done against the cohesive forces 
between the molecules. The way in which these attractive forces enter into 
the gas equation may be seen by considering Fig. 7.3. (b). The molecules 
completely surrounded by other gas molecules are in a uniform field of 
force, whereas the molecules near to or colliding with the container walls 
experience a net inward pull towards the body of the gas. This tends to 
decrease the pressure compared to that which would be exerted by molecules 
in the absence of such attractive forces. 

The total inward pull is proportional to the number of surface-layer 
molecules being pulled, and to the number of molecules in the inner layer 
of the gas that are doing the pulling. Both factors are proportional to the 



Sec. 10] THE KINETIC THEORY 171 

density of the gas, giving a pull proportional to p 2 , or equal to c/> 2 , where 
c is a constant. Since the density is inversely proportional to the volume 
at any given pressure and temperature, the pull may also be written a/V 2 . 
This amount must therefore be added to the pressure to make up for the 
effect of the attractive forces. Then, 

(Y- b)^RT (7.13) 

This is the famous equation of state first given by van der Waals in 1873. 
It provides a good representation of the behavior of gases at moderate 
densities, but deviations become very appreciable at higher densities. The 
values of the constants a and b are obtained from the experimental PVT 
data at moderate densities, or more usually from the critical constants of 
the gas. Some of these values were collected in Table 1.1 on p. 14. 

Equation (7.13) may also be written in the form 

PV-^RT+bP -^+^ 2 (7.14) 

The way in which this equation serves to interpret PV vs. P data may be 
seen from an examination of the compressibility factor curves at different 
temperatures, shown in Fig. 1.5 (p. 15). At sufficiently high temperatures 
the intermolecular potential energy, which is not temperature dependent, 
becomes negligible compared to the kinetic energy of the molecules, which 
increases with temperature. Then the equation reduces to PV -= RT + bP. 
At lower temperatures, the effect of intermolecular forces becomes more 
appreciable. Then, at moderate pressures the ~a\ V term becomes important, 
and there are corresponding declines in the PV vs. P curves. At still higher 
pressures, however, the term +ab/V 2 predominates, and the curves eventually 
rise again. 

10. Collisions between molecules. Now that the oversimplification that 
the molecules of a gas occupy no volume themselves has been abandoned, 
it is possible to consider further the phenomena that depend on collisions 
between the molecules. Let us suppose that all the molecules have a diameter 
d, and consider as in Fig. 7.4 the approach of a molecule A toward another 
molecule B. 

A "collision" occurs whenever the distance between their centers becomes 
as small as d. Let us imagine the center of A to be surrounded by a sphere of 
radius d. A collision occurs whenever the center 'of another molecule comes 
within this sphere. If A is traveling with the average speed c, its "sphere 
of influence" sweeps out in unit time a volume nd^c. Since this volume 
contains n molecules per cc, there are -nnd^c collisions experienced by A per 
second. 

A more exact calculation takes into consideration that only the speed of 



172 



THE KINETIC THEORY 



[Chap. 7 



a molecule relative to other moving molecules determines the number of 
collisions Z x that it experiences. This fact leads to the expression 

Zj -^ V27wd 2 c (7.15) 

The origin of the A/2 factor may be seen by considering, in Fig. 7.5, the 
relative velocities of two molecules just before or just after a collision. The 








2d 




Fig. 7.4. Molecular collisions. 

limiting cases are the head-on collision and the grazing collision. The average 
case appears to be the 90 collision, after which the magnitude of the relative 
velocity is V2c. 

If we now examine the similar motions of all the molecules, the total 





(b) 



(C) 



Fig. 7.5. Relative speeds, (a) Head-on collision. 
(b) Grazing collision, (c) Right-angle collision. 

number of collisions per second of all the n molecules contained in one cc 
of gas is found, from eq. (7.15), to be 



Z n - 



(7.16) 



The factor J is introduced so that each collision is not counted twice (once 
as A hits B, and once as B hits A). 

11. Mean free paths. An important quantity in kinetic theory is the 
average distance a molecule travels between collisions. This is called the 
mean free path. The average number of collisions experienced by one mole- 
cule in one second is, from eq. (7.15)^ -=- \^2irnd 2 c. In this time the 



Sec. 12] 



THE KINETIC THEORY 



173 



molecule has traveled a distance c. The mean free path A is therefore c/Z lt or 

(7.17) 



In order to calculate the mean free path, we must know the molecular 
diameter d. This might be obtained, for example, from the van der Waals 
b ( 4V m ) if the value of Avogadro's Number N were known. So far, our 
development of kinetic theory has provided no method for obtaining this 
number. The theory of gas viscosity as developed by James Clerk Maxwell 
presents a key to this problem, besides affording one of the most striking 
demonstrations of the powers of the kinetic theory of gases. 

12. The viscosity of a gas. The concept of viscosity is first met in problems 
of fluid flow, treated by hydrodynamics and aerodynamics, as a measure of 




-AREA 



Fig. 7.6. Viscosity of fluids. 

the frictional resistance that a fluid in motion offers to an applied shearing 
force. The nature of this resistance may be seen from Fig. 7.6 (a). If a fluid 
is flowing past a stationary plane surface, the layer of fluid adjacent to the 
plane boundary is stagnant; successive layers have increasingly higher 
velocities. The frictional force /, resisting the relative motion of any two 
adjacent layers, is proportional to S, the area of the interface between them, 
and to dvjdr, the velocity gradient between them. This is Newton's Law of 
Viscous Flow, 

dv 

f^viS-j- (7.18) 

dr 

The proportionality constant r\ is called the coefficient of viscosity. It is 
evident that the dimensions of rj are ml~ l t~~ l . In the COS system, the unit is 
g per cm sec, called the poise. 



174 THE KINETIC THEORY [Chap. 7 

The kind of flow governed by this relationship is called laminar or 
streamline flow. It is evidently quite different in character from the effusive 
(or diffusive) flow previously discussed, since it is a massive flow of fluid, in 
which there is superimposed on all the random molecular velocities a com- 
ponent of velocity in the direction of flow. 

An especially important case of viscous flow is the flow through pipes or 
tubes when the diameter of the tube is large compared with the mean free 
path in the fluid. The study of flow through tubes has been the basis for 
many of the experimental determinations cf the viscosity coefficient. The 
theory of the process was first worked out by J. L. Poiseuille, in 1844. 

Consider an incompressible fluid flowing through a tube of circular cross 
section with radius R and length L. The fluid at the walls of the tube is 
assumed to be stagnant, and the rate of flow increases to a maximum at the 
center of the tube [see Fig. 7.6 (b)]. Let v be the linear velocity at any distance 
r from the axis of the tube. A cylinder of fluid of rad'"<; r experiences a 
viscous drag given by eq. (7.18) as 

V dv i r 

Jr --1J/ *L 

For steady flow, this force must be exactly balanced by the force driving 
the fluid in this cylinder through the tube. Since pressure is the force per unit 
area, the driving force is _ 

/, = wVi - J*a) 
where P l is the fore pressure and P% the back pressure. 

Thus, for steady flow, f r f r 



fp _ p \ 

On integration, v =--- ----- \ r -- r 2 + const 

& 



According to our hypothesis, v = when /-=/?; this boundary condition 
enables us to determine the integration constant, so that we obtain finally 



The total volume of fluid flowing through the tube per second is calculated 
by integrating over each element of cross-sectional area, given by 2-nr dr [see 
Fig. 7.6 (c)]. Thus 

dv f\ J ^Pi ~^ 2 )* 4 

~dt^k 2 dr ^ "" " *>^ ( 7 - 19 > 



ZLrj 



Sec. 13] 



THE KINETIC THEORY 



175 



This is Poiseuille's equation. It was derived for an incompressible fluid 
and therefore may be satisfactorily applied to liquids but not to gases. For 
gases, the volume is a strong function of the pressure. The average pressure 
along the tube is (P l + ^ 2 )/2- If ^o i s tne pressure at which the volume is 
measured, the equation becomes 



dV 
dt 



(7.20) 



By measuring the volume rate of flow through a tube of known dimen- 
sions, the viscosity i] of the gas can be determined. Some results of such 
measurements are collected in Table 7.4. 

TABLE 7.4 

TRANSPORT PHENOMENA IN GASES 
(At 0C and 1 aim) 









Thermal 






Gas 


Mean 
Free 
Path A, 

A 


Viscosity 
//, Poise 
x 10~ 6 


Conduc- 
tivity *, 
cal/g 
secC 


Specific 
Heat, c v 
cal/g 


r]C v /K 








x 10 6 






Ammonia 


441 


97.6 


51.3 


0.399 


0.76 


Argon .... 


635 


213 


38.8 


0.0750 


0.41 


Carbon dioxide 


397 


138 


34.3 


0.153 


0.62 


Carbon monoxide 


584 


168 


56.3 


0.177 


0.53 


Chlorine 


287 


123 


18.3 


0.0818 


0.55 


Ethylene 


345 


93.3 


40.7 


0.286 


0.66 


Helium 


1798 


190 


336 


0.743 


0.42 


Hydrogen 
Nitrogen 


1123 
600 


84.2 
167 


406 
58.0 


2.40 
0.176 


0.50 
0.51 


Oxygen 


647 


192 


58.9 


0.155 


0.51 



13. Kinetic theory of gas viscosity. The kinetic picture of gas viscosity 
has been represented by the following analogy: Two railroad trains are 
moving in the same direction, but at different speeds, on parallel tracks. The 
passengers on these trains amuse themselves by jumping back and forth 
from one to the other. When a passenger jumps from the more rapidly 
moving train to the slower one he transports momentum of amount mv, 
where m is his mass and v the velocity of his train. He tends to speed up the 
more slowly moving train when he lands upon it. A passenger who jumps 
from the slower to the faster train, on the other hand, tends to slow it down. 
The net result of the jumping game is thus a tendency to equalize the velocities 
of the two trains. An observer from afar who could not see the jumpers 
might simply note this result as a frictional drag between the trains. 

The mechanism by which one layer of flowing gas exerts a viscous drag 



176 



THE KINETIC THEORY 



[Chap. 7 



on an adjacent layer is exactly similar, the gas molecules taking the role of 
the playful passengers. Consider in Fig. 7.7 a gas in a state of laminar flow 
parallel to the Y axis. Its velocity increases from zero at the plane x to 
larger and larger values of v with increasing x. If a molecule at P crosses to 
g, in one of its free paths between collisions, it will bring to Q, on the 
average, an amount of momentum which is less than that common to the 
position Q by virtue of its distance along the A' axis. Conversely, if a molecule 
travels from Q to P it will transport to the lower, more slowly moving layer, 
an amount of momentum in excess of that belonging to that layer. The net 
result of the random thermal motions of the molecules is to decrease the 

average velocities of the molecules in 
the layer at Q and to increase those 
in the layer at P. This transport of 
momentum tends to counteract the 
velocity gradient set up by the shear 
forces acting on the gas. 

The length of the mean free path 
X may be taken as the average dis- 
tance over which momentum is trans- 
ferred. 8 If the velocity gradient is 
du/dx, the difference in velocity be- 
tween the two ends of the free path 
is X du/dx. A molecule of mass m, 
passing from the upper to the lower 
layer, thus transports momentum 
equal to wA du/dx. On the average, 
one-third of the molecules are moving up and down; if n is the number 
of molecules per cc and c their average speed, the number traveling up and 
down per second per square cm is -J- nc. The momentum transport per 
second is then \nc mX(du/dx)* 

This momentum change with time is equivalent to the frictional force of 
eq. (7.18) which was/^~ r)(du/dx) per unit area. Hence 




Fig. 7.7. 



Kinetic theory of gas 
viscosity. 



du 

~T 
dx 



du 

- - 
dx 



(7.21) 

The measurement of the viscosity thus allows us to calculate the value of 
the mean free path A. Some values obtained in this way are included in 
Table 7.4, in Angstrom units (1 A = 10~ 8 cm). 

8 This is not strictly true, and proper averaging indicates f A should be used. 

9 The factor J obtained here results from the cancellation of two errors in the derivation. 
From eq. (7.9) one should take nc as the molecules moving across unit area but proper 
averaging gives the distance between planes as JA instead of A. 



Sec. 14] THE KINETIC THEORY 177 

By eliminating A between eqs. (7.17) and (7.21), one obtains 

n - -^- (7.22) 

3\/2W 2 

This equation indicates that the viscosity of a gas is independent of its 
density. This seemingly improbable result was predicted by Maxwell on 
purely theoretical grounds, and its subsequent experimental verification was 
one of the great triumphs of the kinetic theory. The physical reason for the 
result is clear from the preceding derivation: At lower densities, fewer 
molecules jump from layer to layer in the flowing gas, but, because of the 
longer free paths, each jump carries proportionately greater momentum. 
For imperfect gases, the equation fails and the viscosity increases with 
density. 

The second important conclusion from eq. (7.22) is that the viscosity of 
a gas increases with increasing temperatuie, linearly with the \/T. This con- 
clusion has been well confirmed by the experimental results, although the 
viscosity increases somewhat more rapidly than predicted by the \/T law. 

14. Thermal conductivity and diffusion. Gas viscosity depends on the 
transport of momentum across a momentum (velocity) gradient. It is a 
typical transport phenomenon. An exactly similar theoretical treatment is 
applicable to thermal conductivity and to diffusion. The thermal conductivity 
of gases is a consequence of the transport of kinetic energy across a tem- 
perature (i.e., kinetic energy) gradient. Diffusion of gases is the transport of 
mass across a concentration gradient. 

The thermal conductivity coefficient K is defined as the heat flow per unit 
time q, per unit temperature gradient across unit cross-sectional area, i.e., by 

c dT 
q =. K- S- 

ax 
By comparison with eq. (7.21), 

dT 1 , de 

K nc/. 

ax 3 ax 

where de/dx is the gradient of e, the average kinetic energy per molecule. 
Now 

de dT de 



where m is the molecular mass and c v is the specific heat (heat capacity per 
gram). It follows that 

K -^ lnmc v ch \pc v cX ^ r\c v (7.23) 

Some thermal conductivity coefficients are included in Table 7.4. It 
should be emphasized that, even for an ideal gas, the simple theory is approxi- 
mate, since it assumes that all the molecules are moving with the same speed, 
c, and that energy is exchanged completely at each collision. 



178 



THE KINETIC THEORY 



[Chap. 7 



The treatment of diffusion is again similar. Generally one deals with the 
diffusion in a mixture of two different gases. The diffusion coefficient D is 
the number of molecules per second crossing unit area under unit con- 
centration gradient. It is found to be 10 

D - -J Vi*2 +- iV a *i 

where X l and X 2 are the mole fractions of the two gases in the mixture. If 
the two kinds of molecules are essentially the same, for example radioactive 
chlorine in normal chlorine, the self-diffusion coefficient is obtained as 

D &c (7.24) 

The results of the simple mean-free-path treatments of the transport 
processes may be summarized as follows: 



Process 


Transport of 


Simple 
Theoretical 
Expression 


CGS Units of 
Coefficient 


Viscosity 
Thermal conductivity . 
Diffusion . 


Momentum mv 
Kinetic energy J/w> 2 
Mass /// 


ry - \ P ck 
K -- \pckc v 
D ~ \Xc 


g/cm sec 
ergs/cm sec degree 
cm 2 /sec 



Now van der Waals' b is given by 



b - 



- 47V 



15. Avogadro's number and molecular dimensions. Equation (7.22) may be 
written, from eq. (7.8), 

Me 2V1RTM 

71 ^ 



(7.25) 



(7.26) 



Let us substitute the appropriate values for the hydrogen molecule, H 2 , 
all in CGS units. 

M= 2.016 6-26.6 

?7-0.93 x 10~ 4 r= 298K 
R = 8.314 x 10 7 

Solving for d, we find d -=-- 2.2 x 10~ 8 cm. 



6 3 
Multiplying these two equations, and solving for d, 



10 For example, see E. H. Kennard, Kinetic Theory of Gases (New York: McGraw-HilJ, 
1938), p. 188. 



Sec. 15] 



THE KINETIC THEORY 



179 



This value may be substituted back into eq. (7.25) to obtain a value for 
Avogadro's Number TV equal to about 10 24 . 

Because of the known approximations involved in the van der Waals 
formula, this value of TV is only approximate. It is nevertheless of the correct 
order of magnitude, and it is interesting that the value can be obtained 
purely from kinetic-theory calculations. Later methods, which will be dis- 
cussed in a subsequent chapter, give the value TV 6.02 x 10 23 . 

We may use this figure to obtain more accurate values for molecular 
diameters from viscosity or thermal conductivity measurements. Some of 
these values are shown in Table 7.5, together with values obtained from 
van der Waals' b, and by the following somewhat different method. 

TABLE 7.5 

MOLECULAR DIAMETERS 

(Angstrom Units) 





From 


From 


From 


From 


Molecule 


Gas 


van der 


Molecular 


Closest 




Viscosity 


Waals' b 


Refraction* 


Packing 


A 


2.86 


2.86 


2.96 


3.83 


CO 


3.80 


3.16 





4.30 


CO 2 


4.60 


3.24 


2.86 





C1 2 


3.70 


3.30 


3.30 


4.65 


He 


2.00 


2.48 


1.48 





H 2 


2.18 


2.76 


1.86 





Kr 


3.18 


3.14 


3.34 


4.02 


Hg 


3.60 


2.38 


_ 





Ne 


2.34 


2.66 





3.20 


N 2 


3.16 


3.14 


2.40 


4.00 


2 


2.96 


2.90 


2.34 


3.75 


H 2 


2.72 


2.88 


2.26 





* The theory of this method is discussed in Section 1 1-18. 

In the solid state the molecules are closely packed together. If we assume 
that these molecules are spherical in shape, the closest possible packing of 
spheres leaves a void space of 26 per cent of the total volume. The volume 
occupied by a mole of molecules is M/p, where M is the molecular weight 
and p the density of the solid. For spherical molecules we may therefore 
write (7r/6)Nd* = Q.14(M/p). Values of d obtained from this equation may 
be expected to be good approximations for the nionatomic gases (He, Ne, 
A, Kr) and for spherical molecules like CH 4 , CC1 4 . The equation is only 
roughly applicable to diatomic molecules like N 2 or O 2 . 

The rather diverse values often obtained for molecular diameters calcu- 
lated by different methods are indications of the inadequacy of a rigid-sphere 
model, even for very simple molecules. 

The extreme minuteness of the molecules and the tremendous size of the 



180 THE KINETIC THEORY [Chap. 7 

Avogadro Number N are strikingly shown by two popular illustrations given 
by Sir James Jeans. If the molecules in a glass of water were turned into 
grains of sand, there would be enough sand produced to cover the whole 
United States to a depth of about 100 feet. A man breathes out about 400 cc 
at each breath, or about 10 22 molecules. The earth's atmosphere contains 
about 10 44 molecules. Thus, one molecule is the same fraction of a breath 
of air as the breath is of the entire atmosphere. If the last breath of Julius 
Caesar has become scattered throughout the entire atmosphere, the chances 
are that we inhale one molecule from it in each breath we take. 

16. The softening of the atom. We noted before that the viscosity of a gas 
increases more rapidly with temperature than is predicted by the \/T law. 
This is because the molecules are not actually hard spheres, but must be 
regarded as being somewhat soft, or surrounded by fields of force. This 
is true even for the atom-molecules of the inert gases. The higher the tem- 
perature, the faster the molecules are moving, and hence the further one 
molecule can penetrate into the field of force of another, before it is repelled 
or bounced away. The molecular diameter thus appears to be smaller at 
higher temperatures. This correction has been embodied in a formula due to 
Sutherland (1893) 



d*--<Il\\\-) (7.27) 

Here d^ and C are constants, d^ being interpreted as the value of d as T 
approaches infinity. 

More recent work has sought to express the temperature coefficient of 
the viscosity in terms of the laws of force between the molecules. Thus here, 
just as in the discussion of the equation of state, the qualitative picture of 
rigid molecules must be modified to consider the fields of force between 
molecules. 

We recall from Chapter 1 that forces may be represented as derivatives 
of a potential-energy function,/^ (<3(7/cV), and a representation of this 
function serves to illustrate the nature of the forces. In Fig. 7.8 we have 
drawn the mutual potential energy of pairs of molecules of several different 
gases. We may imagine the motion of one molecule as it approaches rapidly 
toward another to be represented by that of a billiard ball rolled with con- 
siderable force along a track having the shape of the potential curve. As the 
molecule approaches another it is accelerated at first, but then slowed down 
as it reaches the steep ascending portion of the curve. Finally it is brought 
to a halt when its kinetic energy is completely used up, and it rolls back down 
and out the curve again. Since the kinetic energy is almost always greater 
than the depth of the potential-energy trough, there is little chance of a 
molecule's becoming trapped therein. (If it did, another collision would soon 
knock it out again.) 

This softening of the original kinetic-theory picture of the atom as a 



Sec. 17] 



THE KINETIC THEORY 



181 



hard rigid sphere was of the greatest significance. It immediately suggested 
that the atoms could not be the ultimate building units in the construction 
of matter, and that man must seek still further for an indestructible reality 
to explain the behavior of material things. 

So far in this chapter we have dealt with average properties of large 
collections of molecules : average velocities, mean free paths, viscosity, and 



DISTANCE BETWEEN 
MOLECULES 




-15 



Fig. 7.8. Mutual potential energy of pairs of molecules. 

so on. In what follows, the contributions of the individual molecules to these 
averages will be considered in some detail. 

17. The distribution of molecular velocities. The molecules of a gas in 
their constant motion collide many times with one another, and -these 
collisions provide the mechanism through which the velocities of individual 
molecules are continually changing. As a result, there exists a distribution 
of velocities among the molecules; most have velocities with magnitudes 
close to the average, and relatively few have velocities much above or much 
below the average. 



182 



THE KINETIC THEORY 



[Chap. 7 



I sq cm 



.__f d * 



A molecule may acquire an exceptionally high speed as the result of a 
series of especially favorable collisions. The theory of probability shows 
that the chance of a molecule's experiencing a series of n lucky hits is pro- 
portional to a factor of the form e~ an , where a is a constant. 11 Thus the 
probability of the molecule's having the energy E above the average energy 
is likewise proportional to e~ bE . The exact derivation of this factor may be 
carried out in several ways, and the problems involved in the distribution of 
velocities, and hence of kinetic energies, among the molecules, form one of 
the most important parts of the kinetic theory. 

18. The barometric formula. It is common knowledge that the density of 

the earth's atmosphere decreases with increas- 
ing altitude. If one makes the simplifying 
assumption that a column of gas extending 
upward into the atmosphere is at constant 
temperature, a formula may be derived for 
this variation of gas pressure in the gravita- 
tional field. The situation is pictured in 
Fig. 7.9. 

The weight of a thin layer of gas of thick- 
ness dx and one cm 2 cross section is its mass 
Fig. 7.9. Barometric formula, times the acceleration due to gravity, or pg 

dx, where /> is the gas density. The difference 

in pressure between the upper and lower boundaries of the layer is 
( -dP/dx)dx, equal to the weight of the layer of unit cross section. Thus 

dP = pgdx 

A ^ PM 

ror an ideal gas, p -= 



D _ 

Kl 



Therefore 



RT 



dx 



Integrating between the limits P P Q at x 0, and P P at x --= x, 

P Mgx 



In 



RT 



n r> _ -MgxIRT ("l *)Q\ 

r r e \i .LO) 

Now, Mgx is simply the gravitational potential energy at the point x, 
which may be written as E^ per mole. Then 

P^P Q e~ E IHT (7.29) 

If, instead of the molar energy, we consider that of the individual mole- 
cule, e p , eq. (7.29) becomes 

. P = P e" e * lltr (7.30) 

11 If the chance of one lucky hit is 1/c, the chance P for n in a row is P - (I/c) w . Then 



Sec. 19] THE KINETIC THEORY 183 

The constant k is called the Bohzmann constant. Ft is the gas constant per 
molecule. 



Equation (7.30) is but one special case of a very general expression 
derived by L. Boltzmann in 1886. This states that if A? O is the number of 
molecules in any given state, the number n in a state whose potential energy 
is e above that of the given state is 

n = n^e- hlkT (7.31) 

19. The distribution of kinetic energies. 12 To analyze more closely the 
kinetic picture underlying the barometric formula, let us consider the in- 
dividual gas molecules moving with their diverse velocities in the earth's 
gravitational field. The velocity components parallel to the earth's surface 
(in the y and z directions along which no field exists) are not now of interest 
and only the vertical or x component u need be considered. 

The motion of a molecule with an upward velocity u is just like that of 
a ball thrown vertically into the air. If its initial velocity is w , it will rise 
with continuously decreasing speed, as its kinetic energy is transformed into 
potential energy according to the equation 

mgx i/w/ 2 iww 2 

At the height, x =- u Q 2 /2g, it will stop, and then fall back to earth. 

The gravitational field acts as a device that breaks up the mixture of 
various molecular velocities into a "spectrum" of velocities. The slowest 
molecules can rise only a short distance; the faster ones can rise propor- 
tionately higher. By determining the number of molecules that can reach 
any given height, we can likewise determine how many had a given initial 
velocity component. 

As is to be expected from the physical picture of the process, the dis- 
tribution of kinetic energies k among the molecules must follow an ex- 
ponential law just as the potential energy distribution does. Representing 
the fraction of molecules having a velocity between u and u f du by dnfn^ 
this law may be written from eq. (7.31) as 

- Ae~^ lkT du (7.32) 

"o 

Here A is a constant whose value is yet to be determined. 

20. Consequences of the distribution law. This distribution law is com- 
pletely unaffected by collisions between molecules, since a collision results 
only in an interchange of velocity components between two molecules. 
Expressions exactly similar to eq. (7.32) must also hold for the velocity 

12 The method suggested here is given in detail by K. F. Herzfeld in H. S. Taylor's 
Treatise on Physical Chemistry, 2nd ed. (New York: Van Nostrand, 1931), p. 93. 



184 



THE KINETIC THEORY 



[Chap. 7 



distributions in the y and z directions, since it is necessary only to imagine 
some sort of potential field in these directions in order to analyze the 
velocities into their spectrum. 

I50 r 



OJ 

ca 
I ~ 



(T 
UJ 



50 



1C 



200 400 600 800 1000 2000 

u, METERS /SECOND 
Fig. 7.10. One-dimensional velocity distribution (nitrogen at 0C). 

The constant A 9 in eq. (7.32), may be evaluated from the fact that the 
sum of all the fractions of molecules in all the velocity ranges must be unity. 
Thus, integrating over all possible velocities from oo to + oo, we have 

r+oo _ WM /2JIT 

A e du -~ 1 

J - oo 

mu 2 2 
2kT ^ X 

+oo 

A ' ^ ' ^ 



Letting 



2^7^\i/2 /*+ 

V/ J-c 



Since 

Therefore, eq. (7.32) becomes 



1/2 



(7.33) 

This function is shown plotted in Fig. 7.10. It will be noted that the 
fraction of the molecules with a velocity component in a given range declines 



Sec. 20] 



THE KINETIC THEORY 



185 



at first slowly and then rapidly as the velocity is increased. From the curve 
and from a consideration of eq. (7.33), it is evident that as long as Jmw 2 < kT 
the fraction of molecules having a velocity u falls off slowly with increasing u. 
When %mu 2 = \QkT, the fraction has decreased to e~ 10 , or 5 x 10~ 5 times its 
value at \rn\f 1 = kT. Thus only a very small proportion of any lot of mole- 
cules can have kinetic energies much greater than kT per degree of freedom. 
If, instead of a one-dimensional gas (one degree of freedom of trans- 
lation), a two-dimensional gas is considered, it can be proved 13 that the 
probability of a molecule having a given x velocity component u in no way 
depends on the value of its y component v. The fraction of the molecules 
having simultaneously velocity components between u and u + du, and v 
and v -f- dv, is then simply the product of the two individual probabilities. 



dn I m \ 

^ = \27Tkf) 



(7.34) 



This sort of distribution may be graphically represented as in Fig. 7.11, 
where a coordinate system with u and v axes has been drawn. Any point in 
the (w, v) plane represents a simul- 
taneous value of u and v\ the plane 
is a two-dimensional velocity space 
similar to that used on p. 168. The 
dots have been drawn so as to 
represent schematically the density 
of points in this space, i.e., the 
relative frequency of occurrence of 
sets of simultaneous values of u 
and v. 

The graph bears a striking re- 
semblance to a target that has been 
peppered with shots by a marks- 
man aiming at the bull's-eye. In 

the molecular case, each individual Fig . 1Mm Distribution of points in two- 
molecular-velocity component, u or dimensional velocity space: v x = u; v v - v. 
v, aims at .the value zero. The 

resulting distribution represents the statistical summary of the results. The 
more skilful the marksman, the more closely will his results cluster around 
the center of the target. For the molecules, the skill of the marksman has its 
analogue in the temperature of the gas. The lower the temperature, the better 
the chance a molecular-velocity component has of coming close to zero. 

If, instead of the individual components u and v, the resultant speed c 
is considered, where c 2 = u 2 + v 2 9 it is evident that its most probable value 
is not zero. This is because the number of ways in which c can be made up 

18 For a discussion of this theorem see, for example, J. Jeans, Introduction to the 
Kinetic Theory of Gases (London: Cambridge, 1940), p. 1Q5. 




186 



THE KINETIC THEORY 



[Chap. 7 



from u and v increases in direct proportion with c, whereas at first the prob- 
ability of any value of u or v declines rather slowly with increasing velocity. 
From Fig. 7.11, it appears that the distribution of c, regardless of direc- 
tion, is obtainable by integrating over the annular area between c and 
c | dc, which is 2nc dc. The required fraction is then 

dn m .,.., . 



__ e 



(7.35) 



21. Distribution law in three dimensions. The three-dimensional distribu- 
tion law may now be obtained by a simple extension of this treatment. The 



200 




1000 2000 

C, METERS /SECOND 

Fig. 7.12. Distribution of molecular speeds (nitrogen). 



3000 



fraction of molecules having simultaneously a velocity component between 
u and u + du, v and v + dv, and w and w + dw, is 



dn_ 1 

"" 



m 



\2irkTj 



(7.36) 



Sec. 22] THE KINETIC THEORY 187 

We wish an expression for the number with a speed between c and c + dc, 
regardless of direction, where c 2 = u 2 + v 2 + w 2 . 

These are the molecules whose velocity points lie within a spherical shell 
of thickness dc at a distance c from the origin. The volume of this shell is 
477-cVc, and therefore the desired distribution function is 

* 

This is the usual expression of the distribution equation derived by James 
Clerk Maxwell in 1860. 

The equation is plotted in Fig. 7.12 at several different temperatures, 
showing how the curve becomes broader and less peaked at the higher tem- 
peratures, as relatively more molecules acquire kinetic energies greater than 
the average of f kT. 

22. The average speed. The average value f of any property r of the 
molecules is obtained by multiplying each value of r, r,, by the number of 
molecules n l having this value, adding these products, and then dividing by 
the total number of molecules. Thus 



where 2 n i =~ n o * s the total number of molecules. 

In case n is known as a continuously varying function of r, n(r), instead 
of the summations of eq. (7.38) we have the integrations 



Jo rdn \ r ) 1 Too 

-r - = --L rdn w ( 7 - 39 > 

' dn(r) "o j 



This formula may be illustrated by the calculation of the average mole- 
cular speed c. Using eq. (7.37), we have 

c - - f c dn - 47r (^r\ m I" e -""V^V j c 
n Q J o xlirkT] J o 

The evaluation of this integral can be obtained 14 from 

1 



>~~ ( 

2a* 
Making the appropriate substitutions, we find 

c '=O 1/2 <> 

14 Letting x 2 = z, 

Too if 00 l/C~ a2: \ 00 1 

Jo e ' aXxdx ^2J^ e aZdz = 2 I'^/o = 2a 



Too 

I pux 

J Q e 



d 



188 THE KINETIC THEORY [Chap. 7 

Similarly, the average kinetic energy can readily be evaluated as 



\ & dn 
2/7 Jo 

This yields 



- \kT (7.41) 

23. The equipartition of energy. Equation (7.41) gives the average trans- 
lational kinetic energy of a molecule in a gas. It will be noted that the average 
energy is independent of the mass of the molecule. Per mole of gas, 

/C(t,a,,) - INkT- $RT (7.42) 

For a monatomic gas, like helium, argon, or mercury vapor, this translational 
kinetic energy is the total kinetic energy of the gas. For diatomic gases, like 
N 2 or C1 2 , and polyatomic gases, like CH 4 or N 2 O, there may also be energy 
associated with rotational and vibrational motions. 

A useful model for a molecule is obtained by supposing that the masses 
of the constituent atoms are concentrated at points. As will be seen in 
Chapter 9, almost all the atomic mass is in fact concentrated in a tiny 
nucleus, the radius of which is about 10 13 cm. Since the over-all dimensions 
of molecules are of the order of 10~ 8 cm, a model based on point masses is 
physically most reasonable. Consider a molecule composed of n atoms. In 
order to represent the instantaneous locations in space of A? mass points, we 
should require 3/7 coordinates. The number of coordinates required to locate 
all the mass points (atoms) in a molecule is called the number of its degrees 
of freedom. Thus a molecule of n atoms has 3/7 degrees of freedom. 

The atoms within each molecule move through space as a connected 
entity, and we can represent the translational motion of the molecule as a 
whole by the motion of the center of mass of its constituent atoms. Three 
coordinates (degrees of freedom) are required to represent the instantaneous 
position of the center of mass. The remaining (3/7 3) coordinates represent 
the so-called internal degrees of freedom. 

The internal degrees of freedom may be further subdivided into rotations 
and vibrations. Since the molecule has moments of inertia / about suitably 
chosen axes, it can be set into rotation about these axes. If its angular velocity 
about an axis is (, the rotational kinetic energy is i/o> 2 . The vibratory 
motion, in which one atom in a molecule oscillates about an equilibrium 
separation from another, is associated with both kinetic and potential 
energies, being in this respect exactly like the vibration of an ordinary spring. 
The vibrational kinetic energy is also represented by a quadratic expression, 
*,mv 2 . The vibrational potential energy can in some cases be represented also 
by a quadratic expression, but in the coordinates q rather than in the 
velocities, for example, i/o/ 2 . Each vibrational degree of freedom would then 
contribute two quadratic r terms to the total energy of the molecule. 

By an extension of the derivation leading to eq. (7.41), it can be shown 



Sec. 24] 



THE KINETIC THEORY 



189 



that each of these quadratic terms that comprise the total energy of the 
molecule has an average value of \kT. This conclusion, a direct consequence 
of the Maxwell-Boltzmann distribution law, is the most general expression 
of the Principle of Equipartition of Energy. 

24. Rotation and vibration of diatomic molecules. The rotation of a di- 
atomic molecule may be visualized by reference to the so-called dumbbell 
model in Fig. 7.13, which might represent a molecule such as H 2 , N 2 , HCI, 





(b) 



Fig. 7.13. Dumbbell rotator. 

or CO. The masses of the atoms, m r and w 2 , are concentrated at points, 
distant r x and r 2 , respectively, from the center of mass. The molecule there- 
fore has moments of inertia about the X and Z axes, but not about the Y 
axis on which the mass points lie. 

The energy of rotation of a rigid body is given by 



rot - Uco* 



(7.43) 



where o> is the angular velocity of rotation, and / is the moment of inertia. 
For the dumbbell model, / = w^ 2 + w 2 r 2 2 . 

The distances r x and r 2 from the center of mass are 

m 2 mi 



Thus 



(7.44) 



The quantity 



(7.45) 



is called the reduced mass of the molecule. The rotational motion is equivalent 
to that of a mass p at a distance r from the intersection of the axes. 

Only two coordinates are required to describe such a rotation com- 
pletely; for example, two angles 6 and <f> suffice to fix the orientation of the 
rotator in space. There are thus two degrees of freedom for the rotation of 
a dumbbell-like structure. According to the principle of the equipartition of 



190 



THE KINETIC THEORY 



[Chap. 7 



energy, the average rotational energy should therefore be rot = 
- RT. 

The simplest model for a vibrating diatomic molecule (Fig. 7.14) is the 
harmonic oscillator. From mechanics we know that simple harmonic motion 

occurs when a particle is acted on by a restor- 
ing force directly proportional to its distance 
, t*99$w$9999f{pt , ft from the equilibrium position. Thus 





Fig. 7.14. 



Harmonic oscil- 
lator. 



The constant K is called the force constant. 

The motion of a particle under the influ- 
ence of such a restoring force may be represented by a potential energy 
function U(r). 

f P u \ 

f= -\Sr-)^- lcr 



U(r) - Jicr 2 (7.47) 

This is the equation of a parabola and the potential-energy curve is 
drawn in Fig. 7.15. The motion of the partide, as has been pointed out in 
previous cases, is analogous to that of a ball moving on such a surface. 
Starting from rest at any position r, it has 
only potential energy, U = i/cr 2 . As it rolls 
down the surface, it gains kinetic energy up 
to a maximum at position r 0, the equi- 
librium interatomic distance. The kinetic 
energy is then reconverted to potential 
energy as the ball rolls up the other side of 
the incline. The total energy at any time is 
always a constant, 



U(r) 




E vih 



-r +r ^ 

Fig. 7.15. Potential curve of 
harmonic oscillator. 



It is apparent, therefore, that vibrating 

molecules when heated can take up energy as both potential and kinetic 
energy of vibration. The equipartition principle states that the average 
energy for each vibrational degree of freedom is therefore kT, \kT for the 
kinetic energy plus \kT for the potential energy. 

For a diatomic molecule the total average energy per mole therefore 
becomes 

~ ^tnms + rot + ^vib =- $RT + RT + RT - 



25. Motions of polyatomic molecules. The motions of polyatomic mole- 
cules can also be represented by the simple mechanical models of the rigid 
rotator and the harmonic oscillator. If the molecule contains n atoms, there 



Sec. 25] THE KINETIC THEORY 191 

are (3n 3) internal degrees of freedom. In the case of the diatomic molecule, 
3n 3 -- 3. Two of the three internal coordinates are required to represent 
the rotation, leaving one vibrational coordinate. 

In the case of a triatomic molecule, 3/7 3 6. In order to divide these 
six internal degrees of freedom into rotations and vibrations, we must first 
consider whether the molecule is linear or bent. If it is linear, all the atomic 
mass points lie on one axis, and there is therefore no moment of inertia about 
this axis. A linear molecule behaves like a diatomic molecule in regard to 
rotation, and there are only two rotational degrees of freedom. For a linear 
triatomic molecule, there are thus 3n 3 2 4 vibrational degrees of 
freedom. The average energy of the molecules according to the Equipartition 
Principle would therefore be 

E ~~~ ^trans I ^rot ^ ^vih 

=- 3(\RT) + 2(\RT) ^ 4(RT] 6 1 2 RT per mole 

A nonlinear (bent) triatomic molecule has three principal moments of 
inertia, and therefore three rotational degrees of freedom. Any nonlinear 
polatomic molecule has 3/76 vibrational degrees of freedom. For the 
triatomic case, there are therefore three vibrational degrees of freedom. The 
average energy according to the Equipartition Principle would be 

E 3(1 RT) ~\ 3(1 RT) f 3(RT) 
6RT per mole 

Examples of linear triatomic molecules are HCN, CO 2 , and CS 2 . Bent 
triatomic molecules include H 2 O and SO 2 . 

The vibratory motion of a collection of mass points bound together by 
linear restoring forces [i.e., a polyatomic molecule in which the individual 
atomic displacements obey eq. (7.46)] may be quite complicated. It is always 
possible, however, to represent the complex vibratory motion by means of 
a number of simple motions, the so-called normal modes of vibration. In a 
normal mode of vibration, each atom in the molecule is oscillating with the 
same frequency. Examples of the normal modes for linear and bent triatomic 
molecules are shown in Fig. 7.16. The bent molecule has three distinct 
normal modes, each with a characteristic frequency. The frequencies of 
course have different numerical values in different compounds. In the case 
of the linear molecule, there are four normal modes; two correspond to 
stretching of the molecule (v l9 v 3 ) and two correspond to bending (v 2a , v 2b ). 
The two bending vibrations differ only in that one is in the plane of the 
paper and one normal to the plane (denoted by + and ). These vibrations 
have the same frequency, and are called degenerate vibrations. 

When we described the translational motions of molecules and their 
consequences for the kinetic theory of gases, it was desirable at first to employ 
a very simplified model. The same procedure has been followed in this dis- 
cussion of the internal molecular motions. Thus diatomic molecules do not 



192 



THE KINETIC THEORY 



[Chap. 7 



really behave as rigid rotators, since, at rapid rotation speeds, centrifugal 
force tends to separate the atoms by stretching the bond between them. 




9 6 



'2a 






I/I V 2 1/3 

Fig. 7.16. Normal modes of vibration of triatomic molecules. 

Likewise, a more detailed theory shows that the vibrations of the atoms are 
not strictly harmonic. 

26. The equipartition principle and the heat capacity of gases. According 
to the equipartition principle, a gas on warming should take up energy in 
all its degrees of freedom, \RT per mole for each translational or rotational 
coordinate, and RT per mole for each vibration. The heat capacity at con- 
stant volume, C v --= (3E/DTV, could then be readily calculated from the 
average energy. 

From eq. (7.42) the translational contribution to C v is (f)/?. Since 
R *= 1 .986 cal per degree C, the molar heat capacity is 2.98 cal per degree C. 
When this figure is compared with the experimental values in Table 7.6, it 
is found to be confirmed for the monatomic gases, He, Ne, A, Hg, which 



TABLE 7.6 
MOLAR HEAT CAPACITY C v OF GASES 



Gas 



He, Ne, A, Hg 

H 2 . 

N 2 . . 

O a . . 

CI 2 . . 

H 2 O 

C0 a . . 



Temperature (C) 



-100 



2.98 
4.18 
4.95 
4.98 








100 


400 


600 




2.98 


2.98 


2.98 


2.98 




4.92 


4.97 


4.99 


5.00 




4.95 


4.96 


5.30 


5.42 




5.00 


5.15 


5.85 


6.19 




5.85 


5.88 


6.24 


6.40 







6.37 


6.82 


7.60 




6.75 


7.68 


9.86 


10.90 



Sec. 27] THE KINETIC THEORY 193 

have no internal degrees of freedom. The observed heat capacities of the 
diatomic and polyatomic gases are always higher, and increase with tem- 
perature, so that it may be surmised that rotational and vibrational contri- 
butions are occurring. 

For a diatomic gas, the equipartition principle predicts an average energy 
of (%)RT, or C v ()R -= 6.93. This value seems to be approached at high 
'temperatures for H 2 , N 2 , O 2 , and C1 2 , but at lower temperatures the experi- 
mental C v values fall much below the theoretical ones. For polyatomic gases, 
the discrepancy with the simple theory is even more marked. The equi- 
partition principle cannot explain why the observed C r is less than predicted, 
why C v increases with temperature, nor why the C r values differ for the 
different diatomic gases. The theory is t)ius satisfactory for translational 
motion, but most unsatisfactory when applied to rotation and vibration. 

Since the equipartition principle is a direct consequence of the kinetic 
theory, and in particular of the Maxwell-Boltzmann distribution law, it is 
evident that an entirely new basic theory will be required to cope with the 
heat capacity problem. Such a development is found in the quantum theory 
introduced in Chapter 10. 

27. Brownian motion. In 1827, shortly after the invention of the achro- 
matic lens, the botanist Robert Brown 15 studied pollen grains under his 
microscope and watched a curious behavior. 

While examining the form of these particles immersed in water, I observed many 
of them very evidently in motion; their motion consisting not only of a change of 
place in the fluid, manifested by alterations of their relative positions, but also not 
infrequently of a change in form of the particle itself; a contraction or curvature 
taking place repeatedly about the middle of one side, accompanied by a correspond- 
ing swelling or convexity on the opposite side of the particle. In a few instances the 
particle was seen to turn on its longer axis. These motions were such as to satisfy 
me, after frequently repeated observations, that they arose neither from currents in 
the fluid, nor from its gradual evaporation, but belonged to the particle itself. 

In 1888, G. Gouy proposed that the particles were propelled by collisions 
with the rapidly moving molecules of the suspension liquid. Jean Perrin 
recognized that the microscopic particles provide a visible illustration of 
many aspects of the kinetic theory. The dancing granules should be governed 
by the same laws as the molecules in a gas. 

One striking confirmation of this hypothesis was discovered in Perrin's 
work on the distribution of colloidal particles in a gravitational field, the 
sedimentation equilibrium. By careful fractional centrifuging, he was able 
to prepare suspensions of gamboge 16 particles that were spherical in shape 
and very uniform in size. It was possible to measure the radius of the particles 
either microscopically or by weighing a counted number. If these granules 

15 Brown, Phil. Mag., 4, 161 (1828); 6, 161 (1829); 8, 41 (1830). 

18 Gamboge is a gummy material from the desiccation of the latex secreted by garcinia 
more/la (Indo-China). It is used as a bright yellow water color. 



194 



THE KINETIC THEORY 



[Chap. 7 



behave in a gravitational field like gas molecules, their equilibrium distribu- 
tion throughout a suspension should obey the Boltzmann equation 

(7.48) 



Instead of m we may write $7rr 3 (p p t ) where r is the radius of the particle, 
and p and p t are the densities of the gamboge and of the suspending liquid. 
Then eq. (7.48) becomes 



- Pi) 



RT 



N 



(7.49) 



Fig. 7.17. Sedimentation 
equilibrium. 



By determining the difference in the numbers of particles at heights separated 

by h, it is possible to calculate a value for 
Avogadro's Number N. 

A drawing of the results of Perrin's micro- 
scopic examination of the equilibrium distri- 
bution with granules of gamboge 0.6/* in 
diameter 17 is shown in Fig. 7.17. The relative 
change in density observed in 10/j of this 
suspension is equivalent to that occurring in 
6 km of air, a magnification of six hundred 
million. 

The calculation from eq. (7.49) resulted 
in a value of N 6.5 x 10 23 . This value is 

in good agreement with other determinations, and is evidence that the visible 
microscopic particles are behaving as giant molecules in accordance with 
the kinetic theory. These studies were welcomed at the time as a proof of 
molecular reality. 

28. Thermodynamics and Brownian motion. A striking feature of the 
Brownian motion of microscopic particles is that it never stops, but goes on 
continuously without any diminution of its activity. This perpetual motion 
is not in contradiction with the First Law, for the source of the energy that 
moves the particles is the kinetic energy of the molecules of the suspending 
liquid. We may assume that in any region where the colloid particles gain 
kinetic energy, there is a corresponding loss in kinetic energy by the molecules 
of the fluid, which undergoes a localized cooling. This amounts to perpetual 
motion of the second kind, for the transformation of heat into mechanical 
energy is prohibited by the Second Law, unless there is an accompanying 
transfer of heat from a hot to a cold reservoir. 

The study of Brownian motion thus reveals an important limitation of 
the scope of the Second Law, which also allows us to appreciate its true 
nature. The increase in potential energy in small regions of a colloidal sus- 
pension is equivalent to a spontaneous decrease in the entropy of the region. 
On the average, of course, over long periods of time the entropy of the entire 

17 1 micron (/<) - 10~ 3 mm = I0~ 6 m. 



Sec. 29] 



THE KINETIC THEORY 



195 



system does not change. In any microscopic region, however, the entropy 
fluctuates, sometimes increasing and sometimes decreasing. 

On the macroscopic scale such fluctuations are never observed, and the 
Second Law is completely valid. No one observing a book lying on a desk 
would expect to see it spontaneously fly up to the ceiling as it experienced a 
sudden chill. Yet it is not impossible to imagine a situation in which all the 
molecules in the book moved spontaneously in a given direction. Such a 
situation is only extremely improbable, since there are so many molecules in 
any macroscopic portion of matter. 

29. Entropy and probability. The law of the increase of entropy is thus 
a probability law. When the number of molecules in a system becomes 
sufficiently small, the probability of observing a spontaneous decrease in 
entropy becomes appreciable. 

The relation between entropy and probability may be clarified by con- 
sidering (Fig. 7.18) two different gases, A and /?, in separate containers. 



oo oo 

00 00 O 




00 





~ _ *T. 



B 




A+B 



Fig. 7.18. Increase in randomness and entropy on mixing. 

Mien the partition is removed the gases diffuse into each other, the process 
Continuing until they are perfectly mixed. If they were originally mixed, we 
should never expect them to become spontaneously unmixed by diffusion, 
since this condition would require the simultaneous adjustment of some 10 24 
different velocity components per mole of gas. 

The mixed condition is the condition of greater randomness, of greater 
disorder; it is the condition of greater entropy since it arises spontaneously 
from unmixed conditions. [The entropy of mixing was given in eq. (3.42).] 
Hence entropy is sometimes considered a measure of the degree of disorder 
or of randomness in a system. The system of greatest randomness is also 
the system of highest statistical probability, for there are many arrange- 
ments of molecules that can comprise a disordered system, and much fewer 
for an ordered system. When one considers how seldom thirteen spades 
are dealt in a bridge hand, 18 one can realize how much more probable is the 
mixed condition in a system containing 10 24 molecules. 

Mathematically, the probabilities of independent individual events are 
multiplied together to obtain the probability of the combined event. The 

18 Once in 653,013,559,600 deals, if the decks are well-shuffled and the dealers virtuous. 



196 THE KINETIC THEORY [Chap. 7 

probability of drawing a spade from a pack of cards is 1/4; the probability 
of drawing two spades in a row is (1/4)(12/51); the probability of drawing 
the ace of spades is (1/4)0/13) - 1/52. Thus W 12 = WJV<^ Entropy, on the 
other hand, is an additive function, S 12 = S l + S 2 . This difference enables us 
to state that the relation between entropy S and probability W must be a 
logarithmic one. Thus, 

S - a In W f- b (7.50) 

The value of the constant a may be derived by analyzing from the view- 
point of probability a simple change for which the AS is known from 
thermodynamics. Consider the expansion of one mole of an ideal gas, 
originally at pressure P l in a container of volume V l9 into an evacuated 
container of volume K 2 . The final pressure is P 2 and the final volume, K, 
I V 2 . For this change, 

-* (7.5,, 

When the containers are connected, the probability w l of finding one 
given molecule in the first container is simply the ratio of the volume V l to 
the total volume V l -(- K 2 or vv t --= V^\(V^ ! K 2 ). Since probabilities are 
multiplicative, the chance of finding all TV molecules in the first container, 
I.e., the probability W l of the original state of the system, is 



Since in the final state all the molecules must be in one or the other of 
the containers, the probability W 2 ~ \ N 1. 
Therefore from eq. (7.50), 



Comparison with eq. (7.51) shows that a is equal to k, the Boltzmann 
constant. Thus 

S - k In W + b 

W 

AS - S 2 - S l -.-- k In y/ 2 (7.52) 

W v 

This relation was first given by Boltzmann in 1896. 

For physicochemical applications, we are concerned always with entropy 
changes, and may conveniently set the constant b equal to zero. 19 

The application of eq. (7.52) cannot successfully be made until we have 
more detailed information about the energy states of atoms and molecules. 

19 A further discussion of this point is to be found in Chapter 12. 



Sec. 29] THE KINETIC THEORY 197 

This information will allow us to calculate W and hence the entropy and 
other thermodynamic functions. 

The relative probability of observing a decrease in entropy of AS below 
the equilibrium value may be obtained by inverting eq. (7.52): 

~ -=- *-**'* (7.53) 

rreq 

For one mole of helium, S/k at 273 -^ 4 x 10 25 . The chance of observing 
an entropy decrease one-millionth of this amount is about e~ w ". It is evi- 
dent, therefore, that anyone observing a book flying spontaneously into the 
air is dealing with a poltergeist and not an entropy fluctuation (probably!). 
Only when the system is very small is there an appreciable chance of ob- 
serving a large relative decrease in entropy. 

A further analysis may be made of the driving force of a chemical re- 
action or other change, AF =^ - A// -f T&S. It is made up of two terms, 
the heat of the reaction and the increase in randomness times the tempera- 
ture. The higher the temperature, the greater is the driving force due to the 
increase in disorder. This may be physically clearer in the converse state- 
ment: The lower the temperature, the more likely it is that ordered states 
can persist. The drive toward equilibrium is a drive toward minimum 
potential energy and toward maximum randomness. In general, both can- 
not be achieved in the same system under any given set of conditions. The 
free-energy minimum represents (at constant T and P) the most satisfactory 
compromise that can be attained. 



PROBLEMS 

1. At what speeds would molecules of hydrogen and oxygen have to 
leave the surface of (a) the earth, (b) the moon, in order to escape into 
space? At what temperatures would the average speeds of these molecules 
equal these "speeds of escape"? The mass of the moon can be taken as 
-fa that of the earth. 

2. Calculate the number of (a) ergs per molecule, (b) kcal per mole 
corresponding to one electron volt per molecule. The electron volt is the 
energy acquired by an electron in falling through a potential difference of 
one volt. What is the mean kinetic energy of a molecule at 25C in ev? 
What is A: in ev per C? 

3. The density of nitrogen at 0C and 3000 atm is 0.835 g per cc. Cal- 
culate the average distance apart of the centers of the molecules. How does 
this compare with the molecular diameter calculated from van der Waals' 
b = 39.1 cc per mole? 

4. In the method of Knudsen [Ann. Physik, 29, 179 (1909)], the vapor 



198 THE KINETIC THEORY [Chap. 7 

pressure is determined by the rate at which the substance, under its equi- 
librium pressure, diffuses through an orifice. In one experiment, beryllium 
powder was placed inside a molybdenum bucket having an effusion hole 
0.318 cm in diameter. At 1537K, it was found that 0.00888 g of Be effused 
in 15.2 min. Calculate the vapor pressure of Be at 1537K. 

5. Two concentric cylinders are 10cm long, and 2.00 and 2.20cm in 
diameter. The space between them is filled with nitrogen at 10~ 2 mm pressure. 
Estimate the heat flow by conduction between the two cylinders when they 
differ in temperature by 10C. 

6. At 25 C what fraction of the molecules in hydrogen gas have a kinetic 
energy within kT 10 per cent? What fraction at 500C? What fraction of 
molecules in mercury vapor? 

7. Derive an expression for the fraction of molecules in a gas that have 
an energy greater than a given value E in two degrees of freedom. 

8. Show that the most probable speed of a molecule in a gas equals 
V2kT/m. 

9. Derive the expression (\mc 2 ) %kT from the Maxwell distribution 
law. 

10. In a cc of oxygen at 1 atm and 300K, how many molecules have 
translational kinetic energies greater than 2 electron volts? At 1000K? 

11. What is the mean free path of argon at 25C and a pressure of I atm? 
Of 10- 5 atm? 

12. A pinhole 0.2 micron in diameter is punctured in a liter vessel con- 
taining chlorine gas at 300 K and 1 mm pressure. If the gas effuses into a 
vacuum, how long will it take for the pressure to fall to 0.5 mm? 

13. Perrin studied the distribution of uniform spherical (0.212^ radius) 
grains of gamboge (p = 1.206) suspended in water at 15C by taking counts 
on four equidistant horizontal planes across a cell 100/< deep. The relative 
concentrations of grains at the four levels were 

level: 5/< 35 // 65 /< 95 ju 

concentration: 100 47 22.6 12 

Estimate Avogadro's Number from these data. 

14. Show that the number of collisions per second between unlike mole- 
cules, A and 8, in one cc of gas is 



where the reduced mass, JLI (tn A m Ii )/(m A + m B ). In an equimolar mixture 
of H 2 and I 2 at 500K and 1 atm calculate the number of collisions per sec 
per cc between H 2 and H 2 , H 2 and I 2 , I 2 and I 2 . For H 2 take d =r 2. 18 A, for 

U, d -- 3.76 A. 



Chap. 7] THE KINETIC THEORY 199 

15. The f )rce constant of O 2 is 1 1.8 x 10 5 dynes per cm and r (> 1.21 A. 
Estimate the potential energy per mole at r = 0.8r . 

16. Calculate the moments of inertia of the following molecules: (a) 
NaCl, r -= 2.51 A; (b) H 2 O, 'OH = - 9 57 A, L HOH - 105 3'. 

17. In Fig. 7.18, assume that there are 10 white balls and 10 black balls 
distributed at random between the two containers of equal volume. What is 
the AS between the random configuration and one in which there are 8 white 
balls and 2 black balls in the left-hand container, and 2 whites and 8 blacks 
in the right. Calculate the answer by eq. (7.52) and also by eq. (3.42). What 
is the explanation of the different answers? 

18. In a carefully designed high vacuum system it is possible to reach a 
pressure as low as 10~ 10 mm. Calculate the mean free path of helium at this 
pressure and 25C. 

19. The permeability constant at 20C of pyrex glass to helium is given 
as 6.4 x 10~ 12 cc sec" 1 per cm 2 area per mm thickness per cm of Hg pressure 
difference. The helium content of the atmosphere at sea level is about 
5 x 10~ 4 mole per cent. Suppose a 100 cc round pyrex flask with walls 
2 mm thick was evacuated to 10~ 10 mm and sealed. What would be the 
pressure at the end of one year due to inward diffusion of helium? 

REFERENCES 

BOOKS 

1. Herzfeld, K. F. and H. M. Smallwood, "Kinetic Theory of Ideal Gases," 
in Treatise on Physical Chemistry, vol. II, edited by H. S. Taylor and 
S. Glasstone (New York: Van Nostrand, 1951). 

2. Jeans, J. H., Introduction to the Kinetic Theory of Gases (London: Cam- 
bridge, 1940). 

3. Kennard, E. H., Kinetic Theory of Gases (New York: McGraw-Hill, 1938). 

4. Knudsen, M., The Kinetic Theory of Gases (London: Methuen, 1950). 

5. Loeb, L. B., Kinetic Theory of Gases (New York: McGraw-Hill, 1927). 

ARTICLES 

1. Furry, W. H., Am. J. Phys., 16, 63-78 (1948), "Diffusion Phenomena in 
Gases." 

2. Pease, R. N., J. Chem. Ed., 16, 242-47, 366-73 (1939), "The Kinetic 
Theory of Gases." 

3. Rabi, I. L, Science in Progress, vol. IV (New Haven: Yale Univ. Press, 
1945), 195-204, "Streams of Atoms." 

4. Rodebush, W. H., /. Chem. Ed., 27, 39-43 (1950), "The Dynamics of 
Gas Flow." 

5. Wheeler, T. S., Endeavour, 11, 47-52 (1952), "William Higgins, Chemist." 



CHAPTER 8 

The Structure of the Atom 



1. Electricity. The word "electric" was coined in 1600 by Queen Eliza- 
beth's physician, William Gilbert, from the Greek, r/Aocrpov, "amber." 
It was applied to bodies that when rubbed with fur acquired the property 
of attracting to themselves small bits of paper or pith. Gilbert was un- 
willing to admit the possibiliy of "action at a distance," and in his treatise 
De Magnete he advanced an ingenious theory for the electrical attraction. 

An effluvium is exhaled by the amber and is sent forth by friction. Pearls 
carnelian, agate, jasper, chalcedony, coral, metals, and the like, when rubbed are 
inactive; but is there nought emitted from them also by heat and friction? There 
is indeed, but what is emitted from the dense bodies is thick and vaporous [and thus 
not mobile enough to cause attractions]. 

A breath, then . . . reaches the body that is to be attracted and as soon as it is 
reached it is united to the attracting electric. For as no action can be performed by 
matter save by contact, these electric bodies do not appear to touch, but of necessity 
something is given out from the one to the other to come into close contact therewith, 
and to be a cause of incitation to it. 

Further investigation revealed that materials such as glass, after rubbing 
with silk, exerted forces opposed to those observed with amber. Two varieties 
of electricity were thus distinguished, the vitreous and the resinous. Two 
varieties of effluvia, emanating from the pores of the electrics, were invoked 
in explanation. Electricity was supposed to be an imponderable fluid similar 
in many ways to "caloric." Frictional machines for generating high electro- 
static potentials were devised, and used to charge condensers in the form of 
Leiden jars. 

Benjamin Franklin (1747) considerably simplified matters by proposing 
a one-fluid theory. According to this theory, when bodies are rubbed together 
they acquire a surplus or deficit of the electric fluid, depending on their 
relative attraction for it. The resultant difference in charge is responsible for 
the observed forces. Franklin established the convention that the vitreous 
type of electricity is positive (fluid in excess), and the resinous type is negative 
(fluid in defect). 

In 1791, Luigi Galvani accidentally brought the bare nerve of a partially 
dissected frog's leg into contact with a discharging electrical machine. The 
sharp convulsion of the leg muscles led to the discovery of galvanic elec- 
tricity, for it was soon found that the electric machine was unnecessary and 
that the twitching could be produced simply by bringing the nerve ending 
and the end of the leg into contact through a metal strip. The action was 



Sec. 2] THE STRUCTURE OF THE ATOM 201 

enhanced when two dissimilar metals completed the circuit. Galvani, a 
physician, named the new phenomenon "animal electricity" and believed 
that it was characteristic only of living tissues. 

Alessandro Volta, a physicist, Professor of Natural Philosophy at Pavia, 
soon discovered that the electricity was of inanimate origin; and using 
dissimilar metals in contact with moist paper, he was able to charge an 
electroscope. In 1800 he constructed his famous "pile," consisting of many 
consecutive plates of silver, zinc, and cloth soaked in salt solution. From the 
terminals of the pile the thitherto static-electrical manifestations of shock 
and sparks were obtained. 

The news of Volta's pile was received with an enthusiasm and amazement 
akin to that occasioned by the uranium pile in 1945. In May of 1800, 
Nicholson and Carlyle decomposed water into hydrogen and oxygen by 
means of the electric current, the oxygen appearing at one pole of the pile 
and the hydrogen at the other. Solutions of various salts were soon decom- 
posed, and in 1806-1807, Humphry Davy used a pile to isolate sodium and 
potassium from their hydroxides. The theory that the atoms in a compound 
were held together by the attraction between unlike charges immediately 
gained a wide acceptance. 

2. Faraday's Laws and electrochemical equivalents. In 1813 Michael 
Faraday, then 22 years old and a bookbinder's apprentice, went to the 
Royal Institution as Davy's laboratory assistant. In the following years, 
he carried out the series of researches that were the foundations of electro- 
chemistry and electromagnetism. 

Faraday studied intensively the decomposition of solutions of salts, acids, 
and bases by the electric current. With the assistance of the Rev. William 
Whewell, he devised the nomenclature universally used in these studies: 
electrode, electrolysis, electrolyte, ion, anion, cation. The positive electrode 
is called the anode (oo>, "path"); the negative ion (IOP, "going"), which 
moves toward the anode, is called the anion. The positive ion, or cation, moves 
toward the negative electrode, or cathode. 

Faraday proceeded to study quantitatively the relation between the 
amount of electrolysis, or chemical action produced by the current, and the 
quantity of electricity. The unit of electric quantity is now the coulomb or 
ampere second. The results were summarized as follows: 1 

The chemical power of a current of electricity is in direct proportion to the 
absolute quantity of electricity which passes. . . . The substances into which these 
[electrolytes] divide, under the influence of the electric current, form an exceedingly 
important general class. They are combining bodies, are directly associated with the 
fundamental parts of the doctrine of chemical affinity; and have each a definite 
proportion, in which they are always evolved during electrolytic action. I have 
proposed to call . . . the numbers representing the proportions in which they are 
evolved electrochemical equivalents. Thus hydrogen, oxygen, chlorine, iodine, lead, 

1 Phil. Trans. Roy. Soc., 124, 77 (1834). 



202 THE STRUCTURE OF THE ATOM [Chap. 8 

tin, are ions\ the three former are anions, the two metals are cations, and 1, 8, 36, 
125, 104, 58 are their electrochemical equivalents nearly. 

Electrochemical equivalents coincide, and are the same, with ordinary chemical 
equivalents. I think I cannot deceive myself in considering the doctrine of definite 
electrochemical action as of the utmost importance. It touches by its facts more 
directly and closely than any former fact, or set of facts, have done, upon the 
beautiful idea that ordinary chemical affinity is a mere consequence of the electrical 
attractions of different kinds of matter. . . . 

A very valuable use of electrochemical equivalents will be to decide, in cases of 
doubt, what is the true chemical equivalent, or definite proportional, or atomic 
number [weight] of a body. ... I can have no doubt that, assuming hydrogen as 
1, and dismissing small fractions for the simplicity of expression, the equivalent 
number or atomic weight of oxygen is 8, of chlorine 36, of bromine 78.4, of lead 
103.5, of tin 59, etc , notwithstanding that a very high authority doubles several of 
these numbers. 

The "high authority" cited was undoubtedly Jons Jakob Berzelius, who 
was then using atomic weights based on combining volumes and gas-density 
measurements. Faraday believed that when a substance was decomposed, it 
always yielded one positive and one negative ion. Since the current liberates 
from water eight grams of oxygen for each gram of hydrogen, he concluded 
that the formula was HO and that the atomic weight of oxygen was equal 
to 8. It will be recalled that the work of Avogadro, which held the key to 
this problem, was lying forgotten during these years. 

3. The development of valence theory. Much new knowledge about the 
combinations of atoms was being gained by the organic chemists. Especially 
noteworthy was the work of Alexander Williamson. In 1850 he treated 
potassium alcoholate with ethyl iodide and obtained ordinary ethyl ether. 

At that time, most chemists, using O 8, C 6, were writing alcohol 
as C 4 H 5 OOH, and ether C 4 H 5 O. If O 1 6, C 1 2 were used, the formulas 
would be 



Williamson realized that his reaction could be readily explained on this 
basis as 



r\ i p if i _ i/i i -25 r\ 

K f W ~ K1 * C 2 HJ 

The older system could still be maintained, however, if a two-step reaction 
was postulated : 

C 4 H 5 O-OK KO I C 4 H 5 O 

C 4 H 5 l 4 KO - KI + C 4 H 5 O 

Williamson settled the question by treating potassium ethylate with methyl 
iodide. If the reaction proceeded in two steps, he should obtain equal 
amounts of diethyl and dimethyl ethers: 

C 4 H 5 O pK ------ KO + C 4 H 5 O 

' 



C 2 H 3 l + 'KO - KI + C 2 H 3 O 



Sec. 3] THE STRUCTURE OF THE ATOM 203 

On the other hand, if the oxygen atom held two radicals, a new compound, 
methyl ethyl ether, should be the product : 



4- 3 - KI + 25 

K t- j j- iu+ CH 

The new compound was indeed obtained. This was the first unequivocal 
chemical demonstration that the formulas based on C 12, O -- 16, must 
be correct. The concept of valence was gradually developed as a result of 
such organic-chemical researches. 

It should be mentioned that as early as 1819 two other important criteria 
for establishing atomic weights were proposed. Pierre Dulong and Alexis 
Petit pointed out that, for most solid elements, especially the metals, the 
product of the specific heat and the atomic weight appeared to be a constant, 
with a value of around 6 calories per C. If this relation is accepted as a 
general principle, it provides a guide by which the proper atomic weight can 
be selected from a number of multiples. 

In the same year, Eilhard Mitscherlich published his work on isomor- 
phism of crystals, based on an examination of such series as the alums and 
the vitriols. He found that one element could often be substituted for an 
analogous one in such a series without changing the crystalline form, and 
concluded that the substitute elements must enter into the compound in the 
same atomic proportions. Thus if alum is written KA1(SO 4 ) 2 -12 H 2 O, 
ferric alum must be KFe(SO 4 ) 2 -12 H 2 O, and chrome alum must be 
KCr(SO 4 ) 2 -12 H 2 O. The analyst is thus enabled to deduce a consistent set 
of atomic weights for the analogous elements in the crystals. 

Avogadro's Hypothesis, when resurrected at the 1860 conference, re- 
solved all remaining doubts, and the old problem of how to determine the 
atomic weights was finally solved. 

We now recognize that ions in solution may bear more than one elemen- 
tary charge, and that the electrochemical equivalent weight is the atomic 
weight M divided by the number of charges on the ion z. The amount of 
electricity required to set free one equivalent is called the faraday, and is 
equal to 96,519 coulombs. 

The fact that a definite quantity of electric charge, or a small integral 
multiple thereof, was always associated with each charged atom in solu- 
tion strongly suggested that electricity was itself atomic in nature. Hence, 
in 1874, G. Johnstone Stoney addressed the British Association as 
follows: 

Nature presents us with a single definite quantity of electricity which is inde- 
pendent of the particular bodies acted on. To make this clear, I shall express 
Faraday's Law in the following terms. . . . For each chemical bond which is 
ruptured within an electrolyte a certain quantity of electricity traverses the electrolyte 
which is the same in all cases. 

In 1891, Stoney proposed that this natural unit of electricity should be 



204 THE STRUCTURE OF THE ATOM [Chap. 8 

given a special name, the electron. Its magnitude could be calculated by 
dividing the faraday by Avogadro's Number. 



4. The Periodic Law. The idea that matter was constituted of some ninety 
different kinds of fundamental building blocks was not one that could appeal 
for long to the mind of man. We have seen how during the nineteenth 
century evidence was being accumulated from various sources, especially 
the kinetic theory of gases, that the atom was not merely a minute billiard 
ball, a more detailed structure being required to explain the interactions 
between atoms. 

In 1815, William Prout proposed that all atoms were composed of atoms 
of hydrogen. In evidence for this hypothesis, he noted that all the atomic 
weights then known were nearly whole numbers. Prout's hypothesis won 
many converts, but their enthusiasm was lessened by the careful atomic 
weight determinations of Jean Stas, who found, for example, that chlorine 
had a weight of 35.46. 

Attempts to correlate the chemical properties of the elements with their 
atomic weights continued, but without striking success till after 1860, when 
unequivocal weights became available. In 1865, John Newlands tabulated 
the elements in the order of their atomic weights, and noted that every 
eighth element formed part of a set with very similar chemical properties. 
This regularity he unfortunately called "The Law of Octaves." The suggested 
similarity to a musical scale aroused a good deal of scientific sarcasm, and 
the importance of Newland's observations was drowned in the general 
merriment. 

From 1868 to 1870, a series of important papers by Julius Lothar Meyer 
and Dmitri Mendeleev clearly established the fundamental principles of the 
Periodic Law. Meyer emphasized the periodic nature of the physical pro- 
perties of the elements. This periodicity is illustrated by the well-known 
graph of atomic volume vs. atomic weight. Mendeleev arranged the elements 
in his famous Periodic Table. This Table immediately systematized inorganic 
chemistry, made it possible to predict the properties of undiscovered elements, 
and pointed strongly to the existence of an underlying regularity in atomic 
architecture. 

Closer examination revealed certain defects in the arrangement of ele- 
ments according to their atomic weights. Thus the most careful determina- 
tions showed that tellurium had a higher atomic weight than iodine, despite 
the positions in the Table obviously required by their properties. After 
Sir William Ramsay's discovery of the rare gases (1894-1897), it was found 
that argon had an atomic weight of 39. 88, which was greater than that of 
potassium, 39.10. Such exceptipns to the arrangement by weights suggested 
that the whole truth behind the Periodic Law was not yet realized. 



Sec. 5] THE STRUCTURE OF THE ATOM 205 

5. The discharge of electricity through gases. The answer to this and 
many other questions about atomic structure was to be found in a quite 
unexpected quarter the study of the discharge of electricity through gases. 

William Watson, 2 who proposed a one-fluid theory of electricity at the 
same time as Franklin, was the first to describe the continuous discharge of 
an electric machine through a rarefied gas (1748). 

It was a most delightful spectacle, when the room was darkened, to see the 
electricity in its passage: to be able to observe not, as in the open air, its brushes or 
pencils of rays an inch or two in length, but here the corruscations were of the whole 
length of the tube between the plates, that is to say, thirty-two inches. 

Progress in the study of the discharge was retarded by the lack of suitable 
air pumps. In 1855, Geissler invented a mercury pump that permitted the 
attainment of higher degrees of vacuum. In 1858, Julius Pliicker observed 
the deflection of the negative glow in a magnetic field and in 1869 his student, 
Hittorf, found that a shadow was cast by an opaque body placed between 
the cathode and the fluorescent walls of the tube, suggesting that rays from 
the cathode were causing the fluorescence. In 1876, Eugen Goldstein called 
these rays cathode rays and confirmed the observation that they traveled 
in straight lines and cast shadows. Sir William Crookes (1879) regarded the 
rays as a torrent of negatively ionized gas molecules repelled from the 
cathode. The charged particle theory was contested by many who believed 
the rays were electromagnetic in origin, and thus similar to light waves. This 
group was led by Heinrich Hertz, who showed that the cathode radiation 
could pass through thin metal foils, which would be impossible if it were 
composed of massive particles. 

Hermann von Helmholtz, however, strongly championed the particle 
theory; in a lecture before the Chemical Society of London in 1881 he 
declared : 

If we accept the hypothesis that the elementary substances are composed of 
atoms, we cannot avoid concluding that electricity also, positive as well as negative, 
is divided into definite elementary portions which behave like atoms of electricity. 

6. The electron. In 1895, Wilhelm Roentgen discovered that a very pene- 
trating radiation was emitted from solid bodies placed in the path of cathode 
rays. An experimental arrangement for the production of these "X rays" is 
shown in Fig. 8.1. 

J. J. Thomson in his Recollections and Reflections* has described his first 
work in this field: 

It was a most fortunate coincidence that the advent of research students at the 
Cavendish Laboratory came at the same time as the announcement by Roentgen of 
his discovery of the X rays. I had a copy of his apparatus made and set up at the 
Laboratory, and the first thing I did with it was t6 see what effect the passage of 

2 Phil. Trans. Roy. Soc., 40, 93 (1748); 44, 362 (1752). 

3 G. Bell and Sons, London, 1933. 



206 



THE STRUCTURE OF THE ATOM 



[Chap. 8 



these rays through a gas would produce on its electrical properties. To my great 
delight I found that this made it a conductor of electricity, even though the electric 
force applied to the gas was exceedingly small, whereas the gas when it was not 
exposed to the rays did not conduct electricity unless the electric force were in- 
creased many thousandfold. . . The X rays seemed to turn the gas into a gaseous 
electrolyte. 

I started at once, in the late autumn of 1895, on working at the electric properties 
of gases exposed to Roentgen rays, and soon found some interesting and suggestive 

HIGH VOLTAGE 
SOURCE 





\ 



k ANODE 
(TARGET) 



CATHODE 

////>., 

^X-RAYS 
Fig. 8.1. Production of X-rays. 

results. . . . There is an interval when the gas conducts though the rays have 
ceased to go through it. We studied the properties of the gas in this state, and found 
that the conductivity was destroyed when the gas passed through a filter of glass 
wool. 

A still more interesting discovery was that the conductivity could be filtered out 
without using any mechanical filter by exposing the conducting gas to electric forces. 
The first experiments show that the conductivity is due to particles present in the 
gas, and the second shows that these particles are charged with electricity. The 
conductivity due to the Roentgen rays is caused by these rays producing in the gas 
a number of charged particles. 

7. The ratio of charge to mass of the cathode particles. J. J. Thomson 
next turned his attention to the behavior of cathode rays in electric and 
magnetic fields, 4 using the apparatus shown in Fig. 8.2. 




Fig. 8.2. Thomson's apparatus for determining e\m of cathode 
particles. 

The rays from the cathode C pass through a slit in the anode A, which is a metal 
plug fitting tightly into the tube and connected with the earth; after passing through 
a second slit in another earth-connected metal plug B, they travel between two 
parallel aluminium plates about 5 cm apart; they then fall on the end of the tube 

4 Phil. Mag., 44, 293 (1897). 



Sec. 7] 



THE STRUCTURE OF THE ATOM 



207 



and produce a narrow well-defined phosphorescent patch. A scale pasted on the 
outside of the tube serves to measure the deflection of this patch. 

At high exhaustions the rays were deflected when the two aluminium plates 
were connected with the terminal of a battery of small storage cells; the rays were 
depressed when the upper plate was connected with the negative pole of the battery, 
the lower with the positive, and raised when the upper plate was connected with 
the positive, the lower with the negative pole. 

In an electric field of strength , a particle with charge e will be subject 
to a force of magnitude Ee. The trajectory of an electron in an electric field 
of strength E perpendicular to its direction of motion may be illustrated by 




Fig. 8.3. Deflection of electron in an electric field. 

the diagram in Fig. 8.3. If m is the mass of the electron, the equations of 
motion may be written : 



r 
in ~~ --- Ee 

dt* 



(8.2) 



With / - as the instant the particle enters the electric field, its velocity 
in the y direction is zero at / -= 0. This velocity increases while the electron 
is in the field, while its initial velocity in the x direction, r () , remains constant. 

Integrating eqs. (8.2) we obtain 

eE , 



x =- 



V = 



2m 



(8.3) 



Equations (8.3) define a parabolic path, as is evident when t is eliminated 
from the equations, giving 

* 8 (8.4) 



After the electron leaves the field, it travels along a straight line tangent 
to this parabolic path. In many experimental arrangements, its total path is 



208 THE STRUCTURE OF THE ATOM [Chap. 8 

considerably longer than the length of the electric field, so that the deflection 
in the^ direction experienced while in the field is comparatively small com- 
pared to the total observed deflection. To a good approximation, therefore, 
the parabolic path can be considered as a circular arc of radius R E , with the 
force exerted by the field equal to the centrifugal force on the electron in this 
circular path, 

eE- ^ (8.5) 

K K 

The time required to traverse the field of length / is simply l/v so that 
the deflection in eq. (8.3) becomes 

eE / 2 
2m r 2 

Thus 1 ?* (8.6) 

m rE 

The ratio of charge to mass may be calculated from the deflection in the 
electric field, provided the velocity of the particles is known. This may be 
obtained by balancing the deflection in the electric field by an opposite 
deflection in a magnetic field. This magnetic field is applied by the pole 
pieces of a magnet M mounted outside the apparatus in Fig. 8.2, so that the 
field is at right angles to both the electric field and to the direction of motion 
of the cathode rays. 

A moving charged particle is equivalent to a current of electricity, the 
strength of the current being the product of the charge on the particle and 
its velocity. From Ampere's Law, therefore, the magnitude of the force on 
the moving charge is given by 

/-= evBunO (8.7) 

where is the angle between the velocity vector v and the magnetic induction 
vector B. When the magnetic field is perpendicular to the direction of 
motion, this equation becomes 

/- evB (8.8) 

Figure 8.4 illustrates the directional factors involved. 

The force on the electron due to the magnetic field is always perpendicular 
to its direction of motion, and thus a magnetic field can never change the 
speed of a moving charge, but simply changes its direction. As in eq. (8.5), 
the force may be equated to the centrifugal force on the electron, which in 
this case moves in a truly circular path. Thus 

mv 2 

Bev - (8.9) 

K H 

If now the force due to the, magnetic field exactly balances that due to 
the electric field, the phosphorescent patch in Thomson's apparatus will be 



Sec. 8] 



THE STRUCTURE OF THE ATOM 



209 



brought back to its initial position. When this occurs, evB - Ee and v - EjB. 
When this value is substituted in eq, (8.6) one obtains 





m 



2yE 



(8.10) 



The units in this equation may be taken to be those of the absolute 
practical (MKS) system. The charge e is in coulombs; the electric field E in 
volts per meter; the magnetic induction B in webers per square meter 
(1 weber per meter 2 ~ 10 4 gausses); and lengths and masses are in meters 
and kilograms, respectively. 

MAGNETIC 

FIELD 

B 




Fig. 8.4. Deflection of moving electron in magnetic field. 

Thomson found the experimental ratio of charge to mass to be of the 
order of 10 11 coulombs per kilogram. The most recent value of e/m for the 
electron is e/m 1.7589 x 10 11 coulomb per kilogram 5.273 x 10 17 esu 
per gram. 

The value found for the hydrogen ion, H+, in electrolysis was 1836 times 
less than this. The most reasonable explanation seemed to be that the mass 
of the cathode particle was only j- 8 ~ c that of the hydrogen ion ; this presumption 
was soon confirmed by measurements of e, the charge borne by the particle. 

8. The charge of the electron. In 1898, Thomson succeeded in measuring 
the charge of the cathode particles. Two years before, C. T. R. Wilson had 
shown that gases rendered ionizing by X rays caused the condensation of 
clouds of water droplets from an atmosphere supersaturated with water 
vapor. The ions formed acted as nuclei for the condensation of the water 
droplets. This principle was later used in the Wilson Cloud Chamber to 
render visible the trajectories of individual charged corpuscles, and thus 
made possible much of the experimental development of modern nuclear 
physics. 

Thomson and Townsend observed the rate of fall of a cloud in air and 



210 THE STRUCTURE OF THE ATOM [Chap. 8 

from this calculated an average size for the water droplets. The number of 
droplets in the cloud could then be estimated from the weight of water 
precipitated. The total charge of the cloud was measured by collecting the 
charged droplets on an electrometer. The conditions of cloud formation 
were such that condensations occurred only on negatively charged particles. 
Making the assumption that each droplet bore only one charge, it was now 
possible to estimate that the value of the elementary negative charge was 
e ~ 6.5 x 10~ 10 esu. This was of the same order of magnitude as the charge 
on the hydrogen ion, and thus further evidence was provided that the cathode 
particles themselves were "atoms" of negative electricity, with a mass ~^ 
that of the hydrogen atom. 

The exact proof of this hypothesis of the atomic nature of electricity and 
a careful measurement of the elementary electronic charge were obtained in 
1909 by Robert A. Millikan in his beautiful oil-drop experiments. Millikan 
was able to isolate individual droplets of oil bearing an electric charge, and 
to observe their rate of fall under the combined influences of gravity and 
an electric field. 

A body falls in a viscous medium with an increasing velocity until the 
gravitational force is just balanced by the frictional resistance, after which 
it falls at a constant "terminal velocity," v. The frictional resistance to a 
spherical body is given by Stoke's equation of hydrodynamics as 

f-=-(mYirv (8.11) 

where rj is the coefficient of viscosity of the medium and r the radius of the 
sphere. The gravitational force (weight) is equal to this at terminal velocity, 
so that, if p is the density of the body, and /> that of the fluid medium, 

far*g(p - Po) ^ 67T *l rv ( 8 - 12 ) 

If a charged oil droplet falls in an electric field, it can be brought to rest 
when the upward electric force is adjusted to equal the downward gravita- 
tional force, 

eE = fri*g( P - ft) (8.13) 



Since r may be calculated from the terminal velocity in eq. (8.12), only the 
charge e remains unknown in eq. (8.13). Actually, somewhat better results 
were obtained in experiments in which the droplet was observed falling 
freely and then moving in an electric field. In all cases, the charge on the 
oil droplets was found to be an exact multiple of a fundamental unit charge. 
This is the charge on the electron, whose presently accepted value is 5 

e = (4.8022 0.0001) x 10~ 10 esu 
= (1.6018 0.00004) x 10- 19 coulomb 

5 Millikan's result of 4.774 x 10~ 10 esu was low, owing to his use of an erroneous value 
for the viscosity of air. J. D. Stranathan, The Particles of Modern Physics (Philadelphia: 
BJakiston, 1954), Chap. 2, gives a most interesting account of the measurements of e. 



Sec. 9] THE STRUCTURE OF THE ATOM 21 1 

9. Radioactivity. The penetrating nature of the X rays emitted when 
cathode rays impinged upon solid substances was a matter of great wonder 
and interest for the early workers in the field, and many ingenious theories 
were advanced to explain the genesis of the radiation. It was thought at one 
time that it might be connected with the fluorescence observed from the 
irradiated walls of the tubes. Henri Becquerel therefore began to investigate 
a variety of fluorescent substances to find out whether they emitted pene- 
trating rays. All trials with various minerals, metal sulfides, and other com- 
pounds known to fluoresce or phosphoresce on exposure to visible light 
gave negative results, until he recalled the striking fluorescence of a sample 
of potassium uranyl sulfate that he had prepared 15 years previously. After 
exposure to an intense light, the uranium salt was placed in the dark-room 
under a photographic plate wrapped in "two sheets of thick black paper." 
The plate was darkened after several hours' exposure. 

Becquerel soon found that this amazing behavior had nothing to do with 
the fluorescence of the uranyl salt, since an equally intense darkening could 
be obtained from a sample of salt that had been kept for days in absolute 
darkness, or from other salts of uranium that were not fluorescent. The 
penetrating radiation had its source in the uranium itself, and Becquerel 
proposed to call this new phenomenon radioactivity* 

It was discovered that radioactive materials, like X rays, could render 
gases conducting so that charged bodies would be discharged, and the dis- 
charge rate of electroscopes could therefore be used as a measure of the 
intensity of the radiation. Marie Curie examined a number of uranium com- 
pounds and ores in this way, and found that the activity of crude pitchblende 
was considerably greater than would be expected from its uranium content. 
In 1898, Pierre and Marie Curie announced the separation from pitchblende 
of two extremely active new elements, polonium and radium. 

Three different types of rays have been recognized and described in the 
radiation from radioactive materials. The ft rays are high-velocity electrons, 
as evidenced by their deviation in electric and magnetic fields, and ratio of 
charge to mass. Their velocities range from 0.3 to 0.99 that of light. The 
a rays are made up of particles of mass 4 (O = 16 scale) bearing a positive 
charge of 2 (e 1 scale). They are much less penetrating than ft rays, by a 
factor of about 100. Their velocity is around 0.05 that of light. The y rays 
are an extremely penetrating (about 100 times ft rays) electromagnetic radia- 
tion, undeflected by either magnetic or electric fields. They are similar to 
X rays, but have a much shorter wave length. 

Owing to their large mass, the a particles travel through gases in essen- 
tially straight lines, producing a large amount of ionization along their paths. 
The paths of ft particles are longer than those of a's, but are much more 
irregular on account of the easy deflection of the lighter ft particle. 

The phenomena of radioactivity as well as the observations on the 

6 Compt. rend., 127, 501, March 2 f 1896. 



212 THE STRUCTURE OF THE ATOM [Chap. 8 

electrical discharge in gases provided evidence that electrons and positive 
ions were component parts of the structure of atoms. 

10. The nuclear atom. The problem of the number of electrons contained 
in an atom attracted the attention of Thomson and of C. G. Barkla. From 
measurements of the scattering of light, X rays, and beams of electrons, it 
was possible to estimate that this number was of the same order as the 
atomic weight. To preserve the electrical neutrality of the atom, an equal 
number of positive charges would then be necessary. Thomson proposed an 
atom model that consisted of discrete electrons embedded in a uniform 
sphere of positive charge. 

Lord Rutherford 7 has told the story of the next great development in the 
problem, at the University of Manchester in 1910. 

In the early days I had observed the scattering of a particles, and Dr. Geiger in 
my laboratory had examined it in detail. He found in thin pieces of heavy metal 
that the scattering was usually small, of the order of one degree. One day Geiger 
came to me and said, "Don't you think that young Marsden, whom I am training 
in radioactive methods, ought to begin a small research?" Now 1 had thought that 
too, so I said, "Why not let him see if any a particles can be scattered through a 
large angle?" I may tell you in confidence that 1 did not believe they would be, 
since we knew that the a particle was a very fast massive particle, with a great deal 
of energy, and you could show that if the scattering was due to the accumulated 
effect of a number of small scatterings, the chance of an a particle's being scattered 
backwards was very small. Then I remember two or three days later Geiger coming 
to me in great excitement and saying, "We have been able to get some of the a 
particles coming backwards. . . ." It was quite the most incredible event that has 
ever happened to me in my life. It was almost as incredible as if you fired a 15-inch 
shell at a piece of tissue paper and it came back and hit you. 

On consideration I realized that this scattering backwards must be the result of 
a single collision and when I made calculations I saw it was impossible to get any- 
thing of that order of magnitude unless you took a system in which the greater part 
of the mass of the atom was concentrated in a minute nucleus. . . . 

~ In the experimental arrangement used by Marsden and Geiger, a pencil 
of a particles was passed through a thin metal foil and its deflection observed 
on a zinc sulfide screen, which scintillated whenever struck by a particle. 

Rutherford enunciated the nuclear model of the atom in a paper pub- 
lished 8 in 1911. The positive charge is concentrated in the massive center of 
the atom, with the electrons revolving in orbits around it, like planets around 
the sun. Further scattering experiments indicated that the number of elemen- 
tary positive charges in the nucleus of an atom is equal within the experi- 
mental uncertainty to one-half its atomic weight. Thus carbon, nitrogen, and 
oxygen would have 6, 7, and 8 electrons, respectively, revolving around a 
like positive charge. It follows that the charge on the nucleus or the number 
of orbital electrons may be set equal to the atomic number of the element, 
the ordinal number of the position that it occupies in the periodic table. 

7 Ernest Rutherford, Lecture at Cambridge, 1936, in Background to Modern Science, 
ed. by J. Needham and W. Pagel (London: Cambridge, 1938). 

8 Phil. Mag., 21, 669 (1911). 



Sec. ll] THE STRUCTURE OF THE ATOM 213 

According to the nuclear hypothesis, the a particle is therefore the nucleus 
of the helium atom. It was, in fact, known that a particles became helium 
gas when they lost their energy. 

11. X rays and atomic number. The significance of atomic number was 
strikingly confirmed by the work of H. G. J. Moseley. 9 Barkla had discovered 
that in addition to the general or white X radiation emitted by all the ele- 
ments, there were several series of characteristic X-ray lines peculiar to each 
element. Moseley found that the frequency v of a given line in the character- 
istic X radiation of an element depended on its atomic number Z in such a 
way that 

Vv-=fl(Z-6) (8.14) 

where, for each series, a and b are constant for all the elements. The method 
by which the wave length of X rays is measured by using the regular inter- 
atomic spacings in a crystal as a diffraction grating will be discussed in 
Chapter 13. 

When the Moseley relationship was plotted for the K* X-ray lines of the 
elements, discontinuities in the plot appeared corresponding te missing 
elements in the periodic table. These vacant spaces have since been filled. 
This work provided further convincing evidence that the atomic number 
and not the atomic weight governs the periodicity of the properties of the 
chemical elements. 

12. The radioactive disintegration series. Rutherford in 1898, soon after 
the discovery of radioactivity, observed that the activity from thorium would 
diffuse through paper but not through a thin sheet of mica. The radioactivity 
could also be drawn into an ionization chamber by means of a current of 
air. It was therefore evident that radioactive thorium was continuously pro- 
ducing an "emanation" that was itself radioactive. Furthermore, this emana- 
tion left a deposit on the walls of containers, which was likewise active. 
Each of these activities could be quantitatively distinguished from the others 
by its time of decay. As a result of a large amount of careful research by 
Rutherford, Soddy, and others, it was gradually established that a whole 
series of different elements was formed by consecutive processes of radio- 
active change. 

The number of radioactive atoms that decomposes per second is directly 
proportional to the number of atoms present. Thus 



Jf =e ~" (8 - 15) 

if NQ is the number of radioactive atoms present at t - 0. The constant A is 
9 Phil. Mag., 26, 1024 (1913); 27, 703 (1914). 



214 



THE STRUCTURE OF THE ATOM 



[Chap. 8 



called the radioactive-decay constant', the larger the value of A, the more 
rapid the decay of the radioactivity. 

The exponential decay law of eq. (8.15) is plotted in curve A, Fig. 8.5, 
the experimental points being those obtained from uranium X l9 the first 
product in the uranium series. A sample of uranium or of any of its salts is 
found to emit both a and ft particles. If an iron salt is added to a solution 



100 




60 80 100 
TIME, DAYS 

Fig. 8.5. Radioactive decay and regeneration of UX t . 

of a uranium salt, and the iron then precipitated as the hydroxide, it is found 
that the ft activity is removed from the uranium and coprecipitated with the 
ferric hydroxide. This ft activity then gradually decays according to the 
exponential curve A of Fig. 8.5. The original uranium sample gradually 
regains ft activity, according to curve B. It is apparent that the sum of the 
activities given by curves A and B is always a constant. The amount of UX 1 
(the ft emitter) decomposing per second is just equal to the amount being 
formed from the parent uranium. 



Sec. 12] 



THE STRUCTURE OF THE ATOM 



215 



element is the half-life period r, the time required for the activity to be 
reduced to one-half its initial value. From eq. (8.15), therefore, 



In 2 

r 



0.693 



(8.16) 



The half life of uranium is 4.4 x 10 9 years, whereas that of UX^ is 24.5 
days. Because of the long life of uranium compared to UX t , the number of 
uranium atoms present in a sample is effectively constant over measurable 
experimental periods, and the recovery curve of Fig. 8.5 reaches effectively 
the same initial activity after repeated separations of daughter UX t from the 
parent uranium. 

Many careful researches of this sort by Rutherford, Soddy, A. S. Russell, 
K. Fajans, R. Hahn, and others, are summarized in the complete radioactive 
series, such as that for the uranium family shown in Table 8.1. Examination 

TABLE 8.1 
RADIOACTIVE SERIES URANIUM FAMILY 



Name 


Symbol 
of 


At. No. 
Z 


Mass No. 
A 


Particle 
Emitted 


Half Life 




Element 










Uranium 1 . 


U 


92 


238 


a 


4.56 < I0 9 y 


Uranium Xj 


Th 


90 


234 


ft 


24.1 d 


Uranium Xo 


Pa 


91 


234 


fi,y 


1.14m 


Uranium II . 


U 


92 


234 


a 


2.7 x IC^y 


Ionium 


Th 


90 


230 


a 


8.3 x 10 4 y 


Radium 


Ra 


88 


226 


a 


1590y 


Radon 


Rn 


86 


222 


a 


3.825 d 


Radium A . 


Po 


84 


218 


a 


3.05m 


Radium B . 


Pb 


82 


214 


ft.v 


26.8m 


Radium C . 


Bi 


83 


214 


*,ft,v 


19.7m 


Radium C' (99.96%) . 


Po 


84 


214 


a 


1.5 x 10~ 4 s 


Radium C" (0.04%) 


Tl 


81 


210 


ft 


1.32m 


Radium D . 


Pb 


82 


210 


ft,Y 


22 y 


Radium E . 


Bi 


83 


210 


ft,y 


5.0 d 


Radium F . . 


Po 


84 


210 


a 


140d 


Radium G . 


Pb 


82 


206 



















of the properties of the elements in this table established two important 
general principles. When an atom emits an a particle, its position is shifted 
two places to the left in the periodic table; i.e., its atomic number is decreased 
by two. The emission of a ft particle shifts the position one place to the 
right, increasing the atomic number by one. It is evident, therefore, that the 
source of the ft particles is in the nucleus of the atom, and not in the orbital 
electrons. No marked change in atomic weight is associated with the ft 
emission, whereas a emission decreases the atomic weight by four units. 



216 



THE STRUCTURE OF THE ATOM 



[Chap. 8 



13. Isotopes. An important consequence of the study of the radioactive 
series was the demonstration of the existence of elements having the same 
atomic number but different atomic weights. These elements were called 
isotopes by Soddy, from the Greek taos TOKOS, "the same place" (i.e., in 
the periodic table). 

It was soon found that the existence of isotopes was not confined to the 
radioactive elements. The end product of the uranium series is lead, which, 
from the number of intermediate a particle emissions, should have an 
atomic weight of 206, compared to 207.21 for ordinary lead. Lead from the 
mineral curite (containing 21.3 per cent lead oxide and 74.2 per cent uranium 
trioxide), which occurs at Katanga, Belgian Congo, was shown to have an 
atomic weight of 206.03. This fact provided confirmation of the existence of 
nonradioactive isotopes and indicated that substantially all the lead in curite 
had arisen from the radioactive decay of uranium. The time at which the 
uranium was originally deposited can therefore be calculated from the 
amount of lead that has been formed. The geologic age of the earth obtained 
in this way is of the order of 5 X 10 9 years. This is the time elapsed since the 
minerals crystallized from the magma. 

The existence of isotopes provided the solution to the discrepancies in 
the periodic table and to the problem of nonintegral atomic weights. The 
measured atomic weights are weighted averages of those of a number of 
isotopes, each having a weight that is nearly a whole number. The generality 
of this solution was soon shown by the work of Thomson on positive rays. 

14. Positive-ray analysis. In 1886, Eugen Goldstein, using a discharge 
tube with a perforated cathode, discovered a new type of radiation streaming 




Fig. 8.6. Thomson's apparatus for positive- 
ray analysis. 

into the space behind the cathode, to which he gave the name Kanahtrahlen. 
Eleven years later the nature of these rays was elucidated by W. Wien, who 
showed that they were composed of positively charged particles with ratios 
e/m of the same magnitude as those occurring in electrolysis. It was reason- 
able to conclude that they were free positive ions. 

In 1912, Thomson took up, the problem of the behavior of positive rays 
in electric and magnetic fields, using the apparatus shown in Fig. 8.6. The 



Sec. 14] 



THE STRUCTURE OF THE ATOM 



217 




positive rays, generated by ionization of the gas in a discharge tube A, were 
drawn out as a thin pencil through the elongated hole in the cathode B. 
They were then subjected in the region EE' simultaneously to a magnetic 
and to an electric field. 

This was accomplished by inserting strips of mica insulation (D, D') in 
the soft iron pole pieces of the magnet. Then by connecting E and " to a 
bank of batteries, it was possible to supply an electric field that would act 
parallel to the magnetic field of the magnet. 
The trace of the deflected positive rays was 
recorded on the photographic plate P. 

The effect of the superimposed fields may 
be seen from Fig. 8.7. Consider a positive ion 
with charge e to be moving perpendicular to 
the plane of the paper so that, if undeflected, 
it would strike the origin O. If it is subjected 
somewhere along its path to the action of an 
electric field directed along the positive X 
direction, it will be deflected from O to P, 
the deflection being inversely proportional to 
the radius of curvature of the approximately 
circular path traveled in the electric field 

between the plates at EE' in Thomson's apparatus. The actual magnitude 
of the deflection depends on the dimensions of the apparatus. From 
eq. (8.5) and Fig. 8.3, the deflection may therefore be written, taking /q as a 
proportionality constant, 

x = ^ = ^jr (8.17) 

If instead of the electric field a magnetic field in the same direction acts 
on the moving ion, it will be deflected upwards from O to Q 9 the deflection 
being given from eq. (8.9) by 



Fig. 8.7. Thomson's parabola 
method. 



mv 



(8-18) 

^ ' 



The constants k l are the same in eqs. (8.17) and (8.18) if the electric and 
magnetic fields act over the same length of the ion's path, as is the case in 
Thomson's apparatus. 

If the electric and magnetic fields act simultaneously, the ion will be 
deflected to a point R dependent on its velocity v, and its ratio of charge to 
mass. In general, the individual positive ions in a beam are traveling with 
different velocities, and the pattern they form on a viewing screen may.be 
calculated by eliminating v between eqs. (8.17) and (8.18). Thus 



if C 

*1 ~TT ' X 

Em 



(8.19) 



218 



THE STRUCTURE OF THE ATOM 



[Chap. 8 



This is the equation of a parabola. The important conclusion is thereby 
established that all ions of given ratio of charge to mass will strike the screen 
along a certain parabolic curve. Since the charge e' must be an integral 
multiple of the fundamental electronic charge e, the position of the parabola 
will effectively be determined only by the mass of the positive ion. 

The first evidence that isotopes existed among the stable elements was 
found in Thomson's investigation of neon in 1912. He observed a weak 
parabola accompanying that of Neon 20, which could be ascribed only to a 
Neon 22. 

As a result of the work of A. J. Dempster, F. W. Aston, and others, 
positive-ray analysis has been developed into one of the most precise 
methods for measuring atomic masses. The existence of isotopes has been 
shown to be the rule rather than the exception among the chemical elements. 
Apparatus for measuring the masses of positive ions are known as mass 
spectrographs when a photographic record is obtained, and otherwise as 
mass spectrometers. 

15. Mass spectra The Dempster method. The disadvantage of the para- 
bola method is that the ions of any given e/m are spread out along a curve 

so that the density of the pictures ob- 
tained is low at reasonable times of 
exposure. It was most desirable to make 
use of some method that would bring all 

I 1 ~ II y ions of the same e/m to a sharp focus. 

-L fi*t A ^ S\\ ^ ne wa y ^ doing tn * s devised by 

A. J. Dempster in 1918, is shown in 
Fig. 8.8. The positive ions are obtained 
by vaporizing atoms from a heated fila- 
ment A, and then ionizing them by means 
of a beam of electrons from an "electron 
gun" 10 at B. Alternatively, ions can be 




Fig. 8.8. Dempster's mass spectrom- 
eter (direction focusing). 



formed by passing the electron beam 
through samples of gas. A potential 
difference V between A and the slit C 
accelerates the ions uniformly, so that they issue from the slit with approxi- 
mately the same kinetic energies, 



V - e 



(8.20) 



The region D is a channel between two semicircular pieces of iron, through 
which is passed the field from a powerful electromagnet. The field direction 
is perpendicular to the plane of the paper. The ions emerge from the slit C 
in various directions, but since they all have about the same velocity, they 



10 An electron gun is an arrangement by which electrons emitted from a filament are 
accelerated by an electric field and focused into a beam with an appropriate slit system. 



Sec. 16] 



THE STRUCTURE OF THE ATOM 



219 



are bent into circular paths of about the same radius, given by eq. (8.9) as 
R = mv/Be . Therefore, from eq. (8.20), 



m 
7' 



2V 



(8.21) 



It is apparent that for any fixed value of the magnetic field B, the accelera- 
ting potential can be adjusted to bring the ions of the same m/e' to a focus 



80 



60 



I 

cr 



UJ 

> 



20 



X 124 AND 
SCALE MAGNIFIED 
4O X 



124 126 128 130 132 134 
ATOMIC MASS 

Fig. 8.9. Isotopes of xenon. 



136 



138 



at the second slit F, through which they pass to the electrometer G. The 
electrometer measures the charge collected or the current carried through 
the tube by the ions. This was the method used by Dempster in operating 
the apparatus; it is called "direction focusing." A typical curve of ion current 
vs. the mass number calculated from eq. (8.21) is shown in Fig. 8.9, the 
heights of the peaks corresponding to the relative abundances of the isotopes. 

16. Mass spectra Aston's mass spectrograph. A different method of 
focusing was devised by F. W. Aston in 1919, and used by him in the first 
extensive investigations of the occurrence of stable isotopes. The principle 
of this method may be seen from Fig. 8.10. 

Positive ions are generated in a gas discharge tube (not shown) and drawn 
off through the very narrow parallel slits S l and S 2 . Thus, in contrast with 



220 THE STRUCTURE OF THE ATOM [Chap. 8 

Dempster's system, a thin ribbon of rays of closely defined direction is taken 
for analysis ; the velocities of the individual ions may vary considerably, since 
they have been accelerated through different potentials in the discharge tube. 

The thin beam of positive rays first passes through the electric field 
between parallel plates P l and P 2 . The slower ions experience a greater 
deflection, since they take longer to traverse the field ; the beam is accordingly 
spread out, as well as being deflected as shown. 

A group of these rays, selected by the diaphragm D, next passes between 
the parallel pole pieces of the magnet M. The slower ions again experience 
the greater deflection. If the magnetic deflection $ is more than twice the 




S,S 2 



Fig. 8.10. Aston's mass spectrograph (velocity focusing). 

electric deflection 0, all the ions, regardless of velocity, will be brought to a 
sharp focus at some point on the photographic plate P. Aston's method is 
therefore called "velocity focusing." 

More recent developments in mass spectrometry have combined velocity 
and direction focusing in a single instrument. The design has been refined to 
such an extent that it is possible to determine atomic masses to an accuracy 
of one part in 100,000. The precise determination of atomic weights with 
the mass spectrometer is accomplished by carefully comparing sets of closely 
spaced peaks. Thus one may resolve doublets such as H 2 + and He++, 16 O+ 
and CH 4 +, C lf and D 3 +. n By working with such doublets, instrumental 
errors are minimized. 

Mass spectrometers are finding increasing application in the routine 
analysis of complex mixtures of compounds, especially of hydrocarbons. 
For example, a few tenths of a milliliter of a liquid mixture of isomeric 
hexanes and pentanes can be quantitatively analyzed with a modern mass 
spectrometer, a task of insuperable difficulty by any other method. Hydro- 
carbon isomers do not differ in mass, but each isomer ionizes and decom- 
poses in a different way as a result of electron impact. Therefore each isomer 
yields a characteristic pattern of mass peaks in the spectrometer. Most com- 
mercial mass spectrometers follow the Dempster-type design. 

11 The symbol D stands for deuterium or heavy hydrogen, H 2 , which will be discussed 
in following sections. 



Sec. 17] 



THE STRUCTURE OF THE ATOM 



221 



It may be noted that mass-spectrometer chemistry often seems to have 
little respect for our preconceived notions of allowable ionic species. Thus 
Ha 4 " and D 3 + are observed, and benzene vapor yields some C 6 +, a benzene 
ring completely stripped of its hydrogens. Such ions have, of course, a less 
than ephemeral lifetime, since they take only about a microsecond (10~ 6 sec) 
to traverse the spectrometer tube. 

17. Atomic weights and isotopes. A partial list of naturally occurring 
stable isotopes and their relative abundance is given in Table 8.2. Not all of 
these isotopes were first discovered by positive-ray analysis, one notable 
exception being heavy hydrogen or deuterium, whose existence was originally 
demonstrated from the optical spectrum of hydrogen. 

The isotopic weights in Table 8.2 are not exactly integral. Thus the old 

TABLE 8.2 



Atomic 
Number 
Z 


Element 


Symbol 


Mass 
Number 
A 


Isotopic Physical 
Atomic Weight 
(O lfl - 16)Af 


Relative 
Abundance 
(per cent) 


1 


Hydrogen 


H 
D 


1 

2 


1.008131 
2.01473 


99.985 
0.015 


2 


Helium 


He 


3 


3.01711 


10~ 6 








4 


4.00389 


100 


3 


Lithium 


Li 


6 


6.01686 


7.8 








7 


7.01818 


92.1 


4 

5 


Beryllium 
Boron 


Be 
B 


9 
10 


9.01504 
10.01631 


100 
20 








11 


11.01292 


80 


6 


Carbon 


C 


12 


12.00398 


98.9 








13 


13.00761 


1.1 


7 


Nitrogen 


N 


14 
15 


14.00750 
15.00489 


99.62 
0.38 


8 


Oxygen 


O 


16 
17 


16.000000 
17.00450 


99.76 
0.04 








18 


18.00369 


0.20 


9 


Fluorine 


F 


19 


19.00452 


100 


10 


Neon 


Ne 


20 


19.99881 


90.00 








21 


21.00018 


0.27 








22 


21.99864 


9.73 


15 
16 


Phosphorus 
Sulfur 


P 
S 


31 

32 


30.98457 
31.98306 


100 
95.1 








33 


32.98260 


0.74 








34 


33.97974 


4.2 








36 




0.016 


17 


Chlorine 


Cl 


35 


34.98107 


75.4 








37 


36.97829 


24.6 


82 


Lead 


Pb 


204 




1.5 








206 




23.6 








207 




22.6 








208 


208.060 


52.3 


92 


Uranium 


U 


234 




0.006 








235 




0.720 








238 




99.274 



222 



THE STRUCTURE OF THE ATOM 



[Chap. 8 



hypothesis of Prout is nearly but not exactly confirmed. The nearest whole 
number to the atomic weight is called the mass number of an atomic species. 
A particular isotope is conventionally designated by writing the mass number 



80 
70 
60 
50 
40 
30 
20 
10 




-10 



CURVE FOR LIGHT ELEMENTS 




4He 



02468 



10 12 14 16 18 20 22 24 
MASS NUMBER 
(0) 



z 
O 



<r -4 
u_ 

o -6 

z 

5 " 8 

2 -10 

-12 



- CURVE FOR HEAVY ELEMENTS 





l 



20 40 60 80 100 120 140 160 180 200 220 240 
MASS NUMBER 
(b) 

Fig. 8.11. Packing fraction curves, (a) Curve for light elements, 
(b) Curve for heavy elements. 

as a left- or right-hand superscript to the symbol of the element; e.g., 2 H, 
U 235 , and so on. 

The packing fraction of an isotope is defined by 

atomic weight mass number 



packing fraction = 



mass number 



The curves in Fig. 8.11 show how the packing fraction varies with mass 
number, according to the latest atomic-weight data. The further discussion 
of these curves, whose explanation requires an enquiry into the structure of 
the atomic nucleus, will be postponed till the following chapter. 



Sec. 18] THE STRUCTURE OF THE ATOM 223 

It will be noted that oxygen, the basic reference element for the calcula- 
tion of atomic weights, is itself composed of three isotopes, 16, 17, and 18. 
Chemists have been unable to abandon the convention by which the mixture 
of isotopes constituting ordinary oxygen is assigned the atomic weight 
O ~ 16. Weights calculated on this basis are called chemical atomic weights. 
The physicists prefer to call O 16 16, whence ordinary oxygen becomes 
O -^ 16.0043. This leads to a set of physical atomic weights. 

18. Separation of isotopes. For a detailed discussion of separation 
methods, reference may be made to standard sources. 12 Several of the more 
important procedures will be briefly discussed. 

1. Gaseous diffusion. This was the method used to separate 235 UF 6 from 
238 UF 6 in the plant at Oak Ridge, Tennessee. The fundamental principle 
involved has been discussed in connection with Section 7-8 on the effusion 
of gases. 

The separation factor f of a process of isotope separation is defined as 
the ratio of the relative concentration of a given species after processing to 
its relative concentration before processing. Thus/ (fli7 AI 2 / )/( /7 i/ AI 2) where 
(n l9 A?/) an d (>*2> "2') are tne concentrations of species 1 and 2 before and 
after processing. Uranium 235 occurs in natural uranium to the extent of 
one part in 140 (njn^ -= 1/140). If it is desired to separate 90 per cent 
pure U 235 from U 238 , therefore, the over-all separation factor must be 
/- (9/l)/(l/140) - 1260. 

For a single stage of diffusion the separation factor cannot exceed the 
ideal value a, given from Graham's Law, as a VM 2 /M ly where M 2 and 
M 1 are the molecular weights of the heavy and light components, respec- 
tively. For the uranium hexafluorides, a = A/352/349 1.0043. 

Actually, the value of /for a single stage will be less than this, owing to 
diffusion in the reverse direction, nonideal mixing at the barrier surface, and 
partially nondiffusive flow through the barrier. It is therefore necessary to 
use several thousand stages in a cascade arrangement to effect a considerable 
concentration of 235 UF 6 . The theory of a cascade is very similar to that of 
a fractionating column with a large number of theoretical plates. The 
light fraction that passes through the barrier becomes the feed for the next 
stage, while the heavier fraction is sent back to an earlier stage. 

It may be noted that UF 6 has at least one advantage for use in a process for 
separating uranium isotopes, in that there are no isotopes of fluorine except 19 F. 

2. Thermal diffusion. This method was first successfully employed by 
H. Clusius and G. Dickcl, 13 and the experimental arrangement is often 

12 H. S. Taylor and S. Glasstone, Treatise on Physical Chemistry, 3rd ed. (New York: 
Van Nostrand, 1941); H. D. Smyth, Atomic Energy for Military Purposes (Princeton Univ. 
Press, 1945); F. W. Aston, Mass Spectra and Isotopes, 4th ed. (New York: Longmans, 1942). 

13 Naturmssenschaften, 26, 546 (1938). For the theory of the thermal diffusion separa- 
tion see K. Schafer, Angew. Chem., 59, 83 (1947). The separation depends not only on mass 
but also on difference in intermodular forces. With isotopic molecules the mass effect 
predominates and the lighter molecules accumulate in the warmer regions. 



224 THE STRUCTURE OF THE ATOM [Chap. 8 

called a Clusius column. It consists of a long vertical cylindrical pipe with 
an electrically heated wire running down its axis. When a temperature 
gradient is maintained between the hot inner wire and the cold outer walls, 
the lighter isotope diffuses preferentially from the cold to the warmer regions. 
The separation is tremendously enhanced by the convection currents in the 
tube, which carry the molecules arriving near the warm inner wire upwards 
to the top of the column. The molecules at the cold outer wall are carried 
downwards by convection. 

With columns about 30 meters high and a temperature difference of 
about 600C, Clusius was able to effect a virtually complete separation of 
the isotopes of chlorine, Cl 35 and Cl 37 . 

The cascade principle can also be applied to batteries of thermal diffusion 
columns, but for mass production of isotopes this operation is in general 
less economical than pressure-diffusion methods. 

3. Electromagnetic separators. This method employs large mass spectro- 
meters with split collectors, so that heavy and light ions are collected separ- 
ately. Its usefulness is greatest in applications in which the throughput of 
material is comparatively small. 

4. Separation by exchange reactions. Different isotopic species of the 
same element differ significantly in chemical reactivity. These differences are 
evident in the equilibrium constants of the so-called isotopic exchange 
reactions. If isotopes did not differ in reactivity, the equilibrium constants 
of these reactions would all be equal to unity. Some actual examples follow: 

J S 16 2 + H 2 18 -= i S 18 2 + H 2 16 K =-- 1.028 at 25C 

i3 CO + i2 C Q 2 = 12CO + 13 CO 2 K = 1.086 at 25C 

15 NH 3 (g) + 14 NH 4 +(aq.) = 14 NH 3 (g) + 15 NH 4 +(aq.) K - 1.023 at 25C 

Such differences in affinity are most marked for the lighter elements, for 
which the relative differences in isotopic masses are greater. 

Exchange reactions can be applied to the separation of isotopes. The 
possible separation factors in a single-stage process are necessarily very 
small, but the cascade principle is again applicable. H. C. Urey and H. G. 
Thode concentrated 15 N through the exchange between ammonium nitrate 
and ammonia. Gaseous ammonia was caused to flow countercurrently to a 
solution of NH 4 4 ions, which trickled down columns packed with glass 
helices or saddles. After equilibrium was attained in the exchange columns, 
8.8 grams of 70.67 per cent 15 N could be removed from the system, as nitrate, 
every twelve hours. 

As a result of exchange reactions, the isotopic compositions of naturally 
occurring elements show small but significant variations depending on their 
sources . If we know the equilibrium constant of an exchange reaction over 
a range of temperatures, it should be possible to calculate the temperature 
at which a product was formed, from a measurement of the isotopic ratio 
in the product. Urey has applied this method, based on O 18 : O 16 ratios, to 



Sec. 19] 



THE STRUCTURE OF THE ATOM 



225 



the determination of the temperature of formation of calcium carbonate 
deposits. The exchange equilibrium is that between the oxygen in water and 
in bicarbonate ions. The temperature of the seas in remote geologic eras can 
be estimated to within 1C from the O 18 : O 16 ratio in deposits of the shells 
of prehistoric molluscs. 

19. Heavy hydrogen. The discovery of the hydrogen isotope of mass 2," 
which is called deuterium, and the investigation of its properties comprise 
one of the most interesting chapters in physical chemistry. 

In 1931, Urey, Brickwedde, and Murphy proved the existence of the 
hydrogen isotope of mass 2 by a careful examination of the spectrum of a 
sample of hydrogen obtained as the residue from the evaporation of several 
hundred liters of liquid hydrogen. Deuterium is contained in hydrogen to the 
extent of one part in 4500. 

In 1932, Washburn and Urey discovered that an extraordinary concen- 
tration of heavy water, D 2 O, occurred in the residue from electrolysis of 
water. 14 The production of 99 per cent pure D 2 O in quantities of tons per 
day is now a feasible operation. Some of the properties of pure D 2 O as 
compared with ordinary H 2 O are collected in Table 8.3. 

TABLE 8.3 
PROPERTIES OF HEAVY WATER AND ORDINARY WATER 



Property 



Units 


Ordinary 
Water 


Heavy 
Water 


g/cc 
C 


0.997044 
4.0 


1.104625 
11.6 


C 


0.000 


3.802 


c 


100.00 


101.42 


cal/mole 
cal/mole 


1436 
10,484 
81.5 


1510 
10,743 
80.7 





1.33300 


1.32828 


dynes/cm 
millipoise 


72.75 
13.10 


72.8 
16.85 



Density at 25C .... 
Temperature of maximum density . 
Melting point .... 
Boiling point .... 
Heat of fusion .... 
Heat of vaporization at 25 . 
Dielectric constant 
Refractive index at 20 (Na D line) 
Surface tension (20C) . 
Viscosity (10C) . 



PROBLEMS 

1. An Na 4 " ion is moving through an evacuated vessel in the positive 
x direction at a speed of 10 7 cm per sec. At x 0, y =~ 0, it enters an 
electric field of 500 volts per cm in the positive y direction. Calculate its 
position (;c, y) after 10~ 6 sec. 

2. Make calculations as in Problem 1 except that the field is a magnetic 
field of 1000 gauss in the positive z direction. 

14 The mechanism of the separation of H 1 from H 2 during electrolysis is still obscure. 
For discussions see Eyring, et aL, J. Chem. Phys., 7, 1053 (1939); Urey and Teal, Rev. Mod. 
Phys., 7, 34(1935). 



226 THE STRUCTURE OF THE ATOM [Chap. 8 

3. Calculate the final position of the Na+ ion in the above problems if 
the electric and magnetic fields act simultaneously. 

4. Consider a Dempster mass spectrometer, as shown in Fig. 8.8, with a 
magnetic field of 3000 gauss and a path radius of 5.00 cm. At what accelera- 
ting voltage will (a) H+ ions, (b) Na^ ions be brought to focus at the ion 
collector ? 

5. Radium-226 decays by a particle emission with a half life of 1590 
years, the product being radon-222. Calculate the volume of radon evolved 
from 1 g of radium over a period of 50 years. 

6. The half life of radon is 3.825 days. How long would it take for 90 per 
cent of a sample of radon to disintegrate? How many disintegrations per 
second are produced in a microgram ( 1 0~ 6 g) of radium ? 

7. Derive an expression for the average life of a radioactive atom in 
terms of the half life r. 

8. The half life of thorium-C is 60.5 minutes. How many disintegrations 
would occur in 15 minutes from a sample containing initially 1 mg of Th-C 
(at wt. 212)? 

9. Radioactivity is frequently measured in terms of the curie (c) defined 
as the quantity of radioactive material producing 3.7 X 10 10 disintegrations 
per sec. The millicurie is 10~ 3 c, the microcurie, 10~ 6 c. How many grams of 
(a) radium, (b) radon are there in one curie? 

10. It is found that in 10 days 1.07 x 10~ 3 cc of helium is formed from 
the a particles emitted by one gram of radium. Calculate a value for the 
half life of radium from this result. 

11. The half life of U-238 is 4.56 x 10 9 years. The final decay product is 
Pb-206, the intermediate steps being fast compared with the uranium dis- 
integration. In Lower Pre-Cambrian minerals, lead and uranium are found 
associated in the ratio of approximately 1 g Pb to 3.5 g U. Assuming that all 
the Pb has come from the U, estimate the age of the mineral deposit. 

12. A ft particle moving through a cloud chamber under a magnetic field 
of 10 oersteds traverses a circular path of 18 cm radius. What is the energy 
of the particle in ev? 

REFERENCES 

BOOKS 

1. Born, M., Atomic Physics (London: Blackie, 1951). 

2. Feather, N., Lord Rutherford (London: Blackie, 1940). 

3. Finkelnburg, W., Atomic Physics (New York: McGraw-Hill, 1950). 

4. Rayleigh, Lord, Life ofJ. J. Thomson (Cambridge University Press, 1942). 

5. Richtmeyer, F. K., and E. A. Kennard, Introduction to Modern Physics 
(New York: McGraw-Hill, ,1947). 

6. Semat, H., Introduction to Atomic Physics (New York: Rinehart, 1946). 



Chap. 8] THE STRUCTURE OF THE ATOM 227 

7. Stranathan, J. D., The Particles of Modern Physics (Philadelphia: Blaki- 
ston, 1954). 

8. Tolansky, S., Introduction to Atomic Physics (London: Longmans, 1949). 

9. Van Name, F., Modern Physics (New York: Prentice-Hall, 1952). 

ARTICLES 

1. Birge, R. T., Am. J. Phys., 13, 63-73 (1945), "Values of Atomic Con- 
stants." 

2. Glasstone, S., /. Chem. Ed., 24, 478-81 (1947), "William Prout." 

3. Hooykaas, R., Chymia, 2, 65-80 (1949), "Atomic and Molecular Theory 
before Boyle." 

4. Jauncey, G. E., Am. J. Phys., 14, 226-41 (1946), "The Early Years of 
Radioactivity." 

5. Lemay, P., and R. E. Oesper, Chymia, 1, 171-190 (1948), "Pierre Louis 
Dulong." 

6. Mayne, K. I., Rep. Prog. Phys., 15, 24-48 (1952), "Mass Spectrometry." 

7. Rayleigh, Lord, /. Chem. Soc., 467-75 (1942), "Sir Joseph J. Thomson." 

8. Urey, H. C, Science in Progress, vol. I (New Haven: Yale University 
Press, 1939), 35-77, "Separation of Isotopes." 

9. Winderlich, R., J. Chem. Ed., 26, 358-62 (1949), "Eilard Mitscherlich." 



CHAPTER 9 

Nuclear Chemistry and Physics 



1. Mass and energy. During the nineteenth century, two important prin- 
ciples became firmly established in physics : the conservation of mass and 
the conservation of energy. Mass was the measure of matter, the substance 
out of which the physical world was constructed. Energy seemed to be an 
independent entity that moved matter from place to place and changed it 
from one form to another. 

In a sense, matter contained energy, for heat was simply the kinetic 
energy of the smallest particles of matter, and potential energy was associated 
with the relative positions of material bodies. Yet there seemed to be one 
instance, at Jeast, in which energy existed independently of matter, namely 
in the form of radiation. The electromagnetic theory of Clerk Maxwell 
required an energy in the electromagnetic field and the field traversed empty 
space. Yet no experiments can be performed in empty space, so that actually 
this radiant energy was detected only when it impinged on matter. Now a 
very curious fact was observed when this immaterial entity, light energy, 
struck a material body. 

The observation was first made in 1628 by Johannes Kepler, who noted 
that the tails of comets always curved away from the sun. He correctly- 
assigned the cause of this curvature to a pressure exerted by the sun's rays. 
In 1901 this radiation pressure was experimentally demonstrated in the 
laboratory, by means of delicate torsion balances. Thus the supposedly 
immaterial light exerts a pressure. The pressure implies a momentum asso- 
ciated with the light ray, and a momentum implies a mass. If we return to 
Newton's picture of a light ray as made up of tiny particles, simple calcula- 
tions show that the energy of the particles E is related to their 'mass by the 

equation 

E - c 2 m (9.1) 

where c is the speed of light. 

As a result of Albert Einstein's special theory of relativity (1905) it 
appeared that the relation E c*m was applicable to masses and energies 
of any origin. He showed first of ail that no particle could have a speed 
greater than that of light. Thus the inertial resistance that a body offers to 
acceleration by an applied force must increase with the speed of the body. 
As the speed approaches that of light, the mass must approach infinity. The 
relation between mass and speed v is found to be 



(9.2) 
228 



Sec. 2] NUCLEAR CHEMISTRY AND PHYSICS 229 

When v = the body has a rest mass, m Q . Only at speeds comparable with 
that of light does the variation of mass with speed become detectable. 
Equation (9.2) has been confirmed experimentally by measurements of e/m 
for electrons accelerated through large potential differences. 

If a rapidly moving particle has a larger mass than the same particle 
would have at rest, it follows that the larger the kinetic energy, the larger 
the mass, and once again it turns out that the increase in mass Am and the 
increase in kinetic energy AE are related by A" - c 2 AAW. As we shall see 
in the next section, the Einstein equation E = c 2 m has been conclusively 
checked by experiments on nuclear reactions. 

The situation today is therefore that mass and energy are not two distinct 
entities. They are simply two different names for the same thing, which for 
want of a better term is called mass-energy. We can measure mass-energy in 
mass units or in energy units. In the COS system, the relation between the 
two is: 1 gram ^ c 2 ergs = 9 x 10 20 ergs. One gram of energy is sufficient to 
convert 30,000 tons of water into steam. 

2. Artificial disintegration of atomic nuclei. In 1919, Rutherford found 
that when a particles from Radium C were passed through nitrogen, protons 
were ejected from the nitrogen nuclei. This was the first example of the dis- 
integration of a normally stable nucleus. It was soon followed by the demon- 
stration of proton emission from other light elements bombarded with 
a particles. 

In 1923, P. M. S. Blackett obtained cloud-chamber photographs showing 
that these reactions occurred by capture of the a particle, a proton and a new 
nucleus then being formed. For example, 

7 N 14 + 2 He 4 > ( 9 P) -> iH 1 + 8 17 

This type of reaction does not occur with heavy elements because of the 
large electrostatic repulsion between the doubly charged alpha and the high 
positive charges of the heavier nuclei. 

It was realized that the singly charged proton, 1 H 1 , would be a much 
more effective nuclear projectile, but it was not available in the form of 
high-velocity particles from radioactive materials. J. D. Cockroft and 
E. T. S. Walton 1 therefore devised an electrostatic accelerator. This appara- 
tus was the forerunner of many and ever more elaborate machines for pro- 
ducing high-velocity particles. The protons produced by ionization of 
hydrogen in an electric discharge were admitted through slits to the accel- 
erating tube, accelerated across a high potential difference, and finally 
allowed to impinge on the target. 

The energy unit usually used in atomic and nuclear physics, the electron 
volt, is the energy acquired by an electron in falling through a potential 
difference of one volt. Thus 1 ev = eV ~ 1.602 x 10~ 19 volt coulomb 

1 Proc. Roy. Soc., A 729, 477 (1930); 136, 619 (1932). 



230 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 

(joule) = 1.602 x 10~ 12 erg. The usual chemical unit is the kiiocalorie per 
mole. 

1.602 x 10~ 19 x 6.02 x 10 23 , . , 

! ev ____ ^ 23.05 kcal per mole 

One of the first reactions to be studied by Cockroft and Walton was 
iH 1 -f 3 Li 7 -> 2 2 He 4 

The bombarding protons had energies of 0.3 million electron volt (mev.) 
From the range of the emergent a particles in the cloud chamber, 8.3 cm, 
their energy was calculated to be 8.6 mev each, or more than 17 mev for the 
pair. It is evident that the bombarding proton is merely the trigger that sets 
off a tremendously exothermic nuclear explosion. 

The energies involved in these nuclear reactions are several million times 
those in the most exothermic chemical changes. Thus an opportunity is pro- 
vided for the quantitative experimental testing of the E = c 2 m relation. The 
mass-spectrographic values for the rest masses of the reacting nuclei are 
found to be 

H + Li = 2 He 
1.00812 + 7.01822 2 x 4.00391 

Thus the reaction occurs with a decrease in rest mass Aw of 0.01852 g per 
mole. This is equivalent to an energy of 

0.01852 x 9 x 10 20 - 1.664 x 10 19 erg per mole 

I 664 x 10 19 

or -' - 2.763 x 10~ 5 erg per lithium nucleus 

O.v/^c X L\J 

or 2.763 x 10~ 5 x 6.242 x 10 11 - 17.25 x 10 6 - 17.25 mev 

This figure is in excellent agreement with the energy observed from the 
cloud-chamber experiments. Nor is this an isolated example, for hundreds 
of these nuclear reactions have been studied and completely convincing 
evidence for the validity of the equation E = c 2 m has been obtained. 

It has become rather common to say that a nuclear reaction like this 
illustrates the conversion of mass to energy, or even the annihilation of 
matter. This cannot be true in view of the fact that mass and energy are the 
same. It is better to explain what happens as follows: Rest mass is a par- 
ticularly concentrated variety of energy; Jeans once called it bottled energy. 
When the reaction 1 H 1 + 3 Li 7 -> 2 2 He 4 takes place, a small amount of this 
bottled energy is released ; it appears as kinetic energy of the particles re- 
acting, which is gradually degraded into the random kinetic energy or heat 
of the environment. As the molecules of the environment gain kinetic energy, 
they gain mass. The hotter a substance, the greater is its mass. Thus, in the 
nuclear explosion, the concentrated rest mass (energy) is degraded into the 
heat mass (energy) of the environment. There has been no over-all change 
in mass and no over-all change in energy; mass-energy is conserved. 



Sec. 3] NUCLEAR CHEMISTRY AND PHYSICS 231 

The measurement of the large amounts of energy released in nuclear 
reactions now provides the most accurate known means of determining small 
mass differences. The reverse process of calculating atomic masses from the 
observed energies of nuclear reactions is therefore widely applied in the 
determination of precise atomic weights. A few of the values so obtained 
are collected in Table 9.1 and compared with mass-spectrometer data. The 

TABLE 9.1 
ATOMIC WEIGHTS (O 16 - 16.0000) 



Atom 


Mass Spectrometer 
Value 


Mass-Energy 
Value 


H 1 


1.008141 


1.008142 


H 2 


2.014732 


2.014735 


* H 3 





3.01700 


He 3 





3.01698 


He 4 


4.00386 


4.00387 


*He 6 





6.02047 


Li 6 


6.0145 


6.01686 


Li 7 


7.01818 


7.01822 


C 12 


12.00381 


12.00380 



agreement between the two methods is exact, within the probable error of the 
experiments. It would be hard to imagine a more convincing proof of the 
equivalence of mass and energy. The starred isotopes are radioactive, and 
the only available mass values are those from the E = c 2 m relation. 

3. Methods for obtaining nuclear projectiles. It was at about this point in 
its development that nuclear physics began to outgrow the limitations of 
small-scale laboratory equipment. The construction of machines for the pro- 
duction of enormously accelerated ions, capable of overcoming the repulsive 
forces of nuclei with large atomic numbers, demanded all the resources of 
large-scale engineering. 

One of the most generally useful of these atom-smashing machines has 
been the cyclotron, shown in the schematic drawing of Fig. 9.1, which was 
invented by E. O. Lawrence of the University of California. The charged 
particle is fed into the center of the "dees" where it is accelerated by a strong 
electric field. The magnetic field, however, constrains it to move in a circular 
path. The time required to traverse a semicircle is t ^R^v = (TT/B) (m/e) 
from eq. (8.9); this is a constant for all particles having the same ratio e/m. 
The electric field is an alternating one, chosen so that its polarity changes 
with a frequency twice that of the circular motion of the charged particle. 
On each passage across the dees, therefore, the particle receives a new for- 
ward impulse, and describes a trajectory of ever increasing radius until it is 
drawn from the accelerating chamber of the cyclotron. The 184-in. machine 
at Berkeley, California, will produce a beam of 100 mev deuterons (nuclei of 
deuterium atoms) having a range in air of 140 fe'et. 



232 



NUCLEAR CHEMISTRY AND PHYSICS 



[Chap. 9 



A limit to the energy of ions accelerated in the original type of cyclotron 
is the relativistic increase of mass with velocity; this eventually destroys the 
synchronization in phase between the revolving ions and the accelerating 
field across the dees. This problem has been overcome in the synchro- 
cyclotron, in which frequency modulation, applied to the alternating accel- 
erating potential, compensates for the relativistic defocusing. This modifica- 
tion of the original design of the Berkeley instrument has more than doubled 
the maximum ion energies obtained. 



Iniulotor 



Feed lines 
Oee 



Internal beam 



Electric 
deflector 




Vacuum can 



Lower pole 



Magnet 
pole piece 



Fig. 9.1. Schematic diagram of the cyclotron. (From Lapp and Andrews, 
Nuclear Radiation Physics, 2nd Ed. Prentice-Hall, 1954.) 

The synchrotron employs modulation of both the electric accelerating 
field and the magnetic focusing field. With this principle, it is possible to 
achieve the billion-volt range for protons. The cosmotron, a synchrotron 
completed in 1952 at Brookhaven National Laboratory, accelerates protons 
in a toroidal vacuum chamber with orbits 60 ft in diameter. The C-shaped 
magnets are placed around the vacuum chamber. Pulses of about 10 11 protons 
at 3.6 mev are fired into the chamber. After about 3 x 10 6 revolutions, the 
pulse of protons has reached 3 bev (3000 mev), and is brought to the target. 
A similar machine at Berkeley is designed to produce 10 bev protons. These 
particles are thus well within the energy range of cosmic rays. 

4. The photon. The essential duality in the nature of radiation has already 
been remarked: sometimes it is appropriate to treat it as an electromagnetic 
wave, while at other times a corpuscular behavior is displayed. The particle 
of radiation is called the photon. 

A more detailed discussion of the relation between waves and particles 
will be given in the next chapter. One important result may be stated here. 
A homogeneous radiation of wave length A or frequency v = cjX may be 
considered to be composed of photons whose energy is given by the relation, 



= hv 



(9.3) 



Sec. 4] NUCLEAR CHEMISTRY AND PHYSICS 233 

Here h is a universal constant, called Planck's constant, with the value 
6.624 x 10~ 27 erg sec. A photon has no rest mass, but since e me 2 , its 
mass is m hv/c 2 . 

The corpuscular nature of light was first clearly indicated by the photo- 
electric effect, discovered by Hertz in 1887, and theoretically elucidated by 
Einstein in 1905. Many substances, but notably the metals, emit elections 
when illuminated with light of appropriate wave lengths. A simple linear 
relation is observed between the maximum kinetic energy of the photo- 
electrons emitted and the frequency of the incident radiation. The slope of 
the straight line is found to be Planck's constant h. Thus, 

\mv 2 = hv ~ < (9.4) 

Such an equation can be interpreted only in terms of light quanta, or 
photons, which in some way transmit their energy hv to electrons in the 
metal, driving them beyond the field of attraction of the metal ions. The 
term <f> represents the energy necessary 
to overcome the attractive force tending 
to hold the electron within the metal. 

If a photon (e.g., from X or y rays) 
strikes an electron, an interchange of 
energy may take place during the collision. 
The scattered photon will have a higher 
frequency if it gains energy, a lower fre- 
quency if it loses energy. This is called the 
Compton effect. 

Consider in Fig. 9.2 a photon, with Fig. 9.2. The Compton effect, 
initial energy hv, hitting an electron at 

rest at O. Let hv' be the energy of the scattered photon and let the scattered 
electron acquire a speed v. Then ifm is the mass of the electron, its momentum 
will be mv and its kinetic energy \mv 2 . The scattering angles are a and ft. 
The laws of conservation of energy and of momentum both apply to the 
collision. From the first, 

hv = hv' + \mv 2 

From the second, for the jc and y components of the momentum, 

hv hv' 

r^ cos a + mv cos p 
c c 

hv' . 

= sin a mv sin p 
c 

Eliminating ft from the momentum equations by setting sin 2 ft + cos 2 ft 
= 1, and assuming that v' v <^ v, we find for the momentum imparted to 
the electron, 

2hv . a 

-* (9 ' 5) 




234 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 

Then from the energy equation, by eliminating v, the change in frequency of 
the photon is 

hv* 

& v -- v _ v ' ~ ( 1 _ cos a ) 

me 2 

The predicted angular dependence of the change in frequency has been 
confirmed experimentally in studies of the Compton scattering of X rays by 
electrons in crystals. The Compton effect has also been observed in cloud- 
chamber photographs. 

5. The neutron. In 1930, W. Bothe and H. Becker discovered that a very 
penetrating secondary radiation was produced when a particles from polo- 
nium impinged on light elements such as beryllium, boron, or lithium. They 
believed this radiation to consist of 7 rays of very short wavelength, since no 
track was made in a cloud chamber, and therefore charged particles were 
not being formed. 

In 1932, Frederic and Irene Curie-Joliot found that this new radiation 
had much greater ionizing power after it had passed through paraffin, or 
some other substance having a high hydrogen content, and during its passage 
protons were emitted from these hydrogen-rich materials. 

James Chadwick 2 solved the problem of the new "radiation." He realized 
that it was made up of particles of a new kind, having a mass comparable 
with that of the proton, but bearing no electric charge. These particles were 
called neutrons. Because of its electrical neutrality, forces between the neutron 
and other particles become appreciable only at very close distances of 
approach. The neutron, therefore, loses energy only slowly as it passes 
through matter; in other words, it has a great penetrating power. The 
hydrogen nucleus is most effective in slowing a neutron, since it is of com- 
parable mass, and energy exchange is a maximum between particles of like 
mass, during actual collisions or close approaches. 

The reaction producing the neutron can now be written 

2 He 4 4- 4 Be* - 6 C 12 + ^ 

Neutrons can be produced by similar reactions of other light elements with 
high energy a particles, protons, deuterons, or even y rays, for example : 

1 H 2 + /n>-> 1 H l + /2 1 
H l + 3 Li 7 ~> 4 Be 7 + o/? 1 

Beams of neutrons can be formed by means of long pinholes or slits in 
thick blocks of paraffin, and methods are available for producing beams of 
uniform energy. 3 

Because it can approach close to an atomic nucleus without being electro- 
statically repelled, the neutron is an extraordinarily potent reactant in nuclear 
processes. 

2 Proc. Roy. Soc., A 136, 692 (1932). 

3 E. Fermi, J. Marshall, and L. Marshall, Phys. Rev., 72, 193 (1947); W. Zinn, ibid., 71, 
757 (1947). 



Sec. 6] NUCLEAR CHEMISTRY AND PHYSICS 235 

6. Positron, meson, neutrino. The year 1932 was a successful one for 
nuclear physics, because two new fundamental particles were discovered, 
the neutron and the positron. The latter was detected by Carl D. Anderson 
in certain cloud-chamber tracks from cosmic rays. The positron is the 
positive electron e+. It had previously been predicted by the theoretical work 
of Dirac. In 1933 Frederic and Irene Curie-Joliot found that a shower of 
positrons was emitted when a rays from polonium impinged on a beryllium 
target. When targets of boron, magnesium, or aluminum were used, the 
emission of p6sitrons was observed to continue for some time after the 
particle bombardment was stopped. This was the first demonstration of 
artificial radioactivity. 4 A typical reaction sequence is the following: 

5 B 10 + 2 He 4 -> O n l + 7 N 1:i ; 7 N 13 -> 6 C 13 + e+ 

More than a thousand artificially radioactive isotopes are now known, 
produced in a variety of nuclear reactions. 5 

The positron escaped detection for so long because it can exist only 
until it happens to meet an electron. Then a reaction occurs that annihilates 
both of them, producing a y-ray photon: 

e -\ f e~~ -> hv 
The energy equivalent to the rest mass of an electron is: 

f - me 2 - 9.11 x 10- 28 x (3.00 x 10 10 ) 2 --- 8.20 x 10 7 erg 
If this is converted into a single y-ray photon, the wavelength would be 

A - n,c ~ 9.U X-IO 

The y radiation obtained in 'the annihilation of electron-positron pairs has 
either this wavelength or one-half of it. The latter case corresponds to the 
conversion of the masses of both e + and e~~ into a single y-ray photon. The 
reverse process, the production of an electron-positron pair from an energetic 
photon, has also been observed. 

In 1935, H. Yukawa proposed for the structure of the nucleus a theory 
that postulated the existence of a hitherto unknown kind of particle, which 
would be unstable and have a mass of about 150 (electron ~ 1). From 1936 
to 1938 Anderson's work at Pasadena revealed the existence of particles, 
produced by cosmic rays, which seemed to have many of the properties 
predicted by Yukawa. These particles are the ^-mesons, which may be 
charged plus or minus, have a mass of 209 2, and a half life of 2.2 x 10~ 6 
sec. The particles required by the theory, however, resemble more closely 
the 7r-mesons, discovered in 1947 by the Bristol cosmic-ray group headed by 
C. F. Powell. These have a mass of 275, and decay to /^-mesons, with a 
half life of 2.0 x 10~ 8 sec. Several other particles, with masses of 800 to 

4 C. R. Acad. Set. Paris, 198, 254, 559 (1934). 

5 G. Seaborg and I. Perlman, Rev. Mod. Phys., 20, 585 (1948). 



236 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 

1300, have also been discovered. The theoretical interpretation of the variety 
of particles now known will require some new great advance in fundamental 
theory. 

In order to satisfy the law of conservation of mass-energy in radioactive 
decays, decay of mesons, and similar processes, it is necessary to postulate 
the existence of neutral particles with rest masses smaller than that of the 
electron. These neutrinos have not yet been detected by physical methods, 
since their effects are necessarily small. 

7. The structure of the nucleus. The discovery of the neutron led to an 
important revision in the previously accepted picture of nuclear structure. 
Instead of protons and electrons, it is now evident that protons and neutrons 
are the true building units. These are therefore called nucleons. 

Each nucleus contains a number of protons equal to its atomic number 
Z, plus a number of neutrons /?, sufficient to make up the observed mass 
number A. Thus, A n + Z. 

The binding energy E of the nucleus is the sum of the masses of the 
nucleons minus the actual nuclear mass M. Thus, 

E -=- Zm n + (A - Z)m n M (9.6) 

The proton mass m ir = 1.00815, the neutron mass m n 1.00893 in atomic 
mass units. To convert this energy from grams per mole to mev per nuclcon, it 
must be multiplied by 

C 2 

. _ 934 

W x 10 x 1.602 x 10 ~ 12 

One of the convincing arguments against the existence of electrons as 
separate entities in the nucleus is based on the magnitude of the observed 
binding energies. For example, if the deuteron 1 H 2 were supposed to be 
made up of two protons and an electron, the binding energy would be 
0.001 53 gram per mole. Yet the electron's mass is only 0.00055 gram per 
mole. For the electron to preserve its identity in the nucleus while creating 
a binding energy about three times its own mass would seem to be physically 
most unreasonable. 

We do not yet know the nature of the forces between nucleons. The 
nuclear diameter is given approximately by d 1.4 x 10~ 13 A m cm, A being 
the mass number. The forces therefore must be extremely short-range, unlike 
electrostatic or gravitational forces. The density of nuclear material is around 
10 14 g per cc. A drop big enough to see would weigh 10 7 tons. There is an 
electrostatic repulsion between two protons, but this longer-range (inverse 
square) force is outweighed by the short-range attraction, so that at separa- 
tions around 10~ 13 cm the attraction between two protons is about the same 
as that between two neutrons or a neutron and a proton. According to 
Yukawa's theory, the attractive forces between nucleons are due to a new 
type of radiation field, in which the mesons play a role like that of the 
photons in an ordinary electromagnetic field. 



Sec. 7] 



NUCLEAR CHEMISTRY AND PHYSICS 



237 



A further insight into nuclear forces can be obtained by examining the 
composition of the stable (nonradioactive) nuclei. In Fig. 9.3 the number of 
neutrons in the nucleus is plotted against the number of protons. The line 
has an initial slope of unity, corresponding to a one-to-one ratio, but it 
curves upward at higher atomic numbers. The reason for this fact is that 
the electrostatic repulsion of the protons increases as the nucleus becomes 
larger, since it is a longer range force than the attraction between protons. 
To compensate for this repulsion more neutrons are necessary. Yet there is 



140 
130 
120 
MO 
100 
90 



c/> 

i 70 

t; 60 

UJ 

2 50 
40 
30 
20 
10 






10 20 30 40 50 60 70 80 90 100 110 
PROTONS, Z 

Fig. 9.3. Number of neutrons vs. number of protons in stable nuclei. 

a limit to the number of extra neutrons that can be accommodated and still 
produce added stability, so that the heavier nuclei become less stable. 

This effect is illustrated clearly in Fig. 9.4, which shows the binding 
energy per nucleon as a function of the mass number. Only the stable iso- 
topes lie on this reasonably smooth curve. Natural or artificial radioactive 
elements fall below the curve by an amount that is a measure of their in- 
stability relative to a stable isotope of the same mass number. 

The successive maxima in the early part of the curve occur at the following 
nuclei: He 4 , Be 8 , C 12 , O 16 , Ne 20 . These are all nuclei containing an equal 
number of protons and neutrons, and in fact they are all polymers of He 4 . 
It is possible to say, therefore, that the forces between nucleons become 
saturated, like the valence bonds between atoms. The unit He 4 , two protons 
and two neutrons, appears to be one of exceptional stability. The nuclear 
shell structure is also clearly indicated in the packing-fraction vs. mass number 



238 



NUCLEAR CHEMISTRY AND PHYSICS 



[Chap. 9 



curves of Fig. 8.1 1 . The lower the packing fraction, the greater is the binding 
energy per nucleon. 

Another viewpoint is to consider that there are certain allowed energy 
levels in the nucleus. Each level can hold either two neutrons or two protons. 6 
The upper proton levels become raised in energy owing to the coulombic 



GY PER NUCLEON - MEV 

T) ->l 00 Q 


r 










\ 






FISSIO 
ENERG 


N ^ 

Y 




f 










\ 


1 










\ 


\ 












o: ~ 

UJ 

z 

UJ 

o 
z 5 


\ 












z 

CD 

4 













50 



200 



250 



100 150 

MASS NUMBER 

Fig. 9.4. Binding energy per nucleon as a function of atomic mass 
number. 

repulsion. Of all the stable nuclei, 152 have both n and Z even; 52 have Z 
odd, n even; 55, Z even, n odd; and only 4 have both n and Z odd. The four 
odd-odd nuclei are H 2 , Li 6 , B 10 , N 14 . Not only are the even-even nuclei the 
most frequent, they also usually have the greatest relative abundance. It can 
be concluded that filled nuclear energy levels confer exceptional stability. 

8. Neutrons and nuclei. Since the neutron is an uncharged particle, it is 
not repelled as it approaches a nucleus, even if its energy is very low. We 
often distinguish fast neutrons, with a kinetic energy of > 100 ev, and slow 
neutrons, with energies from 0.01 to 10 ev. If the energies have the same 
magnitude as those of ordinary gas molecules (kT), the neutrons are called 
thermal neutrons. At 300K, kT --= 0.026 ev. 

The interaction of a neutron and a nucleus can be represented by the 
intermediate formation of a compound nucleus which may then react in 
several ways. If the neutron is released again, with the reformation of the 
original nucleus, the process is called scattering. If the neutron is retained 
for some time, although there may be a subsequent decomposition of the 
compound nucleus into new products, the process is called capture or 
absorption . 

* See Section 10-25 and discussion of nuclear spin on p. 247. 



Sec. 8] 



NUCLEAR CHEMISTRY AND PHYSICS 



239 



A quantitative description of the interaction between a nucleus and a 
neutron is given in terms of the effective nuclear cross section, a. Consider 
a beam of neutrons in which the neutron flux is n per cnv 2 per sec. If the 
beam passes through matter in which there are c nuclei of a given kind per 
cc, the number of neutrons intercepted per sec in a thickness fix is given by 



- dn nac dx 



(9.7) 



An initial flux of n is therefore reduced after a distance .v to n x n c cax . 
The scattering cross section a s is distinguished from the absorption cross 
section cr rt , and a a s \ a (l . Nuclear cross sections are generally of the order 
10~ 24 cm 2 , and the whimsical physicists have called this unit the barn. 



500 



CD 

J 100 



o 

UJ 



o 
o: 
o 

j 
< 
i- 
o 



50 



10 




I 



500 



Fig. 9.5. 



5 10 5O 100 

NEUTRON ENERGY-6V 

Nuclear cross section of silver. [From Rainwater, Havens, Wu, 
and Dunning, Phys. Rev. 71, 65 (1947),] 



The cross sections depend on the kinetic energy of the neutrons and may 
be quite different in the low- and high-velocity ranges. The dependence of a 
on energy yields important information about energy levels in the nucleus, 
for when the neutron energy is very close to a nuclear energy level, a 
"resonance" occurs that greatly facilitates capture of the neutron, and 
hence greatly increases the value of a u . For example, for thermal neutrons, 
a^H 1 ) = 0.31 barn, a^H 2 ) - 0.00065 barn. Both ^H 1 and X H 2 have high 
scattering cross sections, and are therefore effective in slowing fast neutrons, 
but many of the thermal neutrons produced would be lost by capture to 
jH 1 dH 1 + Q/2 1 -> 1 H 2 ). It is for this reason that heavy water is a much more 
efficient neutron moderator than light water. In H 2 O a thermal neutron 



240 



NUCLEAR CHEMISTRY AND PHYSICS 



[Chap. 9 



would have, on the average, 150 collisions before capture; in D 2 O, 10 4 ; in 
pure graphite, 10 3 . 

A particularly important scattering cross section is that of cadmium. 
The cadmium nucleus has a resonance level in the thermal neutron region, 
leading to the tremendously high a = 7500 at 0.17ev. Thus a few milli- 
meters of cadmium sheet is practically opaque to thermal neutrons. 

The cross section for silver is shown in Fig. 9.5 as a function of neutron 
energy. The peaks in the curve correspond to definite neutron energy levels 
in the nucleus. The task of the nuclear physicist is to explain these levels, as 
the extranuclear energy levels of the electrons have been explained by 
the Bohr theory and quantum mechanics. (See Chapter 10.) 

9. Nuclear reactions. The different types of nuclear reactions are con- 
veniently designated by an abbreviated notation that shows the reactant 
particle and the particle emitted. Thus an (n, p) reaction is one in which 
a neutron reacts with a nucleus to yield a new nucleus and a proton, e.g., 
7 N 14 + X -* 6 C 14 + iH 1 would be written 7 N 14 (/i,/7) 6 C 14 . 

In Table 9.2 the various nuclear reaction types are summarized. The 

TABLE 9.2 
TYPES OF NUCLEAR REACTIONS 



Reaction 
Type 



n capture 
np 

no. 

n,2n 
p capture 
pn 

pen 



Normal Mass 
Change 



Slightly + 

si 4 light clem, 
si heavy 

Very - 



si 4- light elem. 
heavy 



r*~ 
OH 


~' j 

si light elem. 
4 heavy 


ap 


si 4- except -for 
light clem. 


dp 


Always + 


dn 


Always + 


* 


Always 4- 


yn 


Always 


yp 


Always - 



Dependence 
on Energy of 
Projectile 


Yield 


Type of 
Radio- 
activity 
Usually 
Produced 


Example 


Resonance 


100 per cent 


ft- 


Ag 107 + ! = Ag 108 


Smooth 


High for light 
elements 


ft- 


N 14 f n 1 - C 14 4- H l 


Smooth 


High for light 
elements 


fi- 


Mg* -f n 1 - Ne" 4- He* 


Smooth 


Low 


P 


P" 4- n 1 = P" + 2n 


Resonance 


High 


fi+ 


C 13 4 H 1 - N 18 


Threshold then 
smooth 


High 


^ 


Cu" 4- H 1 = Zn" f- n 1 


Smooth 


High 


Stable 


F" 4- H 1 = O 16 4- He* 


Smooth 


Low 





Be* 4- H 1 = Be 8 4- H 


Smooth 


High for light 
elements 


P 


C 11 4- He* - O 18 + n 1 


Smooth 


High for light 
elements 


Stable 


N 14 4- He 4 = 0" 4- H 1 


Smooth 


High for light 
elements 


0- 


Co" 4- H - Co" -f H 1 


Smooth 


High for light 
elements 


fi+ 


C 11 4- H 1 - N 18 4- H 1 


Smooth 


High for light 
elements 


Stable 


O" 4- H 1 = N 14 4- He* 


Sharp threshold 


Low 


f 


Be* 4- y = Be' + /i l 


Sharp threshold 


Low 





H* 4- Y - H 1 4 /> 



Sec. 10] NUCLEAR CHEMISTRY AND PHYSICS 241 

second column gives the normal rest-mass change for the reaction. A positive 
mass change is equivalent to an endothermic reaction, a negative mass 
change to an exothermic reaction. The next column indicates how the yield 
depends on the energy of the bombarding particle. In most cases there is a 
smooth increase in yield with increasing energy, but for capture processes 
there is a marked resonance effect. 

10. Nuclear fission. Perusal of the binding energy curve in Fig. 9.4 reveals 
that a large number of highly exothermic nuclear reactions are possible, 
since the heavy nuclei toward the end of the periodic table are all unstable 
relative to the nuclei lying around the maximum of the curve. 

In the January 1939 number of Naturwissenschaften, Otto Hahn and 
S. Strassman reported that when the uranium nucleus is bombarded with 
neutrons it may split into fragments, one of which they identified as an 
isotope of barium. About 200 mev of energy is released at each fission. 

It was immediately realized that secondary neutrons would very possibly 
be emitted as a result of uranium fission, making a chain reaction possible. 
The likelihood of this may be seen as follows: Consider the fission of a 
92 U 235 nucleus to yield, typically, a 56 Ba 139 as one of the observed disintegra- 
tion products. If balance is to be achieved between the numbers of protons 
and neutrons before and after fission, the other product would have to be 
36 Kr 98 . This product would be far heavier than any previously known krypton 
isotope, the heaviest of which was 36 Kr 87 , a ft" emitter of 4 hours half-life. 
Now the hypothetical 36 Kr 96 can get back to the proton-neutron curve of 
Fig. 9.3 by a series of ft" emissions, and in fact a large number of new ft" 
emitters have been identified among the fission products. The same result 
can be achieved, however, if a number of neutrons are set free in the fission 
process. Actually, both processes occur. 

The fission process usually consists, therefore, of a disintegration of 
uranium into two lighter nuclei, one of mass number from 82 to 100, and 
the other from 128 to 150, plus a number, perhaps about three, of rapidly 
moving neutrons. In only about one case in a thousand does symmetrical 
fission into two nuclei of approximately equal mass occur. 

To determine which isotope of uranium is principally responsible for 
fission, A. O. Nier and his coworkers separated small samples of U 235 
(0.7 per cent abundance) and U 238 (99.3 per cent) with a mass spectrometer. 
It was found that U 235 undergoes fission even when it captures a slow 
thermal neutron, but U 238 is split only by fast neutrons with energies greater 
than 1 mev. As usual, the capture cross section for slow neutrons is much 
greater than that for fast neutrons, so that U 235 fission is a much more likely 
process than that of U 238 . The process of fission can be visualized by con- 
sidering the nucleus as a drop of liquid. When a neutron hits it, oscillations 
are set up. The positive charges of the protons acquire an unsymmetrical 
distribution, and the resulting repulsion can lead to splitting of the nuclear 
drop. Since U 235 contains an odd number of neutrons, when it gains a 



242 



NUCLEAR CHEMISTRY AND PHYSICS 



[Chap. 9 



neutron considerable energy is set free. This kinetic energy starts the dis- 
turbance within the nucleus that leads to fission. The isotope U 238 already 
contains an even number of neutrons, and the capture process is not so 
markedly exothermic. Therefore the neutron must be a fast one, bringing 
considerable kinetic energy into the nucleus, in order to initiate fission. 
Fission of other heavy elements, such as lead, has been produced by bom- 
bardment with 200 mev deuterons produced by the Berkeley cyclotron. 
Fission can also be induced by y rays with energies greater than about 5 mev 
(photofissiori). 

In Fig. 9.6 are shown the mass distributions of the fission products in 
three different cases that have been carefully studied. When fission is produced 



uj 6 



u. (/) 
O < 




50 



IOO 
MASS NUMBER 



ISO 



Fig. 9.6. Mass distribution of products in three different fission reactions. The 
energies of the particles initiating the fission are: n, thermal; a, 38 mev; rf, 200 mev. 
Note how the distribution becomes more symmetrical as the energy of the incident 
particle increases. (From P. Morrison, "A Survey of Nuclear Reactions" in Experi- 
mental Nuclear Physics, ed. E. Segre, Wiley, 1953.) 

by highly energetic particles, the distribution of masses is quite symmetrical, 
and the most probable split is one that yields two nuclei of equal mass. This 
is the result that would be expected from the liquid-drop model. The un- 
symmetrical splitting that follows capture of slower particles has not yet 
received a satisfactory theoretical explanation, but it is undoubtedly related 
to the detailed shell structure inside the nucleus. 

A nuclear reaction of great interest is spontaneous fission, discovered in 
1940 by Flerov and Petrzhak in the U.S.S.R. It cannot be attributed to cosmic 
radiation or to any other known external cause, and it must be considered to 



Sec. 11] NUCLEAR CHEMISTRY AND PHYSICS 243 

be a new type of natural radioactivity. For example, when about 6 g of Th 232 
were observed for 1000 hr, 178 spontaneous fissions were detected. Sponta- 
neous fission is usually a very rare reaction, but it becomes much more 
frequent in some of the transuranium elements. 

11. The transuranium elements. In 1940, E. McMillan and P. H. Abelson 7 
found that when U 238 is irradiated with neutrons, a resonance capture can 
occur that leads eventually to the formation of two new transuranium 
elements. 

U 238 f" 1 - 



92 
92 U 239 23inin -> 93 Np 239 -f 



The 94 Pu 239 is a weakly radioactive a emitter (r 2.4 x 10 4 years). Its 
most important property is that, like U 235 , it undergoes fission by slow 
neutrons. 

It was shown by G. T. Seaborg 8 and his coworkers that bombardment 
of U 238 with a particles leads by an (a, n) reaction to Pu 241 . This is a ft emitter 
and decays to give 95 Am 241 , an isotope of americium which is a-radioactive 
with 500 years half-life. By 1954, the last of the transuranium elements to have 
been prepared were curium (96), berkelium (97), californium (98), and 
elements (99) and (100). Curium can be made by an (a, ri) reaction on Pu 239 : 

94 Pu 239 f 2 He 4 ^ ^ + 96 Cm 242 

The preparation of new examples of the transuranium elements has been 
facilitated by the technique of using heavy ions accelerated in the cyclotron. 
Thus high energy beams of carbon ions, ( 6 C 12 ) 6 ! , can increase the atomic 
number of a target nucleus by six units in one step. For example, isotopes of 
californium have been synthesized as follows: 9 



92 U 238 + 6 C 12 -> 98 Cf 244 + 6/1 
Element (99) was prepared by: 

92 U 238 + 7 N 14 -> 99 X 247 + 5/i 

12. Nuclear chain reactions. Since absorption of one neutron can initiate 
fission, and more than one neutron is produced at each fission, a branching 
chain can occur in a mass of fissionable material. The rate of escape of 
neutrons from a mass of U 235 , for example, depends on the area of the 
mass, whereas the rate of production of neutrons depends on the volume. 
As the volume of the mass is increased, therefore, a critical point is finally 
reached at which neutrons are being produced more rapidly than they are 
being lost. 

7 Phys. Rev., 57, 1185 (1940). 

8 Science, 104, 379 (1946); Chem. Eng. News, 25, 358 (1947). 

9 A. Ghiorso, S. G. Thompson, K. Street, and G. T. Seaborg, Phys. Rev. 81, 1954 (1951). 



244 



NUCLEAR CHEMISTRY AND PHYSICS 



[Chap. 9 



If two masses of U 235 of subcritical mass are suddenly brought together 
a nuclear explosion can take place. 

For the continuous production of power, a nuclear pile is used. The 
fissionable material is mixed with a moderator such as graphite or heavy 
water, to slow down the neutrons. Control of the rate of fission is effected 
by introducing rods of a material such as cadmium, which absorbs the 
thermal neutrons. The depth to which the cadmium rods are pushed into 
the pile controls the rate of fission. 

The pile also serves as a source of intense beams of neutrons for research 
purposes. As shown in Fig. 9.7, a diagram of the Brookhaven pile, these 



REMOVABLE PLUG 



ION CHAMBER FOR 
PILE CONTROL 




PNEUMATIC TUBE 
RABBIT" 



Fig. 9.7. Diagrammatic sketch of the Brookhaven pile showing the features of 
importance for pile neutron research. (From D. J. Hughes, Pile Neutron Research, 
Addison-Wesley, 1953.) 

beams can be either fast neutrons from the center of the pile, or thermal 
neutrons drawn out through a layer of moderator. 

13. Energy production by the stars. The realization of the immense 
quantities of energy that are released in exothermic nuclear reactions has 
also provided an answer to one of the great problems of astrophysics the 
source of the energy of the stars. At the enormous temperatures prevailing 
in stellar interiors (e.g., around 10 million degrees in the case of our sun) the 
nuclei have been stripped of electrons and are moving with large kinetic- 
theory velocities. Thus the mean thermal kinetic energy of an a particle at 
room temperature is of the order of ^ ev, but at the temperature of the sun 
it has become 10 4 ev. In other words, at stellar temperatures many of the 
nuclei have attained energies comparable with those of the high-velocity 
particles produced on earth by means of the cyclotron and similar devices. 

Nuclei with these high energies will be able to overcome the strong 
electrostatic repulsion between their positive charges and approach one 



Sec. 14] NUCLEAR CHEMISTRY AND PHYSICS 245 

another sufficiently closely to initiate various nuclear reactions. It is these 
so-called thermonuclear reactions that account for the energy production of 
the stars. 

In 1938, Carl von Weizsacker and Hans Bethe independently proposed 
a most ingenious mechanism for stellar-energy production. This is a cycle 
proceeding as follows: 

C 12 + H 1 ->N 13 -\- hv 

N 13 -C 13 I- <?- 

C 13 f H 1 -^ N 14 |- hv 

N u I H 1 ^O ir M hv 

O 15 ^N 15 4-e'- 

N 15 -h H 1 >C 12 -|- He 4 

The net result is the conversion of four H nuclei into one He nucleus through 
the mediation of C 12 and N 14 as "catalysts" for the nuclear reaction; 30mev 
are liberated in each cycle. This carbon cycle appears to be the principal 
source of energy in very hot stars (T> 5 x 10 s K). 

The energy of somewhat cooler stars, like our sun (T ~ 10" K), appears 
to be generated by the proton-proton cycle : 

iH 1 + t H l -- jH- | H | 0.42 mev 
^ + X H 2 -- 2 He 3 4- y f 5.5 mev 
2 He 3 + 2 He 3 - 2 He 4 f 2 1 H 1 f 12.8 mev 

The net result is the conversion of 4 protons to one helium nucleus, with the 
liberation of 24.6 mev plus the annihilation energy of the positron. 

Gamow has estimated 10 that reactions between hydrogen nuclei ^H 1 -| 
t H 2 -> 2 He 3 4- y; 2 jH 2 -> 2 He 4 -1 y) would have an appreciable rate at 
temperatures below 10 6 degrees; reactions of protons with lithium nuclei 
( X H 1 -f 3 Li 6 -^ 2 He 4 -f 2 He 3 ; X H 1 f 3 Li 7 -> 2 2 He 4 ) require about 6 x 10 6 
degrees; reactions such as jH 1 + 5 B 10 - -> 6 C X1 h y require about 10 7 degrees. 

The temperatures attainable by means of uranium or plutonium fission 
are high enough to initiate thermonuclear reactions of the lighter elements. 
The fission reaction acts as a "match" to start the fusion reactions. Easiest 
of all to "ignite" should be mixtures containing tritium, the hydrogen isotope 
of mass 3. 

X H 3 + jH 2 -- 2 He 4 4 O n l f 17.6 mev (y) 
!H 3 f 1 H 1 - 2 He 4 + 19.6 mev (y) 

The tritium can be prepared by pile reactions such as 3 Li (J f A? 1 ^ 2 He 4 f 1 H 3 . 
The isotope Li 6 has an abundance of 7.52 atom per cent. 

14. Tracers. The variety of radioactive isotopes now available has made 
possible many applications in tracer experiments, in which a given type of 
atom can often be followed through a sequence of chemical or physical 

10 George Gamow, The Birth and Death of the Sun (frew York: Penguin, 1945), p. 128. 



246 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 

changes. Stable isotopes can also be used as tracers, but they are not so 
easily followed and are available for relatively few elements. Radioactive 
isotopes can be obtained from four principal sources: (1) natural radio- 
activity; (2) irradiation of stable elements with beams of ions or electrons 
obtained from accelerators such as cyclotrons, betatrons, etc.; (3) pile 
irradiation with neutrons; (4) fission products. In Table 9.3 are listed a few 
of the many available isotopes. 

TABLE 9.3 
ARTIFICIAL RADIOACTIVE ELEMENTS 



Nucleus Activity Half Life 



C 11 p+, y 


21 min 


c 14 p- 


5700 yr 


N 13 /?+, y 


9.9 min 


o 16 p 


125 sec 


Na 22 ft*, y 


3.0 yr 


Na 24 /?-, y 


14.8 hr 


P 32 j j- 


14.3 days 


S 35 jJ- 


87.1 days 


Ca 45 /?- 


152 days 


Fe 59 j3~, y 


46 days 


Co 60 0~, y 


5.3 yr 


Cu 64 ^ + , p~ 


12.8hr 



One of the earliest studies with radioactive tracers used radioactive lead 
to follow the diffusion of lead ions in solid metals and salts. For example, 
a thin coating of radiolead can be plated onto the surface of a sample of 
metallic lead. After this is maintained at constant temperature for a definite 
time, thin slices are cut off and their radioactivity measured with a Geiger 
counter. The self-diffusion constant of Pb in the metal can readily be cal- 
culated from the observed distribution of activity. Many such diffusion 
studies have now been made in metals and in solid compounds. The results 
obtained are of fundamental importance in theories of the nature and prop- 
erties of the solid state. Diffusion in liquids, as well as the permeability of 
.natural and synthetic membranes, can also be conveniently followed by 
radioactive tracer methods. 

The solubility of water in pure hydrocarbons is so low that it is scarcely 
measurable by ordinary methods. If water containing radioactive hydrogen, 
or tritium, ^^ a ft~~ emitter of 12 years half life, is used, even minute amounts 
dissolved in the hydrocarbons are easily measured. 11 

A useful tracer method is isotopic dilution analysis. An example is the 
determination of amino acids in the products of protein hydrolysis. The 
conventional method would require the complete isolation of each amino 
acid in pure form. Suppose, however, a known amount of an amino acid 

11 C. Black, G. G. Joris, and H. S. Taylor, /. Chem. Phys., 16, 537 (1948). 



Sec. 15] NUCLEAR CHEMISTRY AND PHYSICS 247 

labeled with deuterium or carbon-14 is added to the hydrolysate. After 
thorough mixing, a small amount of the given acid is isolated and its activity 
measured. From the decrease in activity, the total concentration of the acid 
in the hydrolysate can be calculated. 

Tracers are used to elucidate reaction mechanisms. One interesting 
problem was the mechanism of ester hydrolysis. Oxygen does not have a 
radioactive isotope of long enough half life to be a useful tracer, but the 
stable O 18 can be used. By using water enriched with heavy oxygen (O*) the 
reaction was $hown to proceed as follows: 

O O 

R C/ + HO*H -, R-C< I R'OH 

X OR' X O*H 

The tagged oxygen appeared only in the acid, showing that the OR group is 
substituted by O*H in the hydrolysis. 12 

Radioactive isotopes of C, Na, S, P, etc., are of great use in investigations 
of metabolism. They supplement the stable isotopes of H, N, and O. For 
^example it has been found that labeled phosphorus tends to accumulate 
preferentially in rapidly metabolizing tissues. This has led to its trial in cancer 
therapy. The results in this case have not been particularly encouraging, but 
it may be possible to find metabolites or dyes that are specifically concen- 
trated in tumor tissues, and then to render these compounds radioactive by 
inclusion of appropriate isotopic atoms. 13 

15. Nuclear spin. In addition to its other properties, the nucleus may 
have an intrinsic angular momentum or spin. All elementary particles (i.e., 
neutrons, protons, and electrons) have a spin of one-half in units of h/27r. 
The spin of the electron will be considered in some detail in the next chapter. 
The spin of the elementary particles can be either plus or minus. If an axis is 
imagined passing through the particle, the sign corresponds to a clockwise or 
counterclockwise spin, although this picture is a very crude one. The spin of 
a nucleus is the algebraic sum of the spins of the protons and neutrons that 
it contains. 

The hydrogen nucleus, or proton, has a spin of one-half. If two hydrogen 
atoms are brought together to form H 2 , the nuclear spins can be either 
parallel ( 1f ) or antiparallel ( 11, ). Thus there are two nuclear spin isomers of 
H 2 . The molecule with parallel spins is called "orthohydrogen," the one 
with antiparallel spins is called "parahydrogen." Since spins almost never 
change their orientation spontaneously, these two isomers are quite stable. 
They have different heat capacities and different molecular spectra. Other 
molecules composed of two identical nuclei having nonzero spin behave 
similarly, but only in the cases of H 2 and D 2 are there marked differences in 
physical properties. 

12 M. Polanyi and A. L. Szabo, Trans. Farad. Soc., 30, 508 (1934). 

13 M. D. Kamen, Radioactive Tracers in Biology (New York: Academic, 1947). 



248 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 

PROBLEMS 

1. What is the Am in g per mole for the reaction H 2 -f t O 2 H 2 O for 
which A// - - 57.8 kcal per mole? 

2. From the atomic weights in Table 9.1, calculate the AE in kcal per 
mole for the following reactions: 

^ + o/;' - ^2, t H 2 f- o* 1 = ^ 1 H 1 f e - X, 2 t H 2 - 2 He 4 

3. Calculate the energies in (a) ev (b) kcal per mole of photons having 
wavelengths of 2.0 A, 1000 A, 6000 A, 1 mm, 1 m. 

4. To a hydrolysate from 10 g of protein is added 100 mg of pure 
CD 3 CHNH 2 COOH (deuterium-substituted alanine). After thorough mixing, 
100 mg of crystalline alanine is isolated which has a deuterium content of 
1.03 per cent by weight. Calculate the per cent alanine in the protein. 

5. A 10-g sample of iodobenzene is shaken with 100 ml of a 1 M KI solu- 
tion containing 2500 counts per min radio-iodine. The activity of the iodo- 
benzene layer at the end of 2 hours is 250 cpm. What per cent of the iodine 
atoms in the iodobenzene have exchanged with the iodide ions in solution? 

6. Calculate the mass of an electron accelerated through a potential of 
2 x 10 8 volts. What would the mass be if the relativity effect is ignored? 

7. Naturally occurring oxygen consists of 99.76 per cent O 16 , 0.04 per 
cent O 17 , and 0.20 per cent O 18 . Calculate the ratio of atomic weights on the 
physical scale to those on the chemical scale. 

8. The work function of a cesium surface is 1.81 volts. What is the 
longest wavelength of incident light that can eject a photoelectron from Cs? 

9. The 77-meson has a mass about 285 times that of the electron; the 
/t-meson has a mass about 215 times that of the electron. The 7r-meson 
decays into a //-meson plus a neutrino. Estimate A for the reaction in ev. 

10. Calculate the energy necessary to produce a pair of light mesons. 
This pair production has been accomplished with the 200-in. California 
cyclotron. 

11. The scattering cross section, a, of lead is 5 barns for fast neutrons. 
How great a thickness of lead is required to reduce the intensity of a neutron 
beam to 5 per cent of its initial value? How great a thickness of magnesium 
with a = 2 barns? 

12. According to W. F. Libby [Science, 109, 22V (1949)] it is probable 
that radioactive carbon- 14 (r = 5720 years) is produced in the upper atmo- 
sphere by the action of cosmic-ray neutrons on N 14 , being thereby main- 
tained at an approximately constant concentration of 12.5 cpm per g of 
carbon. A sample of wood from an ancient Egyptian tomb gave an activity 
of 7.04 cpm per g C. Estimate the age of the wood. 

13. A normal male subject weighing 70.8 kg was injected with 5.09 ml 
of water containing tritium (9' x 10 9 cpm). Equilibrium with body water was 



Chap. 9] NUCLEAR CHEMISTRY AND PHYSICS 249 

reached after 3 hr when a 1-ml sample of plasma water from the subject had 
an activity of 1.8 x 10 5 cpm. Estimate the weight per cent of water in the 
human body. 

14. When 38 Sr 88 is bombarded with deuterons, 38 Sr 89 is formed. The cross 
section for the reaction is 0.1 barn. A SrSO 4 target 1.0 mm thick is exposed 
to a deuteron beam current of 100 microamperes. If scattering of deuterons 
is neglected, compute the number of Sr atoms transmuted in 1.0 hr. The 
Sr 88 is 82.6 per cent of Sr, and Sr 89 is a Remitter of 53-day half life. Compute 
the curies of Sr 89 produced. 

15. When 79 Au 197 (capture cross section a c =-- 10~ 22 cm 2 ) is irradiated with 
slow neutrons it is converted into 79 Au 198 (r --= 2.8 days). Show that in general 
the number of unstable nuclei present after irradiation for a time / is 

o^ (1 _ e ^ 

A 

Here n Q is the number of target atoms and <f> is the slow neutron flux. For the 
case in question, calculate the activity in microcuries of a 100-mg gold sample 
exposed to a neutron flux of 200/cm 2 sec for 2 days. 

16. The conventional unit of quantity of X radiation is the roentgen, r. 
It is the quantity of radiation that produces 1 esu of ions in 1 cc of air at 
STP (1 esu === 3.3 x 10~ 10 coulomb). If 32.5 ev are required to produce a 
single ion pair in air, calculate the energy absorbed in 1 liter of air per 
roentgen. 

17. Potassium-40 constitutes 0.012 per cent of natural K, and K is 0.35 
per cent of the weight of the body. K 40 emits /? and y rays and has r 
4.5 x 10 8 yr. Estimate the number of disintegrations per day of the K 40 in 
each gram of body tissue. 

18. The isotope 89 Ac 225 has r 10 days and emits an a with energy of 
5.80 mev. Calculate the power generation in watts per 100 mg of the isotope. 

REFERENCES 

BOOKS 

1. Baitsell, G. A. (editor), Science in Progress, vol. VI (New Haven: Yale 
Univ. Press, 1949). Articles by H. D. Smyth on Fission; J. A. Wheeler on 
Elementary Particles; E. O. Lawrence on High Energy Physics; G. T. 
Seaborg on Transuranium Elements. 

2. Bethe, H., Elementary Nuclear Theory (New York: Wiley, 1947). 

3. Friedlander, G., and J. W. Kennedy, Introduction to Radiochemistry (New 
York: Wiley, 1949). 

4. Gamow, G., and C. L. Critchfield, Theory of Atomic Nucleus and Nuclear 
Energy Sources (New York: Oxford, 1949). 

5. Goodman, C. (editor), The Science and Engineering of Nuclear Power 
(2 vols) (Boston: Addison-Wesley, 1947, 1949). 



250 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 

6. Halliday, D., Introductory Nuclear Physics (New York: Wiley, 1950). 

7. Hughes, D. J., Pile Neutron Research (Boston: Addison- Wesley, 1953). 

8. Lapp, R. E., and H. L. Andrews, Nuclear Radiation Physics, 2nd ed. 
(New York: Prentice-Hall, 1954). 

9. Libby, W. F., Radiocarbon Dating (Chicago: Univ. of Chicago Press, 
1952). 

ARTICLES 

1. Anderson, C. D., Science in Progress, 7, 236-249 (1951), 'The Elementary 
Particles of Physics." 

2. Curtan, S. C., Quart. Rev., 7, 1-18 (1953), "Geological Age by Means of 
Radioactivity/' 

3. Dunning, J. R., Science in Progress, 7, 291-355 (1951), "Atomic Structure 
and Energy." 

4. Hevesy, G. C., J. Chem. Soc., 1618-1639 (1951), "Radioactive Indicators 
in Biochemistry." 

5. Pryce, M. H. L., Rep. Prog. Phys., 17, 1-35 (1954), "Nuclear Shell 
Structure." 

6. Wilkinson, M. K., Am. J. Phys., 22, 263-76 ( 1 954), "Neutron Diffraction." 



CHAPTER 10 

Particles and Waves 



1. The dual nature of light. It has already been noted that in the history 
of light two different theories were alternately in fashion, one based on the 
particle model and the other on the wave model. At the present time both 
must be regarded with equal respect. In some experiments light displays 
notably corpuscular properties: the photoelectric and Compton effects can 
be explained only by means of light particles, or photons, having an energy 
s hv. In other experiments, which appear to be just as convincing, the 
wave nature of light is manifest: polarization and interference phenomena 
require an undulatory theory. 

This unwillingness of light to fit neatly into a single picture frame has 
been one of the most perplexing problems of natural philosophy. The situa- 
tion recalls the impasse created by the "null result" of the Michelson-Morley 
experiment. This result led Einstein to examine anew one of the most basic 
of physical concepts, the idea of the simultaneity of events in space and time. 
The consequence of his searching analysis was the scientific revolution 
expressed in the relativity theories. 

An equally fundamental enquiry has been necessitated by the develop- 
ments arising from the dual nature of light. These have finally required a 
re-examination of the meaning and limitations of physical measurement 
when applied to systems of atomic dimensions or smaller. The results of 
this analysis are as revolutionary as the relativity theory; they are embodied 
in what is called quantum theory or wave mechanics. Before discussing the 
significant experiments that led inexorably to the new theories, we shall 
review briefly the nature of vibratory and wave motions. 

2. Periodic and wave motion. The vibration of a simple harmonic oscilla- 
tor, discussed on page 190, is a good example of a motion that is periodic 
in time. The equation of motion (/= ma) is md 2 x/dt 2 --== KX. This is a 
simple linear differential equation. 1 It can be solved by first making the sub- 
stitution p dx/dt. Then d*x/dt* =-- dpjdt -= (dp/dx)(dx/dt) = p(dp/dx), and 
the equation becomes p(dp\dx) + (K/W)X 0. Integrating, p 2 +(t</m)x 2 = const. 

The integration constant can be evaluated from the fact that when the 
oscillator is at the extreme limit of its vibration, x = A, the kinetic energy 
is zero, and hence p 0. Thus the constant = (K/m)A 2 . Then 



1 See, for example, Gran vi lie et al., Calculus, p. 383.. 



252 PARTICLES AND WAVES [Chap. 10 



~~df 



V/,4 2 - * 2 

. , X IK 

sin" 1 - / t -\ const 

A *< m 



This integration constant can be evaluated from the initial condition that at 
/ =-- 0, A- ---=-- 0; therefore constant 0. 

The solution of the equation of motion of the simple harmonic oscillator 
is accordingly: 



x -= AsmJ-~t (10.1) 

If we set VK//W --- 2771-, this becomes 

x ^ A sin 2irvt (10.2) 

The simple harmonic vibration can be represented graphically by this 
sine function, as shown in Fig. 10.1. A cosine function would do just as 

well. The constant v is called the 
frequency of the motion; it is the 
number of vibrations in unit time. 
The reciprocal of the frequency, 
r -= ]/v, is called the period of the 
motion, the time required for a single 
vibration. Whenever t -- n(r/2), 
where n is an integer, the displace- 
ment x passes through zero. 
Fig. 10.1. Simple harmonic vibration. The quantity A, the maximum 

value of the displacement, is called 

the amplitude of the vibration. At the position x =- A, the oscillator reverses 
its direction of motion. At this point, therefore, the kinetic energy is zero, 
and all the energy is potential energy E p . At position x 0, all the energy is 
kinetic energy E k . Since the total energy, E = E p + E k , is always a constant, 
it must equal the potential energy at x -= A. On page 190 the potential 
energy of the oscillator was shown to be equal to } 2 KX 2 , so that the total 
energy is 

E - \KA 2 (10.3) 




The total energy is proportional to the square of the amplitude. This im- 
portant relation holds true for all periodic motions. 

The motion of a harmonic oscillator illustrates a displacement periodic 
with time, temporally periodic. If such an oscillator were immersed in a 
fluid medium it would set up a disturbance which would travel through the 



Sec. 3] PARTICLES AND WAVES 253 

medium. Such a disturbance would be not only temporally periodic but also 
spatially periodic. It would constitute what is called a wave. For example, a 
tuning fork vibrating in air sets up sound waves. An oscillating electric dipole 
sets up electromagnetic waves in space. 

Let us consider a simple harmonic wave moving in one dimension, x. If 
one takes an instantaneous "snapshot" of the wave, it will have the form of 
a sine or cosine function. This snapshot is the profile of the wave. If at a 
point x = the magnitude of the disturbance <f> equals 0, then at some 
further point x = A, the magnitude will again be zero, and so on at 2A, 
3A . . . A. This quantity A is called the wavelength. It is the measure of the 
wave's periodicity in space, just as the period T is the measure of its periodicity 
in time. The profile of the simple sine wave has the form: 

<-,4sin27r~ (10.4) 

X. 

Now consider the expression for the wave at some later time /. The idea 
of the velocity of the wave must then be introduced. If the disturbance is 
moving through the medium with a velocity c in the positive x direction, in 
a time t it will have moved a distance ct. The wave profile will have exactly 
the same form as before if the origin is shifted from ;c = to a new origin 
at x = ct. Referred to this moving origin, the wave profile always maintains 
the form of eq. (10.4). To refer the disturbance back to the stationary origin, 
it is necessary only to subtract the distance moved in time t from the value 
of x. Then the equation for the moving wave becomes 

f\ 
<f>^-Asm~(x-ct) (10.5) 

Note that the nature of the disturbance <f> need not be specified: in the case 
of a water wave it is the height of the undulation ; in the case of an electro- 
magnetic wave it is the strength of an electric or magnetic field. 

Now it is evident that c/A is simply the frequency: v = c/A. The number 
of wavelengths in unit distance is called the wave number, k I/A, so that 
eq. (10.5) can be written in the more convenient form: 

<f> = A sin 27r(kx - vt) (10.6) 

3. Stationary waves. In Fig. 10.2, two waves, fa and fa, are shown that 
have the same amplitude, wavelength, and frequency. They differ only in 
that fa has been displaced along the X axis relative to fa by a distance d/2,7rk. 
Thus they may be written 

fa = A sin 2ir(kx vt) 
fa = A sin [2ir(kx vt) + d] 
The quantity d is called the phase of fa relative to fa. 



254 



PARTICLES AND WAVES 



[Chap. 10 



When the displacement is exactly an integral number of wavelengths, 
the two waves are said to be in phase \ this occurs when d = 2?r, 4?r, or any 
even multiple of 77. When d --= 77, 3rr 9 or any odd multiple of n, the two waves 
are exactly out of phase. Interference phenomena are readily explained in 
terms of these phase relationships, for when two superimposed waves of 
equal amplitude are out of phase, the resultant disturbance is reduced to zero. 




Fig. 10.2. Waves differing in phase. 

The expression (10.6) is one solution of the general partial differential 
equation of wave motion, which governs all types of waves, from tidal waves 
to radio waves. In one dimension this equation is 



In three dimensions the equation becomes 



v 



a; 2 



(10.7) 



(10.8) 



The operator V 2 (del squared) is called the Laplacian. 

One important property of the wave equation is apparent upon inspec- 
tion. The disturbance $ and all its partial derivatives appear only in terms 
of the first degree and there are no other terms. This is therefore a linear 
homogeneous differential equation. 2 It can be verified by substitution that if 
fa and <^ 2 are any two solutions of such an equation, then a new solution can 
be written having the form 

i i i i / 1 r\ r>\ 

<p - fli9i ~r #2r2 (10.9) 

where a l and a 2 are arbitrary constants. This is an illustration of the principle 
of superposition. Any number of solutions can be added together in this way 
to obtain new solutions. This is essentially what is done when a complicated 
vibratory motion is broken down into its normal modes (page 191), or when 
a periodic function is represented by a Fourier series. 

An important application of the superposition principle is found in the 
addition of two waves of the form of eq. (10.6) that are exactly the same 

8 Granvilie, he. c//., pp. 372, 377. 



Sec. 4] PARTICLES AND WAVES 255 

except that they are going in opposite directions. Then the new solution 
will be 

</> -^ A sin 2rr(kx vt) + A sin 2v(kx } vt) 

or ^ 2 A sin 27r&jc cos 27rvt (10.10) 

. x \ y xv 
since sin x + sin j 2 sin cos 

This new wave, which does not move either forward or backward, is a 
stationary wave. The waves of the original type [eq. (10.6)] are called pro- 
gressive waves. It will be noted that in the stationary wave represented by 
eq. (10.10), the disturbance <f> always vanishes, irrespective of the value of /, 
for points at which sin 2-nkx = or x 0, iAr, $A% S/v . . . (n/2)k. These 
points are called nodes. The distance between successive nodes is Ik or A/2, 
one-half a wavelength. Midway between the nodes are the positions of 
maximum amplitude, or antinodes. 

Solutions of the one-dimensional type, which have just been discussed, 
will apply to the problem of a vibrating string in the idealized case in which 
there is no damping of the vibrations. In a string of infinite length one can 
picture the occurrence of progressive waves. Consider, however, as in Fig. 
10.2, a string having a certain finite length L. This limitation imposes certain 
boundary conditions on the permissible solutions of the wave equation. If the 
ends of the string are held fixed: at x and at x L, the displacement 
<f) must 0. Thus there must be an integral number of nodes between and 
L, so that the allowed wavelengths are restricted to those that obey the 
equation 

n Y L (10.11) 

where n is an integer. This occurrence of whole numbers is very typical of 
solutions of the wave equation under definite boundary conditions. In order 
to prevent destruction of the wave by interference, there must be an integral 
number of half wavelengths fitted within the boundary. This principle will 
be seen to have important consequences in quantum theory. 

4. Interference and diffraction. The interference of light waves can be 
visualized with the aid of the familiar construction of Huygens. Consider, 
for example, in (a) Fig. 10.3, an effectively plane wave front from a single 
source, incident upon a set of slits. The latter is the prototype of the well 
known diffraction grating. Each slit can now be regarded as a new light 
source from which there spreads a semicircular wave (or hemispherical in 
the three-dimensional case). If the wavelength of the radiation is A, a series 
of concentric semicircles of radii A, 2A, 3A . . . may be drawn with these 
sources as centers. Points on these circles represent the consecutive maxima 
in amplitude of the new wavelets. Now, following Huygens, the new resultant 
wave fronts are the curves or surfaces that are simultaneously tangent to the 



256 



PARTICLES AND WAVES 



[Chap. 10 




a cos oi 



secondary wavelets. These are called the "envelopes" of the wavelet curves 
and are shown in the illustration. 

The important result of this construction is that therfe are a number of 
possible envelopes. The one that moves straight ahead in the same direction 
as the original incident light is called the zero-order beam. On either side of 
this are first-, second-, third-, etc., order diffracted beams. The angles by 
which the diffracted beams deviate from the original direction evidently 
depend on the wavelength of the incident radiation. The longer the wave- 
length, the greater is the diffraction. This is, of course, the basis for the use 
of the diffraction grating in the measurement of the wavelength of radiation. 



l* f ORDER 



ORDER 



2 nd ORDER 



(a) (b) 

Fig. 10.3. Diffraction: (a) Huygens* construction; (b) path difference. 

The condition for formation of a diffracted beam can be derived from a 
consideration of (b) Fig. 10.3, where attention is focused on two adjacent slits. 
If the two diffracted rays are to reinforce each other they must be in phase, 
otherwise the resultant amplitude will be cut down by interference. The 
condition for reinforcement is therefore that the difference in path for the two 
rays must be an integral number of wavelengths. If a is the angle of diffraction 
and a the separation of the slits, this path difference is a cos a and the 
condition becomes 

a cos a = /a (10.12) 

where h is an integer. 

This equation applies to a linear set of slits. For a two-dimensional plane 
grating, there are two similar equations to be satisfied. For the case of light 
incident normal to the grating, 

a cos a = AA 
b cos ft = A 

It will be noted that the diffraction is appreciable only when the spacings 




Sec. 5] PARTICLES AND WAVES 257 

of the grating aj^rftt very much larger than the wavelength of the incident 
light. In order /to obtain diffraction effects with X rays, for example, the 
spacings should be of the order of a few Angstrom units. 3 

Max von Laue, in 1912, realized that the interatomic spacings in crystals 
were probably of the order of magnitude of the wavelengths of X rays. 
Crystal structures should therefore serve as three-dimensional diffraction 
gratings for X rays. This prediction was immediately verified in the critical 
experiment of Friedrich, Knipping, and Laue. A typical X-ray diffraction 
picture is shown in Fig. 13.7 on page 375. The far-reaching consequences of 
Laue's discovery will be considered in some detail in a later chapter. It is 
mentioned here as a demonstration of the wave properties of X rays. 

5. Black-body radiation. The first definite failure of the old wave theory 
of light was not found in the photoelectric effect, a particularly clear-cut case, 
but in the study of black-body radiation. All objects are continually absorbing 
and emitting radiation. Their properties as absorbers or emitters may be 
extremely diverse. Thus a pane of window glass will not absorb much of 
the radiation of visible light but will absorb most of the ultraviolet. A sheet 
of metal will absorb both the visible and the ultraviolet but may be reasonably 
transparent to X rays. 

In order for a body to be in equilibrium with its environment, the radia- 
tion it is emitting must be equivalent (in wavelength and amount) to the 
radiation it is absorbing. It is possible to conceive of objects that "are perfect 
absorbers of radiation, the so-called ideal black bodies. Actually, no sub- 
stances approach very closely to this ideal over an extended range of wave- 
lengths. The best laboratory approximation to an ideal black body is not a 
substance at all, but a cavity. 

This cavity, or hohlraum, is constructed with excellently insulating walls, 
in one of which a small orifice is made. When the cavity is heated, the radia- 
tion from the orifice will be a good sample of the equilibrium radiation within 
the heated enclosure, which is practically ideal black-body radiation. 

There is a definite analogy between the behavior of the radiation within 
such a hohlraum and that of gas molecules in a box. Both the molecules and 
the radiation are characterized by a density and both exert pressure on the 
confining walls. One difference is that the gas density is a function of the 
volume and the temperature, whereas the radiation density is a function of 
temperature alone. Analogous to the various velocities distributed among 
the gas molecules are the various frequencies distributed among the oscilla- 
tions that comprise the radiation. 

At any given temperature there is a characteristic distribution of the gas 
velocities given by Maxwell's equation. The corresponding problem of the 
spectral distribution of black-body radiation, that is, the fraction of the 

8 It is also possible to use larger spacings and work with extremely small angles of 
incidence. The complete equation, corresponding to eq. (10.12), for incidence at an angle 
oto, is 0(cos a cos ao) = M. 



258 



PARTICLES AND WAVES 



[Chap. 10 



total energy radiated that is within each range of wavelength, was first 
explored experimentally (1877-1900) by O. Lummer and E. Pringsheim. 
Some of their results are shown in Fig. 10.4. These curves indeed have a 
marked resemblance to those of the Maxwell distribution law. At high tem- 
peratures the position of the maximum is shifted to shorter wavelengths 
an iron rod glows first dull red, then orange, then white as its temperature 
is raised and higher frequencies become appreciable in the radiation. 



140 




12345 
WAVE LENGTH, MICRONS 

Fig. 10.4. Data of Lummer and Pringsheim on spectral distribution 
of radiation from a black body at three different temperatures. 

When these data of Lummer and Pringsheim appeared, attempts were 
made to explain them theoretically by arguments based on the wave theory 
of light and the principle of equipartition of energy. Without going into the 
details of these efforts, which were uniformly unsuccessful, it is possible to 
see why they were foredoomed to failure. 

According to the principle of the equipartition of energy, an oscillator in 
thermal equilibrium with its environment should have an average energy 
equal to kT, \kT for its kinetic energy and \kT for its potential energy, 
where k is the Boltzmann constant. This classical theory states that the 
average energy depends in no way on the frequency of the oscillator. In a 
system containing 100 oscillators, 20 with a frequency v l of 10 10 cycles per 
sec and 80 with v 2 =-- 10 14 cycles per sec, the equipartition principle predicts 
that 20 per cent of the energy shall be in the low-frequency oscillators and 
80 per cent in the. high-frequency oscillators. 

The radiation within a hohlraum can be considered to be made up of 



Sec. 6] PARTICLES AND WAVES 259 

standing waves of various frequencies. The problem of the energy distribu- 
tion over the various frequencies (intensity / vs. v) apparently reduces to 
the determination of the number of allowed vibrations in any range of 
frequencies. 

The possible high-frequency vibrations greatly outnumber the low- 
frequency ones. The one-dimensional case of the vibrating string can be 
used to illustrate this fact. We have seen in eq. (10.11) that in a string of 
length L, standing waves can occur only for certain values of the wavelength 
given by X = 2L/n. It follows that the number of allowed wavelengths from 
any given value A to the maximum 2L is equal to n -= 2L/A. We wish to find 
the additional number of allowed wavelengths that arise if the limiting wave- 
length value is decreased from X to A dX. The result is obtained by 
differentiation 4 as 

dn = ^dl (10.13) 

A 

This indicates that the number of allowed vibrations in a region from A to 
A dk increases rapidly as the wavelength decreases (or the frequency 
increases). There are many more high-frequency than low-frequency vibra- 
tions. The calculation in three dimensions is more involved 5 but it yields 
essentially the same answer. For the distribution of standing waves in an 
enclosure of volume V, the proper formula is dn (Sir F/A 4 )c/A, or 

dn^%7T-v*dv (10.14) 

c 3 

Since there are many more permissible high frequencies than low fre- 
quencies, and since by the equipartition principle all frequencies have the 
same average energy, it follows that the intensity / of black-body radiation 
should rise continuously with increasing frequency. This conclusion follows 
inescapably from classical Newtonian mechanics, yet it is in complete dis- 
agreement with the experimental data of Lummer and Pringsheim, which 
show that the intensity of the radiation rises to a maximum and then falls 
off sharply with increasing frequency. This abject failure of classical mechani- 
cal principles when applied to radiation was viewed with unconcealed dismay 
by the physicists of the time. They called it the "ultraviolet catastrophe." 

6. Planck's distribution law. The man who first dared to discard classical 
mechanics and the equipartition of energy was Max Planck. Taking this 
step in 1900, he was able to derive a new distribution law, which explained 
the experimental data on black-body radiation. 

Newtonian mechanics (and relativity mechanics too) was founded upon 
the ancient maxim that natura non facit saltum ("nature does not make a 
jump"). Thus an oscillator could be presumed to take up energy continuously 

4 It is assumed that in a region of large L and small A, n is so large that it can be con- 
sidered to be a continuous function of A. 

5 R. H. Fowler, Statistical Mechanics (London: Cambridge, 1936), p. 112. 



260 PARTICLES AND WAVES [Chap. 10 

in arbitrarily small increments. Although matter was believed to be atomic 
in its constitution, energy was assumed to be strictly continuous. 

Planck discarded this precept and suggested that an oscillator, for 
example, could acquire energy only in discrete units, called quanta. The 
quantum theory began therefore as an atomic theory of energy. The magni- 
tude of the quantum or atom of energy was not fixed, however, but depended 
on the oscillator frequency according to 

s - hv (10.15) 

Planck's constant h has the dimensions of energy times time (e.g., 6.62 x 
10-27 er g sec ^ a q uan tity known as action. 

According to this hypothesis it is easy to see qualitatively why the in- 
tensity of black-body radiation always falls off at high frequencies. At fre- 
quencies such that hv ^> kT, the size of the quantum becomes much larger 
than the mean kinetic energy of the atoms comprising the radiator. The 
larger the quantum, the smaller is the chance of an oscillator having the 
necessary energy, since this chance depends on an e~ h ' f * T Boltzmann factor. 
Thus oscillators of high frequency have a mean energy considerably less 
than the kT of the classical case. 

Consider a collection of N oscillators having a fundamental vibration 
frequency v. If these can take up energy only in increments of hv, the allowed 
energies are 0, hv, 2hv, 3hv, etc. Now according to the Boltzmann formula, 
eq. (7.31), if N Q is the number of systems in the lowest energy state, the 
number N { having an energy e { above this ground state is given by 

AT, = TV/** 1 (10.16) 

In the collection of oscillators, for example, 

NI = N e~ hv l kT 

N* = Ne-~ 2hvlkT 

N 3 = N Q e~* hv l kT 
The total number of oscillators in all energy states is therefore 



-tfo 2 

t-O 

The total energy of all the oscillators equals the energy of each level times 
the number in that level. 

E = # 



Sec. 7] PARTICLES AND WAVES 261 

The average energy of an oscillator is therefore 

. E _ 

e ~~ ~N ~ 



According to this expression, the mean energy of an oscillator whose 
fundamental frequency is v approaches the classical value of kT when hv 
becomes much less than kT. 1 Using this equation in place of the classical 
equipartition of energy, Planck derived an energy-distribution formula in 
excellent agreement with the experimental data for black-body radiation. 
The energy density E(v) dv is simply the number of oscillations per unit 
volume between v and v + dv [eq. (10.14)] times the average energy of an 
oscillation [eq. (10.17)]. Hence Planck's Law is 

STT/Z r 3 dv 
E(v) dv - - hv/kT -- (10.18) 

7. Atomic spectra. Planck's quantum theory of energy appeared in 1901. 
Strong confirmation was provided by the theory of the photoelectric effect 
proposed by Einstein in 1905. Another most important application of the 
theory was soon made, in the study of atomic spectra. 

An incandescent gas emits a spectrum composed of lines at definite wave- 
lengths. Similarly if white light is transmitted through a gas, certain wave- 
lengths are absorbed, causing a pattern of dark lines on a bright background 
when the emergent light is analyzed with a spectrograph. These emission 
and absorption spectra must be characteristic of certain preferred frequencies 
in the gaseous atoms and molecules. A sharply defined line spectrum is 
typical of atoms. Molecules give rise to spectra made up of bands, which 
can often be analyzed further into closely packed lines. For example, the 
spectra of atomic hydrogen (H) and of molecular nitrogen (N 2 ) are shown 
in Fig. 10.5a and b. 

In 1885, J. J. Balmer discovered a regular relationship between the fre- 
quencies of the atomic hydrogen lines in the visible region of the spectrum. 
The wave numbers v' are given by 



with Wj = 3, 4, 5 . . . etc. The constant ^ is called the Rydberg constant, 
and has the value 109,677.581 cm" 1 . It is one of the most accurately known 
physical constants. 

6 In eq. (10.17) let e~ x = y, then the denominator S/ = 1 + y 4- y* -f - . . . = 
1/(1 ~ y\ (y< 1). The numerator, Zi>' = y(\ + 2v + 3/ +...)= yl(\ - y) 2 , (y < 1) 
so that eq. (10.17) becomes hvy/(l - y) = Ar/fr*"/** 1 - 1). 

7 When hv < kT, e^/** 1 1 -f (hv/kT). 



262 



PARTICLES AND WAVES 



[Chap. 10 



Other hydrogen series were discovered later, which obeyed the more 
general formula, 



JL.-JL) 

!/ 



(10.19) 



Lyman found the series with 2 = 1 in the far ultraviolet, and others were 
found in the infrared by Paschen (/ 2 = 3), Bracket! (n a 4), and Pfund 




H, 






A. 



Fig. 10.5a. Spectra of atomic hydrogen. (From Herzberg, Atomic Spectra and 
Atomic Structure, Dover, 1944.) 




fcjJK-ftfVl^ 

MIBffP 8 ''' vCTP 






Fig. 10.5b. Spectra of molecular nitrogen. (From Harrison, Lord, and 
Loofbourow, Practical Spectroscopy, Prentice-Hall, 1948.) 

(/? 2 = 5). A great number of similar series have been observed in the atomic 
spectra of other elements. 

8. The Bohr theory. These characteristic atomic line spectra could not be 
explained on the basis of the Rutherford atom. According to this model, 
electrons are revolving around a positively charged nucleus, the coulombic 
attraction balancing the force due to the centripetal acceleration. The classical 
theory of electromagnetic radiation demands that an accelerated electric 
charge must continuously emit radiation. If this continuous emission of 
energy actually occurred, the electrons would rapidly execute a descending 
spiral and fall into the nucleus. The Rutherford atom is therefore inherently 
unstable according to classical mechanics, but the predicted continuous 
radiation does not in fact occur. The fact that the electrons in atoms do not 
follow classical mechanics is also clearly shown by the heat-capacity values 
of gases. The C v for monatomic gases equals f R, which is simply the amount 
expected for the translation of the atom as a whole. It is evident that the 
electrons in the atoms do not take up energy as the gas is heated. 

Niels Bohr, in 1913, suggested that the electrons can revolve around the 
nucleus only in certain definite orbits, corresponding to certain allowed 



Sec. 8] PARTICLES AND WAVES 263 

energy states. Radiation is emitted in discrete quanta whenever an electron 
falls from an orbit of high to one of lower energy, and is absorbed whenever 
an electron is raised from a low to a higher energy orbit. If E ni and E nt are 
the energies of two allowed states of the electron, the frequency of the spectral 
line arising from a transition is 

A 1 

v = = -(E ni -E nt ) (10.20) 

A separate and arbitrary hypothesis is needed to specify which orbits are 
allowed. The simplest orbits of one electron moving in the field of force of 
a positively charged nucleus are the circular ones. For these orbits, Bohr 
postulated the following frequency condition: 8 only those orbits occur for 
which the angular momentum mvr is an integral multiple of h\2-n. 

mvr = n~> n - 1, 2, 3 . . . (10.21) 

ITT 

The integer n is called a quantum number. 

The mechanics of motion of the electron in its circular orbit of radius r 
can be analyzed starting with Newton's equation,/ ma. The force is the 
coulombic attraction between nucleus, with charge Ze, and electron, i.e., 
Ze 2 /r 2 . The acceleration is the centripetal acceleration, v 2 /r. Therefore 
Ze 2 /r 2 mv 2 /r, and 



(10.22) 
mv* 

h 2 
Then, from eq. (10.21) r - n 2 - (10.23) 



In the case of a hydrogen atom Z ==- 1, and the smallest orbit, n 1, 
would have a radius, 

"o = TT~2 - ' 529 A (10 ' 24 ) 

4n 2 me 2 

This radius is of the same order of magnitude as that obtained from the 
kinetic theory of gases. 

It may be noted that the radii of the circular Bohr orbits depend on the 
square of the quantum number. It can now be demonstrated that the Balmer 
series arises from transitions between the orbit with n = 2 and outer orbits; 
in the Lyman series, the lower term is the orbit with n = 1 ; the other series 
are explained similarly. These results are obtained by calculating the energies 
corresponding to the different orbits and applying eq. (10.20). The energy 
level diagram for the hydrogen atom is shown in Fig. 10.6. 

8 It will be seen a little later that this condition is simply another form of Planck's 
hypothesis that h is the quantum of action. 



264 



PARTICLES AND WAVES 



[Chap. 10 



Volts 


n 


cm l 


14 
13.53 






13 




1 1A 


12 




1 1 !"l^ 10,000- 






23.23 Ti 

io - n & ^ "T w 






o> eo r- t- m 0> o t- o^ ^ o > S ^ 


11 




l!r ss "f *i* J" 


10 










^ 30,000- 


9 


_ 


1 






| 40,000- 


8 


- 


S 






50,000- 


7 


- 




6 


(A 


60,000- 




"C 

-^ to eo *! 




5 


c 10 H5 CS 

5 c * *" 

CO 04<=> <T 


**7 oo 

IS 70.000- 




$ 




4 


- 








80,000- 


3 


- 








90,000- 


2 


- 








100,000- 


1 













Fig. 10.6. Energy levels of the H atom. (After G. Herzberg, 
Atomic Spectra, Dover, 1944.) 

The total energy E of any state is the sum of the kinetic and potential 
energies: ^ ^ 

* p r 

Ze* Ze 2 



From eq. (10.22), E = 

/ 

Therefore from eq. (10.23), E == 
The frequency of a spectral line is then 



2r 

Z 2 

l 2 """^ 2 



(10.25) 
(10.26) 



Sec. 9] PARTICLES AND WAVES 265 

Comparison with the experimental eq. (10.19) yields a theoretical value of 
the Rydberg constant for atomic hydrogen of 

.^-~ 109,737 cm- 
ch 3 

This is in excellent agreement with the experimental value. 

This pleasing state of affairs represented a great triumph for the Bohr 
theory and lent some solid support to the admittedly ad hoc hypothesis on 
which it is based. 

Several improvements in the original Bohr theory were made by Arnold 
Sommerfeld. He considered the possible elliptical orbits of an electron 
around the nucleus as one focus. Such orbits are known to be stable con- 
figurations in dynamical systems such as the planets revolving around the 
sun. 

For a circular orbit, the radius r is constant so that only angular momen- 
tum, associated with the variable 6, need be considered. For elliptical orbits 
two quantum numbers are needed, for the two variables r and 0. The 
azimutha! quantum number k was introduced to give the angular momentum 
in units of h\1-n. The principal quantum number n was defined 9 so that the 
ratio of the major axis to the minor axis of the elliptical orbit was n\k. 
Then k can take any value from 1 to /?, the case n k corresponding to 
a circular orbit. 

9. Spectra of the alkali metals. An electron moving about a positively 
charged nucleus is moving in a spherically symmetrical coulombic field of 
force. Besides the hydrogen atom, a series of hydrogenlike ions satisfy this 
condition. These ions include He^, Li++, and Be l+ +, each of which has a 
single electron. Their spectra are observable when electric sparks discharge 
through the vapour of the element (spark spectra). They are very similar in 
structure to the hydrogen spectrum, but the different series are displaced to 
shorter wavelengths, as a consequence of the dependence of frequency on 
the square of the nuclear charge, given by eq. (10.26). 

If an electron is moving in a spherically symmetrical field, the energy 
level is the same for all elliptical orbits of major axis a as it is for the circular 
orbit of radius a. In other words, the energy is a function only of the principal 
quantum number n. All energy levels with the same n are the same, irrespec- 
tive of the value of k, the azimuthal quantum number. For example, if 
n = 3, there are three superimposed levels or terms of identical energy, 
having k = 1, 2, or 3. Such an energy level is said to have a threefold de- 
generacy. Actually, even in hydrogen, a very slight splitting of these degener- 
ate levels is found in the fine structure of the spectra, revealed by spectro- 
graphs of high resolving power. 

9 Derivations and detailed discussions of these aspects of the old quantum theory may 
be found in S. Dushman's article in Taylor's Treatise on Physical Chemistry, 2nd ed., p. 1 170. 



266 



PARTICLES AND WAVES 



[Chap. 10 



EV 

5.37 
5 



UL 



For most of the atoms and ions that may give rise to spectra the electrons 
concerned in the transitions are not moving in spherically symmetric fields. 
Consider, for example, the case of the lithium atom, which is typical of the 
alkali metals. The electron whose transitions are responsible for the observed 
spectrum is the outer, valence, or optical electron. This electron does not 
move in a spherical field, since its position at any instant is influenced by 
the positions of the two inner electrons. If the outer electron is on one side 
of the nucleus, it is less likely that the other two will be there also, because 
of the electrostatic repulsions. Thus the field is no longer spherical, and the 
elliptical orbits can no longer have the same energy as a circular orbit of the 

same n value. The elliptical orbits will have 
different energy levels depending on their 
ellipticity, which is governed by the allowed 
values of the azimuthal quantum number k. 
For each n, there will be n different energy 
levels characterized by different k's. 

The lowest ojr ground state is that for 
which n = 1 and k ^ 1. States with k ~ 1 
are called s states. This is therefore a Is 
state. When n = 2, k can be either 1 or 2. 
States with k = 2 are called p states. We 
therefore have a 2s state and a 2p state. 
Similarly, when n = 3, we have 3s, 3p, and 
3d (k = 3) states; when n = 4, we have 
4s, 4/?, 4J, and 4f(k = 4) states. 

In this discussion there has been a tacit 
assumption that the energy levels of the 
atom are determined solely by the quan- 
tum states of the valence electrons. This 
is actually not true, and all the electrons and even the nucleus should be 
considered in discussing the allowed energy states. Then, instead of the 
quantum number k, which gives the angular momentum of the single 
electron, a new quantum number L must be used that gives the resultant 
angular momentum of all the electrons. According as L ~ 0, 1, 2, 3 . 
etc., we refer to the atomic states as S, P, A F . . . etc. In the case of atoms 
like the alkali metals, which have only one valence electron, it turns out 
that the resultant angular momenta of the inner electrons add vectorially to 
zero. Therefore in this case only ttie single electron need be considered after 
all. 10 Nevertheless we shall use the more proper notation, 5, />, D, F, to 
refer to the energy levels. 

The energy-level diagram for lithium is shown in Fig. 10.7. The observed 

10 The situation becomes more complicated when there are two or more optical electrons. 
An excellent discussion is given by G. H. Herzberg, Atomic Spectra and Atomic Structure 
(New York: Dover Publications, 1944). 




Fig. 10.7. Energy levels and spec- 
tral transitions in the lithium atom. 



Sec. 10] PARTICLES AND WAVES 267 

spectral series arise from the combinations of these terms, as shown in the 
diagram. It will be noted that only certain transitions are allowed; others are 
forbidden. Certain selection rules must be obeyed, as for example in this case 
the rule that AL must be + 1 or 1 . 

Experimentally four distinct series have been observed in the atomic 
spectra of the alkalis. The principal series is the only one found in absorption 
spectra and arises from transitions between the ground state 1*9 and fc the 
various P states. It may be written symbolically : 

v^ 15 - mP 

Absorption spectra almost always arise from transitions from the ground 
state only, since at ordinary temperatures the proportion of atoms in excited 
states is usually vanishingly small, being governed by the exponential Boltz- 
mann factor e ~^ ElkT . At the much higher temperatures required to excite 
emission spectra, some of the higher states are sufficiently populated by 
atoms to give rise to a greater variety of lines. 

Thus in the emission spectra of the alkali metals, in addition to the prin- 
cipal series, three other series appear. These may be written symbolically as 

v 2P - mS the sharp series 

v 2P - mD the diffuse series 

v --= 3D mF the fundamental series 

The names are not notably descriptive, although the lines in the sharp series 
are indeed somewhat narrower than the others. 

10. Space quantization. So far in the discussion of allowed Bohr orbits, 
we have not considered the question of how the orbits can be oriented in 
space. This is because in the absence of an external electric or magnetic 
field there is no way of distinguishing between different orientations, since 
there is no physically established axis of reference. If an atom is placed in a 
magnetic field, however, one can ask how the orbits will be oriented relative 
to the field direction. 

The answer given by the Bohr theory is that only certain orientations 
are allowed. These are determined by the condition that the component of 
angular momentum in the direction of the magnetic field, e.g., in the Z 
direction, must be an integral multiple of h/27r. Thus 

P. - % (10.27) 

where m is the magnetic quantum number. This behavior is called space 
quantization. 

The allowed values of m are 1, 2, 3, etc., up to &, k being the 
azimuthal quantum number, which gives the magnitude of the total angular 



268 



PARTICLES AND WAVES 



[Chap. 10 




k3 

Fig. 10.8. Spatial quantiza- 
tion of angular momentum in a 
magnetic field H. 



momentum in units of h/2ir. An example of space quantization for the case 
k = 3 is illustrated in Fig. 10.8. 

For any value of k, there are 2k allowed orientations corresponding to 
the different values of m. In the absence of an external field, the correspond- 
ing energy level will be 2/r-fold degenerate. 
In the presence of an electric or magnetic 
field this energy level will be split into its 
individual components. This splitting gives 
rise to a splitting of the corresponding spec- 
tral lines. In a magnetic field this is called 
the Zeeman effect', in an electric field, the 
Stark effect. This observed splitting of the 
spectral lines is the experimental basis for the 
introduction into the Bohr theory of space 
quantization and the quantum number m. 

11. Dissociation as series limit. It will be 
noted in the term diagram for lithium that the 
energy levels become more closely packed as 
the height above the ground state increases. 
They finally converge to a common limit 

whose height above the ground level corresponds to the energy necessary to 
remove the electron completely from the field of the nucleus. In the observed 
spectrum, the lines become more and more densely packed and finally 
merge into a continuum at the onset of dissociation. The reason for the 
continuous absorption or emission is that the free electron no longer has 
quantized energy states but can take up kinetic energy of translation 
continuously. 

The energy difference between the series limit and the ground level 
represents the ionization potential I of the atom or ion. Thus the fast ionization 
potential of Li is the energy of the reaction Li+ 4- e -> Li. The second 
ionization potential is the energy of Li+ f + e -> Li+. 

Examples of ionization potentials are given in Table 10.1. The way in 
which the values of / vary with position in the periodic table should be 
noted. This periodicity is very closely related to the periodic character of the 
chemical properties of the elements, for it is the outer electrons of an atom 
that enter into its chemical reactions. Thus the alkali metals have low 
ionization potentials; the inert gases, high ionization potentials. 

12. The origin of X-ray spectra. The origin of the characteristic X-ray 
line series studied by Moseley (see Chapter 8) is readily understood in terms 
of the Bohr theory. The optical spectra are caused by transitions of outer or 
valence electrons, but the X-ray spectra are caused by transitions of the 
inner electrons. X rays are generated when high-velocity particles such as 
electrons impinge upon a suitable target. As the result of such a collision, 
an electron may be driven completely from its orbit, leaving a "hole" in the 



Sec. 13] PARTICLES AND WAVES 269 

TABLE 10.1 
IONIZATION POTENTIALS OF CHEMICAL ELEMENTS (IN ELECTRON VOLTS) 



Element 


First lonizatlon 
Potential 


Second lonization 
Potential 


H 


13.60 




He 


24.58 


54.41 


Li 


5.39 


75.62 


Be 


9.32 


18.21 


B 


8.30 


25.12 


C 


11.27 


24.38 


N 


14.55 


29.61 


O 


13.62 


35.08 


F 


17.42 


34.98 


Ne 


21.56 


40.96 


Na 


5.14 


47.29 


Mg 


7.65 


15.03 


Al 


5.99 


18.82 


Si 


8.15 


16.34 


P 


10.98 


19.65 


S 


10.36 


23.41 


Cl 


12.96 


23.80 


A 


15.76 


27.62 


K 


4.34 


31.81 


Ca 


6.11 


11.87 



target atom. When electrons in outer shells, having larger values of the 
principal quantum number n, drop into this hole, a quantum of X radiation 
is emitted. 

13. Particles and waves. One might go on from here to describe the 
further application of the Bohr theory to more complex problems in atomic 
structure and spectra. Many other quite successful results were obtained, 
but there were also a number of troublesome failures. Attempts to treat 
cases in which more than one outer electron is excited, as in the helium 
spectrum, were in general rather discouraging. 

The Bohr method is essentially nothing more than the application of a 
diminutive celestial mechanics, with coulombic rather than gravitational 
forces, to tiny solar-system models of the atom. Certain quantum conditions 
have been arbitrarily superimposed on this classical foundation. The rather 
capricious way in which the quantum numbers were introduced and adjusted 
always detracted seriously from the completeness of the theory. 

Now there is one branch of physics in which, as we have seen, integral 
numbers occur very naturally, namely in the stationary-state solutions of 
the equation for wave motion. This fact suggested the next great advance in 
physical theory: the idea that electrons, and in fact all material particles, 
must possess wavelike properties. It was already known that radiation 
exhibited both corpuscular and undulatory aspects. Now it was to be shown, 
first theoretically and soon afterwards experimentally, that the same must 
be true of matter. 



270 PARTICLES AND WAVES [Chap. 10 

This new way of thinking was first proposed in 1923 by Due Louis 
de Broglie. In his Nobel Prize Address he has described his approach as 
follows. 11 

. . . When 1 began to consider these difficulties [of contemporary physics] I 
was chiefly struck by two facts. On the one hand the quantum theory of light cannot 
be considered satisfactory, since it defines the energy of a light corpuscle by the 
equation E ---= hv, containing the frequency v. Now a purely corpuscular theory 
contains nothing that enables us to define a frequency; for this reason alone, there- 
fore, we are compelled, in the case of light, to introduce the idea of a corpuscle and 
that of periodicity simultaneously. 

On the other hand, determination of the stable motion of electrons in the atom 
introduces integers; and up to this point the only phenomena involving integers 
in Physics were those of interference and of normal modes of vibration. This fact 
suggested to me the idea that electrons too could not be regarded simply as corpus- 
cles, but that periodicity must be assigned to them also. 

A simple two-dimensional illustration of this viewpoint may be seen in 
Fig. 10.9. There are shown two possible electron waves of different wave- 
lengths for the case of an electron revolving 
around an atomic nucleus. In one case, the 
circumference of the electron orbit is an 
integral multiple of the wavelength of the 
electron wave. In the other case, this condi- 
tion is not fulfilled and as a result the wave 
is destroyed by interference, and the supposed 
state is nonexistent. The introduction of in- 
tegers associated with the permissible states 

Fig. 10.9. Schematic drawing Qf e|ectronic motjon therefore occurs quite 
of an electron wave constrained ~ 

to move around nucleus. The naturally once the electron is given wave 
solid line represents a possible properties. The situation is exactly analogous 
stationary wave. The dashed line with the occurrence of stationary waves on 
shows how a wave of somewhat a vibrating string. The necessary condition 
different wavelength would be for a staWc ^ Qf radius . fc 
destroyed by interference. e 

27rr e =-. nX (10.28) 

A free electron is associated with a progressive wave so that any energy 
is allowable. A bound electron is represented by a standing wave, which can 
have only certain definite frequencies. 

In the case of a photon there are two fundamental equations to be 
obeyed : e ~ hv, and e = me 2 . When these are combined, one obtains 
hv = me 2 or X ---- c/v -= h/mc hip, where p is the momentum of the 
photon. Broglie considered that a similar equation governed the wave- 
length of the electron wave. Thus, 

* * 

A = = - (10.29) 

mv p 

11 L. de Broglie, Matter and Light^ (New York: Dover Publications [1st ed., W. W. 
Norton Co.], 1946). 




Sec. 14] 



PARTICLES AND WAVES 



271 



The original Bohr condition for a stable orbit was given by eq. (10.21) 
as 27Ttnvr e = nh. By combination with eq. (10.28), one again obtains eq. 
(10.29) so that the Broglie formulation gives the Bohr condition directly. 

The Broglie relation, eq. (10.29), is the fundamental one between the 
momentum of the electron considered as a particle and the wavelength of its 
associated wave. Consider, for example, an electron that has been accelerated 
through a potential difference V of 10 kilovolts. Then Ve ~ -i/w 2 , and its 
velocity would be 5.9 x 10 9 cm per sec, about one-fifth that of light. The 
wavelength of such an electron would be 



h 6.62 x 10~ 27 

~ mv ~ (9AI x \Q r )(53~ 



0.12 A 



This is about the same wavelength as that of rather hard X rays. 



H 2 

a F 
22- 

Golf ball 
Baseball 



TABLE 10.2 
WAVELENGTHS OF VARIOUS PARTICLES 



Particle 


Mass 
(g) 


Velocity 
(cm/sec) 


Wavelength 
(A) 


ilectron .... 


9.1 x 10- 28 


5.9 x 10 7 


12.0 


It electron 


9.1 X 10~ 28 


5.9 x 10 8 


1.2 


volt electron . 


9.1 X 10~ 28 


5.9 x 10 9 


0.12 


t proton 
t a particle 
lecule at 200C 


1.67 x 10- 24 
6.6 x 10- 24 
3.3 x 10~ 24 


1.38 x JO 7 
6.9 x 10 6 
2.4 x 10 5 


0.029 
0.015 
0.82 


cle from radium 


6.6 x 10- 24 


1.51 x 10 9 


6.6 X 10~ 5 


bullet .... 


1.9 


3.2 x 10 4 


1.1 x 10~ 23 


ill .... 


45 


3 x 10 3 


4.9 x 10~ 24 


11 .... 


140 


2.5 x 10 3 


1.9 x 10~ 24 



Table 10.2 lists the theoretical wavelengths associated with various 
particles. 12 The wavelengths of macroscopic bodies are exceedingly short, 
so that any wave properties will escape our observation. Only in the atomic 
world does the wave nature of matter become manifest. 

14. Electron diffraction. If any physical reality is to be attached to the 
idea that electrons have wave properties, a 1 .0 A electron wave should be 
diffracted by a crystal lattice in very much the same way as an X-ray wave. 
Experiments along this line were first carried out by two groups of workers, 
who shared a Nobel prize for their efforts. C. Davisson and L. H. Germer 
worked at the Bell Telephone Laboratories in New York, and G. P. Thom- 
son, the son of J. J. Thomson, and A. Reid were at the University of Aber- 
deen. Diffraction diagrams obtained by Thomson by passing beams of 



12 After J. D. Stranathan, The Particles of Modern physics (Philadelphia: Blakiston, 
1942), p. 540. 



272 PARTICLES AND WAVES [Chap. 10 

electrons through thin gold foils are shown in Fig. 10.10. The wave nature 
of the electron was unequivocally demonstrated by these researches. More 
recently, excellent diffraction patterns have been obtained from crystals 
placed in beams of neutrons. 

Electron beams, owing to their negative charge, have one advantage not 
possessed by X rays as a means of investigating the fine structure of matter. 
Appropriate arrangements of electric and magnetic fields can be designed to 
act as "lenses" for electrons. These arrangements have been applied in the 



[The photograph below 
was one of the first ob- 
tained. The one at the 
right is a recent example.] 





Fig. 10.10. Diffraction diagrams obtained by passing beams of electrons 
through thin gold foils. (Courtesy Professor Sir George Thomson.) 

development of electron microscopes capable of resolving images as small as 
20 A in diameter. We could wish for no clearer illustrations of the wave 
properties of electrons than the beautiful electron micrographs of viruses, 
fibers, and colloidal particles that have been obtained with these instruments. 
15. The uncertainty principle. In the development of atomic physics we 
have noted the repeated tendency toward the construction of models of the 
atom and its constituents from building blocks that possess all the normal 
properties of the sticks and stones of everyday life. One fundamental axiom 
of the classical mechanics developed for commonplace occurrences was the 
possibility of simultaneously measuring different events at different places. 
Such measurement appears at first to be perfectly possible because to a first 
approximation the speed of light is infinitely large, and it takes practically 
no time to signal from place to place. More refined measurements must 
consider the fact that this speed is really not infinite, but only 3 x 10 10 cm 
per sec. This speed is indeed large compared with that of a rocket, but not 
compared with that of an accelerated electron. As a result, attempts to apply 
the old mechanics to moving electrons were a failure, and the new relativisitic 
mechanics of Einstein was needed to correct the situation. 



Sec. 15] PARTICLES AND WAVES 273 

In a similar way, in our ordinary macroscopic world, the value of the 
Planck constant h may be considered to be effectively zero. The Broglie 
wavelengths of ordinary objects are vanishingly small, and a batter need 
not consider diffraction phenomena when he swings at an inside curve. If 
we enter into the subatomic world, h is no longer so small as to be negligible. 
The Broglie wavelengths of electrons are of such a magnitude that diffraction 
effects occur in crystal structures. 

One of the fundamental tenets of classical mechanics is that it is possible 
to specify simultaneously the position and momentum of any body. The 
strict determinism of mechanics rested upon this basic assumption. Knowing 
the position and velocity of a particle at any instant, Victorian mechanics 
would venture to predict its position and velocity at any other time, past or 
future. Systems were completely reversible in time, past configurations being 
obtained simply by substituting / for t in the dynamical equations. But, is 
it really possible to measure simultaneously the position and momentum of 
any particle? The possible methods of measurement must be analyzed in 
detail before an answer can be given. 

To measure with precision the position of a very small object, a micro- 
scope of high resolving power is required. With visible light one cannot 
expect to locate objects much smaller than a tenth of a micron. The size of 
the smallest body that can be observed is limited by diffraction effects, which 
begin to create a fuzziness in the image when the object is of the same order 
of magnitude as the wavelength of the incident light. The limit of resolution 
is given according to the well known formula of Abbe as R A/2/4, and 
the maximum value of the numerical aperture A is unity. 

In order to determine the position of an electron to within a few per cent 
uncertainty, radiation of wavelength around 10~ 10 cm or 10~ 2 A would have 
to be used. We shall conveniently evade the technical problems involved in 
the design and manufacture of a microscope using these y rays. With such 
very short rays, there will be a very large Compton effect, and the y ray will 
impart considerable momentum to the electron under observation. This 
momentum is given by eq. (9.5) as mv = 2(hv/c) sin a/2. Since the range in 
scattering angle is from to 7T-/2, corresponding to the aperture of the micro- 
scope (A = 1), the momentum is determined only to within an uncertainty 
of A/? = mv & A/A. On account of diffraction, the error A^ in the determina- 
tion of position is of the order of the wavelength A. 

The product of the uncertainty in momentum times the uncertainty in 
position is therefore of the order of h, 

&p-&q~h (10.30) 

This is the famous uncertainty principle of Werner Heisenberg (1926). It is 
impossible to specify simultaneously the exact position and momentum of a 
particle because our measuring instruments necessarily disturb the object 
being measured. This disturbance is negligible with man-sized objects, but 



274 PARTICLES AND WAVES [Chap. 10 

the disturbance of atom-sized particles cannot be neglected. Herein is the 
essential meaning of the failure of classical mechanics and the success of 
wave mechanics. 13 

16. Waves and the uncertainty principle. Some kind of uncertainty prin- 
ciple is always associated with a wave motion. This fact can be seen very 
clearly in the case of sound waves. Consider the case of an organ pipe, set 
into vibration by depressing a key, whose vibration is stopped as soon as the 
key is released. The vibrating pipe sets up a train of sound waves in the air, 
which we hear as a note of definite frequency. Now suppose the time between 
the depression and the release of the key is gradually shortened. As a result, 
the length of the train of waves is shortened also. Finally the time will come 
when the period during which the key is depressed is actually less than the 
period r of the sound wave, the time required for one complete vibration. 
Once this happens, the frequency of the wave is no longer precisely deter- 
mined, for at least one complete vibration must take place to define the 
frequency. It appears, therefore, that the time and the frequency cannot both 
be fixed at any arbitrary value. If a very small time is chosen, the frequency 
becomes indeterminate. 

When waves are associated with particles, a similar uncertainty principle 
is a necessary consequence. If the wavelength or frequency of an electron 
wave, for example, is to be a definitely fixed quantity, the wave must be 
infinite in extent. Any attempt to confine a wave within boundaries requires 
destructive interference at these boundaries in order to reduce the resultant 
amplitudes there to zero. This interference can be secured only by super- 
imposing waves of different frequencies. It follows that an electron wave of 
perfectly definite frequency, or momentum, must be infinitely extended and 
therefore must have a completely indeterminate position. In order to fix 
the position, superimposed waves of different frequency are required, and 
as the position becomes more closely defined the momentum becomes 
fuzzier. 

The uncertainty relation eq. (10.30) can be expressed not only in terms 
of position and momentum but also for energy and time. Thus, 

A/? A? - AF A/ ?v h (10.31) 

This equation is used to estimate the sharpness of spectral lines. In general, 
lines arising from transitions from the ground state of an atom are sharp. 
This is because the optical electron spends a long time in the ground state 
and thus A", the uncertainty in the energy level, is very small. On the other 
hand, the lifetime of excited states may sometimes be very short, and trans- 
itions between such excited energy levels may give rise to diffuse or broad- 
ened lines as a result of the uncertainty A in the energy levels, which is 

18 Many natural philosophers would, not agree with this statement. See H. Margenau, 
Physics Today, 7, 6 (1954). 



Sec. 17] PARTICLES AND WAVES 275 

reflected in an uncertainty, Av -= A//z, in the frequency of the observed 
line. 14 

17. Zero-point energy. According to the old quantum theory, the energy 
levels of a harmonic oscillator were given by E n nhv. If this were true the 
lowest energy level would be that with n 0, and would therefore have zero 
energy. This would be a state of complete rest, represented by the minimum 
in the potential energy curve in Fig. 7.15. 

The uncertainty principle does not allow such a state of completely 
defined position and completely defined (in this case, zero) momentum. As 
a result, the wave treatment shows that the energy levels of the oscillators 
are given by 

* = ( + I)'"' 0- 32 ) 

Now, even when n 0, the ground state, there is a residual zero-point energy 
amounting to 

=-, \hv (10.33) 

This must be added to the Planck expression for the mean energy of an 
oscillator, which was derived in eq. (10.17). 

18. Wave mechanics the Schrodinger equation. In 1926, Erwin Schr5- 
dinger and W. Heisenberg independently laid the foundations for a distinctly 
new sort of mechanics which was expressive of the wave-particle duality of 
matter. This is called wave or quantum mechanics. 

The starting point for most quantum mechanical discussions is the 
Schrodinger wave equation. We may recall that the general differential 
equation of wave motion in one dimension is given by eq. (10.7) as 



_ 

dx* ~ & ' a/ 2 

where <f> is the displacement and v the velocity. In order to separate the 
variables, let <f> =--- y(x) sin 2-nvt. On substitution in the original equation, 
this yields 

d*W 47T 2 V 2 

T? + ^T-V = (10-34) 

dx* v 2 

This is the wave equation with the time dependence removed. In order 
to apply this equation to a "matter wave," the Broglie relation is introduced, 
as follows: The total energy E is the sum of the potential energy U and the 
kinetic energy p 2 j2m. E = p*/2m + U. Thus, p = [2m(E - (7)] 1/2 , or 
X = hip = h[2m(E U)]~ m . Substituting this in eq. (10.34), one obtains: 

(,0.35) 



14 This is not the only cause of broadening of spectral lines. There is in addition a 
pressure broadening due to interaction with the electric fields of neighboring atoms or 
molecules, and a Doppler broadening, due to motion of the radiating atom or molecule 
with respect to the observer. 



276 PARTICLES AND WAVES [Chap. 10 

This is the famous Schrftdinger equation in one dimension. For three 
dimensions it takes the form 

o2.*| 

W + -r^-(E- U)y> = (10.36) 

rr 

Although the equation has been obtained in this way from the ordinary 
wave equation and Broglie's relation, it is actually so fundamental that it is 
now more usual simply to postulate the equation as the starting point of 
quantum mechanics, just as Newton's/^ ma is postulated as the starting 
point of ordinary mechanics. 

As is usual with differential equations, the solutions of eq. (10.36) for 
any particular set of physical conditions are determined by the particular 
boundary conditions imposed upon the system. Just as the simple wave 
equation for a vibrating string yields a discrete set of stationary-state solutions 
when the ends of the string are held fixed, so in general solutions are obtained 
for the SchrOdinger equation only for certain energy values E. In many cases 
the allowed energy values are discrete and separated, but in certain other 
cases they form a continuous spectrum of values. The allowed energy values 
are called the characteristic, proper, or eigen- values for the system. The 
corresponding wave functions y am called the characteristic functions or 
eigenf unctions. 

19. Interpretation of the y functions. The eigenfunction ip is by nature a 
sort of amplitude function. In the case of a light wave, the intensity of the 
light or energy of the electromagnetic field at any point is proportional to 
the square of the amplitude of the wave at that point. From the point of 
view of the photon picture, the more intense the light at any place, the more 
photons are falling on that place. This fact can be expressed in another way 
by saying that the greater the value of y>, the amplitude of a light wave in 
any region, the greater is improbability of a photon being within that region. 

It is this interpretation that is most useful when applied to the eigen- 
functions of Schrftdinger's equation. They are therefore sometimes called 
probability amplitude functions. If y(x) is a solution of the wave equation for 
an electron, then the probability of finding the electron within the range 
from x to x + dx is given 16 by y> 2 (x)dx. 

The physical interpretation of the eigenfunction as a probability ampli- 
tude function is reflected in certain mathematical conditions that it must 
obey. It is required that y>(x) be single T valued, finite, and continuous for all 
physically possible values of x. It must be single-valued, since the probability 
of finding the electron at any point x must have one and only one value. It 
cannot be infinite at any point, for then the electron would be fixed at exactly 
that point, which would be inconsistent with the wave properties. The require- 
ment of continuity is helpful in the selection of physically reasonable solutions 
for the wave equation. 

15 Since the function y may be a complex quantity, the probability is written more 
generally as ^v>, where $ is the complex conjugate of y>. Thus, e.g., if y> = e~ ix t y> = *'* 



Sec. 20] 



PARTICLES AND WAVES 



277 



20. Solution of wave equation the particle in a box. The problem of 
finding the solution of the wave equation in any particular case may be an 
extremely difficult one. Sometimes a solution can be devised in principle that 
in practice would involve several decades of calculations. The recent develop- 
ment of high-speed calculating machines has greatly extended the range of 
problems for which numerical solutions can be obtained. 

The simplest case to which the wave equation can be applied is that of a 
free particle; i.e., one moving in the absence of any potential field. In this 
case we may set U = and the one-dimensional equation becomes 



87T 2 /M 

-- E V = 

A solution of this equation is readily found 16 to be 
y A sin ( -^ V2mE x \ 



(10.37) 



(10.38) 



where A is an arbitrary constant. This is a perfectly allowable solution as 
long as E is positive, since the sine of a real quantity is everywhere single- 
valued, finite, and continuous. Thus all positive values of E are allowable 




w 



(a) 



(b) 



(0 



Fig. 10.11. Electron in a one-dimensional box. (a) the potential function, 
(b) allowed electron waves, (c) tunnel effect. 

and the free particle has a continuous spectrum of energy states. This con- 
clusion is in accord with the picture previously given of the onset of the 
continuum in atomic spectra as the result of dissociation of an electron from 
the atom. 

What is the effect of imposing a constraint upon the free particle by 
requiring that its motion be confined within fixed boundaries? In three 

16 See, for example, Granville et-aL, op. ciY., p. 390. The solution can be verified by 
substitution into the equation. 



278 PARTICLES AND WAVES [Chap. 10 

dimensions this is the problem of a particle enclosed in a box. The one- 
dimensional problem is that of a particle required to move between set 
points on a straight line. The potential function that corresponds to such a 
condition is shown in (a), Fig. 10.11. For values of x between and a the 
particle is completely free, and U =- 0. At the boundaries, however, the 
particle is constrained by an infinite potential wall over which there is no 
escape; thus U --= oo when x ^ 0, x - a. 

The situation now is similar to thcit of the vibrating string considered at 
the beginning of the chapter. Restricting the electron wave within fixed 
boundaries corresponds to seizing hold of the ends of the string. In order 
to obtain stable standing waves, it is again necessary to restrict the allowed 
wavelengths so that there is an integral number of half wavelengths between 
and a; i.e., n(A/2) a. Some of the allowed electron waves are shown in 
(b), Fig. 10.11, superimposed upon the potential-energy diagram. 

The permissible values of the kinetic energy E n of the electron in a box 
can be obtained from the Broglie relation X - hjmv. 

2 i 

= \m 



(10.39) 

From this equation, two important consequences can be deduced which 
will hold true for the energy of electrons, not only in this special case, but 
quite generally. First of all, it is apparent that as the value of a increases, 
the energy decreases. Other factors being the same, the more room the 
electron has to move about in, the lower will be its energy. The more localized 
is its motion, the higher will be its energy. Remember that the lower the 
energy, the greater the stability of a system. 

Secondly, the integer n is a typical quantum number, which now appears 
quite naturally and without any ad hoc hypotheses. It determines the number 
of nodes in the electron wave. When n 1 there are no nodes. When n 2 
there is a node in the center of the box; when n -- 3 there are two nodes, and 
so on. The value of the energy depends directly on 2 , and therefore rises 
rapidly as the number of nodes increases. 

The extension of the one-dimensional result to a three-dimensional box 
of sides a, b, and c is very simple. The allowed energy levels for the three- 
dimensional case depend on a set of three integers (n l9 2 , %): since there are 
three dimensions, there are three quantum numbers. 

h* In* TV* 3 2 \ 

=o-(-V + 7J + -r) 00.40) 

8/w \ a 2 b* c* / 

This result shows that according to wave mechanics even the trans- 
lational motion of a particle in a box is quantized. Because of the extremely 



Sec. 21] PARTICLES AND WAVES 279 

small value of h 2 these levels lie very closely packed together except in cases 
where the dimensions of the box are vanishingly small. 

If electron waves in one dimension are comparable with vibrations of a 
violin string, those in two dimensions are like the pulsations of a drumhead, 
whereas those in three dimensions are like the vibrations of a block of steel. 
The waves can then have nodes along three directions, and the three quantum 
numbers determine the number of nodes. 

21. The tunnel effect. Let us take a baseball, place it in a well constructed 
box, and nail the lid down tightly. Now any proper Newtonian will assure 
us that the ball is in the box and is going to stay there until someone takes 
it out. There is no probability that the ball will be found on Monday inside 
the box and on Tuesday rolling along outside it. Yet if we transfer our 
attention from a baseball in a box to an electron in a box, quantum mechanics 
predicts exactly this unlikely behavior. 

To be more precise, consider in (c), Fig. 10.11, a particle moving in a 
"one-dimensional box" with a kinetic energy E k . It is confined by a potential- 
energy wall of thickness d and height U Q . Classical mechanics indicates that 
the particle can simply move back and forth in its potential energy well; 
since the potential-energy barrier is higher than the available kinetic energy, 
the possibility of escape is absolutely nil. 

Quantum mechanics tells a different story. The wave equation (10.35) for 
the region of constant potential energy U Q is 

*) 4 (87T 2 w/7* 2 ) (E - U )y> - 



This equation has the general solution 

W -- ^ e ^^!/i)^2m(E'-U n )x 

In the region within the box E ^ U (} and this solution is simply the familiar 
sine or cosine wave of eq. (10.38) written in the complex exponential form. 17 
In the region within the potential-energy barrier, however, U Q > ", so that 
the expression under the square root sign is, negative. One can therefore 
multiply out a V 1 term, obtaining the following result: 

y> -= Ae-V* 1 ^***^'* (10.41) 

This exponential function describes the behavior of the wave function 
within the barrier. It is evident that according to wave mechanics the prob- 
ability of finding an electron in the region of negative energy is not zero, 
but is a certain finite number that falls off exponentially with the distance 
of penetration within the barrier . The behavior of the wave function is shown 
in (c), Fig. 10.11. So long as the barrier is not infinitely high nor infinitely 
wide there is always a certain probability that electrons (or particles in 
general) will leak through. This is called the tunnel effect. 

17 See, for example, Courant and Robbins, What Is Mathematics (New York: Oxford, 
1941), p. 92, for a description of this notation: e io =- cos -f / sin 0. 



280 PARTICLES AND WAVES [Chap. 10 

The phenomenon is not observed with baseballs in boxes or with cars in 
garages, 18 being rendered extremely improbable by the various parameters 
in the exponential. In the world of atoms, however, the effect is a common 
one. One of the best examples is the emission of an a particle in a radio- 
active disintegration. The random nature of this emission is a reflection of 
the fact that the position of the particle is subject to probability laws. 

22. The hydrogen atom. If the translational motion of the atom as a 
whole and the motion of the atomic nucleus are neglected, the problem of 
the hydrogen atom can be reduced to that of a single electron in a coulombic 
field. This is in a sense a modification of the problem of a particle in a three- 
dimensional box, except that now the box is spherical. Also, instead of steep 
walls and zero potential energy within, there is now a gradual rise in potential 
with distance from the nucleus: at r = oo, U 0; at r = 0, U = oo. 

The potential energy of the electron in the field of the nucleus is given by 
U -= e 2 /r. The Schrddinger equation therefore becomes 



In view of the spherical symmetry of the potential field, it is convenient to 
transform this expression into spherical coordinates, 

i a / a^A i a 2 ^ i a / a^A 

r* Or V Or / + r 2 sin 2 6 * M>* + ^sinO ' 00 \ Sm OO/ + 



The polar coordinates r, 0, and <f> have their usual significance (Fig. 7.2, 
page 168). The coordinate r measures the radial distance from the origin; 
is a "latitude"; and <f> a "longitude." Since the electron is moving in three 
dimensions, three coordinates obviously suffice to describe its position at 
any time. 

In this equation, the variables can be separated, since the potential is a 
function of r alone. Let us substitute 



That is, the wave function is a product of three functions, one of which 
depends only on r, one only on 6, and the last only on <f>. We shall skip the 
intervening steps in the solution and the application of the boundary con- 
ditions that permit only certain allowed eigenfunctions to be physically 
meaningful. 19 From our previous experience, however, we shall not be sur- 
prised to find that the final solutions represent a set of discrete stationary 

18 This extreme example is described by G. Gamow in Mr. Tompkins in Wonderland 
(New York: Macmillan, 1940), which is recommended as an introduction to this chapter in 
Physical Chemistry. 

19 For the steps in the solution see, for example, L. Pauling and E. B. Wilson, Introduc- 
tion to Quantum Mechanics (New York: McGraw-Hill, 1935), Chap. V. 



Sec. 22] PARTICLES AND WAVES 281 

energy states for the hydrogen atom, characterized by certain quantum 
numbers, n, /, and m. Nor is it surprising that exactly three quantum numbers 
are required for this three-dimensional motion, just as one sufficed for the 
waves on a string, whereas three were needed for the particle in a box. 

The allowed eigenfunctions are certain polynomials whose properties had 
been extensively studied by mathematicians well before the advent of quan- 
tum mechanics. In order to give them a measure of concreteness, some 
examples of these hydrogen wave functions are tabulated in Table 10.3 for 
the lower values of the quantum numbers n, /, and m. 

TABLE 10.3 
THE HYDROGENLIKE WAVE FUNCTIONS 



K Shell 
n = i, / = o, m = 0: 

/2T\ 3 / 2 



1 /2T\ 3 
--^ ~ 

V-rr \<V 



L Shell 

n = 2, / = 0, m = 0: 



-!=(?)''' (2 -*)*-* 

4X/27T W V <*J 



Y> 2P . - -4 
4V 2- 

= 2, /= 1,/n = 1: 

v = L^ (? ) 3/2 5: e -Zr/2 sin cos 
4V/27T ^o 7 flb 



_ 

4V27 



These quantum numbers can be assigned a significance purely in terms 
of the wave-mechanical picture, but they are also the logical successors to 
the numbers of the old quantum theory. 

Thus n is still called the principal quantum number. It determines the total 
number of nodes in the wave function, which is equal to n 1 . These nodes 
may be either in the radial function R(r), or in the azimuthal function 0(0). 
When the quantum number / is zero, there are no nodes in the function. 
In this case the number of nodes in R(r) equals n 1 . 

The azimuthal quantum number /replaces the k(= I + 1) of old quantum 
theory. The angular momentum is given by Vl(l + 1) H/2iT. Now / can take 
any value from to n 1 ; then / is the number of nodal surfaces passing 
through the origin. 

The magnetic quantum number m still gives the value of the components 



282 



PARTICLES AND WAVES 



[Chap. 10 



of angular momentum along the r axis, since p 0tZ mh/27T 9 exactly as in 
eq. (10.27). The allowed values of m now run from / to {-/, including zero. 
The great advantage of the new theory is that these numbers all arise 
quite naturally from Schrodinger's equation. 



n = 3 




(a) 



mo 6 




Fig. 10.12. (a) Radial part of wave functions for hydrogen atom, (b) Radial 
distribution functions giving probability of finding electrons at a given distance 
from nucleus. (After G. Herzberg, Atomic Spectra, Dover, 1944.) 

23. The radial wave functions. In (a), Fig. 10.12, the radial wave functions 
have been plotted for various choices of n and /. In case / 0, all the nodes 
appear in the radial function. 

The value of ^ 2 (r) is proportional to the probability of finding the electron 
at any particular distance r in some definite direction from the nucleus. More 
important physically is the radial distribution function, 47rr 2 ^ 2 (r), which gives 
the probability of finding the electron within a spherical shell of thickness dr 



Sec. 23] 



PARTICLES AND WAVES 



283 



at a distance r from the nucleus, irrespective of direction. (Compare the 
problem of gas-velocity distribution on page 187.) The radial distribution 
functions are shown in (b), Fig. 10.12. In place of the sharply defined electron 
orbits of the Bohr theory, there is a more diffuse distribution of electric 
charge. The maxima in these distribution curves, however, correspond closely 
with the radii of the old Bohr orbits. Yet there is always a definite probability 




n -2 

/ -o 




> 2, m - 1 



3, m - 



n = 3, m 2 n = 3, m = db 1 n = 3, w = n = 4, m 




Fig. 10.13. Electron clouds of the H atom. (From Herzberg, Atomic Spectra and 
Atomic Structure, Dover, 1944.) 

of finding the electrons much closer to or much farther from the nucleus. 
The strict determinism of position in the classical description has been 
replaced by the probability language of wave mechanics. 

A particularly clear illustration of the wave mechanical representation of 
the hydrogen atom can be obtained from the illustrations in Fig. 10.13. 
Here the intensity of the shading is proportional to the value of y> 2 , the 
probability distribution function. There is a greater probability of an electron 
being in a light-colored region. It should be clearly understood that quantum 
mechanics does not say that the electron itself is smeared out into a cloud. 



284 



PARTICLES AND WAVES 



[Chap. 10 



It is still to be regarded as a point charge. Its position and momentum cannot 
be simultaneously fixed, and all that the theory can predict that has physical 
meaning is the probability that the electron is in any given region. 

A wave function for an electron is sometimes called an orbital. When 
/ = we have an 5 orbital, which is always spherically symmetrical. When 
/ = 1 we have a p orbital. The p orbitals can have various orientations in 
space corresponding to the allowed values of w, which may be 1, 0, or +L 
In Fig. 10.14, the angular parts of the wave functions are represented for s 

z 




Fig. 10.14. Polar representation of absolute values of angular part of wave 
function for the H atom. The j-type function (/ == 0) is spherically symmetrical. 
There are three possible />-type functions, directed along mutually perpendicular 
axes (x, y. z). 

and p orbitals, and the directional 1 character of the p orbitals is very evident. 
It will be shown later that the directional character of certain chemical 
bonds is closely related to the directed orientations of these orbitals. 

24. The spinning electron. There is one aspect of atomic spectra that 
cannot be explained on the basis of either the old quantum theory or the 
newer wave mechanics. This is the multiplicity or multiple! structure of 
spectral lines. Typical of this multiplicity are the doublets occurring in the 
spectra of the alkali metals: for example, in the principal series each line is 
in reality a closely spaced double line. This splitting is revealed immediately 
with a spectroscope of good resolving power. The occurrence of double lines 
indicates that each term or energy level for the optical electron must also be 
split into two. 



Sec. 25] PARTICLES AND WAVES 285 

A satisfactory explanation for the occurrence of multiple energy levels 
was first proposed in 1925 by G. E. Uhlenbeck and S. Goudsmit. They 
postulated that an electron itself may be considered to be spinning on its 
axis. 20 As a result of spin the electron has an inherent angular momentum. 
Along any prescribed axis in space, for example, the direction of a magnetic 
field, the components of the spin angular momentum are restricted to values 
given by sh/2-rr, where s can have only a value of + J or -J. 

In effect, the electron spin adds a new quantum number s to those re- 
quired to describe completely the state of an electron. We now have, therefore, 
the following quantum numbers: 

n the principal quantum number; allowed values 1, 2, 3, . . . 

/ the azimuthal quantum number, which gives the orbital angular 

momentum of the electron; allowed values 0, 1, 2, . . ., n I. 
m the magnetic quantum number, which gives the allowed orientation 

of the "orbits" in an external field; allowed values /, / + 1 

-/ + 2, . . ., + /. 
s the spin quantum number; allowed values \-\ or J. 

25. The Pauli Exclusion Principle. An exact solution of the wave equation 
for an atom has been obtained only in the case of hydrogen; i.e., for the 
motion of a single electron in a spherically symmetric coulombic field. 
Nevertheless, in more complex atoms the energy levels can still be specified 
in terms of the four quantum numbers n, /, m, s 9 although in many cases the 
physical picture of the significance of the numbers will be lost. This is 
especially true of electrons in inner shells, for which a spherically symmetric 
field would be a very poor approximation. On the other hand, the behavior 
of an outer or valence electron is sometimes strikingly similar to that of the 
electron in the hydrogen atom. In any case, the important fact is that the 
four quantum numbers still suffice to specify completely the state of an 
electron even in a complex atom. 

There is a most important principle that determines the allowable quan- 
tum numbers for an electron in an atom and consequently has the most 
profound consequences for chemistry. It is the Exclusion Principle, first 
enunciated by Wolfgang Pauli. In a single atom no two electrons can have 
the same set of four quantum numbers, n, /, m, s. At present this principle 
cannot be derived from fundamental concepts, but it may have its ultimate 
origin in relativity theory. It is suggestive that relativity theory introduces a 
"fourth dimension," so that a fourth quantum number becomes necessary. 

26. Structure of the periodic table. The general structure of the periodic 
table is immediately clarified by the Exclusion Principle. We recall that even 
in a complex atom the energy levels of the electrons can be specified by 



80 No attempt will be made to reconcile this statement with the idea that an electron is 
x>int charge. It is merely 
properties of electrons, ab 



.a point charge. It is merely a convenient pictorial way of speaking of one of the fundamental 
about which the complete story is not yet written. 



286 



PARTICLES AND WAVES 



[Chap. 10 



means of four quantum numbers : , /, m, s. The Exclusion Principle requires 
that no two electrons in an atom can have the same values for all four quan- 
tum numbers. The most stable state, or ground state, of an atom will be that 
in which the electrons are in the lowest possible energy levels that are con- 
sistent with the Exclusion Principle. The structure of the periodic table is a 
direct consequence of this requirement. 

The lowest atomic energy state is that for which the principal quantum 
number n is 1, and the azimuthal quantum number / is 0. This is a Is state. 
The hydrogen atom has one electron and this goes therefore into the \s level. 
The helium atom has two electrons, which may both be accommodated in 
the \s state if they have opposing spins. With two electrons in the \s state, 
there is an inert gas configuration since the shell n 1 or K shell is com- 
pleted. The completed shell cannot add electrons and a large energy would 
be needed to remove an electron. 

Continuing to feed electrons into the lowest lying energy levels, we come 
to lithium with 3 electrons. The first two go into the Is levels, and the third 
electron must occupy a 2s level. The 2s electron is much less tightly bound 
than the Is electrons. The first ionization potential of Li is 5.39 ev, the second 
75.62 ev. This is true because the 2s electron is usually much farther from 
the nucleus than the Is, and besides it is partially shielded from the +3 
nuclear charge by the two Is electrons. A Is electron, on the other hand, is 
held by the almost unshielded +3 nuclear charge. 

The L shell, with n --= 2, can hold 8 electrons two 2s and six 2p electrons, 
the quantum numbers being as follows : 

n I 

2 



m 


-1 



+ 1 



s 
4 

4 
4 
4 



When the L shell is filled, the next electron must enter the higher-lying 
M shell of principal quantum number n = 3. 

A qualitative picture of the stability of the complete octet is obtained by 
considering the elements on either side of neon. 





Z 


Is 


2s 


IP 


3s 


o . 


8 


2 


2 


4 




F . 


9 


2 


2 


5 




Ne . 


10 


2 


2 


6 




Na . 


11 


2 


2 


6 


1 


Mg . . 


12 


2 


2 


6 


2 



The attraction of an electron by the positively charged nucleus is governed 
by Coulomb's Law, but for electrons outside the innermost shell the shielding 
effect of the other electrons must be taken into consideration. For a given 



Sec. 27] PARTICLES AND WAVES 287 

electron, the shielding effect of other electrons is pronounced only if they 
lie in a shell between the given electron and the nucleus. Electrons in the 
same shell as the given electron have little shielding effect. 

Thus in fluorine, the nuclear charge is -f 9; each of the five 2p electrons 
is attracted by this +9 charge minus the shielding of the Is and 2s electrons, 
four in all, resulting in an effective nuclear charge of about 4 5. The 2p 
electrons in fluorine are therefore tightly held, the first ionization potential 
being about 18 volts. If an extra electron is added to the 2p level in fluorine, 
forming the fluoride ion F~, the added electron is also tightly held by the 
effective +5 nuclear charge. The electron affinity of F is 4.12ev; that is, 
F + e->F- + 4.12ev. 

Now suppose one attempted to add another electron to F~ to form F 53 . 
This electron would have to go into the 3s state. In this case, all ten of the 
inner electrons would be effective in shielding the -f 9 nucleus, and indeed 
the hypothetical eleventh electron would be repelled rather than attracted. 
Thus the fluoride ion is by far the most stable configuration and the - 1 
valence of fluorine is explained. If the tendency of one atom to add an 
electron (electron affinity) is of the same magnitude as the tendency of 
another atom to lose an electron (ionization potential) a stable electrovalent 
bond is possible. 

Considering now the sodium atom, we can see that its eleventh electron, 
3s 1 , is held loosely (/ ~ 5.11 ev). It is shielded from the +11 nucleus by 
10 inner electrons. 

If we continue to feed electrons into the allowed levels, we find that the 
3/? level is complete at argon (Is 2 2s 2 2/? 6 3s 2 3/? 6 ), which has the stable s 2 /? 6 octet 
associated with inert gas properties. 

27. Atomic energy levels. In Table 10.4 the assignment of electrons to 
levels is shown for all the elements, in accordance with our best present 
knowledge as derived from chemical and spectroscopic data. 

In the element following argon, potassium with Z = 19, the last electron 
enters the 4s orbital. This is required by its properties as an alkali metal, and 
the fact that its spectral ground state is *S as in Li and Na. We may well 
ask, however, why the 4s orbitals are lower than the 3d orbitals, which pro- 
vide 10 vacant places. The answer to this question should help to clarify the 
structure of the remainder of the periodic table and the properties of the 
elements in the transition series. It may be noted that in this section we are 
speaking of orbitals, or quantum mechanical wave functions y> for the elec- 
trons. The Bohr picture was useful in dealing with the lighter elements (up to 
A) but it gives an inadequate picture of the remainder of the periodic table. 

The reason why the 4s orbital for potassium has a lower energy than a 
3d orbital arises from the fundamental difference in form of s, p, and d 
orbitals. The electron distributions in. the 3s, 3/?, and 3d orbitals for the 
hydrogen atom were shown in (b), Fig. 10.12. The ordinates of the curves 
are proportional to the radial distribution functions, and therefore to the 



TABLE 10.4 
ELECTRON CONFIGURATIONS OF THE ELEMENTS 



Shell: 


K 


L 


A/ 


AT 


Element 


Is 


2s 2p 


3* 3/7 3rf 


45 4p 4d 4f 


1. H 


1 








2. He 


2 








3. Li 


2 


1 






4. Be 


2 


2 






5. B 


2 


2 1 






6. C 


2 


2 2 






7. N 


2 


2 3 






8. 


2 


2 4 






9. F 


2 


2 5 






10. Ne 


2 


2 6 






11. Na 


2 


2 6 


1 




12. Mg 


2 


2 6 


2 




13. Al 


2 


2 6 


2 1 




14. Si 


2 


2 6 


2 2 




15. P 


2 


2 6 


2 3 




16. S 


2 


2 6 


2 4 




17. Cl 


2 


2 6 


2 5 




18. A 


2 


2 6 


2 6 




19. K 


2 


2 6 


2 6 


1 


20. Ca 


2 


2 6 


2 6 


2 


21. Sc 


2 


2 6 


2 6 1 


2 


22. Ti 


2 


2 6 


262 


2 


23. V 


2 


2 6 


263 


2 


24. Cr 


2 


2 6 


265 


1 


25. Mn 


2 


2 6 


265 


2 


26. Fe 


2 


2 6 


266 


2 


27. Co 


2 


2 6 


267 


2 


28. Ni 


2 


2 6 


268 


2 


29. Cu 


2 


2 6 


2 6 10 


1 


30. Zn 


2 


2 6 


2 6 10 


2 


31. Ga 


2 


2 6 


2 6 10 


2 1 


32. Ge 


2 


2 6 


2 6 10 


2 2 


33. As 


2 


2 6 


2 6 10 


2 3 


34. Se 


2 


2 6 


2 6 10 


2 4 


35. Br 


2 


2 6 


2 6 10 


2 5 


36. Kr 


2 


2 6 


2 6 10 


2 6 



Shell: 


K 


L 


Af 


AT 





P 


Q 


Element 








45 4p 4d 4f 


5s 5p 5d 5f 5g 


6s 6p 6d 


Is 


37. Rb 
38. Sr 


2 
2 

2 
2 
2 
2 
2 
2 
2 
2 


8 
8 


18 
18 


2 6 
2 6 


1 

2 


6 


2 




39. Y 
40. Zr 
41. Nb 
42. Mo 
43. Tc 
44. Ru 
45. Rh 
46. Pd 


8 
8 
8 
8 
8 
8 
8 
8 


18 
18 
18 
18 
18 
18 
18 
18 


2 6 1 
262 
264 
265 
2 6 (5) 
267 
268 
2 6 10 


2 
2 
1 
1 
(2) 
1 
1 



TABLE 10.4 (Cont.) 



Shell: 


K 


L 


M 


N 


O 






/> 


(2 


Element 








4s 4j> 4d 4f 


5s 5p 5d . 


V% 


6s 


6p 6d 


7* 


47. Ag 


2 


8 


18 


2 6 10 


1 










48. Cd 


2 


8 


18 


2 6 10 


2 










49. In 


2 


8 


18 


2 6 10 


2 1 










50. Sn 


2 


8 


18 


2 6 10 


2 2 










51. Sb 


2 


8 


18 


2 6 10 


2 3 










52. Te 


2 


8 


18 


2 6 10 


2 4 










53. I 


2 


8 


18 


2 6 10 


2 5 










54. Xe 


2 


8 


18 


2 6 10 


2 6 










55. Cs 


2 


8 


18 


2 6 10 


2 6 




1 






56. Ba 


2 


8 


18 


2 6 10 


2 6 




2 






57. La 


2 


8 


18 


2 6 10 


2 6 1 




2 






58. Ce 


2 


8 


18 


2 6 10 2 


2 6 




2 






59. Pr 


2 


8 


18 


2 6 10 3 


2 6 




2 






60. Nd 


2 


8 


18 


2 6 10 4 


2 6 




2 






61. Pm 


2 


8 


18 


2 6 10 5 


2 6 




2 






62. Sm 


2 


8 


18 


2 6 10 6 


2 6 




2 






63. Eu 


2 


8 


18 


2 6 10 7 


2 6 




2 






64. Gd 


2 


8 


18 


2 6 10 7 


2 6 




2 






65. Tb 


2 


8 


18 


2 6 10 8 


2 6 




2 






66. Dy 


2 


8 


18 


2 6 10 9 


2 6 




2 






67. Ho 


2 


8 


18 


2 6 10 10 


2 6 




2 






68. Er 


2 


8 


18 


2 6 10 11 


2 6 




2 






69. Tu 


2 


8 


18 


2 6 10 13 


2 6 




2 






70. Yb 


2 


8 


18 


2 6 10 14 


2 6 




2 






71. Lu 


2 


8 


18 


2 6 10 14 


2 6 1 




2 






72. Hf 


2 


8 


18 


2 6 10 14 


262 




2 






73. Ta 


2 


8 


18 


2 6 10 14 


263 




2 






74. W 


2 


8 


18 


2 6 10 14 


264 




2 






75. Re 


2 


8 


18 


2 6 10 14 


265 




2 






76. Os 


2 


8 


18 


2 6 10 14 


266 




2 






77. Ir 


2 


8 


18 


2 6 10 14 


267 




2 






78. Pt 


2 


8 


18 


2 6 10 14 


269 




1 






79. Au 


2 


8 


18 


2 6 10 14 


2 6 10 




1 






80. Hg 


2 


8 


18 


2 6 10 14 


2 6 10 




2 






81. Tl 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


1 




82. Pb 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


2 




83. Bi 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


3 




84. Po 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


4 




85. At 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


5 




86. Rn 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


6 




87. Fr 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


6 


1 


88. Ra 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


6 


2 


89. Ac 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


6 1 


2 


90. Th 


2 


8 


18 


2 6 10 14 


2 6 10 




2 


6 2 


2 


91. Pa 


2 


8 


18 


2 6 10 14 


2 6 10 


2 


2 


6 1 


2 


92. U 


2 


8 


18 


2 6 10 14 


2 6 10 


3 


2 


6 1 


2 


93. Np 


2 


8 


18 


2 6 10 14 


2 6 10 


5 


2 


6 


2 


94. Pu 


2 


8 


18 


2 6 10 14 


2 6 10 


6 


2 


6 


2 


95. Am 


2 


8 


18 


2 6 10 14 


2 6 10 


7 


2 


6 


2 


96. Cm 


2 


8 


18 


2 6 10 14 


2 6 10 


1 


2 


6 1 


2 


97. Bk 


2 


8 


18 


2 6 10 14 


2 6 10 


8 


2 


6 1 


2 


98. Cf 


2 


8 


18 


2 6 10 14 


2 6 10 


9 


2 


6 1 


2 



290 PARTICLES AND WAVES [Chap. 10 

probability of finding an electron within a given region. Now, of course, 
these hydrogen wave functions are not a completely accurate picture of the 
orbitals in a more complex atom with many electrons. The approximation 
is satisfactory, however, for valence electrons, which move in the hydrogen- 
like field of a nucleus shielded by inner electrons. 

The 4s and 3p orbitals predict a considerable concentration of the charge 
cloud closely around the nucleus, 21 whereas the 3d orbital predicts an ex- 
tremely low probability of finding the electron close to the nucleus. As a 
result of this penetration of the 4s orbital inward towards the nucleus, a 
4s electron will be more tightly bound by the positive nuclear charge, and 
will therefore be in a lower energy state than a 3d electron, whose orbital 
does not penetrate, and which is therefore more shielded from the nucleus 
by the inner shells. It is true that the most probable position for a 4s electron 
is farther from the nucleus than that for a 3d electron; the penetration effect 
more than makes up for this, since the coulombic attraction decreases as 
the square of r, the distance of the electron from the nucleus. Since 4s lies 
lower than 3d, the nineteenth electron in potassium enters the 4s rather than 
the 3d level, and potassium is a typical alkali metal. 

In Fig. 10.15 the relative energies of the orbitals are plotted as functions 
of the atomic number (nuclear charge). This graph is not quantitatively 
exact, but is designed to show roughly how the relative energy levels of the 
various orbitals change with increasing nuclear charge. The energies are 
obtained from atomic spectra. 

Although the effect is not shown in the figure, it should be noted that the 
energy levels of the s and p orbitals fall steadily with increasing atomic 
number, since the increasing nuclear charge draws the penetrating s and p 
orbitals closer and closer to the nucleus. At low atomic numbers, up to 
Z ^ 20 (Ca), the 3d levels are not lowered, since there are not yet sufficient 
electrons present for the d's to penetrate the electron cloud that surrounds 
and shields the nucleus. As more electrons are added, however, the 3d 
orbitals eventually penetrate the shielding electrons and begin to fall with 
increasing Z. This phenomenon is repeated later with the 4rfand 4f orbitals. 
At high Z, therefore, orbitals with the same principal quantum number tend 
to lie together; at low Z they may be widely separated because of different 
penetration effects. 

Following calcium, the 3d orbitals begin to be filled rather than the 4p. 
One obtains the first transition series of metals, Sc, Ti, V, Cr, Mn, Fe, Co, Ni. 
These are characterized by variable valence and strongly colored compounds. 
Both these properties are associated with the closeness of the 4s and 3d levels, 

21 The distinct difference between this quantum-mechanical picture and the classical 
Bohr orbits should be carefully noted. There are four successive maxima in the y function 
for the 4s orbital, at different distances from the nucleus. The quantum mechanical picture 
of an atom is a nucleus surrounded by a cloud of negative charge. There are differences in 
density of the cloud at different distances from the nucleus. The cloud is the superposition 
of the v> functions for all the orbitals occupied by electrons. 



Sec. 27] 



PARTICLES AND WAVES 



291 



which provide a variable number of electrons for bond formation, and 
possible excited levels at separations corresponding with the energy available 
in visible light (~ 2 ev). 

The filling of the 3d shell is completed with copper, which has the con- 
figuration \s*2s*2p*3s*3p*3d l 4s l . Copper is not an alkali metal despite the 
outer 4s electron, since the 3d level is only slightly below the 4s and Cu++ 
ions are readily formed. 





2s 



Fig. 10.15. Dependence of energies of orbitals on the nuclear charge Z. 

The next electrons gradually fill the 4s and 4p levels, the process being 
completed with krypton. The next element, rubidium, is a typical alkali with 
one 5s electron outside the 4s 2 4p B octet. Strontium, with two 5s electrons, is 
a typical alkaline earth of the Mg, Ca, Sr, Ba series. 

Now, however, the 4d levels become lowered sufficiently to be filled 
before the 5p. This causes the second transition series, which is completed 
with palladium. Silver follows with the copper type structure, and the filling 
of the 5,y and 5p levels is completed with xenon. A typical alkali (Cs) and 
alkaline earth (Ba) follow with one and two 6s electrons. 

The next electron, in lanthanum, enters the 5d level, and one might 
suspect that a new transition series is underway. Meanwhile, however, with 



292 PARTICLES AND WAVES [Chap. 10 

increasing nuclear charge, the 4forbitals have been drastically lowered. The 
4/ levels can hold exactly 14 electrons. 22 As these levels are filled, we obtain 
the 14 rare earths with their remarkably similar chemical properties, deter- 
mined by the common 5s 2 5p 6 outer configuration of their ions. This process 
is complete with lutecium. 

The next element is hafnium, with 5d 2 6s 2 . Its properties are very similar 
to those of zirconium with 4d 2 5s 2 . This similarity in electronic structures was 
predicted before the discovery of hafnium, and led Coster and Hevesy to look 
for the missing element in zirconium minerals, where they found it in 1923. 

Following Hf the 5d shell is filled, and then the filling of the 6p levels is 
completed, the next s 2 p* octet being attained with radon. The long missing 
halogen (85) and alkali (87) below and above radon have been found as 
artificial products from nuclear reactions. 23 They are called "astatine" and 
"francium." 

Radium is a typical alkaline earth metal with two Is electrons. In the 
next element, actinium, the extra electron enters the 6d level, so that the 
outer configuration is 6d l ls 2 ; this is to be compared with lanthanum with 
5d*6s 2 . It was formerly thought that the filling of the 6d levels continued in 
the elements following actinium. As i result of studies of the properties of 
the new transuranium elements it now appears more likely that actinium 
marks the beginning of a new rare-earth group, successive electrons entering 
the 5f shell. Thus the trivalent state becomes more stable compared to the 
quadrivalent state as one proceeds through Ac, Th, Pa, U, Np, Pu, Am, Cm, 
just as it does in the series La, Ce, Pr, Nd, Pm, Sm, Eu, etc. This is true because 
successive electrons added to the/shell are more tightly bound as the nuclear 
charge increases. The actinide "rare earths" therefore resemble the lanthanide 
rare earths rather than the elements immediately above themselves in the 
periodic table. 

PROBLEMS 

1. What is the average energy, , of a harmonic oscillator of frequency 
10 13 sec- 1 at 0, 200, 1000C? What is the ratio ejkT at each temperature? 

2. The K* X-ray line of iron has a wavelength of 1 .932 A. A photon of 
this wavelength is emitted when an electron falls from the L shell into a 
vacancy in the K shell. Write down the electronic configuration of the ions 
before and after emission of this line. What is the energy difference in kcal 
per mole between these two configurations? 

" As follows: 

n 4 
/ 3 

m 3, -2, -1, 0, 4-1, +2, -fl 

* i, i, i, i, t, t, * 

23 For an excellent account, see Glenn T. Seaborg, "The Eight New Synthetic Elements," 
American Scientist, 36, 361 (1948). 



Chap. 10] PARTICLES AND WAVES 293 

3. The fundamental vibration frequency of N 2 corresponds to a wave 
number of 2360 cm" 1 . What fraction of N 2 molecules possess no vibrational 
energy (except their zero-point energy) at 25C ? 

4. The first line in the Lyman series lies at 1216 A, in the Balmer series, 
at 6563 A. In the absorption spectra of a certain star, the Balmer line appears 
to have one-fourth the intensity of the Lyman line. Estimate the temperature 
of the star. 

5. Calculate the ionization potential of hydrogen as the energy required 
to remove the electron from r r - 0.53A to infinity against the coulombic 
attraction of the proton. 

6. An excited energy level has a lifetime of 10~ 10 sec. What is the mini- 
mum width of the spectral line arising in a transition from the ground state 
to this level ? 

7. Calculate the wavelength of a proton accelerated through a potential 
difference of 1 mev. 

8. For a particle of mass 9 x 10~ 28 g confined to a one-dimensional box 
100 A long, calculate the number of energy levels lying between 9 and 10 ev. 

9. Consider an electron moving in a circular path around the lines of 
force in a magnetic field. Apply the Bohr quantum condition eq. (10.21) to 
this rotation. What is the radius of the orbit of quantum number n 1 in 
a magnetic field of 10 5 gauss? 

10. The K al X-ray line is emitted when an electron falls from an L level 
to a hole in the K level. Assume that the Rydberg formula holds for the 
energy levels in a complex atom, with an effective nuclear charge 7! equal 
to the atomic number minus the number of electrons in shells between the 
given electron and the nucleus. On this basis, estimate the wavelength of 
the Af al X-ray line in chromium. The experimental value is 2.285 A. 

11. The wave function for the electron in the ground state of the hydrogen 
atom is y ls = (7ra 3 )~ 1/2 e~ r/a , where a is the radius of the Bohr orbit. 
Calculate the probability that an electron will be found somewhere between 
0.9 and 1.1 a . What is the probability that the electron will be beyond 2 a ? 

12. Write an account of the probable inorganic chemistry of Np, Pu, 
Am, Cm, in view of their probable electron configurations. Compare the 
chemistry of astatine and iodine, francium and cesium. 

REFERENCES 

BOOKS 

1. de Broglie, L., Matter and Light (New York: Dover, 1946). 

2. Heitler, W., Elementary Wave Mechanics (New York: Oxford, 1945). 

3. Herzberg, G., Atomic Spectra and Atomic Structure (New York: Dover 
Publications, 1944). 



294 PARTICLES AND WAVES [Chap. 10 

4. Mott, N. F., Elements of Wave Mechanics (Cambridge: Cambridge Univ. 
Press, 1952). 

5. Pauling, L., and E. B. Wilson, Introduction to Quantum Mechanics (New 
York: McGraw-Hill, 1935). 

6. Pitzer, K. S., Quantum Chemistry (New York: Prentice-Hall, 1953). 

7. Slater, J. C, Quantum Theory of Matter (New York: McGraw-Hill, 1951). 

8. Whittaker, E. T., From Euclid to Eddington, A Study of Conceptions 
of the External World (London: Cambridge, 1949). 

ARTICLES 

1. Compton, A. H., Am. J. Phys. 9 14, 80-84 (1946), "Scattering of X-Ray 
Photons." 

2. de Vault, D., /. Chem. Ed., 21, 526-34, 575-81 (1944), "The Electronic 
Structure of the Atom." 

3. Glockler, G.,J. Chem. Ed., 18, 418-23 (1941), "Teaching the Introduction 
to Wave Mechanics." 

4. Margenau, H., Am. J. Phys., 13, 73-95 (1945); 12, 119-30, 247-68 (1944), 
"Atomic and Molecular Theory Since Bohr." 

5. Meggers, W. F., /. Opt. Soc. Am., 41, 143-8 (1951), "Fundamental 
Research in Atomic Spectra." 

6. Zworykin, V. K., Science in Progress, vol. Ill (New Haven: Yale Univ. 
Press, 1942), pp. 69-107, "Image Formation by Electrons." 



CHAPTER 11 

The Structure of Molecules 



1. The development of valence theory. The electrical discoveries at the 
beginning of the nineteenth century strongly influenced the concept of the 
chemical bond. Indeed, Berzelius proposed in 1812 that all chemical com- 
bination was caused by electrostatic attraction. As it turned out 115 years 
later, this theory happened to be true, though not in the sense supposed by 
its originator. It did much to postpone the acceptance of diatomic structures 
for the common gaseous elements, such as H^ N 2 , and O 2 . It was admitted 
that most organic compounds fitted very poorly into the electrostatic scheme, 
but until 1828 it was widely believed that these compounds were held together 
by "vital forces," arising by virtue of their formation from living things. In 
that year, Wohler's synthesis of urea from ammonium cyanate destroyed this 
distinction between organic and inorganic compounds, and the vital forces 
gradually retreated to their present refuge in living cells. 

Two general classes of compounds came to be distinguished, with an 
assortment of uncomfortably intermediate specimens. The polar compounds, 
of which NaCl was a prime example, could be adequately explained as being 
composed of positive and negative ions held together by coulombic attrac- 
tion. The nature of the chemical bond in the nonpolar compounds, such as 
CH 4 , was completely obscure. Nevertheless, the relations of valence with 
the periodic table, which were demonstrated by Mendeleev, emphasized the 
remarkable fact that the valence of an element in a definitely polar compound 
was usually the same as that in a definitely nonpolar compound, e.g., O in 
K 2 O and (C 2 H 5 ) 2 O. 

In 1904 Abegg pointed out the rule of eight: To many elements in the 
periodic table there could be assigned a negative valence and a positive valence 
the sum of which was eight, for example, Cl in LiCl and C1 2 O 7 , N in NH 3 
and N 2 O 5 . Drude suggested that the positive valence was the number of 
loosely bound electrons that an atom could give away, and the negative 
valence was the number of electrons that an atom could accept. 

Once the concept of atomic number was clearly established by Moseley 
(1913), further progress was possible, for then the number of electrons in an 
atom became known. The special stability of a complete outer octet of 
electrons was soon noticed. For example: He, 2 electrons; Ne, 2 + 8 elec- 
trons; A, 2 4- 8 + 8 electrons. In 1916, W. Kossel made an important con- 
tribution to the theory of the electrovalent bond, and in the same year 
G. N. Lewis proposed a theory for the nonpolar bond. 

Kossel explained the formation of stable ions by a tendency of the atoms 

295 



296 THE STRUCTURE OF MOLECULES [Chap. 11 

to gain or lose electrons until they achieve an inert-gas configuration. Thus 
argon has a completed octet of electrons. Potassium has 2 + 8 + 8 + 1, and 
it tends to lose the outer electron, becoming the positively charged K+ ion 
having the argon configuration. Chlorine has 2 + 8 + 7 electrons and tends 
to gain an electron, becoming Cl with the argon configuration. If an atom 
of Cl approaches one of K, the K donates an electron to Cl, and the resulting 
ions combine as K f Cl:, the atoms displaying their valences of one. The 
extension to other ionic compounds is familiar. 

G. N. Lewis proposed that the links in nonpolar compounds resulted 
from the sharing of pairs of electrons between atoms in such a way as to 
form stable octets to the greatest possible extent. Thus carbon has an atomic 
number of 6; />., 6 outer electrons, or 4 less than the stable neon configura- 
tion. It can share electrons with hydrogen as follows: 

H 

xo 



XO 

H 

Each pair of shared electrons constitutes a single covalent bond. The Lewis 
theory explained why the covalence and electrovalence of an atom are usually 
identical, for an atom usually accepts one electron for each covalent bond 
that it forms. 

The development of the Bohr theory led to the idea that the electrons 
were contained in shells or energy levels at various distances from the nucleus. 
These shells were specified by the quantum numbers. By about 1925, a 
systematic picture of electron shells was available that represented very well 
the structure of the periodic table and the valence properties of the elements. 
The reason why the electrons are arranged in this way was unknown. The 
reason why a shared electron pair constitutes a stable chemical bond was 
also unknown. 

An answer to both these fundamental chemical problems was provided 
by the Pauli Exclusion Principle. Its application to the problem of the 
periodic table was shown in the previous chapter. Its success in explaining 
the nature of the chemical bond has been equally remarkable. 

2. The ionic bond. The simplest type of molecular structure to understand 
is that formed from two atoms, one of which is strongly electropositive (low 
ionization potential) and the other, strongly electronegative (high electron 
affinity). Such, for example, would be sodium and chlorine. In crystalline 
sodium chloride, one cannot speak of an NaCl molecule since the stable 
arrangement is a three-dimensional crystal structure of Na+ and Cl~ ions. 
In the vapor, however, a true NaCl molecule exists, in which the binding is 
almost entirely ionic. 

The attractive force between two ions with charges q and q 2 can be 
represented at moderate distances of separation r by the coulombic force 



Sec. 3] 



THE STRUCTURE OF MOLECULES 



297 



or ty a potential V ~q\q^r. If the ions are brought so close to- 
gether that their electron clouds begin to overlap, a mutual repulsion between 
the positively charged nuclei becomes evident. Born and Mayer have sug- 
gested a repulsive potential having the form U be~ r/a , where a and b are 
constants. 

The net potential for two ions is therefore 



+ be r/a 



(11-1) 



This potential-energy function is plotted in Fig. 11.1 for NaCl, the minimum 
in the curve representing the stable internuclear separation for a Na+Cl~ 




5 10 15 20 

INTERNUCLEAR SEPARATION, r, A 

Fig. 11.1. Potential energy of Na + }- Cl . (The internuclear distance in the stable 
molecule is 2.51 A. Note the long range of the coulombic attraction.) 

molecule. Spectra of this molecule are observed in the vapor of sodium 
chloride. 

3. The covalent bond. One of the most important of all the applications 
of quantum mechanics to chemistry has been the explanation of the nature 
of the covalent bond. The simplest example of such a bond is found in the 
H 2 molecule. Although Lewis, in 1918, declared that this bond consists of a 
shared pair of electrons, it was in 1927 that a real understanding of the 
nature of the binding was provided by the work of W. Heitler and F. London. 

If two H atoms are brought together there results a moderately com- 
plicated system consisting of two 4- 1 charged nuclei and two electrons. If 
the atoms are very far apart their mutual interaction is effectively nil. In 



298 



THE STRUCTURE OF MOLECULES 



[Chap. 1 1 



other words, the potential energy of interaction V ~ when the internuclear 
distance r oo. At the other extreme, if the two atoms are forced very 
closely together, there is a large repulsive force between the two positively 
charged nuclei, so that as r -> 0, U > oo. Experimentally we know that two 
hydrogen atoms can unite to form a stable hydrogen molecule, whose dis- 
sociation energy is 4.48 ev, or 103.2 kcal per mole. The internuclear separation 
in the molecule is 0.74 A. 




-5 - 



0.5 



2.5 



I 1.5 2 

ANGSTROMS 

Fig. 11.2. Potential energy curve for hydrogen molecule. (Note the shorter 
range of the valence forces in H 2 , as compared with the ionic molecule NaCl 
shown in Fig. 11.1.) 

These facts about the interaction of two H atoms are summarized in the 
potential-energy curve of Fig. 11.2. The problem before us is to explain the 
minimum in the curve. This is simply another way of asking why a stable 
molecule is formed, or what is the essential nature of the covalent bond in H 2 . 

The quantum-mechanical problem is to solve the Schrodinger equation 
for the system of two electrons and two protons. Consider the situation in 
Fig. 11.3, where the outer electron orbits overlap somewhat. According to 
quantum mechanics, of course, these orbits are not sharp. There are eigen- 
functions ^(1) for electron (1) and y(2) for electron (2), which determine 
the probability of finding the electrons at any point in space. As long as the 
atoms are far apart, the eigenfunction for electron (1) on nucleus (a) will be 
simply that found on page 281 for the ground state of a hydrogen atom 
namely, y ls (\) - (7ra *)- l/2 e~ r/a : 



Sec. 3] THE STRUCTURE OF MOLECULES 299 

For the two electrons, a wave function is required that expresses the 
probability of simultaneously finding electron (1) on nucleus (a) and electron 
(2) on nucleus (b). Since the combined probability is the product of the two 
individual probabilities, such a function would be a(\)b(2). Here a(\) and 
b(2) represent eigenfunctions for electron (1) on nucleus (a) and electron (2) 
on nucleus (b). 

A very important principle must now be considered. There are no physical 
differences and no way of distinguishing between a system with (1) on (a) 
and (2) on (b) and a system with (2) on 
(a) and (1) on (b). The electrons cannot 
be labeled. The proper wave function 
for the system must contain in itself an 
expression of this fundamental truth. 

To help solve this problem we need 
only recall from page 254 that if ^i 
and y> 2 are two solutions of the wave 
equation, then any linear combination Fig< u 3 , nte ract,on of two h>drogen 
of these solutions is also a solution, e.g., atoms. 

c iVi + C 2 1 /V There are two particular 

linear combinations that inherently express the principle that the electrons 
are indistinguishable. These are 

Vf = fl(l)A(2) + a(2)b(\) 
V_ -a(\)b(2) -a( 




If the electrons are interchanged in these functions, y> + is not changed at all; 
it is called a symmetric function. y_ is changed to y>_, but this in itself does 
not change the electron distribution since it is y>* which gives the probability 
of finding an electron in a given region, and ( 1/>) 2 = y> 2 . The function ^_ 
is called antisymmetric. 

So far the spin properties of the electrons have not been included, and 
this must be done in order to obtain a correct wave function. The electron- 
spin quantum number s, with allowed values of either + 1 or I, determines 
the magnitude and orientation of the spin. We introduce two spin functions 
a and ft corresponding to s - +A and s = \. For the two-electron system 
there are then four possible complete spin functions: 

Spin Function Electron 1 Electron 2 
a(l)a(2) -ft +i 

0)0(2) +J -1 

00 M2) -J -fj 

00)0(2) -t -* 

When the spins have the same direction they are said to be parallel', when 
they have opposite directions, antiparallel. 

Once again, however, the fact that the electrons are indistinguishable 



300 



THE STRUCTURE OF MOLECULES 



[Chap. 11 



forces us to choose linear combinations for the two-electron system which 
are either symmetric or antisymmetric. There are three possible symmetric 
spin functions: 

a(l)a(2) \ 

sym 



There is one antisymmetric spin function : 

<x(l)/3(2) oc(2)/?(l) antisym 

The possible complete wave functions for the H-H system are obtained 
by combining these four possible spin functions with the two possible orbital 
wave functions. This leads to eight functions in all. 

At this point in the argument the Pauli Principle enters in an important 
way. The Principle is stated in a more general form than was used before: 
"Every allowable eigenfunction for a system of two or more electrons must 
be antisymmetric for the interchange of the coordinates of any pair of 
electrons." It will be shown a little later that the prohibition against four 
identical quantum numbers is a special case of this statement. 

As a consequence of the exclusion principle, the only allowable eigen- 
functions are those made up either of symmetric orbitals and antisymmetric 
spins or of antisymmetric orbitals and symmetric spins. There are four such 
combinations for the H-H system: 



rbital 


Spin 


Total Spin 


Term 


+ 0(2 W) 


a( 1)0(2) - a(2)0(l) 


(singlet) 


tZ 


- a(2)b(\) 


1 


1 (triplet) 


3 D 




a( 1)0(2) 1 a(2)0(l)J 







' The term symbol S expresses the fact that the molecular state has a 
total angular momentum of zero, since it is made up of two atomic S terms. 
The multipliu.y of the term, or number of eigenfunctions corresponding 
with it, is added as a left-hand superscript. This multiplicity is always 
2f? + 1 where & is the total spin. 

The way in which the general statement of the exclusion principle reduces 
to that in terms of quantum numbers can readily be seen in a typical example. 
Multiplying out the *X function gives y> = aa(l)6(2) aa(2)6(l). If the 
quantum numbers n, /, m are the same for both electrons, their orbital func- 
tions are identical, a = b, so that y = a<x.(\)afi(2) aa(2)00(l). If the fourth 
quantum number s is also the same for both, either or , the spin 
functions must be either both a or both /?. Then y = 0, that is, the proba- 
bility of such a system is zero, In other words, eigenfunctions that assign 



Sec. 4] THE STRUCTURE OF MOLECULES 301 

identical values of n, /, m, and s to two electrons are outlawed. This result 
was shown in a special case, but it is in fact a completely general consequence 
of the requirement of antisymmetry. 

4. Calculation of the energy in H-H molecule. The next step is to calculate 
the energy for the interaction of two hydrogen atoms using the allowed 
wave functions. The different electrostatic interactions are shown in Fig. 1 1 .3: 

i. electron (1) with electron (2), potential, (7, 2 / r i2 

ii. electron (1) with nucleus (/>), / 2 e 2 /r lb 

iii. electron (2) with nucleus (a\ t/ 3 ^ e z /r 2a 

iv. nucleus (a) with nucleus (/?), (7 4 ^ e 2 /r att 

Note that the interactions of electron (1) with its own nucleus (a) and of 
electron (2) with nucleus (b) are already taken into account by the fact that 
we are starting with two hydrogen atoms. 

The potential for the interaction of two electrons a distance r 12 apart is 
/! -e 2 /r l2 . In order to find the energy of interaction, we must multiply 
this by the probability of finding an electron in a given element of volume 
dv, and then integrate over all of space. Since the required probability is 
ifdv, this gives E --- J U^dv. Since the total potential is U U -f t/ 2 + 
U 3 + t/ 4 , the total energy of interaction of the two hydrogen atoms becomes 

--= J t/yVr (11.2) 

This energy must now be calculated for both the symmetric and the anti- 
symmetric orbital wave functions. Squaring these functions, one obtains 

y 2 -0 2 (l)/>*(2) f <P(2)b\ I) 2a( 1)6(2X2)6(1) 

The f sign is for the X S function, the sign for the 3 2 function. 
The integral in eq. (1 1.2) can therefore be written 

E^2C2A (11.3) 

where C = J Ua\\}b*(2)dv 

A - J Ua(\)b(\)a(2)b(2)dv ( ' ' 

C is called the coulombic energy , and A is called the exchange energy. 

The coulombic energy is the result of the ordinary electrostatic interaction 
between the charges of the electrons and the nuclei. The behavior of this 
coulombic energy as the two hydrogen atoms approach each other can be 
estimated qualitatively as follows, although the actual integration is not too 
difficult if we use the simple Is orbitals for a and b. At large intern uclear 
distances, C is zero. At very small distances C approaches infinity owing to 
the strong repulsions between the nuclei. At intermediate distances where 
the electron clouds overlap there is a net attractive potential since portions 
of the diffuse electron clouds are close to the nuclei and the resulting attrac- 
tion more than compensates for the repulsions between different parts of tL 
diffuse clouds and between the still relatively distant nuclei. The resulting 



302 



THE STRUCTURE OF MOLECULES 



[Chap. 1 1 



dependence of the coulombic energy on the internuclear distance r is shown 
as curve C in Fig. 1 1 .4. 

The depth of the minimum in the coulombic potential energy curve is 
only about 0.6 ev compared to the observed 4.75 ev for the H-H bond. The 
classical electrostatic interaction between two hydrogen atoms is thus com- 
pletely inadequate to explain the strong covalent bond. The solution to the 
problem must be in the specifically quantum mechanical phenomenon of the 
exchange energy A. 

The exchange energy arises from the fact that the electrons are indis- 
tinguishable, and besides considering the interaction of electron 1 on nucleus 

a, we have to consider interactions 
occurring as if electron 1 were on 
nucleus b. Since quantum mechanics 
is expressed in the language of y> 
functions, we even have to consider 
interactions arising between charge 
densities that represent electron 1 on 
both a and b simultaneously. Even 
to try to express the phenomenon in 
terms of artificially labeled electrons 
involves us in difficulties, but it is 
clear qualitatively that "exchange" 
may increase the density of electronic 
charge around the positive nuclei 
and so increase the binding energy. 

Like the coulombic energy, the 
exchange energy is zero when there 
is no overlap. At a position of large 
overlap it may lead to a large attrac- 
tive force and large negative potential 
energy. The exact demonstration of this fact would require the evaluation 
of the integral. When this is done we obtain a curve for the variation of 
A with r. 

The total energy of interaction 2C 2A can now be plotted. It is clear 
that 2C I 2 A leads to a deep minimum in the potential energy curve. This 
is the solution for the symmetric orbital wave function; i.e., the anti- 
symmetric spin function. It is the case, therefore, in which the electron spins 
are antiparallel. The spin of one electron is -| i, and that of the other is i. 
The other curve, 2C-2/4, corresponds to the antisymmetric orbital wave 
function, which requires symmetric spin functions, or parallel spins. The 
two curves are drawn as *S and 3 X in Fig. 11.4. The deep minimum in the 
1 S curve indicates that the Heitler-London theory has successfully explained 
the covalent chemical bond in the hydrogen molecule. The binding energy 
is about 10 per cent coulombic, and 90 per cent exchange energy. 



o -I 

Ul 

_i 
u -2 



-3 



-4 




/EXPERIMENTAL 
" CURVE 



05 10 



15 20 
A 



25 30 35 



Fig. 11.4. 



Heitler-London treatment of 
the H 2 molecule. 



Sec. 5] THE STRUCTURE OF MOLECULES 303 

Since the covalent bond is formed between atoms that share a pair of 
electrons with opposite spins, covalence is often called spin valence. Only 
when the spins are opposed is there an attractive interaction due to the 
exchange phenomenon. If the spins are parallel, there is a net repulsion 
between two approaching hydrogen atoms. It is interesting to note that if 
two H atoms are brought together, there is only one chance in four that 
they will attract each other, since the stable state is a singlet and the repulsive 
state is a triplet. 

The Heitler-London theory is an example of the valence-bond (V.B.) 
approach to molecular structure. 

5. Molecular orbitals. An alternative to the Heitler-London method of 
applying quantum mechanics to molecular problems is the method of mole- 
cular orbitals, developed by Hund, Mulliken, and Lennard-Jones. Instead of 
starting with definite atoms, it assumes the nuclei in a molecule to be held 
fixed at their equilibrium separations, and considers the effect of gradually 
feeding the electrons into the resulting field of force. Just as the electrons in 
an atom have definite orbitals characterized by quantum numbers, n, /, m, 
and occupy the lowest levels consistent with the Pauli Principle, so the elec- 
trons in a molecule have definite molecular orbitals and quantum numbers, 
and only two electrons having opposite spins can occupy any particular 
molecular orbital. In our description of the molecular orbital (M.O.) method 
we shall follow an excellent review by C. A. Coulson. 1 

For diatomic molecules, the molecular quantum numbers include a prin- 
cipal quantum number n, and a quantum number A, which gives the com- 
ponents of angular momentum in the direction of the internuclear axis. 
This A takes the place of the atomic quantum number /. We may have states 
designated <r, TT, <5 . . . as A 0, 1 , 2 . . . . 

6. Homonuclear diatomic molecules. Homonuclear diatomic molecules 
are those that are formed from two identical atoms, like H 2 , N 2 , and O 2 . 
Such molecules provide the simplest cases for application of the M.O. 
method. 

If a hydrogen molecule, H 2 , is pulled apart, it gradually separates into 
two hydrogen atoms, H a and H 6 , each with a single \s atomic orbital. If the 
process is reversed and the hydrogen atoms are squeezed together, these 
atomic orbitals coalesce into the molecular orbital occupied by the electrons 
in H 2 . We therefore adopt the principle that the molecular orbital can be 
constructed from a linear combination of atomic orbitals (L.C.A.O.). Thus 
y y(A : Is) -}- yy(B : Is) 

Since the molecules are completely symmetrical, y must be 1. Then 
there are two possible molecular orbitals : 

v , - y(A : Is) + y(B : Is) 
^ M = y(A : Is) ~ y(B : Is) 
1 Quarterly Reviews, 1, 144 (1947). 



304 



THE STRUCTURE OF MOLECULES 



[Chap. 11 



These molecular orbitals are given a pictorial representation in (a), Fig. 
11.5. The Is A.O.'s are spherically symmetrical (see page 283). If two of 
these are brought together until they overlap, the M.O. resulting can be 
represented as shown. The additive one, vv leads to a building up of charge 



(o) 



J V"(A U)t^(B'U) 





B A^(A 



2Py 2Py 



2P) 



F'ig. 11.5. Formation of molecular orbitals by linear combinations of 
atomic orbitals. 

density between the nuclei. The subtractive one, y u , has an empty space free 
of charge between the nuclei. Both these M.O.'s are completely symmetrical 
about the internuclear axis; the angular momentum about the axis is zero, 
and they are called a orbitals. The first one is designated as a a\s orbital. 
It is called a bonding orbital, for the piling up of charge between the nuclei 
tends to bind them together. The second one is written as a* Is, and is an 
antibonding orbital, corresponding to a net repulsion, since there is no 



Sec. 6] THE STRUCTURE OF MOLECULES 305 

shielding between the positively charged nuclei. Antibonding orbitals will 
be designated with a star. 

A further insight into the nature of these orbitals is obtained if we 
imagine the H nuclei squeezed so tightly together that they coalesce into 
the united nucleus of helium. Then the bonding orbital a\s merges into the 
Is atomic orbital of helium. The antibonding o*\s must merge into the next 
lowest A.O. in helium, the 2s. This 2s level is 19.7 ev above the Lv, and this 
energy difference is further evidence of the antibonding nature of the a* Is. 

The electron configurations of the molecules are built up just as in the 
atomic case, by feeding electrons one by one into the available orbitals. In 
accordance with the Pauli Principle, each M.O. can hold two electrons with 
opposite spins. 

In the case of H 2 , the two electrons enter the o\s orbital. The configura- 
tion is (als) 2 and corresponds to a single electron pair bond between the 
H atoms. 

The next possible molecule would be one with three electrons, He 2 +. 
This has the configuration (orl,s) 2 (cr*l,s) 1 . There are two bonding electrons 
and one antibonding electron, so that a net bonding is to be expected. The 
molecule has, in fact, been observed spectroscopically and has a dissociation 
energy of 3.0 ev. 

If two helium atoms are brought together, the result is (crls) 2 (tf* Is) 2 . 
Since there are two bonding and two antibonding electrons, there is no ten- 
dency to form a stable He 2 molecule. We have now used up all of our avail- 
able M.O.'s and must make some more in order to continue the discussion. 

The next possible A.O.'s are the 2s, and these behave just like the Is 
providing a2s and a*2s M.O.'s with accommodations for four more elec- 
trons. If we bring together two lithium atoms with three electrons each, the 
molecule Li 2 is formed. Thus 

Li[]s*2s l ] 4 Li[\s*2s l ] ->Li 2 [(a\s)*(o*\s)*(a2s)*] 

Actually, only the outer-shell or valence electrons need be considered, and 
the M.O.'s of inner #-shell electrons need not be explicitly designated. 
The Li 2 configuration is therefore written as [KK(a2s)' 2 ] . The molecule 
has a dissociation energy of 1.14ev. The hypothetical molecule Be 2 , with 
eight electrons, does not occur, since the configuration would have to be 
[KK(a2s) 2 (a*2s)*\. 

The next atomic orbitals are the 2/?'s shown in Fig. 10.14. There are 
three of these, p X9 p v , p Z9 mutually perpendicular and with a characteristic 
wasp-waisted appearance. The most stable M.O. that can be formed from 
the atomic p orbitals is one with the maximum overlap along the inter- 
nuclear axis. This M.O. is shown in (b), Fig. 1 1.5, and with the corresponding 
antibonding orbital can be written 

y> = ip(A : 2p x ) + y( B : 2 Px) <*lp 
: 2p x ) - y(B : 2p x ) o*2p 



306 



THE STRUCTURE OF MOLECULES 



[Chap. II 



These orbitals have the same symmetry around the internuclear axis as the 
a orbitals formed from atomic s orbitals. They also have a zero angular 
momentum around the axis. 

The M.O.'s formed from the p v and p z A.O.'s have a distinctly different 
form, as shown in (c), Fig. 1 1.5. As the nuclei are brought together, the sides 
of the p y or p z orbitals coalesce, and finally form two streamers of charge 
density, one above and one below the internuclear axis. These are called 
TT orbitals; they have an angular momentum of one unit. 

We can summarize the available M.O.'s as follows, in order of increasing 
energy: 

crhy < cr*l s < o2s < o*2s < o2p < 7T y 2p 7r z 2p < Tr y *2p rr z *2p < o*2p 

With the good supply of M.O.'s now available, the configurations of 
other homonuclear molecules can be determined, by feeding pairs of electrons 
with opposite spins into the orbitals. 

The formation of N 2 proceeds as follows : 



There are six net bonding electrons, so that it can be said that there is a 
triple bond between the two N's. One of these bonds is a a bond; the other 
two are TT bonds at right angles to each other. 
Molecular oxygen is an interesting case: 

O[\s 2 2s*2p*] f- O(\s 2 2s 2 2p 4 ] -> O 2 [KK(a2s) 2 (o*2s) 2 (o2p) 2 (7r2p)*(TT*2p) 2 ] 

There are four net bonding electrons, or a double bond consisting of a a and 
a TT bond. Note that a single bond is usually a a bond, but a double bond is 
not just two equivalent single bonds, but a a plus a TT. In O 2 , the (n*2p) 
orbital, which can hold 4 electrons, is only half filled. Because of electrostatic 
repulsion between the electrons, the most stable state will be that in which 
the electrons occupy separate orbitals and have parallel spins. Thus these 
two electrons are assigned as (TT y *2p) l (7r.*2p) 1 . The total spin of O 2 is then 
if = 1, and its multiplicity, 2^ + 1 ~ 3. The ground state of oxygen is 3 2. 

TABLE 11.1 
PROPERTIES OF HOMONUCLEAR DIATOMIC MOLECULES 



Molecule 





Binding 
Energy 
(ev) 


Internuclear 
Separation 

(A) 


Vibration 
Frequency 
(sec- 1 ) 


Bonding 
Antibonding 
Electrons 




3.6 


.59 


3.15 x 10 13 


2 




5.5 


.31 


4.92 x 10 13 


4 




7.4 


.09 


7.08 x 10 13 


6 




5.1 


.20 


4.74 x 10 13 


4 




3.0 


.30 


3.40 x 10 13 


2 



Sec. 7] THE STRUCTURE OF MOLECULES 307 

In the M.O. method, all the electrons outside closed shells make a con- 
tribution to the binding energy between the atoms. The shared electron pair 
bond is not particularly emphasized. The way in which the excess of bonding 
over antibonding orbitals determines the tightness of binding may be seen by 
reference to the simple diatomic molecules in Table 11.1. 

7. Heteronuclear diatomic molecules. If the two nuclei in a diatomic 
molecule are different, it is still possible to build up molecular orbitals by 
an L.C.A.O., but now the symmetry of the homonuclear case is lost. Con- 
sider, for example, the molecule HC1. The bond between the atoms is un- 
doubtedly caused mainly by electrons in an M.O. formed from the \s A.O. 
of H and a 3/7 A.O. of Cl. 

The M.O. can be written as 



:\s) + yy(Q\ : 3/7) 

Now y is no longer 1, but there are still a bonding orbital for f-y and an 
antibonding orbital for y. Actually, the chlorine has a greater tendency 
than the hydrogen to hold electrons, and thus the resulting M.O. partakes 
more of the chlorine A.O. than of the hydrogen A.O. 

The larger y, the more unsymmetrical is the orbital, or the more polar 
the bond. Thus in the series HI, HBr, HC1, HF, the value of y increases as 
the halogen becomes more electronegative. 

8. Comparison of M.O. and V.B. methods. Since the M.O. and the V.B. 
methods are the two basic approaches to the quantum theory of molecules, 
it is worth while to summarize the distinctions between them. 

The V.B. treatment starts with individual atoms and considers the inter- 
action between them. Consider two atoms a and b with two electrons (I) 
and (2). A possible wave function is ^, a(\)b(2). Equally possible is 
i/> 2 ^ b(\)a(2), since the electrons are indistinguishable. Then the valence 
bond (Heitler-London) wave function is 



The M.O. treatment of the molecule starts with the two nuclei. If a(\) is 
a wave function for electron (I) on nucleus (a), and b(\) is that for electron (I) 
on nucleus (b\ the wave function for the single electron moving in the field 
of the two nuclei is y>i = c v a(\) + c 2 b(\) (L.C.A.O.). Similarly for the second 
electron, y 2 = ^0(2) + c 2 b(2). The combined wave function is the product 
of these two, or 

VMO ViVa --= c^a(\)a(2) + c* b(\)b(2) + Cl c 2 [a(\)b(2) f- a(2)b(\)} 



Comparing the y VB with the ^MO> we see that VMO g ives a Iar g e wei g ht 
to configurations that place both electrons on one nucleus. In a molecule 
AB, these are the ionic structures A+Br and A~B + . The ^ vn neglects these 
ionic terms. Actually, for most molecules, M.O. considerably overestimates 
the ionic terms, whereas V.B. considerably underestimates them. The true 



308 THE STRUCTURE OF MOLECULES [Chap. 1 1 

structure is usually some compromise between these two extremes, but the 
mathematical treatment of such a compromise is much more difficult. 

9. Directed valence. In the case of polyatomic molecules, a rigorous M.O. 
treatment would simply set up the nuclei in their equilibrium positions and 
pour in the electrons. It is, however, more desirable to preserve the idea of 
definite chemical bonds, and to do this we utilize bond orbitals, or localized 
molecular orbitals. 

For example, in the water molecule, the A.O.'s that take part in bond 
formation are the \s orbitals of hydrogen, and the 2p x and 2p y of oxygen. 
The stable structure will be that in which there is maximum overlap of these 
orbitals. Since p x and p v are at right angles to each other, the situation in 
Fig. 11.6 is obtained. The observed valence angle in H 2 O is not exactly 90 





y 
Fig. 11.6. Formation of a molecular orbital for H 2 O. 

but actually 105. The difference can be ascribed in part 2 to the polar nature 
of the bond; the electrons are drawn toward the oxygen, and the residual 
positive charge on the hydrogens causes their mutual repulsion. In H 2 S the 
bond is less polar and the angle is 92. The important point is the straight- 
forward fashion in which the directed valence is explained in terms of the 
shapes of the atomic orbitals. 

The most striking example of directed valence is the tetrahedral orienta- 
tion of the bonds formed by carbon in aliphatic compounds. To explain 
these bonds, it is necessary to introduce a new principle, the formation of 
hybrid orbitals. The ground state of the carbon atom is \s 2 2s 2 2p*. There are 
two uncoupled electrons 2p x , 2p u , and one would therefore expect the carbon 
to be bivalent. In order to display a valence of four, the carbon atom must 
have four electrons with uncoupled spins. The simplest way to attain this 
condition is to excite or promote o,ne of the 2s electrons into the/? state, and 
to have all the resulting p electrons with uncoupled spins. Then the outer 
configuration would be 2s2/? 3 , with 2,?f 2/? J .J2/? 1/ f 2/^j. This excitation requires 
the investment of about 65 kcal per mole of energy, but the extra binding 
energy of the four bonds that are formed more than compensates for the 
promotion energy, and carbon is normally quadrivalent rather than bivalent. 

If these four 2s2p* orbitals of carbon were coupled with the Is orbitals 

2 A more detailed theory shows that the 2s electrons of the oxygen also take part in the 
bonding, forming hybrid orbitals like those discussed below for carbon. 



Sec. 9] 



THE STRUCTURE OF MOLECULES 



309 



of hydrogen to yield the methane molecule, it might at first be thought that 
three of the bonds would be different from the remaining one. Actually, of 
course, the symmetry of the molecule is such that all the bonds must be 
exactly the same. 

Pauling 3 showed that in a case like this it is possible to form four 
identical hybrid orbitals that are a linear combination of the s and p orbitals. 
These are called tetrahedral orbitals, t l9 / 2 > 'a *4 since they are spatially 
directed to the corners of a regular tetrahedron. One of them is shown in 
(a), Fig. 11.7. In terms of the 2s and 2p orbitals it has the form: y^) = 
\\p(2s) + (V3/2)y(2p x ). The hybrid / orbitals then combine with the Is 
orbitals of hydrogen to form a set of localized molecular orbitals for methane. 





(0) 

Fig. 11.7. Hybrid atomic orbitals for carbon: (a) a single tetrahedral 
orbital; (b) three trigonal orbitals. 

The tetrahedral orbitals are exceptionally stable since they allow the electron 
pairs to avoid one another to the greatest possible extent. 

In addition to the tetrahedral hybrids, the four sp 3 orbitals of carbon 
can be hybridized in other ways. The so-called trigonal hybrids mix the 2s, 
2p x , and 2p y to form three orbitals at angle of 120. These hybrids are shown 
in (b), Fig. 11.7. For example, y - Viy<2s) -f V|y<2/7 x ). The fourth A.O., 
2/? z , is perpendicular to the plane of the others. This kind of hybridization 
is that used in the aromatic carbon compounds like benzene, and also in 
ethylene, which are treated separately in the next section. 

Hybrid orbitals are not restricted to carbon compounds. An interesting 
instance of their occurrence is in the compounds of the transition elements. 
It will be recalled that these elements have a d level that is only slightly 
lower than the outer s level. Cobalt, for example, has an outer configuration 
of 3d 7 4s 2 . The cobaltic ion, Co+++, having lost three electrons, has 3*/ 8 . It is 
noted for its ability to form complexes, such as the hexamminocobaltic ion, 

H.N^ /NH 3 1 
H 3 N Co NH 3 

\NH a 



8 L. Pauling, Nature of the Chemical Bond (Ithaca, N.Y.: Cornell Univ. Press, 1940), 
p. 85. 



310 THE STRUCTURE OF MOLECULES [Chap. 11 

This characteristic can be explained by the fact that there are six low-lying 
empty orbitals, each of which can hold a pair of electrons: 



O OOOi 

Is 2s 2p 3s 3p 3d 4s 4p 

These cPspP orbitals can be filled by taking twelve electrons from six NH 3 
groups, forming the hexamminocobaltic ion with the stable rare gas con- 
figuration. Once again, hybridization takes place, and six identical orbitals 
are formed. Pauling's calculation showed that these orbitals should be 
oriented toward the vertices of an octahedron, and the octahedral arrange- 
ment is confirmed by the crystal structures of the compounds. 

10. Nonlocalized molecular orbitals. It is not always possible to assign 
the electrons in molecules to molecular orbitals localized between two nuclei. 
The most interesting examples of delocalization are found in conjugated and 
aromatic hydrocarbons. 

Consider, for example, the structure of butadiene, usually written 
CH 2 =CH~-CH=CH 2 . The molecule is coplanar, and the C C C bond 
angles are close to 120. The M.O.'s are evidently formed from hybrid 
carbon A.O.'s of the trigonal type. Three of these trigonal orbitals lie in 
a plane and are used to form localized bonds with C and H as follows: 
CH 2 CH CH CH 2 . The fourth orbital is a /?-shaped one, perpendicular 





(a) (b) 

Fig. 11.8. Nonlocalized IT orbital in butadiene. 

to the others. These orbitals line up as shown in (a), Fig. 11.8, for the in- 
dividual atoms. When the atoms are pushed together, the orbitals overlap 
to form a continuous sheet above and below the carbon nuclei as in (b). 
This typical nonlocalized orbital is called a n orbital, and it can hold four 
electrons. 

It is important to note that the four TT electrons are not localized in 
particular bonds, but are free to move anywhere within the region in the 
figure. Since a larger volume is available for the motion of the electrons, 
their energy levels are lowered, just as in the case of the particle in a box. 
Thus delocalization results in an extra binding energy, greater than would 
be achieved in the classical structure of alternating double and single bonds. 
In the case of butadiene, this delocalization energy, often called the resonance 
energy, amounts to about 7 kcal per mole. 



Sec. 11] 



THE STRUCTURE OF MOLECULES 



311 



Benzene and other aromatic molecules provide the most remarkable 
instances of nonlocalized orbitals. The discussion of benzene proceeds very 
similarly to that of butadiene. First the carbon A.O.'s are prepared as trigonal 
hybrids and then brought together with the hydrogens. The localized orbitals 
formed lie in a plane, as shown in (a), Fig. 1 1.9. The p orbitals extend their 
sausage-shaped sections above and below the plane, (b), and when they 
overlap they form two continuous bands, (c), the TT orbitals, above and below 
the plane of the ring. These orbitals hold six mobile electrons, which are 





Fig. 11.9. Localized trigonal orbitals (a) and nonlocalized -n orbitals (c) 

in benzene. 

completely delocalized. The resulting resonance energy is about 40 kcal per 
mole. 

The properties of benzene bear out the existence of these mobile -n elec- 
trons. All the C C bonds in benzene have the same length, 1.39 A compared 
to 1.54 in ethane and 1.30 in ethylene. The benzene ring is like a little loop 
of metal wire containing electrons; if a magnetic field is applied normal to 
the planes of the rings in solid benzene, the electrons are set in motion, and 
experimental measurements show that an induced magnetic field is caused 
that opposes the applied field. 

11. Resonance between valence-bond structures. Instead of the M.O. 
method it is often convenient to imagine that the structure of a molecule is 
made up by the superposition of various distinct valence-bond structures. 
Applying this viewpoint to the case of benzene, one would say that the 
actual structure is formed principally by resonance between the two Kekule" 
structures, 



and 





312 THE STRUCTURE OF MOLECULES 

with smaller contributions from the three Dewar structures, 



[Chap. 11 






According to the resonance theory, the eigenfunction ^ describing the 
actual molecular structure is a linear combination of the functions for 
possible valence bond structures, 



This is an application of the general superposition principle for wave func- 
tions. Each eigenfunction y corresponds to some definite value E for the 
energy of the system. The problem is to determine the values of a l9 a 2 , # 3 , 
etc., in such a way as to make E a minimum. The relative magnitude of these 
coefficients when E is a minimum is then a measure of the contribution to 
the over-all structure of the different special structures represented by 
Vi ^2 Y>3 1* must b e clearly understood that the resonance description 
does not mean that some molecules have one structure and some another. 
The structure of each molecule can only be described as a sort of weighted 
average of the resonance structures. 

Two rules must be obeyed by possible resonating structures: (1) The 
structures can differ only in the position of electrons. Substances that differ 
in the arrangement of the atoms are ordinary isomers and are chemically 
and physically distinguishable as dis^jjact* compounds. (2) The resonating 
structures must have the same number of paired and unpaired electrons, 
otherwise they would have different total spins and be physically distinguish- 
able by their magnetic properties. 

In substituted benzene compounds, the contributions of various ionic 
structures must be included. For example, aniline has the following resonance 
structures : 




H H 



H 



N 





H H 



H H 





covalent 



ionic 



The ionic structures give aniline an additional resonance energy of 7 kcal, 
compared with benzene. The increased negative charge at the ortho and para 
positions in aniline accounts for the fact that the NH 2 group in aniline directs 
positively charged approaching substituents (NO a +, Br+) to these positions. 
The way in which the V.B. method would treat the hydrogen halides is 



Sec. 12] THE STRUCTURE OF MOLECULES 313 

instructive. Two important structures are postulated, one purely covalent 
and one purely ionic : 

H+ :C1:~ and H:C1: 

The actual structure is visualized as a resonance hybrid somewhere between 
these two extremes. Its wave function is 



V ~~ ^covalent + 

The value of a is adjusted until the minimum energy is obtained. Then 
(a 2 /! + a 2 ) 100 is called the per cent ionic character of the bond. For the 
various halides the following results are found : 



Molecule 
% Ionic Character 


HF 
60 


HC1 

17 


HBr 

11 


HI 

5 



The bond in HI is predominantly covalent; in HF, it is largely ionic. The 
distinction between these different bond types is thus seldom clearcut, and 
most bonds are of an intermediate nature. 

The tendency of a pair of atoms to form an ionic bond is measured by 
the difference in their power to attract an electron, or in their electronegativity. 
Fluorine is the most, and the alkali metals are the least, electronegative of 
the elements. The fractional ionic character of a bond then depends upon 
the difference in electronegativity of its constituent atoms. 

12. The hydrogen bond. It has been found that in many instances a 
hydrogen atom can act as if it formed a bond to two other atoms instead of 
to only one. A typical example is the dimer of formic acid, which has the 
structure 

O H O 

/ \ 

H C C H 

\ / 

O H O 

This hydrogen bond is not very strong, usually having a dissociation energy 
of about 5 kcal, but it is extremely important in many structures, such as the 
proteins. It occurs in general between hydrogen and the electronegative 
elements N, O, F, of small atomic volumes. 

We know that hydrogen can form only one covalent bond, since it has 
only the single Is orbital available for bond formation. Therefore the hydro- 
gen bond is essentially an ionic bond. Since the proton is extremely small, 
its electrostatic field is very intense. A typical hydrogen-bonded structure is 
the ion (HF 2 )~, which occurs in hydrofluoric acid and in crystals such as 
KHF 2 . It can be represented as a resonance hybrid of three structures, 

:F: H F- F~ H :F: F- H+ F- 



314 



THE STRUCTURE OF MOLECULES 



[Chap. 11 




The ionic F H f F~ structure is the most important. It is noteworthy that 
electroneg^Hve elements with large ionic radii, e.g., Cl, have little or no 
tendenc^o form hydrogen bonds, presumably owing to their less concen- 
trates electrostatic fields. 

13. Dipole moments. If a bond is formed between two atoms that differ 
in electronegativity, there is an accumulation of negative charge on the 
more electronegative atom, leaving a positive charge on the more electro- 
positive atom. The bond then constitutes an electric dipole, which is by 
definition an equal positive and negative charge, _q, separated by a distance 

r. A dipole, as in (a), Fig. 11.10, is 
characterized by its dipole moment, a 
vector having the magnitude qr and 
the direction of the line joining the 
positive to the negative charge. The 
dimensions of a dipole moment are 
charge times length. Two charges with 
the magnitude of e(4. 80 x 10~ 10 esu) 
separated by a distance of I A would 
have a dipole moment of 4.80 x 10~~ 18 
csu cm. The unit 10~ 18 esu cm is 
called the debye, (d). 

If a polyatomic molecule contains 

two or more dipoles in different bonds, the net dipole moment of the mole- 
cule is the resultant of the vector addition of the individual bond moments. 
An example of this is shown in (b), Fig. 11.10. 

The measurement of tha dipole moments of molecules provides an insight 
into their geometric structure and also into the character of their valence 
bonds. Before we can discuss the determination of dipole moments, however, 
it is necessary to review some aspects of the theory of dielectrics. 

14. Polarization of dielectrics. Consider a parallel-plate capacitor with 
the region between the plates evacuated, and let the charge on one plate be 
-for and on the other a per square centimeter. The electric field within the 
capacitor is then directed perpendicular to the plates and has the magnitude 4 
EQ -- 47TCT. The capacitance is 

q aA A 




Fig. 11.10. (a) Definition of dipole 
moment; (b) vector addition of dipole 
moments in orthodichlorobenzene. 



where A is the area of the plates, rfthe distance, and (/the potential difference 
between them. 

Now consider the space between the plates to be filled with some material 
substance. In general, this substance falls rather definitely into one of two 
classes, the conductors or the insulators. Under the influence of small fields, 
electrons move quite freely through conductors, whereas in insulators or 

4 See, for example, G. P. Harnwell, Electricity and Magnetism (New York: McGraw- 
Hill, 1949), p. 26. 



Sec. 14] 



THE STRUCTURE OF MOLECULES 



315 



dielectrics these fields displace the electrons only slightly from their equi- 
librium positions. 

An electric field acting on a dielectric thus causes a separation of positive 
and negative charges. The field is said to polarize the dielectric. This polarizo- 
tion is shown pictorially in (a), Fig. 11.11. The polarization can occur in twa 
ways: the induction effect and the orientation effect. An electric field always 
induces dipoles in molecules on which it is acting, whether or not they contain 
dipoles to begin with. If the dielectric does contain molecules that are per- 
manent dipoles, the field tends to align these dipoles along its own direction. 
The random thermal motions of the molecules oppose this orienting action. 




I CM 



(a) (b) 

Fig. 11.11. (a) Polarization of a dielectric; (b) definition of the 
polarization vector, P. 

Our main interest is in the permanent dipoles, but before these can be studied, 
effects due to the induced dipoles must be clearly distinguished. 

It is found experimentally that when a dielectric is introduced between 
the plates of a capacitor the capacitance is increased by a factor e, called the 
dielectric constant. Thus if C is the capacitance with a vacuum, the capaci- 
tance with a dielectric is C eC . Since the charges on the capacitor plates 
are unchanged, this must mean that the field between the plates is reduced 
by the factor e, so that E = E Q /e. 

The reason why the field is reduced is clear from the picture of the 
polarized dielectric, for all the induced dipoles are aligned so as to produce 
an over-all dipole moment that cuts down the field strength. Consider in 
(b), Fig. 11.11, a unit cube of dielectric between the capacitor plates, and 
define a vector quantity P called the polarization, which is the dipole moment 
per unit volume. Then the effect of the polarization is equivalent to that 
which would be produced by a charge of -f P on one face and ~P on the 
other face (1 cm 2 ) of the cube. The field in the dielectric is now determined 
by the net charge on the plates, so that 

J-47r(er-P) (11.5) 

A new vector has been defined, called the displacement D, which depends 
only on the charge or, according to D = 47ror. It follows that 

D-J5+4rrP, and DIE = e (11.6) 

It is apparent that in a vacuum, where e = 1, D = E. 



316 THE STRUCTURE OF MOLECULES [Chap. 11 

15. The induced polarization. Let us consider the induced or distortion 
polarization, P I} , produced by an electric field acting on a dielectric that does 
not contain permanent dipoles. 

The first problem to be solved is the magnitude of the dipole moment m 
induced in a molecule by the field acting on it. It may be assumed that this 
induced moment is proportional to the intensity of the field 5 F, so that 

m-aoF (11.7) 

The proportionality constant OQ is called the distortion polarizability of the 
molecule. It is the induced moment per unit field strength, and has the 
dimensions of a volume, since q r/(q/r 2 ) r 3 v. 

At first it might seem that the field acting on a molecule should be simply 
the field E of eq. (1 1.5). This would be incorrect, however, for the field that 
polarizes a molecule is the local field immediately surrounding it, and this is 
different from the average field E throughout the dielectric. For an isotropic 
substance this local field can be calculated 6 to be 

F-B + *?-* (I I J) 

In the absence of permanent dipoles, the polarization or dipole moment 
per unit volume is the number of molecules per cc, , times the average 
moment induced in a molecule, m. Thus, from eqs. (11.7) and (11.8), 

- /7<x ( E + 



Since, from eq. (11.6), E(e 1) ^-- 4*rP D , 

477/700 



3 (H.9) 



This is the Clausius-Mossotti equation. 

Multiplying both sides by the ratio of molecular weight to density M/p, 

4 



B + 2 p 3p 3 

The quantity P M is called the molar polarization. So far it includes only the 
contribution from induced dipoles, and in order to obtain the complete 
molar polarization, a term due to permanent dipoles must be added. -> 

16. Determination of the dipole moment. Having examined the effect of 
induced dipoles on the dielectric constant, we are in a position to consider 

5 This is true only for isotropic substances; otherwise, for example in nonisotropic 
crystals, the direction of the moment may not coincide with the field direction. This dis- 
cussion therefore applies only to gases, liquids, and cubic crystals. 

6 A good derivation is given by Slater and Frank, Introduction to Theoretical Physics 
(New York: McGraw-Hill, 1933), p. 278; also, Syrkin and Dyatkina, The Structure of 
Molecules (New York: Interscience, 1950), p. 471. 



Sec. 16] THE STRUCTURE OF MOLECULES 317 

the influence of permanent dipoles. If the bonds in a molecule are ionic or 
partially ionic, the molecule has a net dipole moment, unless the individual 
bond moments add vectorially to zero. 

It is now possible to distinguish an orientation polarization of a dielectric, 
which is that caused by permanent dipoles, from the distortion polarization, 
caused by induced dipoles. 

There will always be an induced moment. It is evoked almost instanta- 
neously in the direction of the electric field. It is independent of the tempera- 
ture, since if the molecule's position is disturbed by thermal collisions, the 
dipole is at once induced again in the field direction. The contribution to the 
polarization caused by permanent dipoles, however, is less at higher tem- 
peratures, since the random thermal collisions of the molecules oppose the 
tendency of their dipoles to line up in the electric field. 

It is necessary to calculate the average component of a permanent dipole 
in the field direction as a function of the temperature. Consider a dipole with 
random orientation. If there is no field, all orientations are equally probable. 
This fact can be expressed by saying that the number of dipole moments 
directed within a solid angle da) is simply Adw, where A is a constant depend- 
ing on the number of molecules under observation. 

If a dipole moment // is oriented at angle to a field of strength F its 
potential energy 7 is U - //Fcos 0. According to the Boltzmann equation, 
the number of molecules oriented within the solid angle da} is then 

Ae-' ulkT dco = A 



The average value of the dipole moment in the direction of the field, by 
analogy with eq. (7.39), can be written 

A* cos (>lkT 1 cos Oda> 



To evaluate this expression, let [iFjkT x, cos = y; then dw -- 2-n- sin 9 dO 
- 277 dy. 

Thus * 



i (e x e~ x ) 

Since e**dy ~- ---------- 

_ s 



m e x 



-- = coth x -- = L(x) 
p e x e~ x x x 

Here L(x) is called the "Langevin function," in honor of the inventor of this 
treatment. 

7 Harnwell, op. cit., p. 64. 
M 



318 



THE STRUCTURE OF MOLECULES 



[Chap. 11 



In most cases x = [iF/kTis a very small number 8 so that on expanding L(x) 
in a power series, only the first term need be retained, leavingL(x) = x/3, or 

/|2 



The total polarizability of a dielectric is found by adding this contribution 
due to permanent dipoles to the distortion polarizability, and may be written 
a = a o 4- CM 2 /3*r). Instead of eq. (1 1.10), the total polarization is therefore 



This equation was first derived by P. Debye. 



40 



i 30 

O 

NJ 

5 20 

.j 

o 

a. 



3 .0 




1.0 



5.0 



2.0 3.0 4.0 

1/TXlO 3 

Fig. 11.12. Application of the Debye equation to the polarizations of 
the hydrogen ha 1 ides. 

When the Clausius-Mossotti treatment is valid, 9 

e - 1 M / 
PU - - - - ^ 

e + 2 P 

For gases, e is not much greater than 1, so that 

E - 1 M 4n 



= "=- ^ a o 



(11.13) 



8 Values of n range around 10" 18 (esu) (cm). If a capacitor with 1 cm between plates is 

(3 x 10^\ 
airiov " 10 " 17 erg com P ared witn kT = 10 ~ 14 er 



at room temperature. 

9 This is the case only for gases or for dilute solutions of dipolar molecules in non-polar 
solvents. If there is a high concentration of dipolar molecules, as in aqueous solutions, there 
are localized polarization fields that cannot be treated by the Clausius-Mossotti method. In 
other words, the permanent dipoles tend to influence the induced polarization. 



Sec. 17] 



THE STRUCTURE OF MOLECULES 



319 



It is now possible to evaluate both OQ and // from the intercept and slope 
of P M vs. l/T^plots, as shown in Fig. 11.12. The necessary experimental data 
are values of the dielectric constant over a range of temperatures. They are 
obtained by measuring the capacitance of a capacitor using the vapor or 
solution under investigation as the dielectric between the plates. A number 
of dipole-moment values are collected in Table 11.2. 





TABLE 11.2 




DIPOLE MOMENTS 


Compound 


Moment 
(debyes) 


Compound 


HC1 


1.03 


CH 3 Br 


HBr 


0.78 


CH 3 C1 


HI 


0.38 


CH 3 I 


H 2 


1.85 


CH 3 OH 


H 2 S 


0.95 


C 2 H 5 C1 


NH 3 


1.49 


(C 2 H 5 ) 2 


S0 2 


1.61 


C 6 H 5 OH 


C0 2 


0.0 


QH 5 N0 2 


CO 


0.11 


C 6 H 5 .CH 2 C1 



Moment 
(debyes) 




.45 




.85 




.35 




.68 


; 


2.02 




.14 




.70 


4.08 


1.85 



17. Dipole moments and molecular structure. Two kinds of information 
about molecular structure are provided by dipole moments: (1) The extent 
to which a bond is permanently polarized, or its per cent ionic character; 
and (2) an insight into the geometry of the molecule, especially the angles be- 
tween its bonds. Only a few examples of the applications will be mentioned. 10 

The H Cl distance in HC1 is 1.26 A (found by methods described on 
page 334). If the structure were H+C1 , the dipole moment would be 

H - (1.26)(4.80) - 6.05d 

The actual moment of 1.03 suggests therefore that the ionic character of 
the bond is equivalent to a separation of charges of about \e. 

Carbon dioxide has no dipole moment, despite the difference in electro- 
negativity between carbon and oxygen. It may be concluded that the molecule 
is linear, O C O; the moments due to the two C O bonds, which are 
surely present owing to the difference in electronegativity of the atoms, 
exactly cancel each other on vector addition. 

On the other hand, water has a moment of 1.85d, and must have a 
triangular structure (see Fig. 1 1 .6). It has been estimated that each O H 
bond has a moment of 1.60d and the bond angle is therefore about 105, 
as shown by a vector diagram. 

10 R. J. W. LeFevre, Dipole Moments (London: Methiien, 1948) gives many interesting 
examples. 



320 THE STRUCTURE OF MOLECULES [Chap. 11 

A final simple example is found in the substituted benzene derivatives: 






OH OH 




- 1.55 1.70 



The zero moments of /?-dichloro- and sym-trichlorobenzene indicate that 
benzene is planar and that the C -Cl bond moments are directed in the 
plane of the ring, thereby adding to zero. The moment of />di-OH benzene, 
on the other hand, shows that the O H bonds are not in the plane of the 
ring, but directed at an angle to it, thus providing a net moment. 

18. Polarization and refractivity. It may be recalled that one of the most 
interesting results of Clerk Maxwell's electromagnetic theory of light 11 was 
the relationship f /r^ 2 , where n R is the index of refraction. Thus the 
refractive index is related through eq. (11.10) to the molar polarization. 

The physical reason for this relationship can be understood without 
going into the details of the electromagnetic theory. The refractive index of 
a medium is the ratio of the speed of light in a vacuum to its speed in the 
medium, n R - c/c m . Light always travels more slowly through a material 
substance than it does through a vacuum. A light wave is a rapidly alternating 
electric and magnetic field. This field, as any other, acts to polarize the 
dielectric through which it passes, pulling the electrons back and forth in 
rapid alternation. The greater the polarizability of the molecules, the greater 
is the field induced in opposition to the applied field, and the greater therefore 
is the "resistance" to the transmission of the light wave. Thus high polariz 
ability means low c m and high refractive index. We have already seen that 
increasing the polarization increases the dielectric constant. The detailed 
theory leads to the Maxwell relation, e -= n n 2 . 

This relation is experimentally confirmed only under certain conditions: 

(1) The substance contains no permanent dipoles. 

(2) The measurement is made with radiation of very long wavelength, in 
the infrared region. 

(3) The refractive index is not measured in the neighborhood of a wave- 
length where the radiation is absorbed. 

The first restriction arises from the fact that dielectric constants are 
measured at low frequencies (500 to 5000 kc), whereas refractive indices are 
measured with radiation of frequency about 10 12 kc. A permanent dipole 
cannot line up quickly enough to follow an electric field alternating this 
rapidly. Permanent dipoles therefore contribute to the dielectric constant 
but not to the refractive index. 

The second restriction is a result of the effect of high frequencies on the 

11 G. P. Harnwell, opt cit., p. 579. 



Sec. 19] THE STRUCTURE OF MOLECULES 321 

induced polarization. With high-frequency radiation (in the visible) only the 
electrons in molecules can adjust themselves to the rapidly alternating electric 
fields; the more sluggish nuclei stay practically in their equilibrium positions. 
With the lower-frequency infrared radiation the nuclei are also displaced. 

It is customary, therefore, to distinguish, in the absence of permanent 
dipoles, an electronic polarization P K and an atomic polarization P A . The 
total polarization, P A \ P K , is obtained from dielectric-constant measure- 
ments or infrared determinations of the refractive index. The latter are hard 
to make, but sometimes results with visible light can be successfully extra- 
polated. The electronic polarization P K can be calculated from refractive 
index measurements with visible light. Usually P A is only about 10 per cent 
of P E , and may often be neglected. 

When the Maxwell relation is satisfied, we obtain from eq. (11.10) the 

Lorenz-Lorentz equation: 

n 2 t AY 

vri'7 = />A/ (1L14) 

The quantity at the left of eq. (11.14) is often called the molar refraction 
R M . When the Maxwell relation holds, R M - P M . 

It will be noted that the molar refraction R M has the dimensions of 
volume. It can indeed be shown from simple electrostatic theory 12 that a 
sphere of conducting material of radius r, in an electric field F, has an induced 
electric moment of m = r 3 / 7 . According to this simple picture, the molar 
refraction should be equal to the true volume of the molecules contained in 
one mole. A comparison of some values of molecular volume obtained in 
this way from refractive index measurements with those obtained from 
van der Waal's b was shown in Table 7.5. 

19. Dipole moments by combining dielectric constant and refractive index 
measurements. The Lorenz-Lorentz equation also provides an alternative 
method of separating the orientation and the distortion polarizations, and 
thereby determining the dipole moment. A solution of the dipolar compound 
in a nonpolar solvent e.g., nitrobenzene in benzene-- is prepared at various 
concentrations. The dielectric constant is measured and the apparent molar 
polarization calculated from eq. (1J.10). This quantity is made up of the 
distortion polarizations of both solute and solvent plus the orientation 
polarization of the polar solute. The molar polarizations due to distortion 
can be set equal to the molar refractions R M , calculated from the refractive 
indices of the pure liquids. When these R M are subtracted from the total 
apparent P M , the remainder is the apparent molar orientation polarisation 
for the solute alone. This polarization is plotted against the concentration in 
the solution and extrapolated to zero concentration. 13 A value is obtained in 

12 Slater and Frank, op. r/7., p. 275. 

13 E. A. Guggenheim, Trans. Faraday Soc., 47, 573 (1951), gives an improved method 
for extrapolation. 



322 THE STRUCTURE OF MOLECULES [Chap. 1 i 

this way from which the effect of dipole interaction has been eliminated. 
From eq. (11.13), therefore, it is equal to (4n/3)N([i*/3kT) and the dipole 
moment of the polar solute can be calculated. 

20. Magnetism and molecular structure. The theory for the magnetic 
properties of molecules resembles in many ways that for the electric polariza- 
tion. Thus a molecule can have a permanent magnetic moment and also a 
moment induced by a magnetic field. 

Corresponding to eq. (11.6), we have 

B H + 4nI (11.15) 

where B is the magnetic induction, H is the field strength, and / is the in- 
tensity of magnetization or magnetic moment per unit volume. These 
quantities are the magnetic counterparts of the electrical D, /?, and P. In a 
vacuum B H, but otherwise B -= e'H, where t', the permeability, is the 
magnetic counterpart of the dielectric constant F. Usually, however, mag- 
netic properties are discussed in terms of 

~ X (H.16) 

where % is called the magnetic susceptibility per unit volume of the medium. 
(Electric susceptibility would be P/E.) 

The susceptibility per mole is % M - (M/p)x- The magnetig^fffialogue of 
eq. (11. 13) is 

(1U7) 

where a is the induced moment and JU M is the permanent magnetic dipole 
moment. Just as before, the two effects can be experimentally separated by 
temperature-dependence measurements. 

An important difference from the electrical case now appears, in that 
> or XM> can b e either positive or negative. If % M is negative, the medium 
is called diamagnetic; if % M is positive, it is called paramagnetic. For iron, 
nickel, and certain alloys, % M is positive and much larger than usual, by a 
factor of about a million. Such substances are called ferromagnetic. From 
eq. (11.15) it can be seen that the magnetic field in diamagnetic substances 
is weaker than in a vacuum, whereas in paramagnetic substances it is 
stronger. 

An experimental measurement of susceptibility can be made with the 
magnetic balance. The specimen is suspended so that it is partly inside and 
partly outside a strong magnetic field. When the magnet is turned on, a 
paramagnetic substance tends to be drawn into the field region, a dia- 
magnetic tends to be pushed out of the field. From the weight required to 
restore the original balance point, the susceptibility is calculated. 



Sec. 20] THE STRUCTURE OF MOLECULES 323 

The phenomenon of diamagnetism is the counterpart of the distortion 
polarization in the electrical case. The effect is exhibited by all substances 
and is independent of the temperature. A simple interpretation is obtained 
if one imagines the electrons to be revolving around the nucleus. If a mag- 
netic field is applied, the velocity of the moving electrons is changed, pro- 
ducing a magnetic field that, in accordance with Lenz's Law, is opposed in 
direction to the applied field. The diamagnetic susceptibility is therefore 
always negative. 

When paramagnetism occurs, the diamagnetic effect is usually quite over- 
shadowed, amounting to only about 10 per cent of the total susceptibility. 
Paramagnetism is associated with the orbital angular momentum and the 
spin of uncoupled electrons, i.e., those that are not paired with others having 
equal but opposite angular momentum and spin. 

An electron revolving in an orbit about the nucleus is like an electric 
current in a loop of wire, or a turn in a solenoid. The resultant magnetic 
moment is a vector normal to the plane of the orbit, and proportional to 
the angular momentum p of the revolving electron. In the MKS system of 
units (charge in coulombs) the magnetic moment is (e/2m)p (weber meters). 14 
Since p can have only quantized values, m^l-n, where m l is an integer, the 
allowed values of the magnetic moment are m^eh^nm). It is evident, there- 
fore, that there is a natural unit of magnetic moment, eh/47rm. It is called the 
Bohr magneton. 

The ratio of magnetic moment to angular momentum is called the gyro- 
magnetic ratio, R . For the orbital motion of an electron, R g e/2m. The 
spinning electron also acts as a little magnet. For electron spin, however, 
R g = e/m. Since the intrinsic angular momentum of an electron can have 
only quantized values %(h/2ir), the magnetic moment of an unpaired 
electron is eh/fam, or one Bohr magneton. 

In the case of molecules, only the contributions due to spin are very 
important. This is true because there is a strong internal field within a mole- 
cule. In a diatomic molecule, for example, this field is directed along the 
internuclear axis. This internal field holds the orbital angular momenta of 
the electrons in a fixed orientation. They cannot line up with an external 
magnetic field, and thus the contribution they would normally make to the 
susceptibility is ineffective. It is said to be quenched. There remains only the 
effect due to the electron spin, which is not affected by the internal field. 
Thus a measurement of the permanent magnetic moment of a molecule tells 
us how many unpaired spins there are in its structure. 

There have been many applications of this useful method, 15 of which 
only one can be mentioned here. Let us consider two complexes of cobalt, 

14 A derivation is given by C. A. Coulson, Electricity (New York: Interscience, 1951), 
p. 91. In electrostatic units the magnetic moment is (e/2mc)p t where c is the speed of light 
i/i vacuo. 

16 P. W. Selwood, Magnetochemistry (New York: Interscience, 1943). 



324 



THE STRUCTURE OF MOLECULES 



[Chap. 11 



{Co(NH 3 ) 6 } C1 3 and K 3 {CoF 6 }. Two possible structures may be suggested for 
such complexes, one covalent and one ionic, as follows: 

3d 4s 4p Unpaired Spins 

Covalent . 11 11 11 11 ft 11 ft ft ft 

Ionic . . 11 t t t t .. 4 

The hexammino complex is obviously covalent, but the structure of the 
hexafluoro complex is open to question. It is found that the hexammino 
complex has zero magnetic moment, whereas the {CoF 6 } r ^ complex has a 
moment of 5.3 magnetons. The structures can thus be assigned as follows: 



NR, 



H 3 N 



Co 



NFi, 



NFL 



C1= 



and 



21. Nuclear paramagnetism. In addition to the magnetism due to the 
electrons in an atom there is also magnetism due to the nuclei. We may 
consider a nucleus to be composed of protons and neutrons, and both these 
nucleons have intrinsic angular momenta or spins, and hence act as ele- 
mentary magnets. In most nuclei these spins add to give a nonzero resultant 
nuclear spin. It was first predicted that the magnetic moment of the proton 
would be 1 nuclear magneton, ehl^-nM, where M is the proton mass. Actually, 
however, the proton has a magnetic moment of 2.79245 nuclear magnetons, 
and the neutron moment is -1.9135. The minus sign indicates that the 
moment behaves like that of a negatively charged particle. Since M is almost 
2000 times the electronic mass m, nuclear magnetic moments are less than 
electronic magnetic moments by a factor of about 1000. 

The existence of nuclear magnetism was first revealed in the hyperfine 
structure of spectral lines. As an example consider the hydrogen atom, a 
proton with one orbital electron. The nucleus can have a spin / i4, and 
the electron can have a spin S = i. The nuclear and the electron spins can 
be either parallel or antiparallel to each other, and these two different align- 
ments will differ slightly in energy, the parallel state being higher. Thus the 
ground state of the hydrogen atom will in fact be a closely spaced doublet, 
and this splitting is observed in the atomic spectra of hydrogen, if a spectro- 
graph of high resolving power is employed. The spacing between the two 
levels, A -- hv, corresponds to a frequency v of 1420 megacycles. After the 
prediction of the astrophysicist van der Hulst, an intense emission of radia- 
tion at this frequency was observed from clouds of interstellar dust. The 
study of this phenomenon is an important part of the rapidly developing 
subject of radioastronomy, which is providing much information about 
hitherto uncharted regions of our universe. 



Sec. 21] 



THE STRUCTURE OF MOLECULES 



325 



If a nucleus with a certain magnetic moment is placed in a magnetic 
field, we can observe the phenomenon of space quantization (see page 267). 
The component of the moment in the direction of the field is quantized, and 
for each allowed direction there will be a slightly different energy level. For 
readily accessible magnetic fields, the* frequencies v A//i for transitions 
between two such levels also lie in the microwave range of radio frequencies. 

TEST TUBE 

WITH SAMPLE 



TRANSMITTER 
COIL 




Fig. 11.13. Simplified apparatus for basic nuclear magnetic resonance 
experiment. (Drawing courtesy R. H. Varian.) 

For example, at a field of 7050 gauss, the frequency for protons is 30 mega- 
cycles. The earlier attempts to detect these transitions were unsuccessful, 
but in 1946 E. M. Purcell and Felix Bloch independently developed the 
method of nuclear magnetic resonance. 

The principle of this method is shown in Fig. 11.13. The field H of the 
magnet is variable from to 10,000 gauss. This field produces an equi- 
distant splitting of the nuclear energy levels which arise as a result of space 
quantization. The low-power radio-frequency transmitter operates at, for 
example, 30 megacycles. It causes a small oscillating magnetic field to be 
applied to the sample. This field induces transitions between the energy 



326 



THE STRUCTURE OF MOLECULES 



[Chap. 1 1 



levels, by a resonance effect, when the frequency of the oscillating field equals 
that of the transitions. When such transitions occur in the sample, the 
resultant oscillation in magnetic field induces a voltage oscillation in the 
receiver coil, which can be amplified and detected. 

Figure 11.14 shows an oscillographic trace of these voltage fluctuations 
over a very small range of magnetic fields (38 milligauss) around 7050 
gauss, with ethyl alcohol as the sample. Note that each different kind of 
proton in the molecule CH 3 -CH 2 -OH appears at a distinct value of H. The 
reason for this splitting is that the different protons in the molecule have a 
slightly different magnetic environment, and hence a slightly different 




CH 2 
ETHYL ALCOHOL 



OH 



Fig. 11.14. Proton resonance under high resolution at 30 me and 7050 gauss. 
Total sweep width 38 milligauss. Field decreases linearly from left to right. 

resonant frequency. The areas under the peaks are in the ratio 3:2: 1, 
corresponding to the relative number of protons in the different environments. 
Each peak also has a fine structure. The structural information that can be 
provided by this method is thus almost unbelievably detailed, and a new and 
deep insight into the nature of the chemical bond is provided. Applications 
have been made to problems ranging from isotope analysis to structure 
determinations. 

22. Electron diffraction of gases. One of the most generally useful methods 
for measuring bond distances and bond angles has been the study of the 
diffraction of electrons by gases and vapors. The wavelength of 40,000 volt 
electrons is 0.06 A, about one-tenth the order of magnitude of interatomic 
distances in molecules, so that diffraction effects are to be expected. The fact 
that the electron beam and the electrons in the scattering atoms both are 
negatively charged greatly enhances the diffraction. 

On page 256 diffraction by a set of slits was discussed in terms of the 
Huygens construction. In the same way, if a collection of atoms at fixed 
distances apart (i.e., a molecule) is placed in a beam of radiation, each atom 
can be regarded as a new source of spherical wavelets. From the interference 
pattern produced by these wavelets, the spatial arrangement of the scatter- 
ing centers can be determined. The experimental apparatus for electron 



Sec. 22] 



THE STRUCTURE OF MOLECULES 



327 



diffraction is illustrated in Fig. 11.15. The type of pattern found is a series 
of rings similar to those in Fig. 10.10 but somewhat more diffuse. 

The electron beam traverses a collection of many gas molecules, oriented 
at random to its direction. It is most interesting that maxima and minima 



HOT 
FILAMENT- 
ELECTRON" 
SOURCE 


2 VAPOR 
SUPPLY 


| -| ,-T ,'?..-_- 


<*--PHUIUUKAHMIU 

PLATE 


7^* j | v^J J ~~~-~- 


ACCELERATING 1] 
VOLTAGE HTO VACUUM 

''PUMPS 



Fig. 11.15. Schematic diagram of electron diffraction apparatus. 

are observed in the diffraction pattern despite the random orientation of the 
molecules. This is because the scattering centers occur as groups of atoms 
with the same definite fixed arrangement within every molecule. A collection 
of individual atoms, e.g., argon gas, would give no diffraction rings. Diffrac- 
tion by gases was treated theoretically (for X rays) by Debye in 1915, but 
electron-diffraction experiments were not carried out till the work of Wierl 
in 1930. 

We can show the essential features of the diffraction theory by considering 
the simplest case, that of a diatomic molecule. 16 The molecule is represented 
in Fig. 11.16 with one atom A, at the origin, and the other B, a distance r 
away. The electron beam enters along Y'A 
and the diffracted beam, scattered through 
an angle 0, is picked up at P on a photo- 
graphic film, a distance R from the origin. 
The angles a and <f> give the orientation of 
AB to the primary beam. 

The interference between the waves 
scattered from A and B depends on the 
difference between the lengths of the paths 
which they traverse. This path difference 
is 6 - AP CB - BP. The difference in Fig . n 16 Scattering of electrons 
phase between the two scattered waves is by a diatomic molecule. 




In order to add waves that differ in phase and amplitude, it is convenient 
to represent them in the complex plane and to add vectorially. 17 In our case 
we shall assume for simplicity that the atoms A and B are identical. Then the 
resultant amplitude at P is A --= A -f /V"' 2 ^. A^ called the atomic scatter- 
ing factor, depends on the number of electrons in the atom. The intensity of 

18 The treatment follows that given by M. H. Pirenne, The Diffraction of X rays and 
Electrons by Free Molecules (London: Cambridge, 1946), p. 7. 

17 See Courant and Robbins, What Is Mathematics ? (New York: Oxford, 1941), p. 94. 



328 THE STRUCTURE OF MOLECULES [Chap. 11 

radiation is proportional to the square of the amplitude, or in this case to 
AA, the amplitude times its complex conjugate. Thus 



A 2 /O I ZirioJ* i ,.27n<5/A\ 

^o v^ ~r c -re ; 

- 2/* 2 ( 1 + cos -^- j - 4^ 2 cos 2 

It is now necessary to express 6 in terms of r, /, 0, a, and (/>. This can 
be done by referring to Fig. 11.16. We see first of all that CB -~ r sin a 
sin <f>. Then BP VR 2 + r 2 2rR sin a sin (6 + </). Since r is a few 
Angstroms while R is several cm, r <; /?, so that r 2 is negligible and the 
square root can be expanded 18 to yield BP R r sin <x sin (0 -f- <). 
Then we have 6 = AP CB #P r sin a [sin (0 + </>) sin $ - 2r sin 
0/2 sin a cos [< + (0/2)]. 

In order to obtain the required formula for the intensity of scattering of 
a randomly oriented group of molecules, it is necessary to average the 
expression for the intensity at one particular orientation (a, <f>) over all 
possible orientations. The differential element of solid angle is sin a den d<f>, 
and the total solid angle of the sphere around AB is 4-rr. Hence the required 
average intensity becomes 

, 4 ^o 2 r r jo r L^ 2 \] , M 

lav ~ cos 2 2-rr - sin - sin a cos + - I sin a da. dd> 

4n Jo Jo L A 2 \ 07 J 

On integration, 19 f av - 2 A 2 i\ + 

A n "' ' ( 1L18 ) 

47T . 

where x = sin - 

18 From the binomial theorem, (1 f x) 1 / 2 = 1 + x Jx 2 + . . . . 

19 Let 

/<, = L L cos2 (A cos ft) dp sin a da. 

IT > > 

where 

A = ~. - sin - sin a and p $ -f 0/2 

Then since cos 2 p = (1 f cos 2/?)/2, we obtain 

7 - = VJo Jo (y + cos ( 2 



where / is the Bessel function of order zero (see Woods, Advanced Calculus, p. 282). This 
can now be integrated by introducing the series expansion of 






2 
( " I 



Sec. 23] 



THE STRUCTURE OF MOLECULES 



329 



In Fig. 11.17, }\A is plotted against x, and the maxima and minima in 
the intensity are clearly evident. 

In a more complex molecule with atoms j, k (having scattering factors 
Aj, A k ) a distance r )k apart, the resultant intensity would be 



(11.19) 



This is called the "Wierl equation." The summation must be carried out 
over all pairs of atoms in the molecule. 



5 
4 

C\j O "5 

\ 
~ 2 

I 




27T 



67T 



87T I07T 



Fig. 11.17. Scattering curve for diatomic molecule plot of eq. (11.18). 

In the case of the homonuclear diatomic molecule already considered, 
eq. (1 1.19) becomes 

sin AT 22 



- A A A A A 

si i/i j - -|- /i i/l 2 



A A 

^ 



4- A A 

' ^12 



Sm 



Since r u =-- r 22 =- 0, and (sin x)/x -> 1 as x > 0, and r l2 =- r 21 =^ r, this 
reduces to eq. (1 1.18). 

23. Application of Wierl equation to experimental data. The scattering 
angles of maximum intensity are calculated from the positions of the dark 
rings on the picture and the geometry of the apparatus and camera. This 
gives an experimental scattering curve, whose general form resembles that of 
the theoretical curve shown in Fig. 11.17, although the positions of the 
maxima depend, of course, on the molecule being studied. Then a particular 
molecular structure is assumed and the theoretical scattering curve corre- 
sponding to it is calculated from eq. (11.19). For example, in the benzene 
structure there are three different carbon-carbon distances, six between ortho 
positions, six between meta positions, and three between para positions. 
Therefore the r }k terms consist of 6r cc , 6(V3 r cc ), and 3(2 r cc ). The positions 



where in our case x ~ 2B sin a, with B = (2wr/A) sin 6/2. The required integral is given in 
Pierce's tables (No. 483) as 

C* - r/i f 



: 



The series that results is that for (sin x)/x. (Pierce No. 772.) 



330 



THE STRUCTURE OF MOLECULES 



[Chap. 11 



of hydrogen atoms are generally ignored because of their low scattering 
power. 

It is often sufficiently accurate to substitute the atomic number Z for the 
atomic scattering factor A. For benzene, the Wierl equation would then 
become 

7(61) 6 sin xr 3 sin 2xr 6 sin A/3 xr 
Z* = ~~x7~ ~ f ~~^x7~ + ^~Vlxr~~ 

This function is plotted for various choices of the parameter r, the inter- 
atomic distance, until the best agreement with the experimental curve is 
obtained. In other cases bond angles also enter as parameters to be adjusted 
to obtain the best fit between the observed and calculated curves. It may be 
noted that only the positions of the maxima and not their heights are used. 

TABLE 11.3 
THE ELECTRON DIFFRACTION OF GAS MOLECULES 



Molecule 



NaCl 
NaBr 
Nal 



Bond Distance 

(A) 



Diatomic Molecules 

Molecule 



2.51 - 0.03 
2.64 i_ 0.01 
2.90 0.02 



C1 2 
Br 2 



Bond Distance 

(A) 



2.01 -b 0.03 
2.28 0.02 
2.65 0.10 



Polyatomic Molecules 



Molecule 



Configuration 



Bond 



Bond Distance 









1 


CdI 2 


Linear 


Cd I 


2.60 0.02 


HgCl 2 


Linear 


Hg-Cl 


2.34 0.01 


BC1 3 


Planar 


B Cl 


.73 0.02 


SiF 4 


Tetrahedral 


Si F 


.54 0.02 


SiCl 4 


Tetrahedral 


| Si Cl 


2.00 0.02 


P 4 


Tetrahedral 


i P P 


2.21 0.02 


C1 2 O 


Bent, 115 JL 4 


1 Cl O 


.68 0.03 


S0 2 


Bent, 124 -h 15 


i s-o 


.45 0.02 


CH 2 


Planar 


C 


.15 0.05 


C0 2 


Linear 


c o 


.13 0.04 


QH 6 

_ _ _ i 


Planar 

- - - A 


c c 


1.390 0.005 



Some results of electron diffraction studies are collected in Table 11.3. 
As molecules become more complieated, it becomes increasingly difficult to 
determine an exact structure, since usually only a dozen or so maxima are 
visible, which obviously will not permit the exact calculation of more than 
five or six parameters. Each distinct interatomic distance or bond angle 



Sec. 24] THE STRUCTURE OF MOLECULES 331 

constitutes a parameter. It is possible, however, from measurements on 
simple compounds, to obtain quite reliable values of bond distances and 
angles, which may be used to estimate the structures of more complex 
molecules. 

Some interesting effects of resonance on bond distances have been 
observed. For example, the C Cl distance in CH 3 C1 is 1.76 A but in 
CH 2 =CHCi it is only 1.69 A. The shortening of the bond is ascribed to 
resonance between the following structures: 

Cl: Cl^ 

/' .. / 

H 2 O=C and H 2 C C 

\ \ 

H H 

The C Cl bond in ethylene chloride is said to have about 18 per cent double 
bond character. 

24. Molecular spectra. Perhaps the most widely useful of all methods 
for investigating molecular architecture is the study of molecular spectra. It 
affords information about not only the dimensions of molecules but also the 
possible molecular energy levels. Thus, other methods pertain to the ground 
state of the molecule alone, but the analysis of spectra also elucidates the 
nature of excited states. 

It has been mentioned that the spectra of atoms consist of sharp lines, 
and those of molecules appear to be made up of bands in which a densely 
packed line structure is sometimes revealed under high resolving power. 

Spectra arise from the emission or absorption of definite quanta of radia- 
tion when transitions occur between certain energy levels. In an atom the 
energy levels represent different allowed states for the orbital electrons. A 
molecule too can absorb or emit energy in transitions between different 
electronic energy levels. Such levels would be associated, for example, with 
the different 'molecular orbitals discussed on pages 303-311. In addition 
there are two other possible ways in which a molecule can change its energy 
level, which do not occur in atoms. These are by changes in the vibrations 
of the atoms within the molecule and by changes in the rotational energy of 
the molecule. These energies, like the electronic, are quantized, so that only 
certain distinct levels of vibrational and rotational energy are permissible. 

In the theory of molecular spectra it is customary, as a good first approxi- 
mation, to consider that the energy of a molecule can be expressed simply 
as the sum of electronic, vibrational, and rotational contributions. Thus, 

E - Zf elee -} vib + rot (11.20) 

This complete separation of the energy into three distinct categories is not 
strictly correct. For example, the atoms in a rapidly rotating molecule 
are separated by centrifugal forces, which thus affect the character of the 



332 THE STRUCTURE OF MOLECULES [Chap. 11 

vibrations. Nevertheless, the approximation of eq. (11.20) suffices to 
explain many of the observed characteristics of molecular spectra. 

It will be seen in the following discussions that the separations between 
electronic energy levels are usually much larger than those between vibra- 
tional energy levels, which in turn are much larger than those between 
rotational levels. The type of energy-level diagram that results is shown in 
Fig. 1 1.18. Associated with each electronic level there is a series of vibrattonal 



J' 



J' 



J'- 







Fig. 11.18. Energy-level diagram for a molecule. Two electronic levels 
A and B, with their vibrational levels (v) and rotational levels (J) 

levels, each of which is in turn associated with a series of rotational levels. 
The close packing of the rotational levels is responsible for the banded 
structure of molecular spectra. 

Transitions between different electronic levels give rise to spectra in the 
visible or ultraviolet region; these are called electronic spectra. Transitions 
between vibrational levels within the same electronic state are responsible 
for spectra in the near infrared (< 20/^), called vibration-rotation spectra. 
Finally, spectra are observed in the far infrared (> 20^) arising from transi- 
tions between rotational levels belonging to the same vibrational level; these 
are called pure rotation spectra. 



Sec. 25] THE STRUCTURE OF MOLECULES 333 

25. Rotational levels far-infrared spectra. The model of the rigid rotator, 
described on page 189, may be used for the interpretation of pure rotation 
spectra. The calculation of the allowed energy levels for such a system is a 
straightforward problem in quantum mechanics. The SchrCdinger equation 
in this case is very similar to that for the motion of the electron about the 
nucleus in the hydrogen atom, except that for a diatomic molecule it is a 
question of the rotation of two nuclei about their center of mass. We recall 
that the rotation of a dumbbell model is equivalent to the rotation of the 
reduced mass // at a distance r from the rotation axis. For a rigid rotator the 
potential energy U is zero, so that the wave equation becomes 

V ^0 (11.21) 

Without too great difficulty this equation can be solved exactly. 20 It is 
then found that the eigenfunction y is single valued, continuous, and finite, 
as is required for physical meaning, only for certain values of the energy E y 
the allowed eigenvalues. These are 

WA./ 10 _/(/+!) 

trai ' fcrV* ~~"M*r ( } 

Here / is the moment of inertia of the molecule and the rotational quantum 
number J can have only integral values, 0, 1,2, 3, etc. 

The value of J gives the allowed values of the rotational angular momen- 
tum /?, in units of h/2n: p =-- (h/27r)Vj(J + 1) ^ (h/2n) J. This is exactly 
similar to the way in which the quantum number / in the hydrogen-atom 
system, and the corresponding A in molecules, determine the orbital angular 
momenta of electrons. 

The selection rule for rotational levels is found to be A/ = or 1. 
Thus an expression for AE for the rigid-rotator model is readily derived 
from eq. (11.22). Writing B =-- h/Kir'*!, we obtain for two levels with 
quantum numbers J and J': A ^ hv hB[J(J f 1) -J \J' + I)]. Since 
v -= (A//0, and7 -J' = 1, 

v = 2fl/ (11.23) 

The spacing between energy levels increases linearly with 7, as shown in 
Fig. 11.18. The absorption spectra due to pure rotation arise from transitions 
from each of these levels to the next higher one. By means of a spectrograph 
of good resolving power, the absorption band will be seen to consist of a 
series of lines spaced an equal distance apart. From eq. (11.23) this spacing 
is AT --- v v ----- 2B. 

Pure rotation spectra occur only when the m9lecule possesses a permanent 

20 K. S. Pitzer, Quantum Chemistry (New York: Prentice-Hall, 1953), p. 53. An 
approximate formula is obtained directly from the Bohr hypothesis that the angular 
momentum is quantized in units of h/2ir. Thus /co = Jh/2*, and the kinetic energy 
rV. 



334 THE STRUCTURE OF MOLECULES [Chap. 1 1 

dipole moment. This behavior has been elucidated by quantum mechanical 
arguments, but it can be understood also in. terms of the classical picture 
that radiation is produced when a rotating dipole sends out into space a 
train of electromagnetic waves. If a molecule has no dipole, its rotation 
cannot produce an alternating electric field. 

We have discussed only the problem of the diatomic rotator. The rota- 
tional energy levels of polyatomic molecules are considerably more complex, 
but do not differ much from the diatomic case in the principles involved. 

26. Internuclear distances from rotation spectra. The analysis of rotation 
spectra can give accurate values of the moments of inertia, and hence inter- 
nuclear distances and shapes of molecules. Let us consider the example of 
HC1. 

Absorption by HC1 has been observed in the far infrared, around 
A = 50 microns or v --= 200cm" 1 . The spacing between successive lines is 
A/ 20.1 to 20.7 cm" 1 . Analysis shows that the transition from / = to 
J 1 corresponds to a wave number of v' I/A = 20.6 cm" 1 . The frequency 
is therefore 

v - ~ - (3.00 x 10 10 )(20.6) - 6.20 x 10 11 sec' 1 

A 

The first rotational level, / = 1, lies at an energy of 

hv - (6.20 x 10 U )(6.62 x 10~ 27 ) - 4.10 x 10~ 15 erg 
Fromeq. (11.22), 

= 1^=4.10X10- 

so that /= 2.72 x 10- 40 gcm 2 

Since / --- jur 2 , where // is the reduced mass, we can now determine the inter- 
nuclear distance r. For HC1, 



72 x i()- 40 \ 1/2 



(2 
l 



27. Vibrational energy levels. Investigations in the far infrared are difficult 
to make, and a much greater amount of useful information has been obtained 
from the near-infrared spectra, arising from transitions between different 
vibrational energy levels. 

The simplest model for a vibrating molecule is that of the harmonic 
oscillator, whose potential energy is given by U -= J/cjt 2 , the equation of a 
parabola. The Schrftdinger equation is therefore: 

= (11.24) 



Sec. 27] THE STRUCTURE OF MOLECULES 335 

The solution to this equation can be obtained exactly by quite simple 
methods. 21 The result has already been mentioned as a consequence of 
uncertainty-principle arguments (page 275), being 

vib = ( + i)** (11.25) 

The energy levels are equally spaced, and the existence of a zero point energy, 
EQ ~= \hv$ when v = 0, will be noted. The selection rule for transitions 
between vibrationai energy levels is found to be Ai? i I . 

Actually, the harmonic oscillator is not a very good model for molecular 
vibrations except at low energy levels, near the bottom of the potential- 
energy curve. It fails, for example, to represent the fact that a molecule may 
dissociate if the amplitude of vibration becomes sufficiently large. The sort 
of potential-energy curve that should be used is one like that pictured for 
the hydrpgen molecule in Fig. 1 1.2 on page 298. 

Two heats of dissociation may be defined by reference to this curve. The 
xpectroscopic heat of dissociation, D e , is the height from the asymptote to the 
minimum. The chemical heat of dissociation, Z) , is measured from the ground 
state of the molecule, at v = 0, to the onset of dissociation. Therefore, 

D e = + !Av (11-26) 

In harmonic vibration the restoring force is directly proportional to the 
displacement r. The potential-energy curve is parabolic and dissociation can 
never take place. Actual potential-energy curves, like that in Fig. 1 1.2, corre- 
spond to anharmonic vibrations. The restoring force is no longer directly 
proportional to the displacement. The force is given by dU/dr, the slope 
of the potential curve, and this decreases to zero at large values of r, so that 
dissociation can occur as the result of vibrations of large amplitude. 

The energy levels corresponding to an anharmonic potential-energy curve 
can be expressed as a power series in (v f- i), 

v ib = hv[(v \ 1) - x e (v + i) 2 + y f (v + I) 3 - . . .] (1 1.27) 

Considering only the first anharmonic term, with anharmonicitv constant, x e : 

v ib MM |) -/iwt,(r f i) 2 (11.28) 

The energy levels are not evenly spaced, but lie more closely together as the 
quantum number increases. This fact is illustrated in the levels superimposed 
on the curve in Fig. 1 1 .2. Since a set of closely packed rotational levels is 
associated with each of these vibrationai levels, it is sometimes possible to 
determine with great precision the energy level just before the onset of the 
continuum, and so to calculate the heat of dissociation from the vibration- 
rotation spectra. 

As an example of near-infrared spectra, let us consider some observations 
with hydrogen chloride. There is an intense absorption band at 2886cm" 1 . 

21 Pauling and Wilson, he. cit., p. 68. A student might well study this as a typical 
quantum mechanical problem, since it is about the simplest one available. 



336 THE STRUCTURE OF MOLECULES [Chap, n 

This arises from transitions from the state with v = to that with v = 1, or 
Ay = +1. In addition, there are very much weaker bands at higher fre- 
quencies, corresponding to Ai? =-= +2, +3, . . . etc., which are not com- 
pletely ruled out for an anharmonic oscillator. 
For the v = 1 band in HCl, we have, therefore, 

v = (2886) x 3 x 10 10 - 8.65 x 10 13 sec" 1 

as the fundamental vibration frequency. This is about one hundred times the 
rotation frequency found from the far-infrared spectra. 

The force constant of a harmonic oscillator with this frequency, from 
eq. (10.2), would be K --=-- 4n*v 2 p = 4.81 x 10 5 dynes per cm. If the chemical 
bond is thought of as a spring, the force constant is a measure of its tightness. 

Potential-energy curves of the type shown in Fig. 11.2 are so generally 
useful in chemical discussions that it is most convenient to have an analytical 
expression for them. An empirical function that fits very well is that suggested 
by P. M. Morse: 

I/- D,(l *-'-'>)* (11.29) 

Here /? is a constant that can be evaluated in terms of molecular parameters 
as ft - v V2n*[i/D e . 

28. Microwave spectroscopy. Microwaves are those with a wavelength in 
or around the range from 1 mm to 1 cm. Their applications were rapidly 
advanced as a result of wartime radar research. In recent years, radar tech- 
niques have been applied to spectroscopy, greatly extending the accuracy 
with which we can measure small energy jumps within molecules. 

In ordinary absorption spectroscopy, the source of radiation is usually a 
hot filament or high-pressure gaseous-discharge tube, giving in either case a 
wide distribution of wavelengths. This radiation is passed through the 
absorber and the intensity of the transmitted portion at diffeient wavelengths 
is measured after analysis by means of a grating or prism. In microwave 
spectroscopy, the source is monochromatic, at a well defined single wave- 
length which can, however, be rapidly varied (fiequency modulation). It is 
provided by an electronically controlled oscillator employing the recently 
developed klystron or magnetron tubes. After passage through the cell con- 
taining the substance under investigation, the microwave beam is picked up 
by a receiver, often of the crystal type, and after suitable amplification is fed 
to a cathode-ray oscillograph acting as detector or recorder. The resolving 
power of this arrangement is 100,000 times that of the best infrared grating 
spectrometer, so that wavelength measurements can be made to seven 
significant figures. 

One of the most thoroughly investigated of microwave spectra has been 
that of the "umbrella" inversion of the ammonia molecule, the vibration in 
which the nitrogen atom passes back and forth through the plane of the 
three hydrogen atoms. The rotational fine structure of this transition has 
been beautifully resolved, over 40 lines having been catalogued for 14 NH 3 



Sec. 29] 



THE STRUCTURE OF MOLECULES 



337 



and about 20 for 15 NH 3 . Such measurements provide an almost embarrassing 
wealth of experimental data, permitting the construction of extremely detailed 
theories for the molecular energy levels. 

Pure rotational transitions in heavier molecules are inaccessible to ordi- 
nary infrared spectroscopy because, in accord with eq. (11.22), the large 
moments of inertia would correspond to energy levels at excessively long, 
wavelengths. Microwave techniques have made this region readily accessible. 
From the moments of inertia so obtained, it is possible to calculate inter- 
nuclear distances to better than _t0.002 A. A few examples are shown in 
Table 11.4. 

TABLE 11.4 
INTERNUCIEAR DISTANCES FROM MICROWAVE SPECTRA 



Molecule 


Distance (A) 


Molecule 

i 


Distance (A) 


C1CN 


C Cl 1.630 


I ocs 


C .161 




C N 1.163 




C S .560 


BrCN 


C Br 1 .789 


N. 2 O 


N N .126 




C N 1.160 




N O .191 






S0 2 


S .433 



By observing the spectra under the influence of an electric field (Stark 
effect) the dipole moments of gas molecules can be accurately determined. 
Microwave measurements also afford one of the best methods for finding 
nuclear spins. 

29. Electronic band spectra. The energy differences A between electronic 
states in a molecule are in general much larger than those between successive 
vibrational levels. Thus the corresponding electronic band spectra are 
observed in the visible or ultraviolet region. The A's between molecular 
electronic levels ape usually of the same order of magnitude as those between 
atomic energy levels, ranging therefore from 1 to 10 ev. 

In Fig. 11.19 are shown the ground state of a molecule (Curve A), and 
two distinctly different possibilities for an excited state. In one (Curve B), 
there is a minimum in the potential energy curve, so that the state is a stable 
configuration for the molecule. In the other (Curve C), there is no minimum, 
and the state is unstable for all internuclear separations. 

A transition from ground state to unstable state would be followed 
immediately by dissociation of the molecule. Such transitions give rise to 
a continuous absorption band in the observed spectra. Transitions between 
different vibrational levels of two stable electronic states also lead to a band 
in ihe spectra, but in this case the band can be analyzed into closely packed 
lines corresponding to the different upper and lower vibrational and rota- 
tional levels. The task of the spectroscopist is to measure the wavelengths of 
the various lines and interpret them in terms of the energy levels from which 



338 



THE STRUCTURE OF MOLECULES 



[Chap. 1 1 



they arise. There is obviously a wealth of experimental data here, which 
should make possible a profound knowledge of the structure of molecules. 
There is a general rule, known as the Franck-Condon principle, which is 
helpful in understanding electronic transitions. An electron jump takes place 
very quickly, much more quickly than the period of vibration of the atomic 
nuclei (~ 10~ 13 sec), which are heavy and sluggish compared with electrons. 
It can therefore be assumed that the positions and velocities of the nuclei are 
virtually unchanged during transitions, 22 which can thus be represented by 
vertical lines drawn on the potential energy curves, Fig. 11.19. 




Fig. 11.19. Transitions between electronic levels in molecules. 

By applying the Franck-Condon principle it is possible to visualize how 
transitions between stable electronic states may sometimes give rise to dis- 
sociation. For example, in Curve A of Fig. 11.19, the transition XX' leads 
to a vibrational level in the upper state that lies above the asymptote to the 
potential energy curve. Such a transition will lead to dissociation of the 
molecule. 

If a molecule dissociates from an excited electronic state, the fragments 
formed, atoms in the diatomic case, are not always in their ground states. 
In order to obtain the heat of dissociation into atoms in their ground states, 
it is therefore necessary to subtract the excitation energy of the atoms. For 

22 It may be noted that the vertical line for an electronic transition is drawn from a 
point on the lower curve corresponding with the midpoint in the internuclear vibration. 
This is done because according to quantum mechanics the maximum in \p in the ground 
state lies at the mid-point of the vibration. This is not true in higher vibrational states, 
for which the maximum probability lies closer to the extremes of the vibration. Classical 
theory predicts a maximum probability at the extremes of the vibration. 



Sec. 30] THE STRUCTURE OF MOLECULES 339 

example, in the ultraviolet absorption spectrum of oxygen there is a series 
of bands corresponding to transitions from the ground state to an excited 
state. These bands converge to the onset of a continuum at 1759 A, equiva- 
lent to 7.05 ev. The two atoms formed by the dissociation are found to be 
a normal atom (3P state) and an excited atom (1 D state). The atomic spec- 
trum of oxygen reveals that this 1 D state lies 1 .97 ev above the ground state. 
Thus the heat of dissociation of molecular oxygen into two normal atoms 
(O 2 - 2 O (3P) ) is 7.05 - 1 .97 - 5.08 ev or 1 17 kcal per mole. 

30, Color and resonance. The range of wavelengths from the red end of 
the visible spectrum at 8000 A to the near ultraviolet at 2600 A corresponds 
with a range of energy jumps from 34 to 1 14 kcal per mole/ A compound 
with an absorption band in the visible or near ultraviolet must therefore 
possess at least one electronic energy level from 34 to 114 kcal above the 
ground level. This is not a large energy jump compared with the energy of 
binding of electrons in an electron pair bond. It is therefore not surprising 
that most stable chemical compounds are actually colorless. In fact, the 
appearance of color indicates that one of the electrons in the structure is 
loosely held and can readily be raised from the ground molecular orbital to 
an excited orbital. 

For example, molecules containing an unpaired electron (odd molecules 
and free radicals) are usually colored (NO 2 , CIO 2 , triphenylmethyl, etc.). 
Groups such as NO 2 , C==O, or N N often confer color on a mole- 
cule since they contain electrons, in 7r-type orbitais, that are readily raised 
to excited orbitais. 

In other cases, resonance gives rise to a series of low-lying excited levels. 
The ground state in the benzene molecule can be assigned an orbital written 
as y A + y Ry where A and B denote the two Kekule structures shown on 
page 311. The first excited state is then y> A y B . This state lies 115 kcal 
above the ground level, and the excitation of an electron into this state is 
responsible for the near-ultraviolet absorption band of benzene around 
2600 A. 

In a series of similar molecules such as benzene, naphthalene, anthracene, 
etc., the absorption shifts toward longer wavelength as the molecule becomes 
longer. The same effect is observed in the conjugated polyenes; butadiene 
is colorless but by the time the chain contains about twelve carbon atoms, 
the compounds are deeply colored. This behavior can be explained in terms 
of the increasing delocalization of the ^-electrons as the length of the mole- 
cule increases. Let us recall the simple expression for the energy levels of an 
electron in a box, eq. (10.39), E n = /zV/8m/ 2 , where /is the length of the box. 
In a transition from n^ to n 2 the energy jump is (/i 2 /8m/ 2 ) (nf w 2 2 ). Thus 
not only the value of the energy but also the size of the energy jump falls 
markedly with increasing /. Now the molecular orbitals in organic molecules 
are of course not simple potential boxes, but the situation is physically very 
similar. Anything that increases the space in which the 7r-electron is free to 



340 



THE STRUCTURE OF MOLECULES 



[Chap. 11 



move tends to decrease the energy gap between the ground state and excited 
states, and shifts the absorption toward the red. 

Most dyes have structures that consist of two resonating forms. For 
instance, the phenylene blue ion is 



/ vw v 




+NH 




NH 2 



In this and similar cases, the transition responsible for the color can be 
ascribed to an electron jump between a y A + y> B anc * a y> A y B orbital. 

31. Raman spectra. If a beam of light is passed through a medium, a 
certain amount is absorbed, a certain amount transmitted, and a certain 



N 



Fig. 11.20. Raman spectrum of O 2 excited by Hg 2537-A line. (From Herzberg, 
Molecular Spectra and Molecular Structure, Van Nostrand, 1950.) 

amount scattered. The scattered light can be studied by observations per- 
pendicular to the direction of the incident beam. Most of the light is scattered 
without change in wavelength (Rayleigh scattering); but there is in addition 
a small amount of scattered light whose wavelength has been altered. If the 
incident light is monochromatic, e.g., the Na D line, the scattered spectrum 
will exhibit a number of faint lines displaced from the original wavelength. 
An example is shown in Fig. 11.20. 

This effect was first observed by C. V. Raman and K. S. Krishnan in 
1928. It is found that the Raman displacements, Av, are multiples of vibra- 
tional and rotational quanta characteristic of the scattering substance. There 
are therefore rotational and vibration-rotational Raman spectra, which are 
the counterparts of the ordinary absorption spectra observed in the far and 
near infrared. Since the Raman spectra are studied with light sources in the 
visible or ultraviolet, they provide a convenient means of obtaining the same 
sort of information about molecular structure as is given by the infrared 
spectra. In many cases, the two methods supplement each other, since vibra- 
tions and rotations that are not observable in the infrared (e.g., from mole- 
cules without permanent dipoles) may be active in the Raman. 



Sec. 32] 



THE STRUCTURE OF MOLECULES 



341 



32. Molecular data from spectroscopy. Table 1 1.5 is a collection of data 
derived from spectroscopic observations on a number of molecules. 

TABLE 11.5 
SPECTROSCOPIC DATA ON THE PROPERTIES OF MOLECULES* 

Diatomic Molecules 



Molecule 


Equilibrium 
Inter nuclear 
Separation, 
r. (A) 


Heat of 
Dissociation, 
Do (ev) 


Fundamental 
Vibration 
Frequency 
(a>, cm" 1 ) 


Moment of 
Inertia, 
(g cm* x 10-") 


Cl, 


1.989 


2.481 


564.9 


114.8 


CO 


1.1284 


9.144 


2168 


14.48 


H, 


0.7414 


4 777 


4405 


0.459 


HD 


0.7413 


4.513 


3817 


0.611 


D, 


0.7417 


4556 


3119 


0.918 


HBr 


1.414 


3.60 


2650 


3.30 


HC1" 


1.275 


4.431 


2989 


2.71 


I. 


2.667 


1.542 


214.4 


748 


Li, 


2.672 


1 14 


351.3 


41.6 


N, 


1.095 


7 384 


2360 


13.94 


NaCl 


251 


425 


380 




NH 


1.038 


34 


3300 


1.68 


0, 


1 2076 


5.082 


1580 


19.14 


OH 


0.971 


4.3 


3728 


1 48 













Triatomic Molecules 



Molecule. 
X-Y-Z 


Internuclear 
Separation 
(A) 


Bond 
Angle 


Moments of Inertia 
(g cm* x 10 *) 


Fundamental Vibration 
Frequencies (cm^ 1 ) 








(deg) 
















ffv 


r v* 




'.i 


'71 


J c 


(U l 


>i 


W| 


o~~c o 


1.162 


1 162 


180 




71 67 




1320 


668 


2350 


H O H 


096 


096 


105 


1 024 


1 920 


2947 


3652 


1595 


3756 


D O D 


0.96 


0.96 


105 


1.790 


3812 5752 


2666 


1179 


2784 


H S H 


1.35 


1.35 


92 


2667 


3076 


5.845 


2611 


1290 


2684 


S -0 


1.40 


1.40 


120 


123 


732 


85.5 


1151 524 


1361 


N -N- O 


1.15 


1 23 


180 




66.9 





1285 589 


2224 



* From G. Herzberg, Molecular Spectra and Molecular Structure, Vols. I and II (New York: D. Van Nostrand 
Co., 1950). 

In this chapter we have not discussed the spectra of polyatomic molecules, 
one of the most active branches of modern spectroscopy. It is possible, how- 
ever, to evaluate moments of inertia and vibration frequencies for polyatomic 
molecules by extensions of the methods described for diatomic molecules. 
Generally the high-frequency vibrations are those that stretch the bonds, and 
the lower frequencies are bond-bending vibrations. 

It is often possible to characterize a given type of chemical bond by a 
bond vibration frequency, which is effectively constant in a large number of 
different compounds. For example, the stretching frequency of the O-O 
bond is 1706 in acetone, 1715 in acetaldehyde, 1663 in actetic acid, and 
1736 in methyl acetate. 

The approximate constancy of these bond or group frequencies is the basis 
for the widespread application of infrared spectroscopy to the structure 

H ^termination r\f* n<\i/ /"\ranir rrmnnnHc anrl th* HftnilpH cnPftrilTTI nrOV1flP.<5 



342 



THE STRUCTURE OF MOLECULES 



[Chap. 11 



a method for characterizing a new compound which is as reliable as 
the finger-printing of a suspect citizen. Some typical bond frequencies are 
summarized in Table 1 1 .6. 

TABLE 11.6 
BOND-FREQUENCY INTERVALS FOR INFRARED SPECTRA OF GASES* 



Group 



H-- < 
H N< 



H C=C< 

I 

H S 

N^C 

c-c 



Frequency Interval 



3500-3700 
3300-3500 
3300-3400 
3000-3100 



2550-2650 
2200-2300 
2170-2270 





Group 


Frequency Interval 
cm" 1 




O=-C< 


1700-1850 




>c=c< 


1550-1650 




s-=c< 


1500-1600 




F C 


1100-1300 




Cl C 


700-800 




Br C 


500-600 




I C 


400-500 



* After B. Bak, Elementary Introduction to Molecular Spectra (Amsterdam: North 
Holland Publ. Co., 1954). 

33. Bond energies. In discussions of structure, thermodynamics, and 
chemical kinetics, it is often necessary to have some quantitative information 
about the strength of a certain chemical bond. The measure of this strength 
is the energy necessary to break the bond, the so-called bond energy. The 
energy of a bond between two atoms, A B, depends on the nature of the 
rest of the molecule in which the bond occurs. There is no such thing as a 
strictly constant bond energy for A B that persists through a varied series 

TABLE 11.7 
BOND ENERGIES (KCAL/MOLE) 



Elements 


Hydrides 


Chlorides 


H H 


103.2 


Li H 58 


C Cl 


78 


Li Li 


26 


C H 98.2 


Cl 


49 


C C 


80 


N H 92.2 


Si Cl 


87 


N N 


37 


O H 109.4 


P Cl 


77 


O O 


34 


P H 77 


I Cl 


49.6 


Cl Cl 


57.1 


S H 87(?) 






I I 


35.6 


Cl H 102.1 










Br H 86.7 










Single 


Double 


Triple 


C C 




80 


145 




198 


N N 




37 







225 


O O 




34 


117 







C N 




66 







209 



Sec. 33] THE STRUCTURE OF MOLECULES 343 

of compounds. Nevertheless, it is possible to strike an average from which 
actual A B bonds do not deviate too widely. 

Pauling has reduced a large amount of experimental data to a list of 
normal covalent single-bond energies. 23 If the actual bond is markedly 
polarized (partial ionic character), or if through resonance it acquires some 
double-bond character, its energy may be considerably higher than the norm. 

Values from a recent compilation 24 are given in Table 1 1 .7. These values 
are obtained by a combination of various methods: (1) spectroscopy, (2) 
thermochemistry, and (3) electron impact. The electron impact method 
employs a mass spectrometer and gradually increases the energy of the 
electrons from the ion gun until the molecule is broken into fragments. 

An instance of the application of thermochemical data is the following 
determination of the O H bond strength : 

H 2 ~ 2 H AH = 103.4 kcal (spectroscopic) 

O 2 = 2 O AH 1 18.2 (spectroscopic) 

H 2 f- * O 2 -= H 2 AH - - 57.8 (calorimetric) 

2 H + O = H 2 O AH - 220.3 

This is A// for the formation of 2 O H bonds, so that the bond strength is 
taken as 220.3/2 - 110 kcal. 

PROBLEMS 

1. Write down possible resonance forms contributing to the structures of 
the following: CO 2 , CH 3 COO-, CH 2 .CH-CH:CH 2 , CH 3 NO 2 , C 6 H 5 C1, 
C 6 H 5 NH 2 , naphthalene. 

2. On the basis of molecular orbital theory, how would you explain the 
following? The binding energy of N 2 4 is 6.35 and that of N 2 7.38 ev, whereas 
the binding energy of O 2 + is 6.48 and that of O 2 , 5.08 ev. 

3. The following results are found for the dielectric constant e of gaseous 
sulfur dioxide at 1 atm as a function of temperature: 

K 267.6 297.2 336.9 443.8 

e . 1.009918 1.008120 1.005477 1.003911 

Estimate the dipole moment of SO 2 , assuming ideal gas behavior. 

4. M. T. Rogers 25 found the following values for the dielectric constant e 
and density p of isopropyl cyanide at various mole fractions X in benzene 
solution at 25C: 

X . . . 0.00301 0.00523 0.00956 0.01301 0.01834 0.02517 
e . . . 2.326 2.366 2.442 2.502 2.598 2.718 

P . . . 0.87326 0.87301 0.87260 0.87226 0.87121 0.87108 

For pure C 3 H 7 NC, p = 0.7 '6572, refractive index n D = 1.3712; for pure 

23 For a full discussion: L. Pauling, op. cit., p. 53. 

24 K. S. Pitzer, J. Am. Chem. Soc., 70, 2140 (1948). 
26 J. Am. Chem. Soc.> 69, 457 (1947). 



344 THE STRUCTURE OF MOLECULES [Chap. 11 

benzene, p 0.87345, n D -=- 1.5016. Calculate the dipole moment /i of 
isopropyl cyanide. 

5. Chlorobenzene has /t -- 1.55 d, nitrobenzene // = 3.80 d. Estimate the 
dipole moments of: metadinitrobenzene, orthodichlorobenzene, metachloro- 
nitrobenzene. The observed moments are 3.90, 2.25, 3.40 d. How would you 
explain any discrepancies? 

6. The angular velocity of rotation o> 27rv rot where v roi is the rotation 
frequency of a diatomic rotor. The angular momentum is (h/27r)Vj(J | 1). 
Calculate the rotation frequency of the HC1 molecule for the state with 
/ --=--- 9. Calculate the frequency of the spectral line corresponding to the 
transition J ^ 9 to / - 8. 

7. In the far infrared spectrum of HBr is a series of lines having a separa- 
tion of 16.94cm *. Calculate the moment of inertia and the internuclear 
separation in HBr from this datum. 

8. In the near infrared spectrum of carbon monoxide there is an intense 
band at 2144cm" 1 . Calculate (a) the fundamental vibration frequency of 
CO; (b) the period of the vibration; (c) the force constant; (d) the zero-point 
energy of CO in cal per mole. 

9. Sketch the potential-energy curve for the molecule Li 2 according to 
the Morse function, given D - 1.14 ev, v ~- 351.35 cm" 1 , r f 2.672 A. 

10. The Schumann-Runge bands in the ultraviolet spectrum of oxygen 
converge to a well defined limit at 1759 A. The products of the dissociation 
are an oxygen atom in the ground state and an excited atom. There are two 
low-lying excited states of oxygen, 1 D and 1 S at 1.967 and 4.190 volts above 
the ground state. By referring to the dissociation data in Table 4.4, page 81, 
decide which excited state is formed, and then calculate the spectroscopic 
dissociation energy of O 2 into two O atoms in the ground state. 

11. In a diffraction investigation of the structure of CS 2 with 40-kv 
electrons, Cross and Brockway 26 found four sharp maxima ( f+) each 
followed by a weak maximum ( 4 ) and a deep minimum ( ), at the following 
values of 4-77/A (sin 0/2) 

4.713 6.312 7.623 8.698 10.63 11.63 12.65 14.58 15.54 16.81 

I f- -I -I \ \- -f - -}-+ 4- + + 

CS 2 is a linear molecule. Calculate the C S distance from these data, 
using the approximation that the scattering factor is equal to the atomic 
number Z. 

12. With data from Table 11.5, draw to scale the first five rotational 
levels in the molecule NaCl. At what frequency would the transition J = 4 
to 75 be observed? In NaCl vapor at 1000C what would be the relative 
numbers of molecules in the states with J = 0, J = 1, and J = 2. 

* J. Chem. Phys., 3, 821 (1935). 



Chap. 11] THE STRUCTURE OF MOLECULES 345 

13. In ions of the first transition series, the paramagnetism is due almost 
entirely to the unpaired spins, being approximately equal to /* 2v / S(S -f- 1 ) 
magnetons where S is the total spin. On this basis, estimate // for K 13 , Mn f 2 , 
Co+ 2 , and Cu+. 

REFERENCES 

BOOKS 

1. Bates, L. F., Modern Magnetism (London: Cambridge, 1951). 

2. Bottcher, C. J. F.., Theory of Electric Polarisation (Amsterdam: Elsevier, 
1952). 

3. Burk, R. E., and O. Grummitt (editors), Chemical Architecture (New 
York: Interscience, 1948). 

4. Coulson, C. A., Valence (New York: Oxford, 1952). 

5. Debye, P., Polar Molecules (New York: Dover, 1945). 

6. Gaydon, A. G., Dissociation Energies (London: Chapman and Hall, 
1952). 

7. Gordy, W., W. V. Smith, and R. F. Trambarulo, Microwave Spectra- 
scopy (New York: Wiley, 1953). 

8. Herzberg, G., Infrared and Raman Spectra (New York: Van Nostrand, 
1945). 

9. Herzberg, G., Molecular Spectra and Molecular Structure (New York: 
Van Nostrand, 1950). 

10. Ketelaar, J. A. A., Chemical Constitution (Amsterdam: Elsevier, 1953). 

11. Palmer, W. G., Valency, Classical and Modern (Cambridge, 1944). 

12. Pauling, L., The Nature of the Chemical Bond (Ithaca: Cornell Press, 
1940). 

13. Pitzer, K. S., Quantum Chemistry (New York: Prentice-Hall, 1953). 

14. Rice, F. O., and E. Teller, The Structure of Matter (New York: Wiley, 
1949). 

ARTICLES 

1. Condon, E. U., Am. J. Phys., 75, 365-74 (1947), "The Franck-Condon 
Principle and Related Topics." 

2. Klotz, I. M., /. Chem. Ed., 22, 328-36 (1945), "Ultraviolet Absorption 
Spectroscopy." 

3. Mills, W. H., J. Chem. Soc., 1942, 457-66 (1942), "The Basis of Stereo- 
chemistry." 

4. Pake, G. E., Am. J. Phys., 18, 438-73 (1950), "Nuclear Magnetic 
Resonance." 

5. Pauling, L., /. Chem. Soc., 1461-67 (1948), "The Modern Theory of 
Valency." 

6. Purcell, E. M., Science, 118, 431-36 (1953), "Nuclear Magnetic 
Resonance." 



346 THE STRUCTURE OF MOLECULES [Chap. 11 

7. Selwood, P. W., /. Chem. Ed., 79, 181-88 (1942), "Magnetism and 
Molecular Structure." 

8. Spurr, R., and L. Pauling, J. Chem. Ed., 18, 458-65 (1941), "Electron 
Diffraction of Gases." 

9. Sugden, S., J. Chem. Soc., 328-33 (1943), "Magnetochemistry." 

10. Thompson, H. W., J. Chem. Soc. 9 183-92 (1944), "Infrared Measure- 
ments in Chemistry." 

11. Wilson, E. B., Ann. Rev. Phys. Chem., 2, 151-76 (1951), "Microwave 
Spectroscopy of Gases". 



CHAPTER 12 

Chemical Statistics 



1. The statistical method. If you take a deck of cards, shuffle it well, and 
draw a single card at random, it is not possible to predict what the card will 
be, unless you happen to be a magician. Nevertheless, a -number of significant 
statements can be made about the result of the drawing. For example: the 
probability of drawing an ace is one in thirteen ; the probability of drawing 
a spade is one in four; the probability of drawing the ace of spades is one in 
fifty-two. Similarly, if you were to ask an insurance company whether a 
certain one of its policyholders was going to be alive 10 years from now, the 
answer might be: "We cannot predict the individual fate of John Jones, but 
our actuarial tables indicate that the chances are nine out of ten that he will 
survive." 

We are familiar with many statements of this kind and call them "statisti- 
cal predictions." In many instances it is impossible to foretell the outcome 
of an individual event, but if a large number of similar events are considered, 
a statement based on probability laws becomes possible. An example from 
physics is found in the disintegration of radioactive elements. No one can 
determine a priori whether an isolated radium atom will disintegrate within 
the next 10 minutes, the next 10 days, or the next 10 centuries. If a milligram 
of radium is studied, however, we know that very close to 2.23 x 10 10 atoms 
will explode in any 10-minute period. 

Some applications of statistical principles to chemical systems were dis- 
cussed in Chapter 7. It was pointed out that since the atoms and molecules 
of which matter is composed are extremely small, any large-scale body con- 
tains an enormous number of elementary particles. It is impossible to keep 
track of so many individual particles. Any theory that attempts to interpret 
the behavior of macroscopic systems in terms of atoms and molecules must 
therefore rely heavily on statistical considerations. But just because a system 
does contain so very many particles, its actual behavior will be practically 
indistinguishable from.that predicted by statistics. If a man tossed 10 coins, 
the result might deviate widely from 50 per cent heads; if he tossed a thous- 
and, the percentage deviation would be fairly small; but if some tireless 
player were to toss 10 23 coins, the result would be to all intents and purposes 
exactly 50 per cent heads. 

We have seen already that from the molecular-kinetic point of view the 
Second Law of Thermodynamics is a statistical law. It expresses the drive 
toward randomness or disorder in a system containing a large number of 
particles. Applied to an individual molecule it has no meaning, for in this 

347 



348 CHEMICAL STATISTICS [Chap. 12 

case any distinction between heat (disordered energy) and work (ordered 
energy) disappears. Even for intermediate cases, such as colloidal particles 
in Brownian motion, the Second Law is inapplicable, since the particles 
contain only about 10 6 to 10 9 atoms. 

Now that the structures and energy levels of atoms and molecules have 
been considered, in Chapters 8 through 11, it is possible to see how the 
behavior of macroscopic systems is determined by these atomic and mole- 
cular parameters. We shall confine our attention to systems in equilibrium, 
which are usually treated by thermodynamics. This is not, however, a necessary 
restriction for the statistical method, which is competent to handle also 
situations in which the system is changing with time. These are some- 
times called "rate processes," and include transport phenomena, such as 
diffusion and thermal conductivity, as well as the kinetics of chemical 
reactions. 

Statistical thermodynamics is still a very young science, and many funda- 
mental problems remain to be solved. Thus the only systems that have been 
treated at all accurately are ideal gases and perfect crystals. Imperfect gases 
and liquids present unsurmounted difficulties. 

2. Probability of a distribution. The discussion of statistical thermo- 
dynamics upon which we are embarking will not be distinguished for its 
mathematical precision, nor will any attempt be made to delve into the 
logical foundations of the subject. 1 

The general question to be answered is this: given a macroscopic physical 
system, composed of molecules (and/or atoms), and knowing from quantum 
mechanics the allowed energy states for these molecules, how will we dis- 
tribute the large number of molecules among the allowed energy levels? The 
problem has already been discussed for certain special cases, the answers 
being expressed in the form of "distribution laws," for example, the Maxwell 
distribution law for the kinetic energies of molecules, the Planck distribution 
law for the energies of harmonic oscillators. We wish now to obtain a more 
general formulation. 

The statistical treatment is based on an important principle: the most 
probable distribution in a system can be taken to be the equilibrium dis- 
tribution. In a system containing a very large number of particles, deviations 
from the most probable distribution need not be considered in defining the 
equilibrium condition. 2 

We first require an expression for the probability P of a distribution. 
Then the expression for the maximum probability is obtained by setting the 
variation of P equal to zero, subject to certain restraining conditions imposed 
on the system. 

1 For such treatments, see R. H. Fowler and E. A. Guggenheim, Statistical Thermo- 
dynamics (London: Cambridge, 1939); and R. C. Tolman, Statistical Mechanics (New 
York: Oxford, 1938). 

a See J. E. Mayer and M. Mayer, Statistical Mechanics (New York: Wiley, 1940), for 
a good discussion of this point. 



Sec. 3] CHEMICAL STATISTICS 349 

The method of defining the probability may be illustrated by an example 
that is possibly familiar to some students, the rolling of dice. The probability 
of rolling a certain number n will be defined as the number of different ways 
in which n can be obtained, divided by the total number of combinations 
that can possibly occur. There are six faces on each of two dice so that the 
total number of combinations is 6 2 36. There is only one way of rolling 
a twelve; if the dice are distinguished as a and b y this way can be designated 
as a(6) b(6). Its probability P(\2) is equal to one in 36. For a seven, there 
are six possibilities: 

a(6)-b(\) a(l)-b(6) 

a(5)-b(2) a(2)~b(5) 

a(4)-b(3) 



Therefore, P(7) = H \ =- J. 

Just as with the dice, the probability of a given distribution of molecules 
among energy levels could be defined as the number of ways of realizing the 
particular distribution divided by the total number of possible arrangements. 
For a given system, this total number is some constant, and it is convenient 
to omit it from the definition of the probability of the system. The new 
definition therefore is: the probability of a distribution is equal to the 
number of ways of realizing the distribution. 

3. The Boltzmann distribution. Let us consider a system that has a total 
energy E and contains n identical particles. Let us assume that the allowed 
energy levels for the particles (atoms, molecules, etc.) are known from 
quantum mechanics and are specified as e l9 2 , % " " ' K> ' ' ' etc - How will 
the total energy E be distributed among the energy levels of the n particles? 

For the time being, we shall assume that each particle is distinguishable 
from all the others and that there are no restrictions on how the particles 
may be assigned to the various energy levels. These assumptions lead to the 
"classical" or Boltzmann distribution law. It will be seen later that this law 
is only an approximation to the correct quantum mechanical distribution 
laws, but the approximation is often completely satisfactory. 

Now the n distinguishable particles are assigned to the energy levels in 
such a way that there are n t in level e l9 n 2 in 2 , or in general n K in level e K . 
The probability of any particular distribution, characterized by a particular 
set of occupation numbers , is by definition equal to the number of ways of 
realizing that distribution. Since permuting the particles within a given 
energy level does not produce a new distribution, the number of ways of 
realizing a distribution is the total number of permutations !, divided by 
the number of permutations of the particles within each level, ^ ! n 2 \ . . .n K ! . . . 
The required probability is therefore 

(.2..) 



. n K \ 



N 



350 



CHEMICAL STATISTICS 



[Chap. 12 



As an example of this formula, consider four particles a, b, c, d distributed 
so that two are in e l9 none in 2 an d one ea h i n a an d 4- The possible 
arrangements are as follows : 



l 


% 


*3 


4 


ab 




C 


d 


ab 





d 


c 


ac 





b 


d 


ac 





d 


b 


ad 


. 


b 


c 


ad 





c 


b 


be 





a 


d 


be 





d 


a 


bd 





a 


c 


bd 





c 


a 


cd 





a 


b 


cd 





b 


a 



There are twelve arrangements as given by the formula [0! = 1]: 



,2 



2!0! 1! 1! 2- 1 1 1 

Note that interchanges of the two particles within level s l are not significant. 
Returning to eq. (12.1), the equilibrium distribution is the one for which 
this probability is a maximum. The maximum is subject to two conditions, 
the constancy of the number of particles and the constancy of the total 
energy. These conditions can be written 



= n 



Y F 

I, n K e K - E 



2 ' 2) 



By taking the logarithm of both sides of eq. (12.1), the continued product 
is reduced to a summation. 



In 



In n\ 



n K l 



The condition for a maximum in P is that the variation of P, and hence of 
In P, be zero. Since In A?! is a constant, 



Stirling's formula 3 for the factorials of large numbers is 
In n\ = n In n n 



(12.3) 



(12.4) 



3 For derivation see D. Widder, Advanced Calculus (New York: Prentice-Hall, 1947), 
f>. 317. 



Sec. 3] CHEMICAL STATISTICS 351 

Therefore eq. (12.3) becomes 

^ 2 n K ^ n n K ~ ^ ^ w^ = 
or 2 In 77^^ - (12.5) 



The two restraints in eq. (12.2), since n and E are constants, can be 
written 

(5/2^2 (5 i,- = 

AIT v ji n < 12 - 6 ) 

oE ^= Z, G K on K 

These two equations are multiplied by two arbitrary constants, 4 a and /?, 
and added to eq. (12.5), yielding 

S a a/i^ + S /?e A , a/ijr + Z\nn K Sn K - (12.7) 

The variations 6n K may now be considered to be perfectly arbitrary (the 
restraining conditions having been removed) so that for eq. (12.7) to hold, 
each term in the summation must vanish. As a result, 

In n K + a f fte K ~ 
or n K -=e~*e-*** (12.8) 

This equation has the same form as the Boltzmann distribution law 
previously obtained and suggests that the constant ft equals \jkT. It could 
have been calculated anew. Thus 

n K ^ e -*e~*K lkT (12.9) 

It is convenient at this point to make one extension of this distribution 
law. It is possible that there may be more than one state corresponding with 
the energy level e K . If this is so, the level is said to be degenerate and should 
be assigned a statistical weight g K , equal to the number of superimposed 
levels. The distribution law in this more general form is accordingly 

e-**t kT (12.10) 



The constant a is evaluated from the condition 

Zn K = n 
whence S e~ *g K e~ ** lkT = n 



Therefore eq. (12.10) becomes 

-e K lkT 
" 



* This is an application of Lagrange*s method of undetermined multipliers, the stan- 
dard treatment of constrained maxima problems. See, for example, D. Widder, Advanced 



Calculus, p. 113 



352 CHEMICAL STATISTICS [Chap. 12 

This is the Boltzmann distribution law in its most general form. The 
expression 2 gK e ~'* lkT * n ^ e denominator of eq. (12. 1 1) is very important in 
statistical mechanics. It is called the partition function, and will be denoted 
by the symbol 

-*** (12-12) 



The average energy e of a particle is given by (see eq. 7.38) 



= 



or g = kT* (12.13) 

oT 

4. Internal energy and heat capacity. It is now possible to make use of 
the distribution law to calculate the various functions of thermodynamics. 
Thermodynamics deals not with individual particles, but with large-scale 
systems containing very many particles. The usual thermodynamic measure 
is the mole, 6.02 x 10 23 molecules. 

Instead of considering a large number of individual particles, let us 
consider a large number of systems, each containing a mole of the substance 
being studied. The average energy of these systems will be the ordinary 
internal energy E. We again use eq. (12.13), except that now a whole system 
takes the place of each particle. If the allowed energies of the whole system 
are E l9 2 , . . . E K , the average energy will be 



Writing Z-S&t*-**'* 71 (12.14) 



then, E = kT*--- (12.15) 

oT 

We may call Z the molar partition function to distinguish it from the molecular 
partition function/. It is also called the sum-over-states 
From eq. (12.15) the heat capacity at constant volume is 



5T 






5. Entropy and the Third Law. Equation (12.16) can be employed to 
calculate the entropy in terms of the molar partition function Z. Thus : 



Sec. 5] 



CHEMICAL STATISTICS 



353 



Integrating by parts, we find 



5 = 



: ) +*r 

/ v *>o 



T /a In Z 



dT 



E 
T 



(12.17) 



In this equation, only 5 and \k lnZ| T=0 are temperature-independent 
terms. The constant term, 5 , the entropy at the absolute zero, is therefore 

5 = *lnZ| T . = *In ft (12.18) 

Here g Q is the statistical weight of the Jowest possible energy state of the 
system. Equation (12.18) is the statistical-mechanical formulation of the 
Third Law of Thermodynamics. 

If we consider, for example, a perfect crystal at the absolute zero, there 
will usually be one and only one equilibrium arrangement of its constituent 
atoms, ions, or molecules. In other words, the statistical weight of the lowest 
energy state is unity: the entropy at 0K becomes zero. This formulation 
ignores the possible multiplicity of the ground state due to nuclear spin. If 
the nuclei have different nuclear-spin orientations, there will be a residual 
entropy at 0K. In chemical problems such effects are of no importance, 
since in any chemical reaction the nuclear-spin entropy would be the same 
on both sides of the reaction equation. It is thus conventional to set 5 -= 
for the crystalline elements and hence for all crystalline solids. 

Many statistical calculations on this basis have been quantitatively 
checked by experimental Third-Law values based on heat-capacity data. 
Examples are given in Table 12.1. 

TABLE 12.1 

COMPARISON OF STATISTICAL (SPECTROSCOPIC) AND THIRD-LAW (HEAT-CAPACITY) 

ENTROPIES 





Entropy as Ideal Gas at 1 atm, 298.2K 


Gas 


Statistical 


Third Law 


N 2 


45.78 


45.9 


2 


49.03 


49.1 


a, 


53.31 


53.32 


H 2 


31.23 


29.74 


HCl 


44.64 


44.5 


HBr 


47.48 


47.6 


HI 


49.4 


49.5 


H 2 O 


45.10 


44.28 


N 2 O 


52.58 


51.44 


NH 8 


45.94 


45.91 


CH 4 


44.35 


44.30 


C 2 H 4 


52.47 


52.48 



$54 CHEMICAL STATISTICS [Chap. 12 

In certain cases, however, it appears that even at absolute zero the 
particles in a crystal may persist in more than one geometrical arrangement. 
An example is crystalline nitrous oxide. Two adjacent molecules of N 2 O can 
be oriented either as (ONN NNO) or as (NNO NNO). The energy difference 
A between these alternative configurations is so slight that their relative 
probability e * EIRT is practically unity even at low temperatures. By the time 
the crystal has been cooled to the extremely low temperature at which even 
a minute A might produce a reorientation, the rate of rotation of the 
molecules within the crystal has become vanishingly slow. Thus the random 
orientations are effectively "frozen." As a result, heat-capacity measure- 
ments will not include a residual entropy S Q equal to the entropy of mixing 
of the two arrangements. From eq. (3.42) this would amount to 

5 - -R S X, In X, - R(\ In J + In i) - R In 2 - 1.38 eu 

It is found that the entropy calculated from statistics is actually larger by 
1.14eu than the Third-Law value, which is within the experimental uncer- 
tainty of iO.25 eu in S Q . A number of examples of this type have been 
carefully studied. 5 

If the substance at temperatures close to 0K is not crystalline, but a 
glass, there is also a residual entropy owing to the randomness characteristic 
of vitreous structures. 

Another instance of a residual entropy of mixing at 0K arises from the 
isotopic constitution of the elements. This effect can usually be ignored since 
in most chemical reactions the isotopic ratios change very slightly. 

As a result of this discussion, we shall set S Q ~- in eq. (12.18), obtaining 

S ~+*lnZ (12.19) 

6. Free energy and pressure. From the relation A = E TS and eqs. 
(12.15) and (12.19), the work function becomes 

A = -kT\nZ (12.20) 

The pressure, @A/dV) T , is then 

P = kT*^ (12.2.) 

The Gibbs free energy is simply F = A -\- PV, and from AF the equi- 
librium constants for a reaction can be calculated. 

Expressions have now been obtained that enable us to calculate all 
thermodynamic properties of interest, once we know how to evaluate the 
molar partition function Z. 

7. Evaluation of molar partition functions. The evaluation of the molar 
partition function Z has not yet been accomplished for all types of systems, 
which is of course hardly surprising, for the function Z contains in itself the 

5 For the interesting case of ice, see L. Pauling, /. Am. Chem. Soc., 57, 2680 (1935). 



Sec. 7] CHEMICAL STATISTICS 355 

answer to all the equilibrium properties of matter. If we could calculate Z 
from the properties of individual particles, we could then readily calculate 
all the energies, entropies, free energies, specific heats, and so forth, that 
might be desired. 

In many cases, it is a good approximation to consider that E K , an energy 
of the system, can be represented simply as the sum of energies E K of non- 
interacting individual particles. This would be the case, for example, of a 
crystal composed of independent oscillators, or of an almost perfect gas in 
which the intermolecular forces were negligible. In such instances we can 
write 

EK ^ i(0 + 2 (2) -I- * 3 (3) f . . . e N (N) (12.22) 

This expression indicates that particle (1) occupies an energy level e l9 particle 
(2) an energy level F 2 , etc. Each different way of assigning the particles to 
the energy levels determines <* distinct state of the system E K . 

The molar partition function, or sum over the states E K , then becomes 



(The statistical weights g K are omitted for convenience in writing the ex- 
pressions.) The second summation must be taken over all the different ways 
of assigning the particles to the energy levels E K . It can be rewritten as 



e 

Since each particle has the same set of allowed energy levels, this sum is 
equal 6 to 

(2 e - 8 * lkT ) N 

K 

Thus we find that Z =--/* 

The relation Z /' v applies to the case in which rearranging the particles 
among the energy levels in eq. (12.22) actually gives rise to different states 
that must be included in the summation for Z. This is the situation in a 
perfect crystal, the different particles (oscillators) occupying distinct localized 
positions in the crystal structure. 

In the case of a gas, on the other hand, each particle is free to move 
throughout the whole available volume. States in the gas that differ merely 

* It may be rather hard to see this equality at first. Consider therefore a simple case in 
which there are only two particles (1) and (2) and two energy levels f t and e 2 . The ways of 
assigning the particles to the levels are: 

i = i (0 Ma (2), 2 = *i (2) + 2 (1), E* = i (1) t- FI (2), 
The sum over states is: 

Z = e-W* -f e- E *l* T 4- e ~ 
which is equal to 



*l kT -f e~* 
Now it is evident that this is identical with 

f n = (S*-**/** 1 ) 1 = (e~*il kT -f e- 



356 CHEMICAL STATISTICS [Chap. 12 

by the interchange of two particles are not distinguishable and should be 
counted only once. If each level in eq. (12.22) contains only one particle, 7 
the number of permutations of the particles among the levels is AH We there- 
fore divide the expression for Z by this factor, obtaining for the ideal gas 
case, Z -(!/#!)/*. 

Thus the relations between /and Z in the two extreme cases are 

Ideal crystals Z -/* 

1 (12.23) 

Ideal gases Z = N J N 

Intermediate kinds of systems, such as imperfect gases and liquids, are much 
more difficult to evaluate. 

In proceeding to calculate the partition functions for an ideal gas, it is 
convenient to make use of a simplifying assumption. The energy of a mole- 
cule will be expressed as the sum of translational, rotational, vibrational, and 
electronic terms. Thus 

= ? trans + *rot +" f vlb + *elec (12.24) 

It follows that the partition function is the product of corresponding terms, 

/" ftr&nsfiotf \lbfelec (12.25) 

The simplest case to be considered is that of the monatomic gas, in which 
there are no rotational or vibrational degrees of freedom; except at very high 
temperatures the electronic excitation is usually negligible. 

8. Monatomic gases translational partition function. In Section 10-20 it 
was shown that the translational energy levels for a particle in a one- 
dimensional box are given by 



The statistical weight of each level is unity, g n = 1. Therefore the molecular 
partition function becomes 



~*!* ml *} 

The energy levels are so closely packed together that they can be considered 
to be continuous, and the summation can be replaced by an integration, 



7 When the volume is large and the temperature not very low, there will be many more 
energy levels than there are particles. This will be evident on examination of eq. (10.39) for 
the levels of a particle in a box. Since there is no housing shortage, there is no reason for the 
particles to "double-up" and hence the assumption of single occupancy is a good one. For 
a further discussion, see Tolman, he. cit., pp. 569-572. 



Sec. 8] CHEMICAL STATISTICS 357 



(12.26) 

For three degrees of translational freedom this expression is cubed, and 
since / 3 K, we obtain 



(,2.27) 



This is the molecular partition function for translation. 
The molar partition function is 



(12 28) 
\ ( } 



The energy is therefore 



This is, of course, the simple result to be expected from the equipartition 
principle. 

The entropy is evaluated from eq. (12.19), using the Stirling formula, 

AM = (N/e) N . It follows that 



Nlf 



The entropy is therefore 



(12.29) 
ATT 

This is the famous equation that was first obtained by somewhat un- 
satisfactory arguments by Sackur and Tetrode (1913). As an example, let 
us apply it to calculate the entropy of argon at 273.2K and at one atmosphere 
pressure. Then 

R = 1.98 cal per C 77 = 3.1416 

* = 2.718 m-6.63 X 10" 23 g 

V = 22,414 cc k - 1.38 x 10~ 18 erg per C 

# = 6.02 x 10 23 7- 273.2 

h = 6.62 x 10~ 27 ergsec 



358 CHEMICAL STATISTICS [Chap. 12 

On substituting these quantities into eq. (12.29), the entropy is found to 
be 36.2 cal per deg mole. 

9. Diatomic molecules rotational partition function. The energy levels 
for diatomic molecules, according to the rigid-rotator model, were given by 
eq. (11. 22) as 

J(J_+ l)/r 

fr<)t ^ " ~87T 2 / 

If the moment of inertia / is sufficiently high, these energy levels become so 
closely spaced as to be practically continuous. This condition is, in fact, 
realized for all diatomic molecules except H 2 , HD, and D 2 . Thus for F 2 , 
/ - 25.3 x 10~ 40 gcm 2 ; for N 2 , 13.8 x lO^ 40 ; but for H 2 , / - 0.47 x 1Q- 40 . 
These values are calculated from the interatomic distances and the masses 
of the molecules, since / = //r 2 . 

Now the multiplicity of the rotational levels requires some consideration. 
The number of ways of distributing J quanta of rotational energy between 
two axes of rotation equals 2J -f 1, for in every case except J there are 
two possible alternatives for each added quantum. The statistical weight of 
a rotational level J is therefore 2J + 1 . 

The rotational partition function now becomes 

/ rot -= E (27 4 \)e /<>+ !>*//"' (12 .30) 

Replacing the summation by an* integration, since the levels are closely 
spaced, we obtain 



One further complication remains. In homonuclear diatomic molecules 
(N 14 N 14 , C1 35 C1 35 , etc.) only all odd or all even /'s are allowed, depending on 
the symmetry properties of the molecular eigenfunctions. If the nuclei are 
different (N 14 N 15 , HC1, NO, etc.) there are no restrictions on the allowed 
7's. A symmetry number a is therefore introduced, which is either a = 1 
(heteronuclear) or a = 2 (homonuclear). Then 



^ot ~ - ah2 (12.32) 

As an example of the application of this equation, consider the calcula- 
tion of the entropy of F 2 at 298.2K, assuming translational and rotational 
contributions only. From eq. (12.29), the translational entropy is found to 
be 36.88 eu. Then the rotational part is 



Sec. 10] CHEMICAL STATISTICS 359 

Note that the rotational energy is simply RT in accordance with the equi- 
partition principle. Substituting / 25.3 x lO' 40 , S rot 8.74 eu. Adding 
the translational term, we have 

S =-- S mt f 5 tran8 - 8.74 h 36.88 - 45.62 eu 

This compares with a total entropy of 5 I 298 - 48.48 eu. The vibrational 
contribution at 25C is therefore small. 

10. Polyatomic molecules rotational partition function. The partition 
function in eq. (12.32) holds also for linear polyatomic molecules, with a - 2 
if the molecule has a plane of symmetry (such as O C O), and a - 1 if 
it has not (such as N-^N-- O). 

For a nonlinear molecule, the classical rotational partition function has 
been found to be 



/r 



rot 



In this equation A, /?, C are the three principal moments of inertia of the 
molecule. The symmetry number a is equal to the number of equivalent 
ways of orienting the molecule in space. For example: H 2 O, a 2; NH 3 , 
a-3;CH 4 , o- 12;C 6 H 6 , a == 12. 

11. Vibrational partition function. In evaluating a partition function for 
the vibrational degrees of freedom of a molecule, it is often sufficient to use 
the energy levels of the harmonic oscillator, which from eq. (11.25) are 

fvib -~ 0' f i)** (12.34) 

At low temperatures vibrational contributions are usually small and this 
approximation is adequate. For reasonably exact calculations at higher tem- 
peratures the anharmonicity of the vibrations must be considered. Some- 
times the summation for f can be made by using energy levels obtained 
directly from molecular spectra. 

The partition function corresponding to eq. (12.34) would be, for each 
vibrational degree of freedom, 

f = J e -( p +W' v i kT -__- e -i' v W' y e -rWkT 

V V 

/ vil) -f-"" m '(l -<,-*'/)-! (12.35) 

The total vibrational partition function is the product of terms such as eq. 
(12.35), one for each of the normal modes of vibration of the molecule, 

Aib-'TFAvib < 12 - 36 > 

i 

For the purposes of tabulation and facility in calculations, the vibrational 
contributions can be put into more convenient forms. 

The vibrational energy, from eqs. (12.15), (12.23), and (12.35), is 



360 



CHEMICAL STATISTICS 



[Chap. 12 



Now Nhv/2 is the zero point energy per mole , whence, writing hvjkT = x, 

^-^4 (12-37) 

(12.38) 



Then the heat capacity 

JRx* 

2(cosh x 1) 
From eq. (12.20), since for the vibrational contribution 8 A F, 



( I --.; C,, v 



Finally the contribution to the entropy is 

* ^0 * ''0 

o 



(12.39) 



(12.40) 



T T 

An excellent tabulation of these functions has been given by J. G. Aston. 9 
A much less complete set of values is given in Table 12.2. If the vibration 

TABLE 12.2 
THERMODYNAMIC FUNCTIONS OF A HARMONIC OSCILLATOR 



hv 




(E - o) 


(F-E ) 


hv 




(E - ) 


-(F-Ej 


* kf 


v 


T 


T 


X kT 


V 


T 


T 


0.10 


.985 


.891 


4.674 


1.70 


.571 


0.7551 


0.4008 


0.15 


.983 


.842 


3.917 


1.80 


.528 


0.7070 


0.3591 


0.20 


.981 


.795 


3.394 


1.90 


.484 


0.6640 


0.3219 


0.25 


.977 


.749 


2.999 


2.00 


.439 


0.6221 


0.2889 


0.30 


.972 


.704 


2.683 


2.20 


.348 


0.5448 


0.2333 


0.35 


.967 


.660 


2.424 


2.40 


.256 


0.4758 


0.1890 


0.40 


.961 


.616 


2.206 


2.60 


.164 


0.4145 


0.1534 


0.45 


.954 


.574 


2.017 


2.80 


.074 


0.3603 


0.1246 


0.50 


.946 


.532 


1.853 


3.00 


0.9860 


0.3124 


0.1015 


0.60 


.929 


.450 


1.581 


3.50 


0.7815 


0.2166 


0.0610 


0.70 


.908 


.372 


1.364 


4.00 


0.6042 


0.1483 


0.0367 


0.80 


.884 


.297 


1.186 


4.50 


0.4571 


0.1005 


0.0223 


0.90 


.858 


.225 


1.037 


5.00 


0.3393 


0.0674 


0.0133 


.00 


.830 


.157 


0.9120 


5.50 


0.2477 


0.0449 


0.0081 


.10 


.798 


.091 


0.8044 


6.00 


0.1782 


0.0296 


0.0050 


.20 


.765 


.028 


0.7128 


6.50 


0.1266 


0.0195 


0.0030 


.30 


.729 


0.9678 


0.6321 


7.00 


0.0890 


0.0127 


0.0018 


.40 


.692 


0.9106 


0.5628 


8.00 


0.0427 


0.0053 


0.0006 


.50 


.653 


0.8561 


0.5016 


9.00 


0.0199 


0.0022 


0.0004 


.60 


.612 


0.8043 


0.4481 


10.00 


0.0090 


0.0009 


0.0001 



8 This is evident from eq. (12.21) since /vib is not a function of K, P =-- 0, F = A + 

py=A. 

9 H. S. Taylor and S. Glass tone, Treatise on Physical Chemistry, 3rd ed., vol. 1 , p. 655 
(New York: Van Nostrand, 1942). 



Sec. 12] CHEMICAL STATISTICS 361 

frequency is obtainable from spectroscopic observations, these tables can be 
used to calculate the vibrational contributions to the energy, entropy, free 
energy, and heat capacity. 

12. Equilibrium constant for ideal gas reactions. From the relation 
AF RTln K v , the equilibrium constant can be calculated in terms of 
the partition functions. From eqs. (12.20) and (12.23), A = ~Ar7'lirZ = 
kTln(f y /N\). From the Stirling formula, N! - (N/e) N , and since for an 
ideal gas,F - A+PY= A f RT, we find that F = - RT In (f/N). Let us write 



J yint ~ ,3 -J Y 

where / int denotes the internal partition functions, / rot / vib / elcc , and /' is 

the partition function per unit volume; i.e., f/V. Then, the free energy 
' 



The standard free energy F is the F at unit pressure of one atmosphere. 
The volume of a mole of ideal gas under standard conditions of 1 atm 
pressure is V RT/l. The standard free energy is accordingly 10 

F - RT \nfkT 
For a typical reaction aA + bB ^-- cC \ dD, 



J AJ 



Therefore, K, - 



v fi o i*f b 
J AJ B 

Fromeq. (4.12), 

** rrr: /C c (/v/ ) 

If the concentration terms in K c are expressed in units of molecules per cc 
rather than the more usual moles per cc, we obtain the more concise 
expression, 

ft c f d 

/'A/'B 

This equation can easily be given a simple physical interpretation. Con- 
sider a reaction A -> B, then K c ' ^ /B'//A'- The partition function is the sum 
of the 'probabilities e~ efkT of all the different possible states of the molecules 
(/= e" elkT ). The equilibrium constant is therefore the ratio of the total 
probability of the occurrence of the final state to the total probability of the 
occurrence of the initial state. 

13. The heat capacity of gases. The statistical theory that has now been 
outlined provides a very satisfactory interpretation of the temperature 
dependence of the heat capacity of gases. 

The translational energy is effectively nonquantized. It makes a constant 
contribution C v = $/?, for all types of molecules. 

10 Note that k is in units of cc atm/C. 



362 



CHEMICAL STATISTICS 



[Chap. 12 



Except in the molecules H 2 , HD, and D 2 , the rotational energy quanta 
are small compared to kT at temperatures greater than about 80K. There 
is therefore a constant rotational contribution of C v = R for diatomic and 
linear polyatomic molecules or C r $R for nonlinear polyatomic molecules. 
For example, with nitrogen at 0C, Af lot =-- 8 x 10~ 16 erg compared to 
AT - 377 x 10 16 erg. At temperatures below 80K the rotational heat 



2.00 

LJ 
O 

o 1.50 

>* 
_i 
< 
o 

>? 

o 

o i.OO 



i 
o 



o 



2 
O 
O 




V 1.0 2.0 3.0 

"/hi, 
Fig. 12.1. Heat capacity contribution of a harmonic oscillator. 

capacity can be calculated from the partition function in eq. (12.30) and 
the general formula, eq. (12.16). 

The magnitude of the quantum of vibrational energy hv is usually quite 
large compared to kT at room temperatures. For example, the fundamental 
vibration frequency in N 2 is 2360 cm" 1 , corresponding to f vib of 46.7 x 10~ 14 
erg, whereas at 0C kT 3.77 x 10~ 14 . Such values are quite usual and the 
vibrations therefore make relatively small contributions to low-temperature 
energies, entropies, and specific heats. The data in Table 7.6 (page 192) 
confirm this conclusion. In Fig. 12.1, the heat-capacity curve for a typical 



Sec. 14] CHEMICAL STATISTICS 363 

harmonic oscillator is shown as a function of 7/0,,, where O v ---- hvfk is 
called the characteristic temperature of the vibration. As the temperature is 
raised, vibrational excitation becomes more and more appreciable. If we 
know the fundamental vibration frequencies of a molecule, we can determine 
from Fig. 12.1 or Table 12.2 the corresponding contribution to C r at any 
temperature. The sum of these contributions is the total vibrational heat 
capacity. 

14. The electronic partition function. The electronic term in the partition 
function is calculated directly from eq. (12.12) and the observed spectro- 
scopic data for the energy levels. Often the smallest quantum of electronic 
energy is so large compared to kT that at moderate temperatures the elec- 
tronic energy acquired by the gas is negligible. In other cases, the ground 
state may be a multiplet, but have energy differences so slight that it may be 
considered simply as a degenerate single level. 

There are, however, certain intermediate cases in which the multiplet 
splitting is of the order of kT at moderate temperatures. A notable example 
is NO, with a doublet splitting of around 120 crn^ 1 or 2.38 x 10~ 14 erg. An 
electronic contribution to the heat capacity is well marked in NO. Complica- 
tions arise in these cases, however, owing to an interaction between the 
rotational angular momentum of the nuclei (quantum number J) and the 
electronic angular momentum (quantum number A). The detailed analysis is 
therefore more involved than a simple separation of the internal energy into 
vibrational, rotational, and electronic contributions would indicate. 11 

15. Internal rotation. When certain polyatomic molecules are studied, it 
is found that the strict separation of the internal degrees of freedom into 
vibration and rotation is not valid. Let us compare, for example, ethyfene 
and ethane, CH 2 CH 2 and CH 3 CH 3 . 

The orientation of the two methylene groups in C 2 H 4 is fixed by the 
double bond, so that there is a torsional or twisting vibration about the 
bond but no complete rotation. In ethane, however, there is an internal 
rotation of the methyl groups about the single bond. Thus one of the vibra- 
tional degrees of freedom is lost, becoming an internal rotation. This 
rotation would not be difficult to treat if it were completely free and un- 
restricted, but such is not the case. There are potential-energy barriers, 
amounting to about 3000 calories per mole, which must be overcome before 
rotation occurs. The maxima in energy occur at positions where the hydrogen 
atoms on the two methyl groups are directly opposite to one another, the 
minima at positions where the hydrogens are "staggered." 

The theoretical treatment of the problems of restricted internal rotation 
is still incomplete, but good progress is being made. 12 

16. The hydrogen molecules. Since the moment of inertia of the hydrogen 
molecule, H 2 , is only 0.47 x 10~ 40 gcm 2 , the quantum of rotational energy 

11 Fowler and Guggenheim, op. cif. t p. 102. 

12 J. G. Aston, loc. cit., p. 590. 



364 CHEMICAL STATISTICS [Chap. 12 

is too large for a classical treatment. To evaluate the partition function, the 
complete summation must her carried out. When this was first done, using 
eq. (12.30), modified with a symmetry number a = 2, the calculated specific 
heats were in poor agreement with the experimental values. It was later 
realized that the discrepancy must be a result of the existence of the two 
nuclear-spin isomers for H 2 . 

The proton (nucleus of the H atom) has a nuclear spin / -= % in units of 
/J/27T. The spins of the two protons in the H 2 molecule may either parallel or 
oppose each other. These two spin orientations give rise to the two spin 
isomers: 

ortho H 2 spins parallel resultant spin -= 1 

para H 2 spins antiparallel resultant spin 

Spontaneous transitions between the ortho and para states are strictly 
prohibited. The ortho states are associated with only odd rotational levels 
(J I, 3, 5 . . .), and para states have only even rotational levels (J = 
0, 2, 4 . . .). The nuclear-spin weights are g NS - 3 for ortho, corresponding 
to allowed directions 4-1,0, -1, andg NS 1 for para, whose resultant spin 13 
is 0. At quite high temperatures (~ 0C), therefore, an equilibrium mixture 
of hydrogen consists of three parts ortho and one part para. At quite low 
temperatures (around 80K, liquid-air temperature) the equilibrium con- 
dition is almost pure para hydrogen, with the molecules in the lowest rota- 
tional state, J = 0. 

The equilibrium is attained very slowly in the absence of a suitable 
catalyst, such as oxygen adsorbed on charcoal, or other paramagnetic sub- 
stance. It is thus possible to prepare almost pure/?-H 2 by adsorbing hydrogen 
on oxygenated charcoal at liquid-air temperatures, and then warming the 
gas in the absence of catalyst. 

The calculated heat capacities of pure />-H 2 , pure o-H 2 and of the 1:3 
normal H 2 , are plotted in Fig. 12.2. Mixtures of o- and/?-H 2 are conveniently 
analyzed by measuring their thermal conductivities, since these are pro- 
portional to their heat capacities. 

A similar situation arises with deuterium, D 2 . The nuclear spin of the 
D atom is 1. The possible resultant values for D 2 are therefore 0, 1, and 2. 
Of these, / = and 2 belong to the ortho modification and / ~ 1 is the para. 
The weights (2/ } 1) are 1 + 5 = 6, and 3, respectively. The high-tempera- 
ture equilibrium mixture therefore contains two parts ortho to one part para. 

In the molecule HD, which is not homonuclear, there are no restrictions 
on the allowed rotational energy levels. The partition function of eq. (12.30) 
is directly applicable. 

Other diatomic molecules composed of like nuclei with nonzero nuclear 
spins may also be expected to exist in both para and ortho modifications. 

13 Compare the spatial quantization of the orbital angular momentum of an electron, 
page 268. 



Sec. 17] 



CHEMICAL STATISTICS 



365 



Any thermodynamic evidence for such isomers would be confined to ex- 
tremely low temperatures, because their rotational energy quanta are small. 
The energy levels are so close together that in calculating heat capacities it 
is unimportant whether all odds or all evens are taken. It is necessary only 



< 3.00 




Fig. 12.2. 



100 200 300 

DEGREES KELVIN 

Heat capacities of pure para-hydrogen, pure ortho-hydrogen, 
and 3-o to \~p normal hydrogen. 



to divide the total number of levels by a =- 2. Spectroscopic observations, 
however, will often reveal an alternating intensity in rotational lines caused 
by the different nuclear-spin statistical weights. 

17. Quantum statistics. In deriving the Boltzmann statistics, we assumed 
that the individual particles were distinguishable and that any number of 
particles could be assigned to one energy level. We know from quantum 
mechanics that the first of these assumptions is invalid. The second assump- 
tion is also incorrect if one is dealing with elementary particles or particles 
composed of an odd number of elementary particles. In such cases, the 
Pauli Exclusion Principle requires that no more than one particle can go 
into each energy level. If the particles considered are composed of an even 
number of elementary particles, any number can be accommodated in a 
single energy level. 

Two different quantum statistics therefore arise, which are characterized 
as follows: 



Name 

(1) Fermi-Dirac 

(2) Bose-Einstein 



Obeyed by 

Odd number of elementary 
particles (e.g., electrons, 
protons) 

Even number of elementary 
particles (e.g., deuterons, 
photons) 



Restrictions on n K 

Only one particle per 
state, n K < g K 

Any number of particles 
per state 



It is interesting to note that photons follow the Bose-Einstein statistics, 
indicating that they are complex particles and recalling the formation of 
electron-positron pairs from X-ray photons. 



366 CHEMICAL STATISTICS [Chap. 12 

A schematic illustration of the two types of distribution would be 

O00O O O O 

F.D. B.E. 

Distribution laws are calculated for these two cases by exactly the same 
sort of procedure as was used for the Boltzmann statistics. 14 The results are 
found to be very similar, 

< 12 - 42 > 



F.D. case + 
B.E. case 

Now in almost every case the exponential term is very large compared to 
unity, and the Boltzmann statistics are a perfectly good approximation for 
almost all practical systems. This can be seen by using the value of e* =f/n 
from eq. (12.10). The condition for the Boltzmann approximation is then 

e/kT f 

---- ^ 



1, or 

n 

Using the translational partition function /in eq. (12.27), we have 
e tl1fT (27rmkTj^V 

^ ->' (12 - 43) 

This condition is obviously realized for a gas at room temperature. It is 
interesting to note, however, the circumstances under which it would fail. 
If n/V, proportional to the density, became very high, the classical statistics 
would eventually become inapplicable. This is the situation in the interior of 
the stars, and forms the basis of R. H. Fowler's brilliant contribution to 
astrophysics. A more mundane case also arises, namely in the electron gas 
in metals. We shall consider this in the next chapter, with only a brief 
mention here. A metallic crystal, to a first approximation, may be considered 
as a regular array of positive ions, permeated by a gas of Mobile electrons. 
In this case the density term in eq. (12.43) is exceptionally high and in 
addition the mass term m is lower by about 2 x 10 3 than in any molecular 
case. Thus the electron gas will not obey Boltzmann statistics; it must indeed 
follow the Fermi-Dirac statistics since electrons obey the Pauli Principle. 

PROBLEMS 

1. In the far infrared spectrum of HC1, there is a series of lines with a 
spacing of 20.7 cm" 1 . In the near infrared spectrum, there is an intense band 
at 3.46 microns. Use these data to calculate the entropy of HC1 as an ideal 
gas at 1 atm and 298 K. 

14 For these calculations, see, for example, Tolman, op. cit,, p. 388. 



Chap. 12] 



CHEMICAL STATISTICS 



367 



2. Estimate the equilibrium constant of the reaction C1 2 --- 2 Cl at 
1000K. The fundamental vibration frequency of C1 2 is 565 cm" 1 and the 
equilibrium C1-C1 distance is 1.99 A. Compare with the experimental value 
in Table 4.5. 

3. The isotopic composition of zinc is: 64 Zn 50.9 per cent; 68 Zn 27.3 per 
cent; 67 Zn 3.9 per cent; 68 Zn 17.4 per cent; 70 Zn 0.5 per cent. Calculate the 
entropy of mixing per mole of zinc at 0K. 

4. Thallium forms a monatomic vapor. The normal electronic state of 
the atom is 2 P 1/2 but there is a 2 P^/ 2 state lying only 0.96 ev. above the ground 
state. The statistical weights of the state.s are 2 and 4, respectively. Plot a 
curve showing the variation with temperature of the contribution to the 
specific heat of the vapor caused by the electronic excitation. 

5. In a star whose temperature is 10 6 K, calculate the density of material 
at which the classical statistics would begin to fail. 

6. Calculate the equilibrium constant of the reaction H 2 f D 2 2 HD 
at 300K given: 



H 2 


HD 


D 2 


4371 


3786 


3092 


0.5038 


0.6715 


1.0065 


0.458 


0.613 


0.919 



) e , cm" 1 ..... 
Reduced mass, /i, at. wt. units 
Moment of inertia, /, g cm 2 x 10 40 



7. In Problem 4.10, heat-capacity data were listed for a calculation of the 
Third-Law entropy of nitromethane. From the following molecular data, 
calculate the statistical entropy S 298 . Bond distances (A): N O 1.21; 
CN, 1.46; C H, 1.09. Bond angles: O N O 127; H C N 109J. 
From these distances, calculate the principal moments of inertia, / = 67.2, 
76.0, 137.9 x 10~ 40 gcm 2 . The fundamental vibration frequencies 15 in cm" 1 
are: 476, 599, 647, 921, 1097, 1153, 1384, 1413, 1449, 1488, 1582, 2905, 
3048 (2). One of the torsional vibrations has become a free rotation around 
the CN bond with / = 4.86 x 10 40 . 

8. Calculate the equilibrium constant K p at 25C for O 2 1H + O 2 16 - 
2 O 16 O 1H . The nuclear spins of O 18 and O 16 are both zero. The vibration fre- 
quencies are given by v = (l/27r)(/c/ 1/2 , where K is the same for all three 
molecules. For O 2 10 , v 4.741 x 10 13 sec" 1 . The equilibrium internuclear 
distance, 1.2074 A, does not depend on the isotopic species. 

9. The ionization potential of Na is 5.14 ev. Calculate the degree of dis- 
sociation, Na -= Na+ + e, at 10 4 K and 1 atm. 

15 A. J. Wells and E. B. Wilson, /. Chem. Phys., 9, 314 (1941). 



368 CHEMICAL STATISTICS [Chap. 12 

REFERENCES 

BOOKS 

1. Born, M., Natural Philosophy of Cause and Chance (New York: Oxford, 
1949). 

2. Dole, M., Introduction to Statistical Thermodynamics (New York: 
Prentice-Hall, 1954). 

3. Gurney, R. W., Introduction to Statistical Mechanics (New York: 
McGraw-Hill, 1949). 

4. Khinchin, A. I., Statistical Mechanics (New York: Dover, 1949). 

5. Lindsay, R. B., Physical Statistics (New York: Wiley, 1941). 

6. Rushbrooke, G. S., Introduction to Statistical Mechanics (New York: 
Oxford, 1949). 

7. Schrddinger, E., Statistical Thermodynamics (Cambridge, 1946). 

8. Ter Haar, D., Elements of Statistical Mechanics (New York: Rinehart, 
1954). 

ARTICLES 

1. Bacon, R. H., Am. J. Phys., 14, 84-98 (1946), "Practical Statistics for 
Practical Physicists/' 

2. Eyring, H., and J. Walter, /. Chem. Ed., 18, 73-78 (1941), "Elementary 
Formulation of Statistical Mechanics." 



CHAPTER 13 

Crystals 



1. The growth and form of crystals. The symmetry of crystalline forms, 
striking a responsive chord in our aesthetic nature, has fascinated many 
men, from the lapidary polishing gems for a royal crown to the natural 
philosopher studying the structure of matter. Someone once said that the 
beauty of crystals lies in the planeness of their faces. It was also the measure- 
ment and explanation of these plane faces that first demanded scientific 
attention. 

In 1669, Niels Stensen (Steno), Professor of Anatomy at Copenhagen 
and Vicar Apostolic of the North, compared the interfacial angles in various 
specimens of quartz rock crystals. An interfacial angle may be defined as the 
angle between lines drawn perpendicular to two faces. Steno found that the 
corresponding angles (in different crystals) were always equal. After the 
invention of the contact goniometer in 1780, this conclusion was checked and 
extended to other substances, and the constancy of interfacial angles has 
been called the "first law of crystallography." 

It was a most important principle, for out of a great number of crystalline 
properties it isolated one that was constant and unchanging. Different crystals 
of the same substance may differ greatly in appearance, since corresponding 
faces may have developed to diverse extents as the crystals were growing. 
The interfacial angles, nevertheless, remain the same. 

We can consider that a crystal grows from solution or melt by the de- 
position onto its faces of molecules or ions from the liquid. If molecules are 
deposited preferentially on a certain face, this face will not extend rapidly in 
area, compared with faces at angles to it on which deposition is less frequent. 
The faces with the largest area are therefore those on which added molecules 
are deposited most slowly. 

Sometimes an altered rate of deposition can completely change the form, 
or habit, of a crystal. A well known case is sodium chloride, which grows 
from pure water solution as cubes, but from 15 per cent aqueous urea 
solution as octahedra. It is believed that urea is preferentially adsorbed on 
the octahedral faces, preventing deposition of sodium and chloride ions, and 
therefore causing these faces to develop rapidly in area. 

The real foundations of crystallography may be said to date from the 
work of the Abbe Rene Just Haiiy, Professor of the Humanities at the 
University of Paris. In 1784, he proposed that the regular external form of 
crystals was a reflection of an inner regularity in the arrangement of their 
constituent building units. These units were believed to be little cubes or 

369 



370 



CRYSTALS 



[Chap. 13 



polyhedra, which he called the molecules integrates of the substance This 
picture also helped to explain the cleavage of crystals along uniform planes. 
The Haiiy model was essentially confirmed, 128 years later, by the work of 
Max von Laue with X-ray diffraction, the only difference being in a more 
advanced knowledge of the elementary building blocks. 

2. The crystal systems. The faces of 
crystals, and also planes within crystals, can 
be characterized by means of a set of three 
noncoplanar axes. Consider in Fig. 13.1 three 
axes having lengths a, b, and c, which are cut 
by the plane ABC, making intercepts OA, OB, 
and OC. If a, b, c, are chosen as unit lengths, 
the lengths of the intercepts may be expressed 
as OAja, OBjh, OC/c. The reciprocals of these 
Fig. 13.1. Crystal axes. lengths will then be a/OA, b/OB, c/OC. Now it 

has been established that it is always possible 

to find a set of axes on which the reciprocal intercepts of crystal faces are 
small whole numbers. Thus, if //, k, /are small integers: 




OA 



oc 



This is equivalent to the law of rational intercepts, first enunciated by Haiiy. 
The use of the reciprocal intercepts (hkl) as indices defining the crystal faces 
was first proposed by W. H. Miller in 1839. If a face is parallel to an axis, 




(001) 






(III) (211) 

Fig. 13.2. Miller indices. 

the intercept is at oo, and the Miller index becomes l/oo or 0. The notation 
is also applicable to planes drawn within the crystal. As an illustration 
of the Miller indices, some of the planes in a cubic crystal are shown in 
Fig. 13.2. 



Sec. 3] 



CRYSTALS 



371 



According to the set of axes used to represent their faces, crystals may 
be divided into seven systems. These are summarized in Table 13.1. They 
range from the completely general set of three unequal axes (a, b, c) at three 
unequal angles (a, /?, y) of the triclinic system, to the highly symmetrical set 
of three equal axes at right angles of the cubic system. 

TABLE 13.1 
THE SEVEN CRYSTAL SYSTEMS 



System 


Axes 


Angles 


Example 




Cubic 


a - b = c 


a = ft = y = 90 


Rock salt 




Tetragonal 
Orthorhombic 
Monoclinic 


a -- b\ c 
a\b\ c 
a\ b\ c 


OL-ft=y-- 90 

a _ ft =- y =, 90 


White tin 
Rhombic sulfur 
Monoclinic sulfur 




Rhombohedral 
Hexagonal 
Triclinic 


a b c 
a = b\ c 
a', b\ c 


a - ft y I 90 
a -_= ft = 90 ;y 120 

a y= ^ ^ y ^ 90 


Calcite 
Graphite 
Potassium dichromate 



3. Lattices and crystal structures. Instead of considering, as Haiiy did, 
that a crystal is made of elementary material units, it is helpful to introduce 
a geometrical idealization, consisting only of a regular array of points in 
space, called a lattice. An example in two dimensions is shown in Fig. 13.3. 



o 1 




Fig. 13.3. Two-dimensional lattice with unit cells. 

The lattice points can be connected by a regular network of lines in 
various ways. Thus the lattice is broken up into a number of unit cells. Some 
examples are shown in the figure. Each cell requires two vectors, a and b, 
for its description. A three-dimensional space lattice can be similarly divided 
into unit cells that require three vectors for their description. 

If each point in a space lattice is replaced by an identical atom or group 
of atoms there is obtained a crystal structure. The lattice is an array of points; 
in the crystal structure each point is replaced by a material unit. 

In 1848, A. Bravais showed that all possible space lattices could be 



372 



CRYSTALS 



[Chap. 13 



assigned to one of only 14 classes. 1 The 14 Bravais lattices are shown in 
Fig. 13.4. They give the allowed different translational relations between 
points in an infinitely extended regular three-dimensional array. The choice 
of the 14 lattices is somewhat arbitrary, since in certain cases alternative 
descriptions are possible. 




TRICLINIC 



A 



2 SIMPLE 3 SIDE-CENTERED 
MONOCLINIC MONOCLINIC 





\ 



4. SIMPLE 5 END-CENTERED 

ORTHORHOMBIC ORTHORHOMBIC 



6. FACE-CENTERED 7 BODY- 
ORTHORHOMBIC CENTERED 

ORTHORHOMBIC 




9 RHOMBOHEDRAL 10 SIMPLE 



II BODY-CENTERED 



8. HEXAGONAL 



TETRAGONAL TETRAGONAL 






Fig. 13.4. The fourteen Bravais lattices. 

4. Symmetry properties. The word "symmetry" has been used in referring 
to the arrangement of crystal faces. It is now desirable to consider the nature 
of this symmetry in more detail. If an actual crystal of a substance is studied, 
some of the faces may be so poorly developed that it is difficult or impossible 
to see its full symmetry just by looking at it. It is necessary therefore to 

1 A lattice that contains body-, face-, or end-centered points can always be reduced to 
one that does not (primitive lattice). Thus the face-centered cubic can be reduced to a 
primitive rhombohedral. The centered lattices are chosen when possible because of their 
higher symmetry. 



Sec. 4] CRYSTALS 373 

consider an ideal crystal in which all the faces of the same kind are developed 
to the same extent. It is not only in face development that the symmetry of 
the crystal is evident but also in all of its physical properties, e.g., electric 
and thermal conductivity, piezoelectric effect, and refractive index. 

Symmetry is described in terms of certain symmetry operations, which 
are those that transform the crystal into an image of itself. The symmetry 
operations are imagined to be the result of certain symmetry elements: axes 
of rotation, mirror planes, and centers of inversion. The possible symmetry 
elements of finite figures, i.e., actual crystals, are shown in Fig. 13.5 with 
schematic illustrations. 

(a) T (b) 



MM/ v- 




i' 

Fig. 13.5. Examples of symmetry elements: (a) mirror plane m; (b) rotation 
axes; (c) symmetry center 1 ; (d) twofold rotary inversion axis 2. 

The possible combinations of these symmetry elements that can occur in 
crystals have been shown to number exactly 32. These define the 32 crystallo- 
graphic point groups* which determine the 32 crystal classes. 

The symbols devised by Hermann and Mauguin are used to represent the 
symmetry elements. An axis of symmetry is denoted by a number equal to 
its multiplicity. The combination of a rotation about an axis with reflection 
through a center of symmetry is called an "axis of rotary inversion"; it is 
denoted by placing a bar above the symbol for the axis, e.g., 2, 3. The center 
of symmetry alone is then T. A mirror plane is given the symbol m. 

All crystals necessarily fall into one of the seven systems* but there are 
several classes in each system. Only one of these, called the holohedral class, 
possesses the complete symmetry of the system. For example, consider two 
crystals belonging to the cubic system, rock salt (NaCl) and iron pyrites 
(FeS 2 ). Crystalline rock salt, Fig. 13.6, possesses the full symmetry of the 
cube: three 4-fold axes, four 3-fold axes, six 2-fold axes, three mirror planes 
perpendicular to the 4-fold axes, six mirror planes perpendicular to the 2-fold 
axes, and a center of inversion. The cubic crystals of pyrites might at first 
seem to possess all these symmetry elements too. Closer examination reveals, 

2 A set of symmetry operations forms a. group when the consecutive application of any 
two operations in the set is equivalent to an operation belonging to the set (law of multi- 
plication). It is understood that the identity operation, leaving the crystal unchanged, is 
included in each set; that the operations are reversible; and that the associative law holds, 
A(BC) = (AB)C. 



374 



CRYSTALS 



[Chap. 13 



however, that the pyrites crystals have characteristic striations on their faces, 
as shown in the picture, so that all the faces are not equivalent. These crystals 
therefore do not possess the six 2-fold axes with the six planes normal to 
them, and the 4-fold axes have been reduced to 2-fold axes. 

In other cases, such departures from full symmetry are only revealed, as 
far as external appearance goes, by the orientation of etch figures formed by 
treating the surfaces with acids. Sometimes the phenomenon of pyro- 
electricity provides a useful symmetry test. When & crystal that contains no 
center of symmetry is heated, a difference in potential is developed across 
its faces. This can be observed by the resultant electrostatic attraction 
between individual crystals. 





CD 

(a) (b) 

Fig. 13.6. (a) Rock salt, (b) Pyrites. 

All these differences in symmetry are caused by the fact that the full 
symmetry of the point lattice has been modified in the crystal struc- 
ture, as a result of replacing the geometrical points by groups of atoms. 
Since these groups need not have so high a symmetry as the original 
lattice, classes of lower than holohedral symmetry can arise within each 
system. 

5. Space groups. The crystal classes are the various groups of symmetry 
operations of finite figures, i.e., actual crystals. They are made up of opera- 
tions by symmetry elements that leave at least one point in the crystal 
invariant. This is why they are called point groups. 

In a crystal structure, considered as an infinitely extended pattern in 
space, new types of symmetry operation are admissible, which leave no 
point invariant. These are called space operations. The new symmetry opera- 
tions involve translations in addition to rotations and reflections. Clearly 
only an infinitely extended pattern can have a space operation (translation) 
as a symmetry operation. 

The possible groups of symmetry operations of infinite figures are called 
space groups. They may be considered to arise from combinations of the 



Sec. 6] CRYSTALS 375 

14 Bravais lattices with the 32 point groups. 3 A space group may be visualized 
as a sort of crystallographic kaleidoscope. If one structural unit is introduced 
into the unit cell, the operations of the space group immediately generate 
the entire crystal structure, just as the mirrors of the kaleidoscope produce 
a symmetrical pattern from a few bits of colored paper. 

The space group expresses the sum total of the symmetry properties of 
a crystal structure, and mere external form or bulk properties do not suffice 
for its determination. The inner structure of the crystal must be studied and 
this is made possible by the methods of X-ray diffraction. 

6. X-ray crystallography. At the University of Munich in 1912, there was 
gathered a group of physicists interested in both crystallography and the 




Fig. 13.7. A Laue photograph taken with X-rays. (From Lapp and Andrews, 
Nuclear Radiation Physics, 2nd Ed., Prentice-Hall, 1953.) 

behavior of X rays. P. P. Ewaid and A. Sommerfeld were studying the 
passage of light waves through crystals. At a colloquium discussing some 
of this work, Max von Laue pointed out that if the wavelength of the radia- 
tion became as small as the distance between atoms in the crystals, a diffrac- 
tion pattern should result. There was some evidence that X rays should have 
the right wavelength, and W. Friedrich agreed to make the experimental test. 
On passing an X-ray beam through a crystal of copper sulfate, there was 
obtained a diffraction pattern like that in Fig. 13.7, though not nearly so 

3 A good example of the construction of space groups is given by Sir Lawrence Bragg, 
The Crystalline State (London: G. Bell & Sons, 1933), p. 82. The spjice-group notation is 
described in International Tables for the Determination of Crystal Structures, Vol. I. There 
are exactly 230 possible crystallographic space groups. 



376 



CRYSTALS 



[Chap. 13 



distinct in these first trials. The wave properties of X rays were thus definitely 
established and the new science of X-ray crystallography began. 

Some of the consequences of Laue's great discovery have already been 
mentioned, and on page 257 the conditions for diffraction maxima from a 
regular three-dimensional array of scattering centers were found to be 

cos (a 00) hh 

cos08-/? )-*A (13.1) 

cos (y - y ) - tt 

If monochromatic X rays are used, there is only a slim chance that the 
orientation of the crystal is fixed in such a way as to yield diffraction maxima. 
The Laue method, however, uses a continuous spectrum of X radiation with 
a wide range of wavelengths. This is the so-called white radiation, conveniently 
obtained from a tungsten target at high voltages. In this case, at least some 
of the radiation is at the proper wavelength to experience interference effects, 
no matter what the orientation of crystal to beam. 

7. The Bragg treatment. When the news of the Munich work reached 
England, it was immediately taken up by W. H. Bragg and his son W. L. 




Fig. 13.8. Bragg scattering condition. 

Bragg who had been working on a corpuscular theory of X rays. W. L. Bragg, 
using Laue-type photographs, analyzed the structures of NaCl, KC1, and 
ZnS (1912, 1913). In the meantime (1913), the elder Bragg devised a spectrom- 
eter that measured the intensity of an X-ray beam by the amount of ioniza- 
tion it produced, and he found that the characteristic X-ray line spectrum 
could be isolated and used for crystallographic work. Thus the Bragg method 
uses a monochromatic (single wavelength) beam of X rays. 

The Braggs developed a treatment of X-ray scattering by a crystal that 
was much easier to apply than Laue's theory, although the two are essentially 
equivalent. It was shown that the scattering of X rays could be represented 
as a "reflection" by successive planes of atoms in the crystal. Consider, in 
Fig. 13.8, a set of parallel planes in the crystal structure and a beam of 
X rays incident at an angle 0. Some of the rays will be "reflected" from the 
upper layer of atoms, the angle of reflection being equal to the angle of inci- 
dence. Some of the rays will be absorbed, and some will be "reflected" from 



Sec. 8] CRYSTALS 377 

the second layer, and so on with successive layers. All the waves "reflected" 
by a single crystal plane will be in phase. Only under certain strict conditions 
will the Waves "reflected" by different underlying planes be in phase with 
one another. The condition is that the path difference between the waves 
scattered from successive planes must be an integral number of wavelengths, 
nk. If we consider the "reflected" waves at the point P, this path distance Tor 
the first two planes is 6 = "AB + ~BC. Since triangles AOB and COB are 
congruent, AB BC and d 2 AB. Therefore d 2d sin 0. The condition 
for reinforcement or Bragg "reflection" is thus 

/7A-2</sin0 (13.2) 

According to this viewpoint, there are different orders of "reflection" 
specified by the values n = 1 , 2, 3 . . . The second order diffraction maxima 
from (100) planes may then be regarded as a "reflection" due to a set of 
planes (200) with half the spacing of the (100) planes. 

The Bragg equation indicates that for any given wavelength of X rays 
there is a lower limit to the spacings that can give observable diffraction 
spectra. Since the maximum value of sin is 1, this limit is given by 

" A -- 

~ 2 sin max " 2 

8. The structures of NaCl and KC1. Among the first crystals to be studied 
by the Bragg method were sodium and potassium chlorides. A single crystal 
was mounted on the spectrometer, as shown in Fig. 13.9, so that the X-ray 

IX-RAY BEAM 

L-V/V//J 

SLIT SYSTEM 




IONIZATION 
CHAMBER 

**" ^^J^^ 
DIVIDED 
SCALE 

/TO ELECTROMETER 
Fig. 13.9. Bragg X-ray spectrometer. 

beam was incident on one of the important crystal faces, (100), (1 10), or (1 1 1). 
The apparatus was so arranged that the "reflected" beam entered the ioniza- 
tion chamber, which was filled with methyl bromide. Its intensity was 
measured by the charge built up on an electrometer. 

The experimental data are shown plotted in Fig. 13.10 as "intensity of 
scattered beam" vs. "twice the angle of incidence of beam to crystal." As 
the crystal is rotated, successive maxima "flash out" as the angles are passed 



378 



CRYSTALS 



[Chap. 13 



conforming to the Bragg condition, eq. (13.2). In these first experiments the 
monochromatic X radiation was obtained from a palladium target. Both the 
wavelength of the X rays and the structure of the crystals were unknown to 
begin with. 

It was known, of course, from external form, that both NaCl and KC1 
could be based on a cubic lattice, simple, body-centered, or face-centered. 
By comparing the spacings calculated from X-ray data with those expected 
for these lattices, a decision could be made as to the proper assignment. 



(100) 
(MO) 
(IN) 


/ 


I 






A 










KCl 






/ 






/\ 














A 








A 




(100) 
(HO) 
(Ml) 






L 




A 






/^^ 




NoCl 








A. 






^\ 








> 


\ 




A 











0* 5 10 15 20 25 30 35 40 45 
Fig. 13.10. Bragg spectrometer data, / vs. 20. 

The general expression for the spacing of the planes (hkl) in a cubic 
lattice is 



-" Vh*-+k*n i 

When this is combined with the Bragg equation, we obtain 
sin 2 6 = (A 2 /4a 2 )(// 2 + k 2 + I 2 ) 

Thus each observed value of sin can be indexed by assigning to it the 
value of (hkl) for the set of planes responsible for the "reflection." For a 
simple cubic lattice, the following spacings are allowed: 

(hkl) . . . 100 110 111 200 210 211 220 221,300 etc. 
h 2 + k 2 + 1 2 . .1 2 3 4 5 6 8 9 etc. 

If the observed X-ray pattern from a simple cubic crystal was plotted as 
intensity vs. sin 2 we would obtain a series of six equidistant maxima, with 
the seventh missing, since there is no set of integers hkl such that h 2 + k 2 + I 2 
7. There would then follow seven more equidistant maxima, with the 15th 
missing; seven more, the 23rd missing; four more, the 28th missing; and so on. 



Sec. 8] 



CRYSTALS 



379 



In Fig. 13.11 (a) we see the (100), (110), and (111) planes for a simple 
cubic lattice. A structure may be based on this lattice by replacing each 
lattice point by an atom. If an X-ray beam strikes such a structure at the 
Bragg angle, sin" 1 (A/20), the rays scattered from one (100) plane will be 
exactly in phase with the rays from successive (100) planes. The strong 
scattered beam may be called the "first-order reflection from the (100) 
planes." A similar result is obtained for the (1 10) and (111) planes. We shall 
-*- a ^ 




Fig. 13.11. Spacings in cubic lattices: (a) simple cubic; (b) body-centered cubic; 
(c) face-centered cubic. 

obtain a diffraction maximum from each set of planes (hkl), since for any 
given (hkl) all the atoms will be included in the planes. 

Fig. 13.11 (b) shows a structure based on a body-centered cubic lattice. 
The (110) planes, as in the simple-cubic case, pass through all the lattice 
points, and a strong first-order (1 10) reflection will occur. In the case of the 
(100) planes, however, we find a different situation. Exactly midway between 
any two (100) planes, there lies another layer of atoms. When X rays scattered 
from the (100) planes are in phase and reinforce one another, the rays 
scattered by the interleaved atomic planes will be retarded by half a wave- 
length, and hence will be exactly out of phase with the others. The observed 
intensity will therefore be the difference between the scattering from the two 
sets of planes. If the atoms all have identical scattering powers, the resultant 
intensity will be reduced to zero by the destructive interference, and no 



380 CRYSTALS [Chap. 13 

first-order (100) reflection will appear. If, however, the atoms are different, 
the first-order (100) will still appear, but with a reduced intensity given 
by the difference between the scatterings from the two interleaved sets of 
planes. 

The second-order diffraction from the (100) planes, occurring at the 
Bragg angle with n ^ 2 in eq. (13.2), can equally well be expressed as the 
scattering from a set of planes, called the (200) planes, with just half the 
spacing of the (100) planes. In the body-centered cubic structure, all the atoms 
lie in these (200) planes, so that all the scattering is in phase, and a strong 
scattered beam is obtained. The same situation holds for the (111) planes: 
the first-order (111) will be weak or extinguished, but the second-order (111), 
i.e. the (222) planes, will give strong scattering. If we examine successive 
planes (hkl) in this way, we find for the body-centered cubic structure the 
results shown in Table 13.2, in which planes missing due to extinction are 
indicated by dotted lines. 

TABLE 13.2 
CALCULATED AND OBSERVED DIFFRACTION MAXIMA 

300 
(hkl) . ... 100 110 111 200 210 211 220 211 310 



/,2 + p 4. 72 ! 2 3 4 5 6 8 9 10 



simple cubic . | | | | | | | | | 

body-centered cubic : | | | III 



face-centered cubic 



Sodium Chloride 



200 220 222 400 420 422 440 600 620 

Potassium Chloride .1 I I I I I I 422 I 



In the case of the face-centered cubic structure, Fig. 13.1 1 (c), reflections 
from the (100) and (110) planes are weak or missing, and the (111) planes 
give intense reflection. The results for subsequent planes are included in 
Table 13.2. 

In the first work on NaCl and KC1, the X-ray wavelength was not known, 
so that the spacings corresponding to the diffraction maxima could not be 
calculated. The values of sin 2 0, however, can be used directly. The observed 
maxima are compared in Table 13.2 with those calculated for the different 
cubic lattices. 

The curious result is now not^d that apparently NaCl is face centered 



Sec. 8] 



CRYSTALS 



381 




Fig. 13.12. Sodium chloride 
structure. 



while KC1 is simple cubic. The reason why the KC1 structure behaves toward 
X rays like a simple cubic array is that the scattering powers of K + and Cl~ 
ions are indistinguishable since they both have an argon configuration with 
18 electrons. In the NaCl structure the difference in scattering power of the 
Na+ and Cl~ ions is responsible for the deviation from the simple cubic 
pattern. 

The observed maxima from the (111) face of NaCl include a weak peak 
at an angle of about 10, in addition to the stronger peak at about 20, 
corresponding to that observed with KC1. These results are all explained by 
the NaCl structure shown in Fig. 13.12, which consists of a face-centered 
cubic array of Na+ ions and an interpenetrating face-centered cubic array of 
Cl~ ions. Each Na+ ion is surrounded by six 
equidistant Cl~ ions and each Cl~ ion by 
six equidistant Na+ ions. The (100) and (1 10) 
planes contain an equal number of both 
kinds of ions, but the (111) planes consist of 
either all Na f or all Cl~ ions. Now if X rays 
are scattered from the (111) planes in NaCl, 
whenever scattered rays from successive Na+ 
planes are exactly in phase, the rays scattered 
from the interleaved Cl~ planes are retarded 
by half a wavelength and are therefore exactly 

out of phase. The first-order (111) reflection is therefore weak in NaCl since 
it represents the difference between these two scatterings. In the case of KC1, 
where the scattering powers are the same, the first-order reflections are 
altogether extinguished by interference. Thus the postulated structure is in 
complete agreement with the experimental X-ray evidence. 

Once the NaCl structure was well established, it was possible to calculate 
the wavelength of the X rays used. From the density of crystalline NaCl, 
p = 2.163 g per cm 3 , the molar volume is M/p =--- 58.45/2.163 = 27.02 cc per 
mole. Then the volume occupied by each NaCl unit is 27.02 : (6.02 x 10 23 ) 
= 44.88 x 10~ 24 cc. In the unit cell of NaCl, there are eight Na+ ions at the 
corners of the cube, each shared between eight cubes, and six Na+ ions at 
the face centers, each shared between two cells. Thus, per unit cell, there are 
8/8 + 6/2 = 4 Na + ions. There is an equal number of Cl~ ions, and there- 
fore four NaCl units per unit cell. The volume of the unit cell is there- 
fore 4 x 44.88 x 10" 24 = 179.52 (A) 3 . The interplanar spacing for the 
(200) planes is \a =- J179.52 173 =r 2.82 A. Substituting this value and the 
observed diffraction angle into the Bragg equation, A = 2(2.82) sin 5 58'; 
A - 0.586 A. 

Once the wavelength has been measured in this way, it can be used to 
determine the interplanar spacings in other crystal structures. Conversely, 
crystals with known spacings can be used to measure the wavelengths of 
other X-ray lines. The most generally useful target material is copper, with 



382 



CRYSTALS 



[Chap. 13 



A -- 1.537 A (A^), a convenient length relative to interatomic distances. 
When short spacings are of interest, molybdenum (0.708) is useful, and 
chromium (2.285) is often employed for study of longer spacings. 

The Bragg spectrometer method is generally applicable but is quite time 
consuming. Most crystal structure investigations have used photographic 
methods to record the diffraction patterns. Improved spectrometers have 
been developed recently in which a Geiger-counter tube replaces the electrom- 
eter and ionization chamber. 

9. The powder method. The simplest technique for obtaining X-ray diffrac- 
tion data is the powder method, first used by P. Debye and P. Scherrer. 
Instead of a single crystal with a definite orientation to the X-ray beam, a 



CYLINDRICAL 
" CAMERA 




POWDER 
SPECIMEN 



X-RAY 
BEAM 




FILM 




Fig. 13.13. The powder method. Powder picture of sodium chloride, Cu-K a 
radiation, (c). (Courtesy Dr. Arthur Lessor, Indiana University.) 

mass of finely divided crystals with random orientations is used. The experi- 
mental arrangement is illustrated in (a), Fig. 13.13. The powder is contained 
in a thin-walled glass capillary, or deposited on a fiber. Polycrystalline metals 
are studied in the form of fine wires. The sample is rotated in the beam to 
average as well as possible the orientations of the crystallites. 

Out of the many random orientations of the little crystals, there will be 
some at the proper angle for X-ray reflection from each set of planes. 
The direction of the reflected beam is limited only by the requirement that 
the angle of reflection equal the angle of incidence. Thus if the incident angle 
is 0, the reflected beam makes an angle 20 with the direction of the incident 
beam, (b), Fig. 13.13. This angle 26 may itself be oriented in various directions 
around the central beam direction, corresponding to the various orientations 
of the individual crystallites. For each set of planes, therefore, the reflected 
beams outline a cone of scattered radiation. This cone, intersecting a 



Sec. 10] CRYSTALS 383 

cylindrical film surrounding the specimen, gives rise to the observed dark 
lines. On a flat plate film, the observed pattern consists of a series of con- 
centric circles. A typical X-ray powder picture is shown in (c), Fig. 13.13. 
It may be compared with the electron-diffraction picture obtained by 
G. P. Thomson from a polycrystalline gold foil (page 272). 

After obtaining a powder diagram, the next step is to index the lines, 
assigning each to the responsible set of planes. The distance x of each line 
from the central spot is measured carefully, usually by halving the distance 
between the two reflections on either side of the center. If the film radius is 
r, the circumference 2nr corresponds to a scattering angle of 360. Then, 
x/2irr = 2(9/360. Thus is calculated and, from eq. (13.2), the interplanar 
spacing. 

The spacing data are often used, without further calculation, to identify 
solids or analyze solid mixtures. Extensive tables are available 4 that facilitate 
the rapid identification of unknowns. 

To index the reflections, one must know the crystal system to which the 
specimen belongs. This system can sometimes be determined by microscopic 
examination. Powder diagrams of monoclinic, orthorhombic, and triclinic 
crystals may be almost impossible to index. For the other systems straight- 
forward methods are available. Once the unit-cell size is found, by calculation 
from a few large spacings (100, 110, 111, etc.), all the interplanar spacings 
can be calculated and compared with those observed, thus completing the 
indexing. Then more precise unit-cell dimensions can be calculated from 
high-index spacings. The general formulae giving the interplanar spacings 
are straightforward derivations from analytical geometry. 5 

10. Rotating-crystal method. The rotating-single-crystal method, with 
photographic recording of the diffraction pattern, was developed by E. 
Schiebold around 1919. It has been, in one form or another, the most widely 
used technique for precise structure investigations. 

The crystal, which is preferably small and well formed, perhaps a needle 
a millimeter long and a half-millimeter wide, is mounted with a well defined 
axis perpendicular to the beam which bathes the crystal in X radiation. The 
film may be held in a cylindrical camera, and the crystal is rotated slowly 
during the course of the exposure. In this way, successive planes pass through 
the orientation necessary for Bragg reflection, each producing a dark spot 
on the film. Sometimes only part of the data is recorded on a single film, by 
oscillating through some smaller angle rather than rotating through 360. 
An especially useful method employs a camera that moves the film back and 
forth with a period synchronized with the rotation of the crystal. Thus the 
position of a spot on the film immediately indicates the orientation of the 
crystal at which the spot was formed (Weissenberg method). 

We cannot give here a detailed interpretation of these several varieties 

4 J. D. Hanawalt, Ind. Eng. Chem. Anal., 10, 457 (1938). 

6 C. W. Bunn, Chemical Crystallography (New York: Oxford, 1946), p. 376. 



384 



CRYSTALS 



[Chap. 13 




, 

Fig. 13.14. Rotation picture of zinc oxine dihydrate Weisscnberg method. 
(Courtesy Prof. L. L. Merritt, Indiana University.) 

of rotation pictures. 6 A typical example is shown in Fig. 13.14. Methods 
have been developed for indexing the various spots and also for measuring 
their intensities. These data are the raw material for crystal-structure 
determinations. 

11. Crystal-structure determinations: the structure factor. The problem of 
reconstructing a crystal structure from the intensities of the various X-ray 
diffraction maxima is analogous in some ways to the problem of the forma- 
tion of an image by a microscope. According to Abbe's theory of the micro- 
scope, the objective gathers various orders of light rays diffracted by the 
specimen and resynthesizes them into an image. This synthesis is possible 
because two conditions are fulfilled in the optical case: the phase relation- 
ships between the various orders of diffracted light waves are preserved at 
all times, and optical glass is available to focus and form an image with 
radiation having the wavelength of visible light. We have no such lenses for 
forming X-ray images (compare, however, the electron microscope), and the 
way in which the diffraction data are necessarily obtained (one by one) 
means that all the phase relationships are lost. The essential problem in 
determining a crystal structure is to regain this lost information in some way 
or other, and to resynthesize the structure from the amplitudes and phases 
of the diffracted waves. 

We shall return to this problem in a little while, but first let us see how 
the intensities of the various spots on an X-ray picture are governed by the 
crystal structure. 7 The Bragg relation fixes the angle of scattering in terms of 

' See Bragg, he. cit., p. 30. Also Bunn, he. c//., p. 137. 

7 This treatment follows that given by M. J. Buerger in X-Ray Crystallography (New 
York: Wiley, 1942), which.should be consulted for more details. 



Sec. 11] 



CRYSTALS 



385 



the interplanar spacings, which are determined by the arrangement of points 
in the crystal lattice. In an actual structure, each lattice point is replaced by 
a group of atoms. It is primarily the arrangement and composition of this 
group that controls the intensity of the scattered X rays, once the Bragg 
condition has been satisfied. 

As an example, consider in (a), Fig. 13.15, a lattice in which each point 
has been replaced by two atoms (e.g., a diatomic molecule). Then if a set of 




Fig. 13.15. X-ray scattering from a typical structure. 

lattice planes is drawn through the black atoms, another parallel but slightly 
displaced set can be drawn through the white atoms. When the Bragg con- 
dition is met, as in (b), Fig. 13.15, the reflections from all the black atoms 
are in phase, and the reflections from all the white atoms are in phase. 
The radiation scattered from the blacks is slightly out of phase with that 
from the whites, so that the resultant amplitude, and therefore intensity, is 
diminished by interference. 

The problem now is to obtain a general expression for the phase 



386 CRYSTALS [Chap. 13 

difference. An enlarged view of the structure (two-dimensional) is shown in 
(c), Fig. 13.15, with the black atoms at the corners of a unit cell with sides 
a and /?, and the whites at displaced positions. The coordinates of a black 
atom may be taken as (0, 0) and those of a white as (x, y). A set of planes 
(hk) is shown, for which it is assumed the Bragg condition is being fulfilled; 
these are actually the (32) planes in the figure. Now the spacings a/h along 
a and b/k along b correspond to positions from which scattering differs in 
phase by exactly 360 or 2rr radians, i.e., scattering from these positions is 
exactly in phase. The phase difference between these planes and those going 
through the white atoms is proportional to the displacement of the white 
atoms. The phase difference P x for displacement v in the a direction is given 
by x/(a/h) --= PJ2ir, or P x 2irh(x/a). The total phase difference for dis- 
placement in both a and b directions becomes 

/>, -f Py - 2* 

By extension to three dimensions, the total phase change that an atom at 
(xyz) in the unit cell contributes to the plane (hkl) is 



We may recall (page 327) that the superposition of waves of different 
amplitude and phase can be accomplished by vectorial addition. If /j and 
/ 2 are the amplitudes of the waves scattered by atoms (1) and (2), and P l 
and P 2 are the phases, the resultant amplitude is F f\? lPl 4-/ 2 ^ /J ". For 
any number of atoms, 

^' (13-4) 



When this is combined with eq. (13.3), there is obtained an expression for 
the resultant amplitude of the waves scattered from the (hkl) planes by all 
the atoms in a unit cell: 



F(hkl) = ZJ K <?****!*+ w* * '*/') (13.5) 

The expression F(hkl) is called the structure factor of the crystal. Its 
value is determined by the exponential terms, which depend on the positions 
of the atoms, and by the atomic scattering factors f K , which depend on the 
number and distribution of the electrons in the atom, and on the scattering 
angle 0. 

The intensity of scattered radiation is proportional to the absolute value 
of the amplitude squared, \F(hkl)\ 2 . The crystal structure problem now 
becomes that of obtaining agreement between the observed intensities and 
those calculated from a postulated structure. Structure-factor expressions 
have been tabulated for all the space groups. 8 

8 International Tables for the Determination of Crystal Structures (1952). It is usually 
possible to narrow the choice of space groups to two or three by means of study of missing 
reflections (hkl) and comparison with the tables. 



Sec. 12] CRYSTALS 387 

As an example of the use of the structure factor let us calculate F(hkl) 
for the 100 planes in a face-centered cubic structure, eg., metallic gold. In 
this structure there are four atoms in the unit cell (Z 4), which may be 
assigned coordinates (xja, v/b, z/c) as follows: (000), (J i 0), (i J), and 
(0 i i). Therefore, from eq."(13.5) 



-- f Au (2 h 2^) 
since e" 1 cos TT + / sin TT --- 1 

Thus the structure factor vanishes and there is therefore zero intensity of 
scattering from the (100) set of planes. This is almost a trivial case, since 
inspection of the face-centered cubic structure immediately reveals that there 
is an equivalent set of planes interleaved midway between the 100 planes, so 
that the resultant amplitude of the scattered X rays must be reduced to zero 
by interference. In more complicated instances, however, it is essential to 
use the structure factor to obtain a quantitative estimation of the scattering 
intensity expected from any set of planes (hkl) in any postulated crystal 
structure. 

12. Fourier syntheses. An extremely useful way of looking at a crystal 
structure was proposed by Sir William Bragg when he pointed out that it 
may be regarded as a periodic three-dimensional distribution of electron 
density, since it is the electrons that scatter the X rays. Any such density 
function may be expressed as a Fourier series, a summation of sine and 
cosine terms. 9 It is more concisely written in the complex exponential form. 
Thus the electron density in a crystal may be represented as 






p(xyz) - \ \ A pQr e **P*!* ^ '////ft i /r) ( { 3 6) 

p _ oo q uj r or 

It is not hard to show 10 that the Fourier coefficients A wr are equal to 
the structure factors divided by the volume of the unit cell. Thus 

p(xyz) 1 SSX F(hkl)e"^^ rla ^ vlb ^ lf} (13.7) 

This equation expresses the fact that the only Fourier term that contributes 
to the X-ray scattering by the set of planes (hkl) is the one with the coefficient 
F(hkl), which appears intuitively to be the correct formulation. 

Equation (13.7) summarizes the whole problem involved in structure 
determinations, since in a very real sense the crystal structure is simply 
p(xyz). Positions of individual atoms are peaks in the electron density 
function />, and interatomic regions are valleys in the plot of p. Thus if we 
knew the F(hkiy$ we could immediately plot the crystal structure. All we 
know, however, are the intensities, which are proportional to \F(hkl)\ 2 . As 

9 See, for example, Widder, Advanced Calculus, p. 324. 
10 Bragg, op. cit., p. 221. 



388 



CRYSTALS 



[Chap. 13 



stated earlier, we know the amplitudes but we have necessarily lost the 
phases in taking the X-ray pattern. 

A trial structure is now assumed and the intensities are calculated. If the 
assumed arrangement is even approximately correct, the most intense 
observed reflections should have large calculated intensities. The observed 
F's for these reflections may be put into the Fourier series with the calculated 
signs. 11 The graph of the Fourier summation will give new positions for the 




VAAX \ \ \ 




Fig. 13.16. Fourier map of electron density in glycylglycine projected 
on base of unit cell: (a) 40 terms; (b) 100 terms; (c) 200 terms. 

atoms, from which new f's can be calculated, which may allow more of the 
signs to be determined. Gradually the structure is refined as more and more 
terms are included in the synthesis. In Fig. 13.16 are shown three Fourier 
summations for the structure of glycylglycine. As additional terms are in- 
cluded in the summation, the resolution of the structure improves, just as 
the resolution of a microscope increases with objectives that catch more and 
more orders of diffracted light. 

Sometimes a heavy atom can be introduced into the structure, whose 
position is known from symmetry arguments. The large contribution of the 
heavy atom makes it possible to determine the phases of many of the F's. 

11 The complete Fourier series is rarely used; instead, various two-dimensional pro- 
jections are preferred. 



Sec. 13] 



CRYSTALS 



389 



This was the method used with striking success by J. M. Robertson in his 
work on the phthalocyanine structures, 12 and in the determination of the 
structure of penicillin. This last was one of the great triumphs of X-ray 
crystallography, since it was achieved before the organic chemists knew the 
structural formula. 

13. Neutron diffraction. Not only X-ray and electron beams, but also 
beams of heavier particles may exhibit diffraction patterns when scattered 
from the regular array of atoms in a crystal. Neutron beams have proved to 
be especially useful for such studies. The wavelength is related to the mass 
and velocity by the Broglie equation, X -- hjmv. Thus a neutron with a 
speed of 3.9 x 10 5 cm sec" 1 (kinetic energy 0.08 ev) would have a wave- 
length of 1 .0 A. The diffraction of electron rays or X rays is caused by their 
interaction with the orbital electrons of the atoms in the material through 
which they pass; the atomic nuclei contribute practically nothing to the 
scattering. The diffraction of neutrons, on the other hand, is primarily 
caused by two other effects: (a) nuclear scattering due to interaction of the 
neutrons with the atomic nuclei, (b) magnetic scattering due to interaction of 
the magnetic moments of the neutrons with permanent magnetic moments 
of atoms or ions. 

In the absence of an external magnetic field, the magnetic moments of 
atoms in a paramagnetic crystal are arranged at random, so that the magnetic 
scattering of neutrons by such a crystal is also random. It contributes only 
a diffuse background to the sharp maxima occurring when the Bragg con- 
dition is satisfied for the nuclear 
scattering. In ferromagnetic materials, 
however, the magnetic moments are 
regularly aligned so that the resultant 
spins of adjacent atoms are parallel, 
even in the absence of an external 
field. In antiferromagnetic materials, 
the magnetic moments are also regu- 
larly aligned, but in such a way that 
adjacent spins are always opposed. 
The neutron diffraction patterns dis- 
tinguish experimentally between these 
different magnetic structures, and indi- 
cate the direction of alignment of spins 
within the crystal. 

For example, manganous oxide, 
MnO, has the rock-salt structure (Fig. 13.12), and is antiferromagnetic. The 
detailed magnetic structure as revealed by neutron diffraction is shown in 
Fig. 13.17. The manganous ion, Mn+ 2 , has the electronic structure 3s 2 3p B 3d*. 

12 J. Chem. Soc. (London), 1940, 36. For an account of the work on penicillin, see 
Research, 2, 202 (1949). 




CHEMICAL 

UNIT 

CELL 



Fig. 13.17. Magnetic structure of MnO 
as found by neutron diffraction. Note 
that the "magnetic unit cell" has twice the 
length of the "chemical unit cell." [From 
C. G. Shull, E. O. Wollan, and W. A. 
Strauser, Phys. Rev., 81, 483 (1951).] 



390 



CRYSTALS 



[Chap. 13 



The five 3c/ electrons are all unpaired, and the resultant magnetic moment 
is 2V%(jf I 1) = 5.91 Bohr magnetons. If we consider Mn +2 ions in 
successive (111) planes in the crystal, the resultant spins are oriented so 
that they are alternately positively and negatively directed along the [100] 
direction. 

Another useful application of neutron diffraction has been the location 
of hydrogen atoms in crystal structures. It is usually impossible to locate 
hydrogen atoms by means of X-ray or electron diffraction, because the small 
scattering power of the hydrogen is completely overshadowed by that of 
heavier atoms. The hydrogen nucleus, however, is a strong scatterer of 
neutrons. Thus it has been possible to work out the structures of such com- 
pounds as UH 3 and KHF 2 neutron-diffraction analysis. 13 

14. Closest packing of spheres. Quite a while before the first X-ray struc- 
ture analyses, some shrewd theories about the arrangement of atoms and 





(a) 



(b) 



/ / 











(c; 







(c) 



(d) 



Fig. 13.18. (a) Hexagonal closest packing; (b) cubic closest packing (edge cut 
away to show closest packing normal to cube diagonals); (c) plan of hexagonal 
closest packing; (d) plan of cubic closest packing. 

molecules in crystals were developed from purely geometrical considerations. 
From 1883 to 1897, W. Barlow proposed a number of structures based on 
the packing of spheres. 

There are two different ways in which spheres of the same size can be 
packed together so as to leave a minimum of unoccupied volume, in each 
case 26 per cent voids. They are the hexagonal-closest-packed (hep) and the 

13 S. W. Peterson and H. A. Levy, /. Chem. Phys., 20, 704 (1952). 



Sec. 14] 



CRYSTALS 



391 



cubic-closest-packed (ccp) arrangements depicted in Fig. 13.18. In ccp the 
layers repeat as ABC ABC ABC . . ., and in hep the order is AB AB AB 
... It will be noted that the ccp structure may be referred to a face-centered- 
cubic unit cell, the (ill) planes being the layers of closest packing. 

The ccp structure is found in the solid state of the inert gases, in crystal- 
line methane, etc. symmetrical atoms or molecules held together by van 
der Waals forces. The high-temperature forms of solid H 2 , N 2 , and O 2 occur 
in hep structures. 

The great majority of the typical metals crystallize in the ccp, the hep, 
or a body-centered-cubic structure. Some examples are collected in Table 
13.3. Other structures include the following: 14 the diamond-type cubic of 

TABLE 13.3 
STRUCTURES OF THE METALS 



Cubic Closest Packed 
(fee) or (ccp) 


Hexagonal Closest Packed 
(hep) 


Body-Centered Cubic 
(bcc) 


Ag yFe 
Al Ni 
Au Pb 
ocCa Pt 


aBe Os 

yCa aRu 
Cd flSc 
aCe aTi 


Ba Mo 
aCr Na 
Cs Ta 
aFe Ti 


PCO Sr 


aCo aTl 


^Fe V 


Cu Th 


/?Cr Zn 
Mg aZr 


K ^W 
Li pZr 



grey tin and germanium; the face-centered tetragonal, a distorted fee, of 
y-manganese and indium; the rhombohedral layered structures of bismuth, 
arsenic, and antimony; and the body-centered tetragonal of white tin. It 
will be noted that many of the metals are polymorphic (allotropic), with two 
or more structures depending on conditions of temperature and pressure. 

The nature of the binding in metal crystals will be discussed later. For 
the present, we may think of them as a network of positive metal ions 
packed primarily according to geometrical requirements, and permeated by 
mobile electrons. This so-called electron gas is responsible for the high 
conductivity and for the cohesion of the metal. 

The ccp metals, such as Cu, Ag, Au, Ni, are all very ductile and malle- 
able. The other metals, such as V, Cr, Mo, W, are harder and more brittle. 
This distinction in physical properties reflects a difference between the struc- 
ture types. When a metal is hammered, rolled, or drawn, it deforms by the 
gliding of planes of atoms past one another. These slip planes are those that 
contain the most densely packed layers of atoms. In the ccp structure, the 
slip planes are therefore usually the (111), which occur in sheets normal to 

14 For descriptions see R. W. G. Wyckoff, Crystal Structures (New York: Interscience, 
1948). 



392 CRYSTALS [Chap. 13 

all four of the cube diagonals. In the hep and other structures there is only 
one set of slip planes, e.g., those perpendicular to the hexagonal axis. Thus 
the ccp metals are characteristically more ductile than the others, since they 
have many more glide ways. 

15. Binding in crystals. The geometrical factors, seen in their simplest 
form in the closest packed structures of identical spheres, are always very 
important in determining the crystal structure of a substance. Once they 
are satisfied, other types of interaction must also be considered. Thus, 
for example, when directed binding appears, closest packing cannot be 
achieved. 

Two different theoretical approaches to the nature of the chemical bond 
in molecules have been described in Chapter 11. In the method of atomic 
orbitals, the point of departure is the individual atom. Atoms are brought 
together, each with the electrons that "belong to it," and one considers the 
effect of an electron in one atomic orbital upon that in another. In the second 
approach, the electrons in a molecule are no longer assigned possessively to 
the individual atoms. A set of nuclei is arranged at the proper final distances 
and the electrons are gradually fed into the available molecular orbitals. 

For studying the nature of binding in crystals, these two different treat- 
ments are again available. In one case, the crystal structure is pictured as an 
array of regularly spaced atoms, each possessing electrons used to form 
bonds with neighboring atoms. These bonds may be ionic, covalent, or 
intermediate in type. Extending throughout three dimensions, they hold the 
crystal together. The alternative approach is once again to consider the nuclei 
at fixed positions in space and then gradually to pour the electron cement 
into the periodic array of nuclear bricks. 

Both these methods yield useful and distinctive results, displaying com- 
plementary aspects of the nature of the crystalline state. We shall call the 
first treatment, growing out of the atomic-orbital theory, the bond model of 
the solid state. The second treatment, an extension of the method of mole- 
cular orbitals, we shall call, for reasons to appear later, the band model of 
the solid state. 

16. The bond model. If we consider that a solid is held together by 
chemical bonds, it is useful to classify the bond types. Even though the 
available classifications are as usual somewhat frayed at the edges, the 
following categories may be distinguished: 

(1) The van der Waals bonds. These bonds are the result of forces between 
inert atoms or essentially saturated molecules. These forces are the same as 
those responsible for the a term in the van der Waals equation. Crystals held 
together in this way are sometimes called molecular crystals. Examples 
are nitrogen, carbon tetrachloride, benzene. The molecules tend to pack 
together as closely as their geometry allows. The binding between the mole- 
cules in van der Waals structures represents a combination of factors such 
as dipole-dipole and dipole-polarization interactions, and the quantum 



Sec. 16] 



CRYSTALS 



393 



mechanical dispersion forces, first elucidated by F. London, which are often 
the principal component. 15 

(2) The ionic bonds. These bonds are familiar from the case of the NaCl 
molecule in the vapor state (page 297). In a crystal, the coulombic interaction 
between oppositely charged ions leads to a regular three-dimensional struc- 
ture. In rock salt, each positively charged Na f ion is surrounded by six 
negatively charged Cl ions, and each Cl is surrounded by six Na j . There 
are no sodium-chloride molecules unless one wishes to regard the. entire 
crystal as a giant molecule. 

The ionic bond is spherically symmetrical and undirected; an ion will be 
surrounded by as many oppositely charged ions as can be accommodated 






ID 



(o) (b) 

Fig. 13.19. (a) Diamond structure; (b) graphite structure. 

geometrically, provided that the requirement of over-all electrical neutrality 
is satisfied. 

(3) The covalent bonds. These bonds, we recall, are the result of spin 
valence (page 303), the sharing between atoms of two electrons with anti- 
parallel spins. When extended through three dimensions, they may lead to 
a variety of crystal structures, depending on the valence of the constituent 
atoms, or the number of electrons available for bond formation. 

A good example is the diamond structure in (a), Fig. 13.19. The structure 
can be based on two interpenetrating face-centered cubic lattices. Each point 
in one lattice is surrounded tetrahedrally by four equidistant points in the 
other lattice. This arrangement constitutes a three-dimensional polymer of 
carbon atoms joined together by tetrahedrally oriented sp 3 bonds. Thus the 
configuration of the carbon bonds in diamond is similar to that in the 
aliphatic compounds such as ethane. The C C bond distance is 1.54 A in 
both diamond and ethane. Germanium, silicon, and grey tin also crystallize 
in the diamond structure. 

The same structure is assumed by compounds such as ZnS (zinc blende), 
Agl, A1P, and SiC. In all these structures, each atom is surrounded by four 
unlike atoms oriented at the corners of a regular tetrahedron. In every case 
the binding is primarily covalent. It is interesting to note that it is not neces- 
sary that each atom provide the same number of valence electrons; the 

15 See Chapter 14, Sect. 10. 



394 



CRYSTALS 



[Chap. 13 




Fig. 13.20. Structure of 
selenium 



structure can occur whenever the total number of outer-shell electrons is just 
four times the total number of atoms. 

There is also a form of carbon, actually the more stable allotrope, in 
which the carbon bonds resemble those in the aromatic series of compounds. 

This is graphite, whose structure is shown in 
(b), Fig. 13.19. Strong bonds operate within 
each layer of carbon atoms, whereas much 
weaker binding joins the layers; hence the 
slippery and flaky nature of graphite. The 
C C distance within the layers of graphite 
is 1.34 A, identical with that in anthracene. 

Just as in the discussion of the nature 
of binding in aromatic hydrocarbons (page 

311), we can distinguish two types of electrons within the graphite struc- 
ture. The a electrons are paired to form localized-pair (sp 2 ) bonds, and 
the 77 electrons are free to move throughout the planes of the C 6 rings. 

Atoms with a spin valence of only 2 cannot form regular three-dimen- 
sional structures. Thus we have the interesting structures of selenium (Fig. 
13.20), and tellurium, which consist 
of endless chains of atoms extending 
through the crystal, the individual 
chains being held together by much 
weaker forces. Another way of solving 
the problem is illustrated by the struc- 
ture of rhombic sulfur, Fig. 13.21. 
Here there are well defined, puckered, 
eight-membered rings of sulfur atoms. 
The bivalence of sulfur is maintained 
and the S 8 "molecules" are held 
together by van der Waals attractions. 
Elements like arsenic and antimony 
that in their compounds display a 
covalence of 3 tend to crystallize in 
structures that contain well defined 
layers of atoms. 

(4) 77?^ intermediate-type bonds. 
Just as in individual molecules, these 
bonds arise from resonance between 
covalent and ionic contributions. 
Alternatively, one may consider the 

polarization of one ion by an oppositely charged ion. An ion is said to be 
polarized when its electron "cloud" is distorted by the presence of the 
oppositely charged ion. The larger an ion the more readily is it polarized, 
and the smaller an ion the rhore intense is its electric field and the greater 




Fig. 13.21. 



Structure of rhombic 
sulfur. 



Sec. 17] 



CRYSTALS 



395 




Hg. 13.22, Structure of ice. 



its polarizing power. Thus in general the larger anions are polarized by the 
smaller cations. Even apart from the size effect, cations are less polarizable 
than anions because their net positive charge tends to hold their electrons 
in place. The structure of the ion is also important: rare-gas cations such 
as K+ have less polarizing power than transition cations such as Ag+, 
since their positive nuclei are more effectively shielded. 

The effect of polarization may be seen in the structures of the silver 
halides. AgF, AgCl, and AgBr have the-rock-salt structure, but as the anion 
becomes larger it becomes more strongly polarized by the small Ag+ ion. 
Finally, in Agl the binding has very little ionic character and the crystal has 
the zinc-blende structure. It has been 
confirmed spectroscopically that crystal- 
line silver iodide is composed of atoms 
and not ions. 

(5) The hydrogen bond. The hydrogen 
bond, discussed on page 313, plays an 
important role in many crystal struc- 
tures, e.g., inorganic and organic acids, 
salt hydrates, ice. The structure of ice is 
shown in Fig. 13.22. The coordination 
is similar to that in wurtzite, the hexago- 
nal form of zinc sulfide. Each oxygen is 

surrounded tetrahedrally by four nearest neighbors at a distance of 2.76 A. 
The hydrogen bonds hold the oxygens together, leading to a very open 
structure. By way of contrast, hydrogen sulfide, H 2 S, has a ccp structure, 
each molecule having twelve nearest neighbors. 

(6) 77?? metallic bond. The bond model has also been extended to metals. 
According to this picture, the metallic bond is closely related to the ordinary 
covalent electron-pair bond. Each atom in a metal forms covalent bonds by 
sharing electrons with its nearest neighbors. It is found that there are more 
orbitals available for bond formation than there are electrons to fill them. 
As a result the covalent bonds resonate among the available interatomic 
positions. In the case of a crystal this resonance extends throughout the 
entire structure, thereby producing great stability. The empty orbitals permit 
a ready flow of electrons under the influence of an applied electric field, 
leading to metallic conductivity. 

Structures such as those of selenium and tellurium, and of arsenic and 
antimony, represent transitional forms in which the electrons are much 
more localized because the available orbitals are more completely filled. 
In a covalent crystal like diamond the four .s/; 3 tetrahedral orbitals are 
completely filled. 

17. The band model. It was in an attempt to devise an adequate theory 
for metals that the band model had its origin. The high thermal and electrical 
conductivities of metals focused attention on the electrons as the important 



396 



CRYSTALS 



[Chap. 13 



entities in their structures. If we use as a criterion the behavior of the elec- 
trons, three classes of solids may be distinguished: 

(1) Conductors or metals, which offer a low resistance to the flow of 
electrons, an electric current, when a potential difference is applied. The 
resistivity of metals increases with the temperature. 

(2) Insulators, which have a high electric resistivity. 

(3) Semiconductors, whose resistivity is intermediate between that of 
typical metals and that of typical insulators, and decreases, usually ex- 
ponentially, with the temperature. 

The starting point of the band theory is a collection of nuclei arrayed in 
space at their final crystalline internuclear separations. The total number 



3s 












2p 






- 




- 


2s 

Is 


- 









- 



3s 



2p 

2 

I 



-a - 



(a) 



(b) 



Fig. 13.23. Energy levels in sodium: (a) isolated atoms; (b) section of crystal. 

of available electrons is poured into the resultant field of force, a regularly 
periodic field. What happens? 

Consider in Fig. 13.23 the simplified model of a one-dimensional struc- 
ture. For concreteness, let us think of the nuclei as being those of sodium, 
therefore bearing a charge of + 1 1 . The position of each nucleus will repre- 
sent a deep potential-energy well for the electrons, owing tp the large electro- 
static attraction. If these wells were infinitely deep, the electrons would all 
fall into fixed positions on the sodium nuclei, giving rise to l^Zs^/^S-s 1 
configurations, typical of isolated sodium atoms. This is the situation shown 
in (a), Fig. 13.23. But the wells are not infinitely deep, or in other words the 
potential-energy barriers separating the electrons on different nuclei are not 
infinitely high. The actual situation is more like the one shown in (b), Fig. 
13.23. Now the possibility of a quantum mechanical leakage of electrons 
through the barriers must be considered. Otherwise expressed, there will be 
a resonance of electrons between the large number of identical positions. 
There is always a possibility of an electron on one nucleus slipping through 
to occupy a position on a neighboring nucleus. We are thus no longer con- 
cerned with the energy levels of single sodium atoms but with levels of the 



Sec. 17] CRYSTALS 397 

crystal as a whole. Then the Pauli Principle comes into play, and tells us 
that no more than two electrons can occupy exactly the same energy level. 
Once the possibility of electrons moving through the structure is admitted, 
we can no longer consider the energy levels to be sharply defined. The sharp 
Is energy level in an individual sodium atom is broadened in crystalline 
sodium into a band of closely packed energy levels. A similar situation arists 
for the other energy levels, each becoming a band of levels as shown in (b), 
Fig. 13.23. 

Each atomic orbital contributes one level to a band. In the lower bands 
(Is, 2s, 2p) there are therefore just enough levels to accommodate the number 
of available electrons, so that the bands are completely filled. If an external 
electric field is applied, the electrons in the filled bands cannot move under 
its influence, for to be accelerated by the field they would have to move into 
somewhat higher energy levels. This is impossible for electrons in the interior 
of a filled band, since all the levels above them are already occupied, and the 
Pauli Principle forbids their accepting additional tenants. Nor can the elec- 
trons at the very top of a filled band acquire extra energy, since there are no 
higher levels for them to move into. Very occasionally, it is true, an electron 
may acquire a relatively terrific jolt of energy and be knocked completely 
out of its band into a higher unoccupied band. 

So much for the electrons in the lower bands. The situation is very 
different in the uppermost band, the 3s, which is only half filled. An electron 
in the interior of the 3s band still cannot be accelerated because the levels 
directly above are already filled. Electrons toward the top of the band, how- 
ever, can readily move up into unfilled levels within the band. This is what 
happens when an electric field is applied and a current flows. It will be 
noticed from the diagram that the topmost band has actually broadened 
sufficiently to overlap the tops of the potential-energy barriers, so that these 
electrons can move quite freely through the crystal structure. 

According to this idealized model in which the nuclei are always arranged 
at the points of a perfectly periodic lattice, there would indeed be no resist- 
ance at all offered to the flow of an electric current. The resistance arises from 
deviations from perfect periodicity. An important loss of periodicity is caused 
by the thermal vibrations of the lattice nuclei. These vibrations destroy the 
perfect resonance between the electronic energy levels and cause a resistance 
to the free flow of electrons. As would be expected, the resistance therefore 
increases with the temperature. Another illustration of the same principle is 
found in the increased resistance that results when an alloying constituent is 
added to a pure metal, and the regular periodicity of the structure is dimin- 
ished by the foreign atoms. 

At this point the reader may well be thinking that this is a pretty picture 
for a univalent metal such as sodium, but what of magnesium with its two 
3s electrons and therefore completely filled 3s bands ? Why isn't it an insulator 
instead of a metal? The answer is that in this, and similar cases, detailed 



398 



CRYSTALS 



[Chap. 13 



calculations show that the 3p band is low enough to overlap the top of the 
35 band, providing a large number of available empty levels. 

Thus conductors are characterized either by partially filled bands or by 
overlapping of the topmost bands. Insulators have completely filled lower 
bands with a wide energy gap between the topmost filled band and 
the lowest empty band. These models are represented schematically in 
Fig. 13.24. 

The energy bands in solids can be studied experimentally by the methods 
of X-ray emission spectroscopy. 16 For example, if an electron is driven out of 
the \s level in sodium metal (Fig. 13.23b) the K a X-ray emission occurs when 
an electron from the 3^ band falls into the hole in the Is level. Since the 3s 





FILLED IMPURITY 
LEVELS 

EMPTY IMPURITY 
LEVELS 



(a) 

Fig. 13.24. Band models of solid types: (a) insulator; (b) metal; 
(c) semiconductor. 

electron can come from anywhere within the band of energy levels, the X rays 
emitted will have a spread of energies (and hence frequencies) exactly corre- 
sponding with the spread of allowed energies in the 3s band. The following 
widths (in ev) were found for the conduction bands in a few of the solids 
investigated : 

Li Na Be Mg Al 

4.1 3.4 14.8 7.6 13.2 

18. Semiconductors. Band models for semiconductors are also included 
in Fig. 13.24. These models possess, in addition to the normal bands, narrow 
impurity bands, either unfilled levels closely above a filled band or filled levels 
closely below an empty band. The extra levels are the result of either foreign 
atoms dissolved in the structure or a departure from the ideal stoichiometric 
composition. Thus zinc oxide normally contains an excess of zinc, whereas 
cuprous oxide normally contains an excess of oxygen. Both these compounds 
behave as typical semiconductors. Their conductivities increase approxi- 
mately exponentially with the temperature, because the number of conduc- 
tion electrons depends on excitation of electrons into or out of the impurity 
levels, and excitation is governed by an e ~* E/HT Boltzmann factor. 



16 A review by N. F. Mott gives further references. Prog. Metal Phys. 3, 76-1 14 (1952), 
" Recent Advances in the Electron Theory of Metals." 



Sec. 19] CRYSTALS 399 

If the energy gap between the filled valence band and the empty con- 
duction band is narrow enough, a crystal may be a semiconductor even 
in the absence of effects due to impurities. Germanium with an energy gap 
of 0.72 ev and grey tin with O.lOev are examples of such intrinsic semi- 
conductors. 

19. Brillouin zones. The band theory of the crystalline state leads to a 
system of allowed energy levels separated by regions of forbidden energy. 
In other words, electron waves having a forbidden energy cannot pass through 
the crystal. Those familiar with radio circuits would say that the periodic 
crystal structure acts as a band-pass filter for electron waves. 

In this simple picture we have not considered the variety of periodic 
patterns that may be encountered by an electron wave, depending on the 
direction of its path through the crystal. When this is done, it is found that 
special geometrical requirements are imposed on the band structure, so that 
it is not necessarily the same for all directions in space. Now we can see 
qualitatively an important principle. If an electron wave with an energy in 
a forbidden region were to strike a crystal, it could not be transmitted, but 
would instead be strongly scattered or "reflected" in the Bragg sense. The 
Bragg relation therefore defines the geometric structure of the allowed 
energy bands. This principle was first enunciated by Leon Brillouin, and the 
energy bands constructed in this way are called the Brillouin zones of 
the crystal. 

The quantitative application of the zone theory is still in its early stages. 
Qualitatively it is clear that the properties of crystals are determined by the 
nature of the zones and the extent to which they are filled with electrons. 
This interpretation is especially useful in elucidating the structures of metal 
alloys. 

20. Alloy systems electron compounds. If two pure metals crystallize in 
the same structure, have the same valence and atoms of about the same size, 
they may form a continuous series of solid solutions without undergoing 
any changes in structure. Examples are the systems Cu-Au and Ag-Au. 

When these conditions are not fulfilled, a more complicated phase diagram 
will result. An example is that for the brass system, copper and zinc. Pure 
copper crystallizes in a face-centered-cubic structure and dissolves up to about 
38 per cent zinc in this a phase. Then the body-centered-cubic ft phase super- 
venes. At about 58 per cent zinc, a complex cubic structure begins to form, 
called "y brass," which is hard and brittle. At about 67 per cent Zn, the hexa- 
gonal closest packed e phase arises, and finally there is obtained the r\ phase 
having the structure of pure zinc, a distorted hep arrangement. 

It is most interesting that a sequence of structure changes very similar to 
these is observed in a wide variety of alloy systems. Although the com- 
positions of the phases may differ greatly, the /?, y, and e structures are quite 
typical. W. Hume-Rothery was the first to show that this regular behavior 
was related to a constant ratio of valence electrons to atoms for each phase. 



400 



CRYSTALS 



[Chap. 13 



Examples of these ratios are shown in Table 13.4. The transition metals Fe, 
Co, Ni follow the rule if the number of their valence electrons is taken as 
zero. 17 In all these cases, the zone structure determines the crystal structure, 
and the composition corresponding to each structure is fixed by the number 
of electrons required to fill the zone. Such alloys are therefore sometimes 
called electron compounds. 

TABLE 13.4 
ELECTRON COMPOUNDS ILLUSTRATING HUME-ROTHERY RULE 



Alloy* 



Electrons 


Atoms 


Ratio 



ft Phases (Ratio 3/2 



CuZn 


1 -h 2 


2 


AgCd 


1 + 2 


2 


CuBe 


1 + 2 


2 


AuZn 


1 4-2 


2 


Cu 3 Al 


34-3 


4 


Cu 6 Sn 


54-4 


6 


CoAl 


04-3 


2 


FeAl 


04-3 


2 



3:2 
3:2 
3:2 
3:2 
6:4 
9:6 
3:3 
3:2 



y Phases (Ratio 21/13) 



Cu 6 Zn 8 



Fe 5 Zn 21 
Cu 9 Ga 4 
Cu 9 Al 4 
Cu 31 Sn 8 





54-2x8 


13 


21 :13 




54-2x8 


13 


21 :13 




+ 2 x 21 


26 


42 : 26 




943x4 


13 


21 : 13 




9-1-3x4 


13 


21 :13 




31 4-4 x 8 


39 


63:39 



e Phases (Ratio 7/4) 



CuZn 3 
AgCd 3 
Cu 3 Sn 
Cu 3 Ge 
Au 6 Al 3 
AgsAl, 


14-2x3 
14-2x3 
34-4 
344 
54-3x3 
54-3x3 


4 
4 
4 
4 
8 
8 


7:4 
7:4 
7:4 
7:4 
14:8 
14:8 


* The alloy composition is variable within a certain range, but the nominal compositions 
listed always fall within the range. 



The body-centered /? brass structure illustrates another interesting 
property of some alloy systems, the order-disorder transition. At low 
temperatures, the structure is ordered; the copper atoms occupy only the 

17 Pauling has pointed out that it seems to be unreasonable to say that iron, which is 
famous for its great strength, contributes nothing to the bonding in iron alloys. From 
magnetic moments and other data he concludes that iron actually contributes between 
5 and 6 bonding electrons per atom [/. Am. Chem. Soc., 69, 542 (1947)]. 



Sec. 21] 



CRYSTALS 



401 



body-centered positions. At higher temperatures, the various positions are 
occupied at random by copper and zinc atoms. 

21. Ionic crystals. The binding in most inorganic crystals is predomi- 
nantly ionic in character. Therefore, since coulombic forces are undirected, 
the sizes of the ions play a most important role in determining the final 
structure. Several attempts have been made to calculate a consistent set of 
ionic radii, from which the internuclear distances in ionic crystals could be 
estimated. The first table, given by V. M. Goldschmidt in 1926, was modified 
by Pauling. These radii are listed in Table 13.5. 

TABLE 13.5 
IONIC CRYSTAL RADII (A)* 



o 



0.60 
0.31 
0.20 
0.15 
1.40 
1.36 



Me 



a+ 0.95 


K+ 


1.33 


Rb+ 


1.48 


Cs+ .69 


g f + 0.65 


Ca 4 " 4 " 


0.99 


Sr++ 


1.13 


Ba + + .35 


1 3 + 0.50 


Sc 34 ^ 


0.81 


Y 34 " 


0.93 


La 3+ .15 


4 + 0.41 


Ti 44 " 


0.68 


Zr 4 + 


0.80 


Ce 44 ^ .01 


1.84 


Cr* 4 * 


0.57 


Mo* 4 ^ 


0.62 




1.81 


Cu+ 


0.96 


Ag 4 " 


1.26 


Au+ .37 




Zn + + 


0.74 


Cd++ 


0.97 


Hg++ .10 




Se~~ 


1.98 


Te 


2.21 






Br- 


1.95 


I- 


2.16 





* From L. Pauling, The Nature of the Chemical Bond, 2nd ed. (Ithaca: Cornell Univ. 
Press, 1940), p. 346. 

First, let us consider ionic crystals having the general formula CA. They 
may be classified according to the coordination number of the ions; i.e., the 
number of ions of opposite charge surrounding a given ion. The CsCl struc- 
ture, body centered as shown in Fig. 13.25, has eightfold coordination. The 
NaCi structure (Fig. 13.12) has sixfold co- 
ordination. Although zinc blende (Fig. 13.19a) 
is itself covalent, there are a few ionic crys- 
tals, e.g., BeO, with this structure which has 
fourfold coordination. The coordination 
number of a structure is determined primarily 
by the number of the larger ions, usually the 
anions, that can be packed around the smaller 
ion, usually the cation. It should therefore 
depend upon the radius ratio, /* C ation/ r anion 
r c /r A . The critical radius ratio is that obtained 

when the anions packed around a cation are in contact with both the cation 
and with one another. 

Consider, for example, the structure of Fig. 13.25. If the anions are at 
the cube corners and have each a radius a, when they are exactly touching, 
the unit cube has a side 2a. The length of the cube diagonal is then \/3 2a, 
and the diameter of the empty hole in the center of the cube is therefore 




Fig. 13.25. The cesium 
chloride structure. 



402 



CRYSTALS 



[Chap. 13 



V/3 2a 2a --= 2a(\/3 1). The radius of the cation exactly filling this 
hole is thus a(\/3 1), and the critical radius ratio becomes r c /r A = 
a(\/3 ])/a ----- 0.732. By this simple theory, whenever the ratio falls below 
0.732, the structure can no longer have eightfold coordination, and indeed 
should go over to the sixfold coordination of NaCl. 

In the sixfold coordination, a given ion at the center of a regular octa- 
hedron is surrounded by six neighbors at the corners. The critical radius 
ratio for this structure may readily be shown to be \/2 - 1 0.414. The 
next lower coordination would be threefold, at the corners of an equilateral 
triangle, with a critical ratio of 0.225. 

The structures and ionic-radius ratios of a number of CA compounds 
are summarized in Table 13.6. The radius-ratio rule, while not infallible, 
provides the principal key to the occurrence of the different structure types. 



TABLE 13.6 
STRUCTURES AND RADIUS RATIOS OF CA IONIC CRYSTALS 



Cesium 
Chloride 
Structure - 



Sodium Chloride Structure 



Zinc Blende 

or Wurtzite 

Structure 



Theoretical Range 



CsCl 
CsBr 
Csl 



).732 


0.732-0.414 


0.414-0.225 


0.93 


KF 


0.98 


Rbl 


0.69 


NaBr 


0.49 


ZnS 


0.40 


0.87 
0.78 


BaO 
RbF 


0.94 
0.92 


BaSe 
KBr 


0.68 
0.68 


MgO 
CaTe 


0.46 
0.45 


MgTe 
BeO 


0.29 
0.22 




RbCl 


0.82 


SrS 


0.61 


LiF 


0.44 


BeS 


0.17 




SrO 


0.81 


BaTe 


0.61 


Nal 


0.44 


BeSe 


0.16 




CsF 
RbBr 
BaS 


0.80 
0.76 
0.73 


Kl 
SrSe 
CaS 


0.60 
0.57 
0.58 


MgS 
MgSe 
LiBr 


0.36 
0.33 
0.31 


BeTe 


0.14 




KC1 


0.73 


NaCl 


0.53 


LiCl 


0.30 








CaO 


0.71 


SrTe 


0.51 


Lil 


0.28 








NaF 


0.70 


CaSe 


0.50 











The structures of CA 2 ionic crystals are found to be governed by the 
same coordination principles. Four common structures are shown in Fig. 
13.26. In fluorite each Ca++ is surrounded by eight F~ ions at the corners of 
a cube, and each F~ is surrounded by four Ca +4 ~ at the corners of a tetra- 
hedron. This is an example of 8 : 4 coordination. The structure of rutile 
illustrates a 6 : 3 coordination, and that of cristobalite a 4 : 2 type. Once 
again the coordination is determined primarily by the radius ratio. 

The cadmium-iodide structure illustrates the result of a departure from 
typically ionic binding. The iodide ion is easily polarized, and one can 
distinguish definite CdI 2 groups forming a layerlike arrangement. 



Sec. 22] 



CRYSTALS 



403 




(d) 



Fig. 13.26. CA 2 structures: (a) fluonte; (b) rutile; (c) ft cnstobalite; 
(d) cadmium iodide. 

22. Coordination polyhedra and Pauling's Rule. Many inorganic crystals 
contain oxygen ions; their size is often so much larger than that of the cations 
that the structure is largely determined by the way in which they pack to- 
gether. The oxygens are arranged in coordination polyhedra around the 
cations, some common examples being the following: 

Around B: 3 O\s at corners of equilateral triangle 
Si, Al, Be, B, Zn : 4 O's at corners of tetrahedron 
Al, Ti, Li, Cr: 6 O's at corners of octahedron 

For complex structures, Pauling has given a general rule that determines 
how these polyhedra can pack together. Divide the valence of the positive 
ion by the number of surrounding negative ions; this gives the fraction of 
the valence of a negative ion satisfied by this positive ion. For each negative 
ion, the sum of the contributions from neighboring positive ions should 
equal its valence. This rule simply expresses the requirement that electro- 
static lines of force, starting from a positive ion, must end on a negative ion 
in the immediate vicinity, and not be forced to wander throughout the 
structure seeking a distant terminus. 

As an example of the application of the rule, consider the silicate group, 
(SiO 4 ). The valence of the positive ion, Si +4 , is +4. Therefore each O ion 
has one valence satisfied by the Si+ 4 ion, i.e., one-half of its total valence of 
two. It is therefore possible to join each corner of a silicate tetrahedron to 
another silicate tetrahedron. It is also possible for the silicates to share edges 
and faces, although these arrangements are less favorable energetically, since 
they bring the central Si+ 4 ions too close together. 

In the (A1O 6 ) octahedron, only a valence of \ for each O is satisfied 



404 



CRYSTALS 



[Chap. 13 



by the central Al+ 3 ion. It is therefore possible to join two aluminum octa- 
hedra to each corner of a silicate tetrahedron. 

The various ways of linking the silicate tetrahedra give rise to a great 
diversity of mineral structures. The following classification was given by 
W. L. Bragg: 

(a) Separate SiO 4 groups 

(b) Separate Si O complexes 

(c) Extended Si O chains 

(d) Sheet structures 

(e) Three-dimensional structures 

An example from each class is pictured in Fig. 13.27. In many minerals, 
other anionic groups and cations also occur, but the general principles that 




(S.0 4 ) 4 " 




(S.0 3 )< 

(TETRAHEDRA SHARING CORNERS) 



(Si 2 o 7 r 



(0) 





(Si 2 5 ) z 



(Si 4 0,,) 6 " 

(DOUBLE CHAIN) 
(C) 




Fig. 13.27. Silicate structures: (a) isolated groups; (b) hexagonal-type sheets; 
(c) extended chains; (d) three-dimensional framework. (After W. L. Bragg, The 
Atomic Structure of Minerals, Cornell University Press, 1937.) 

govern the binding remain the same. The structural characteristics are 
naturally reflected in the physical properties of the substances. Thus the 



Sec. 23] CRYSTALS 405 

chainlike architecture is found in the asbestos minerals, the sheet arrange- 
ment in micas and talcs, and the feldspars and zeolites are typical three- 
dimensional polymers. 

23. Crystal energy the Born-Haber cycle. The binding energy in a purely 
ionic crystal can be calculated via ordinary electrostatic theory. The potential 
energy of interaction of two oppositely charged ions may be written 

-z&e 2 be 2 
U=- Y~ + 7n (13.8) 

where r is the internuclear separation and ze the ionic charge. 

In calculating the electrostatic energy of a crystal, we must take into 
account not only the attraction between an ion and the oppositely charged 
ions coordinated around it, but also the repulsions between ions of like sign 
at somewhat larger separations, then attractions between the unlike ions 
once removed, and so on. Therefore, for each ion the electrostatic interaction 
will be a sum of terms, alternately attractive and repulsive, and diminishing 
in magnitude owing to the inverse-square law. For any given structure this 
summation amounts to little more than relating all the different internuclear 
distances to the smallest distance r. Thus, corresponding with eq. (13.8) for 
an ionic molecule, there is obtained for the potential energy of an ionic 
crystal per mole 

u, -<"?" + (,> 

The constant A, which depends on the type of crystal structure, is called the 
Madelung constant. 1 * If e is in esu and if is in kcal per mole, one has the 
following typical A values: NaCl structure, A ----- 1.74756; CsCl, 1.76267; 
rutile, 4.816. 

At the equilibrium internuclear distance r , the energy is a minimum, so 
that (dU/dr) ft 0. Hence for the case z l ----- z 2 , 

ANe 2 z 2 nBe 2 



ANz*t 

I AM 

(13.10) 
n 

The value of the exponent n in the repulsive term can be estimated from 
the compressibility of the crystal, since work is done against the repulsive 
forces in compressing the crystal. Typical values of n range from 6 to 12, 
indicative of the rapid rise in repulsion as the internuclear separation is 
narrowed. 

18 J. Sherman, Chem. Rev., 77, 93 (1932). 



406 CRYSTALS [Chap. 13 

The so-called crystal energy is obtained from eqs. (13.9) and (13.10) as 



This is the heat of reaction of gaseous ions to yield the solid crystal. For 
example, for rock salt: 

Na-(g)-| Cl-(g)- NaCl(c) f c 

Calculated values of E c can be compared with other thermochemical 
quantities by means of the Born-Haber cycle. For the typical case of NaCl, 
this has the form: 

NaCl (c) ------ E - c - -> Na *(g) + Cl" (g) 



. 

Na (c) + C1 2 (g) - -- - -> Na (g) 4 Cl (g) 

The energetic quantities entering into the cycle are defined as follows, all 
per mole: 

EC the crystal energy 

Q the standard heat of formation of crystalline NaCl 
S = heat of sublimation of metallic Na 
/ the ionization potential of Na 
A - the electron affinity of Cl 
D = the heat of dissociation of C1 2 (g) into atoms 
For the cyclic process, by the First Law of Thermodynamics: 

c - S f / f iD -A - Q (13.12) 

All the quantities on the right side of this equation can be evaluated, at 
least for alkali-halide crystals, and the value obtained for the crystal energy 
can be compared with that calculated from eq. (13.11). The ionization 
potentials / are obtained from atomic spectra, and the dissociation energies 
D can be accurately determined from molecular spectra. Most difficult to 
measure are the electron affinities A. 19 

A summary of the figures obtained for various crystals is given in Table 
13.7. When the calculated crystal energy deviates widely from that obtained 
through the Born-Haber cycle, one may suspect nonionic contributions to 
the crystal binding. 

24. Statistical thermodynamics of crystals: the Einstein model. If one 
could obtain an accurate partition function for a crystal, it would then be 
possible to calculate immediately all its thermodynamic properties by making 
use of the general formulas of Chapter 12. 

For one mole of a crystalline substance, containing N atoms, there are 

19 See, for example, P. P. Sutton and J. E. Mayer, J. Chem. Phys., 2, 146 (1934); 3, 20 
(1935). 



Sec. 24] 



CRYSTALS 



407 



TABLE 13.7 

THE BORN-HABER CYCLE 

(Energy Terms in Kilocalories per Mole) 



Crystal 


~Q 


/ 


5 


D 


A 


E e 


E c * 


NaCl 


99 


117 


26 


54 


88 


181 


190 


NaBr 


90 


117 


26 


46 


80 


176 


181 


Nal 


77 


117 


26 


34 


71 


166 


171 


KC1 


104 


99 


21 


54 


88 


163 


173 


KBr 


97 


99 


21 


46 


80 


160 


166 


Kl 


85 


99 


21 


34 


71 


151 


159 


RbCl 


105 


95 


20 


54 


88 


159 


166 


RbBr 


99 


95 


20 


46 


80 


157 


161 


Rbl 


87 


95 


20 


34 


71 


148 


154 



* Calculated, Eq. (13.11). 

3W degrees of freedom. Except when there is rotation of the atoms within 
the solid, we can consider that there are 3jV vibrational degrees of freedom, 
since 3W 6 is to all intents and purposes still 3N. The precise determina- 
tion of 3N normal modes of vibration for such a system would be an im- 
possible task, and it is fortunate that some quite simple approximations give 
sufficiently good answers. 

First of all, let us suppose that the 3N vibrations arise from independent 
oscillators, and then that these are harmonic oscillators, which is a good 
enough approximation at low temperatures, when the amplitudes are small. 
The model proposed by Einstein in 1906 assigned the same frequency v to 
all the oscillators. 

The crystalline partition function according to the Einstein model is, 
from eqs. ( 12.35) and (12.23), 



(13.14) 
(13.15) 
(13.16) 
(13.17) 



z = * 

It follows immediately that, 

E - E - 3Nhv(e hv/kT ~ I)" 1 

S=3m[ f *l kT --~\n(l e 



Cy ^= 



3yVA:rin(l 

/2V 



hv 



Particularly interesting is the predicted temperature variation of C v . We 
recall that Dulong and Petit, in 1819, noted that the molar heat capacities 
of the solid elements, especially the metals, were usually around 3R = 6 
calories per degree. Later measurements showed that this figure was merely 



408 CRYSTALS [Chap. 13 

a high-temperature limiting value, approached by different elements at 
different temperatures. 

If we expand the expression in eq. (13.17) and simplify somewhat, 20 we 
obtain 

______ (13 ig\ 

^ ' } 



When r is large, this expression reduces to C v 37*. For smaller T's, a 
curve like the dotted line in Fig. 13.28 is obtained, the heat capacity being 
a universal function of (v/T). The frequency v can be determined from one 
experimental point at low temperatures and then the entire heat-capacity 
curve can be drawn for the substance. The agreement with the experimental 
data is good except at the lowest temperatures. It is clear that the higher 
the fundamental vibration frequency v, the larger is the quantum of vibra- 
tional energy, and the higher the temperature at which C v attains the 
classical value of 3R. For example, the frequency for diamond is 2.78 x 
10 13 sec- 1 , but for lead it is only 0.19 x 10 13 sec" 1 , so that C v for diamond 
is only about 1.3 at room temperature, but C v for lead is 6.0. The elements 
that follow Dulong and Petit's rule are those with relatively low vibration 
frequencies. 

25. The Debye model. If, instead of a single fundamental frequency, a 
spectrum of vibration frequencies is taken for the crystal, the statistical 
problem becomes somewhat more complicated. One possibility is to assume 
that the frequencies are distributed according to the same law as that given 
on page 261 for the distribution of frequencies in black-body radiation. 
This problem was solved by P. Debye. 

Instead of using eq. (13.14), the energy must be obtained by averaging 
over all the possible vibration frequencies v t of the solid, from to V M the 
maximum frequency. This gives 



M hv 
~ ~~~' 3N o ***- 1 = i?o 7^-\ (13- 19) 

Since the frequencies form a virtual continuum the summation is replaced 
by an integration, by using the distribution function for the frequencies 
found in eq. (10.14) (multiplied by $ since we have one longitudinal and two 
transverse vibrations, instead of the two transverse of radiation). Thus 

dn ^f(v)dv = 1277- ^ r 2 dv (13.20) 

c^ 

where c is now the velocity of elastic waves in the crystal. Then eq. (13.19) 
becomes 

E ~ E "* "-^ dv (13 - 21) 



Recalling that cosech x = 2y(e" - -), and e* = 1 + x + (x/2!) + (*/3!) + . . . . 



Sec. 25] 



CRYSTALS 



409 



Before substituting eq. (13.20) in (13.21) we eliminate c by using eq. 
(10.14), since when n = 3N 9 v = v M9 for each direction of vibration, 

4n 3 a _ 47T 3 9N 2 



Then eq. (13.21) becomes 



Jo 



By differentiation with respect to T y 



C v = 



TV vV' 
Jo (e*"" 



lkT dv 



~r kT*v~* 
Let us set x = Hv/kT, whereupon eq. (13.23) becomes 



(13.22) 



(13.23) 



krv r 

v M 7 Jo 



v - o* (13 ' 24) 

The Debye theory predicts that the heat capacity of a solid as a function 
of temperature should depend only on the characteristic frequency V M . If 




Fig. 13.28. The molar heat capacity of solids. (After F. Seitz, The Modern 
Theory of Solids, McGraw-Hill, 1940.) 

the heat capacities of different solids are plotted against kTjhv M , they should 
fall on a single curve. Such a plot is shown in Fig. 13.28, and the confirma- 
tion of the theory appears to be very good. Debye has defined a characteristic 
temperature, & D hv M /k, and some of these characteristic temperatures are 
listed in Table 13.8 for various solids. The theory of Debye is really adequate 
for isotropic solids only, and further theoretical work will be necessary 



410 



CRYSTALS 



[Chap. 13 



before we have a comprehensive theory applicable to crystals with more 
complicated structures. 

TABLE 13.8 
DEBYE CHARACTERISTIC TEMPERATURES 



Substance 


OD 


Substance 


BD 


Substance 


0D 


Na 


159 


Be 


1000 


Al 


398 


K 

Cu 


100 
315 


Mg 
Ca 


290 
230 


Ti 
Pb 


350 
88 


Ag 
Au 


215 
180 


Zn 
Hg 


235 
96 


Pt 
Fe 


225 
420 


KC1 
NaCl 


227 
281 


AgCl 
AgBr 


183 
144 


CaF 2 
FeS 2 


474 
645 



The application of eq. (13.24) to the limiting cases of high and very low 
temperatures is of considerable interest. When the temperature becomes 
large, e hvlkT becomes small, and the equation may readily be shown to reduce 
to simply C v ~- 3/?, the Dulong and Petit expression. When the temperature 
becomes low, the integral may be expanded in a power series to show that 

C v - aT* (13.25) 

This r 3 law holds below about 30 K and is of great use in extrapolating 
heat-capacity data to absolute zero in connection with studies based on the 
Third Law of Thermodynamics (cf. page 90). 

PROBLEMS 

1. Show that a face-centered-cubic lattice can also be represented as a 
rhombohedral lattice. Calculate the rhombohedral angle a. 

2. To the points in a simple orthorhombic lattice add points at \ \ 0, 
\ \\ I.e., at the centers of a pair of opposite faces in each unit cell. Prove 
that the resulting arrangement of points in space is not a lattice. 

3. Prove that the spacing between successive planes (hkl) in a cubic 
lattice is a/Vh* f k 2 -f~ 7 2 where a is the side of the unit cell. 

4. The structure of fluorite, CaF 2 , is cubic with Z --- 4, a Q 5.45 A. The 
Ca++ ions are at the corners and face centers of the cube. The F~ ions are at 
(Hi, Hi, HI, Hi, *ft, if*. Hi. Hi)- Calculate the nearest 
distance of approach of Ca Ca, F F, Ca F. Sketch the arrangement of 
ions in the planes 100, 110, 111. 

5. MgO has the NaCl structure and a density of 3.65 g per cc. Calculate 
the values of (sin 0)/A at which scattering occurs from the planes 100, 110, 
111,210. 



Chap. 13] 



CRYSTALS 



411 



6. Nickel crystallizes in the fee structure with a Q 3.52 A. Calculate the 
distance apart of nickel atoms lying in the 100, 1 10, and 1 1 1 planes. 

7. A Debye-Scherrer powder picture of a cubic crystal with radiation of 
X -~ 1.539 A displayed lines at the following scattering angles: 





I 








j 


i 


No. of 'me 


1 


2 | 3 


4 5 

1 


6 


7 


8 


9 


0,deg 


13.70 


15.89 


22.75 


26.91 


28.25 


33.15 


37.00 


37.60 


41.95 


Intensity 


w 


vs 


s 


vw 


m 


w 


w 


m 


m 



Note: w weak; s strong; m medium; v very. 

Index these lines. Calculate a (} for the crystal. Identify the crystal. Explain 
the intensity relation between lines 5 and 4 in terms of the structure factor. 

8. Calculate the atomic volume for spheres of radius 1 A in ccp and hep 
structures. Give the unit cell dimensions, a Q for cubic, a Q and r for hexagonal. 

9. Show that the void volume for spheres in both ccp and hep is 25.9 per 
cent. What would be the per cent void in a bcc structure with corner atoms 
in contact with the central atom? 

10. White tin is tetragonal with a (} b Q 5.819 A, and c () 3.175 A. 
Tin atoms are at 000, i J , i j, i |. Calculate the density of the crystal. 
Grey tin has the diamond structure with a {} -- 6.46 A. Describe how the tin 
atoms must move in the transformation from grey to white tin. 

11. In a powder picture of lead with Cu K a radiation (X 1.539 A) the 
line from the 531 planes appeared at sin 0.9210. Calculate a and the 
density of lead. 

12. The Debye characteristic temperature of copper is ( H ) =-- 315 U K. Cal- 
culate the entropy of copper at 0C and 1 atm assuming that a 4.95 x 
10~ 5 deg" 1 , Po 7.5 x 10 7 atm l , independent of the temperature. 

13. Calculate the proton affinity of NH 3 from the following data (i.e., the 
A for reaction NH 3 + H f NH 4 f ). NH 4 F crystallizes in the ZnO type 
structure whose Madelung constant is 1.64. The Born repulsion exponent 
for NH 4 F is 8, the interionic distance is 2.63 A. The electron affinity of 
fluorine is 95.0 kcal. The ionization potential of hydrogen is 31 1.9 kcal. The 
heats of formation from the atoms are: NH 3 279.6; N 2 --= 225; H 2 - 
104.1 ; F 2 - 63.5 kcal. The heat of reaction \ N 2 (g) | 2 H 2 (g) -f J F 2 (g) - 
NH 4 +F~(c)is 11 1.9 kcal. 

REFERENCES 

BOOKS 

1. Barrett, C. S., Structure of Metals (New York: McGraw-Hill, 1952). 

2. Bragg, W. H., and W. L. Bragg, The Crystalline State, vol. I (London: 
Bell, 1934). 

3. Buerger, M. J., X-Ray Crystallography (New York: Wiley, 1942). 



412 CRYSTALS [Chap. 13 

4. Bunn, C. W., Chemical Crystallography (New York: Oxford, 1945). 

5. Evans, R. C., Crystal Chemistry (London: Cambridge, 1939). 

6. Hume-Rothery, W., Atomic Theory for Students of Metallurgy (London: 
Institute of Metals, 1947). 

7. Kittel, C., Introduction to Solid-State Physics (New York: Wiley, 1953). 

8. Lonsdale, K., Crystals and X-Rays (New York: Van Nostrand, 1949). 

9. Phillips, F. C., An Introduction to Crystallography (New York: Long- 
mans, 1946). 

10. Wells, A. F., Structural Inorganic Chemistry (New York: Oxford, 1950). 

11. Wilson, A. H., Semiconductors and Metals (London: Cambridge, 1939). 

12. Wooster, W. A., Crystal Physics (London: Cambridge, 1938). 

ARTICLES 

1. Bernal, J. D., /. Chem. Soc., 643-66 (1946), "The Past and Future of 
X-Ray Crystallography." 

2. DuBridge, L. A., Am. J. Phys., 16, 191-98 (1948), "Electron Emission 
from Metal Surfaces." 

3. Frank, F. C., Adv. Phys., /, 91-109 (1952), "Crystal Growth and Disloca- 
tions." 

4. Fuoss, R. M., J. Chem. Ed., 19, 190-93, 231-35 (1942), "Electrical Pro- 
perties of Solids." 

5. Lonsdale, K., Endeavour, 6, 139-46 (1947), "X-Rays and the Carbon 
Atom." 

6. Robertson, J. M., /. Chem. Soc., 249-57 (1945), "Diffraction Methods in 
Modern Structural Chemistry." 

7. Sidhu, S. S., Am. J. Phys. 9 16, 199-205 (1948), "Structure of Cubic 
Crystals." 

8. Smoluchowski, R., and J. S. Koehler, Ann. Rev. Phys. Chem., 2, 187-216 
(1951), "Band Theory and Crystal Structure." 

9. Weisskopf, V. F., Am. J. Phys., 11, 111-12 (1943), "Theory of the Elec- 
trical Resistance of Metals." 



CHAPTER 14 

Liquids 



1. The liquid state. The crystalline and the gaseous states of matter have 
already been surveyed in some detail. The liquid state remains to be con- 
sidered. Not that every substance falls neatly into one of these three classifi- 
cations there is a variety of intermediate forms well calculated to perplex 
the morphologist : rubbers and resins, glasses and liquid crystals, fibers and 
protoplasm. 

Gases, at least in the ideal approximation approached at high tempera- 
tures and low densities, are characterized by complete randomness on the 
molecular scale. The ideal crystal, on the other hand, is one of nature's most 
orderly arrangements. Because the extremes of perfect chaos and perfect 
harmony are both relatively simple to treat mathematically, the theory of 
gases and crystals is at a respectably advanced stage. Liquids, however, 
representing a peculiar compromise between order and disorder, have so far 
defied a comprehensive theoretical treatment. 

Thus in an ideal gas, the molecules move independently of one another 
and interactions between them are neglected. The energy of the perfect gas 
is simply the sum of the energies of the individual molecules, their internal 
energies plus their translational kinetic energies; there is no intermolecular 
potential energy. It is therefore possible to write down a partition function 
such as that in eq. (12.23), from which all the equilibrium properties of the 
gas are readily derived. 

In a crystalline solid, translational kinetic energy is usually negligible. 
The molecules, atoms, or ions vibrate about equilibrium positions to which 
they are held by strong intermolecular, interatomic, or interionic forces. In 
this case too, an adequate partition function, such as that in eq. (13.13), can 
be obtained. 

In a liquid, on the other hand, the situation is much harder to define. 
The cohesive forces are sufficiently strong to lead to a condensed state, but 
not strong enough to prevent a considerable translational energy of the in- 
dividual molecules. The thermal motions introduce a disorder into the liquid 
without completely destroying the regularity of its structure. It has therefore 
not yet been possible to devise an acceptable partition function for liquids. 

It should be mentioned that in certain circles it is now considered in- 
delicate to speak of individual molecules in condensed systems, such as 
liquids or solids. As James Kendall once put it, we may choose to imagine 
that "the whole ocean consists of one loose molecule and the removal of a 
fish from it is a dissociation process." 

413 

p 



414 LIQUIDS [Chap. 14 

In studying liquids, it is often helpful to recall the relation between 
entropy and degree of disorder. Consider a crystal at its melting point. The 
crystal is energetically a more favorable structure than the liquid to which 
it melts. It is necessary to add energy, the latent heat of fusion, to effect the 
melting. The equilibrium situation, however, is determined by the free-energy 
difference, AF = A// JAS. It is the greater randomness of the liquid, 
and hence its greater entropy, that finally makes the T&S term large enough 
to overcome the A// term, so that the crystal melts when the following 
condition is reached : 



The sharpness of the melting point is noteworthy. There does not in 
general appear to be a continuous gradation of properties between liquid 




CRYSTAL LIQUID GAS 

Fig. 14.1. Two-dimensional models. 

and crystal. The sharp transition is due to the extremely rigorous geometrical 
requirements that must be fulfilled by a crystal structure. It is not possible 
to introduce small regions of disorder into the crystal without at the same 
time seriously disturbing the structure over such a long range that the 
crystalline arrangment is destroyed. Two-dimensional models of the gaseous, 
liquid, and crystalline states are illustrated in Fig. 14.1. The picture of the 
liquid was constructed by J. D. Bernal by introducing around "atom" A 
only five other atoms instead of its normal close-packed coordination of six. 
Every effort was then made to draw the rest of the circles in the most ordered 
arrangement possible, with the results shown. The one point of abnormal 
coordination among some hundred atoms sufficed to produce the long-range 
disorder believed to be typical of the liquid state. We see that if there is to 
be any abnormal coordination at all, there has to be quite a lot of it. Herein 
probably lies an explanation of the sharpness of melting. When the thermal 
motions in one region of a crystal suffice to destroy the regular structure, the 
irregularity rapidly spreads throughout the entire specimen; thus disorder in 
a crystal may be contagious. 

These remarks should not be taken to imply that all crystals are ideally 
perfect, and admit of no disorder at all. It is only that the amount of disorder 
allowed is usually very limited. When the limit is exceeded, complete melting 
of the crystal occurs. There are two types of defect that occur in crystal 
structures. There may be vacant lattice positions or "holes," and there may 
be interstitial positions occupied by atoms or ions. 



Sec. 2] LIQUIDS 415 

It is sometimes convenient to classify liquids, like crystals, from a rather 
chemical standpoint, according to the kind of cohesive forces that hold them 
together. Thus there are the ionic liquids such as molten salts, the liquid 
metals consisting of metal ions and fairly mobile electrons, liquids such as 
water held together mainly by hydrogen bonds, and finally molecular liquids 
in which the cohesion is due to the van der Waals forces between essentially 
saturated molecules. Many liquids fall into this last group, and even when 
other forces are present, the van der Waals contribution may be large. The 
nature of these forces will be considered later in this chapter. 

2. Approaches to a theory for liquids. From these introductory remarks 
it may be evident that there are three possible ways of essaying a theory of 
the liquid state, two cautious ways and one direct way. 

The cautious approaches are by way of the theory of gases and the 
theory of solids. The liquid may be studied as an extremely imperfect gas. 
This is a reasonable viewpoint, since above the critical point there is no 
distinction at all between liquid and gas, and the so-called "fluid state" of 
matter exists. On the other hand, the liquid may be considered as similar to 
a crystal, except that the well-ordered arrangement of units extends over 
a short range only, five or six molecular diameters, instead of over the whole 
specimen. This is sometimes called "short-range order and long-range dis- 
order." This is a reasonable viewpoint, since close to the melting point the 
density of crystal and liquid are very similar; the solid usually expands about 
10 per cent in volume, or only about 3 per cent in intermodular spacing, 
when it melts. It should be realized too that whatever order exists in a liquid 
structure is continuously changing because of thermal motions of the in- 
dividual molecules; it is the time average of a large number of different 
arrangements that is reflected in the liquid properties. 

The imperfect-gas theory of liquids would be suitable close to the critical 
point; the disordered-crystal theory would be best near the melting point. 
At points between, they might both fail badly. A more direct approach to 
liquids would abandon these flanking attacks and try to develop the theory 
directly from the fundamentals of intermolecular forces and statistical 
mechanics. This is a very difficult undertaking, but a beginning has been 
made by Max Born, J. G. Kirkwood, and others. 

We shall consider first some of the resemblances between liquid and 
crystal structures, as revealed by the methods of X-ray diffraction. 

3. X-ray diffraction of liquids. The study of the X-ray diffraction of 
liquids followed the development of the method of Debye and Scherrer for 
powdered crystals. As the particle size of the powder decreases, the width 
of the lines in the X-ray pattern gradually increases. From particles around 
100 A in diameter, the lines have become diffuse halos, and with still further 
decrease in particle size the diffraction maxima become blurred out altogether. 

If a liquid were completely amorphous, i.e., without any regularity of 
structure, it should also give a continuous scattering of X rays without 



416 



LIQUIDS 



[Chap. 14 



maxima or minima. This was actually not found to be the case. A typical 
pattern, that obtained from liquid mercury, is shown in (a), Fig. 14.2, as a 
microphotometer tracing of the photograph. This reveals the maxima and 
minima better than the unaided eye. One or two or sometimes more intensity 
maxima appear, whose positions often correspond closely to some of the 
larger interplanar spacings that occur in the crystalline structures. In the 
case of the metals, these are the close-packed structures. It is interesting that 
a crystal like bismuth, which has a peculiar and rather loose solid structure, 

3i 



t 2 




V 5 10 15 20 25 3.0 

(d) DISTANCE FROM CENTRAL SPOT ON FILM-Cm 




5 



10 II T2 



Fig. 14.2. 



4 5 6 7 8 9 
(b) f- ANGSTROM UNITS 

(a) Photometric tracing of liquid-mercury picture; (b) radial 
distribution function for liquid mercury. 



is transformed on melting into a close-packed structure. We recall that 
bismuth is one of the few substances that contract in volume when melted. 

The fact that only a few maxima are observed in the diffraction patterns 
from liquids is in accord with the picture of short-range order and increasing 
disorder at longer range. In order to obtain the maxima corresponding to 
smaller interplanar spacings or higher orders of diffraction, the long-range 
order of the crystal must be present. 

The diffraction maxima observed with crystals or liquids should be dis- 
tinguished from those obtained by the X-ray or electron diffraction of gases. 
The latter arise from the fixed positions of the atoms within the molecules. 
The individual molecules are far apart and distributed at random. In deriving 
on page 327 the diffraction formula for gases, we considered only a single 
molecule and averaged over all possible orientations in space. With both 
solids and liquids the diffraction maxima arise from the ordered arrangement 



Sec. 4] LIQUIDS 417 

of the units (molecules or atoms) in the condensed three-dimensional struc- 
ture. Thus gaseous argon, a monatomic gas, would yield no maxima, but 
liquid argon displays a pattern similar to that of liquid mercury. 

It is possible to analyze the X-ray diffraction data from liquids by using 
the Bragg relation to calculate spacings. A more instructive approach, how- 
ever, is to consider a liquid specimen as a single giant molecule, and then to 
use the formulas, such as eq. (11.19), derived for diffraction by single 
molecules. A simple theory is obtained only in the case of monatomic liquids, 
such as the metals and group O elements. 

The arrangement of atoms in such a liquid is described by introducing 
the radial distribution function g(r). Taking the center of one atom as origin, 
this g(r) gives the probability of finding the center of another atom at the 
end of a vector of length r drawn from the origin. The chance of finding 
another atom between a distance r and r \ dr, irrespective of angular 
orientation, is therefore 47rr 2 g(r)dr (cf. page 187). It is now possible to 
obtain, for the intensity of scattered X radiation, an expression similar to 
that in eq. (11.19), except that instead of a summation over individual 
scattering centers, there is an integration over a continuous distribution of 
scattering matter, specified by g(r). Thus 



1(0) 



f 4rrr Wr) ^ dr (14.1) 

Jo fir 



sin I - I 

\^/ 



As before, /^ = 

* A 

By an application of Fourier's integral theorem, this integral can be 
inverted, 1 yielding 



By use of this relationship it is possible to calculate a radial-distribution 
curve, such as that plotted in (b), Fig. 14.2, from an experimental scattering 
curve, such as that in (a), Fig. 14.2. The regular coordination in the close- 
packed liquid-mercury structure is clearly evident, but the fact that maxima 
in the curve are rapidly damped out at larger interatomic distances indicates 
that the departure from the ordered arrangement becomes greater and 
greater as one travels outward from any centrally chosen atom. 

4. Results of liquid-structure investigations. X-ray diffraction data from 
liquids are not sufficiently detailed to permit complete structure analyses like 
those of crystals. This situation is probably inevitable because the diffraction 
experiments reveal only an average or statistical structure, owing to the 
continual destruction and reformation of ordered arrangements by the 
thermal motions of the atoms or molecules in the liquid. 

One view, however, proposed by G. W. Stewart (around 1930), is that 

1 See, for example, H. Bateman, Partial Differential Equations of Mathematical Physics 
(New York: Dover Publications, 1944), p. 207. 



418 LIQUIDS [Chap. 14 

there are actually large regions in a liquid that are extremely well ordered. 
These are called cybotactic groups, and are supposed to contain up to 
several hundred molecules. These islands of order are dispersed in a sea of 
almost completely disordered molecules, whose behavior is essentially that 
of a very dense gas. There is a dynamic equilibrium between the cybotactic 
groups and the unattached molecules. This picturesque model is probably 
unsuitable for the majority of liquids and it is usually preferable to think of 
the disorder as being fairly well averaged throughout the whole structure. 

The results with liquid metals have already been mentioned. They appear 
to have approximately close-packed structures quite similar to those of the 
solids, with the interatomic spacings expanded by about 5 per cent. The 
number of nearest neighbors in a close-packed structure is twelve. In liquid 
sodium, each atom is found to have on the average ten nearest neighbors. 

One of the most interesting liquid structures is that of water. J. Morgan 
and B. E. Warren 2 have extended and clarified an earlier discussion by 
Bernal and Fowler. They studied the X-ray diffraction of water over a range 
of temperatures, and obtained the radial distribution curves. 

The maximum of the large first peak occurs at a distance varying from 
about 2.88 A at 1.5C to slightly over 3.00 A at 83C. The closest spacing 
in ice is at 2.76 A. It might at first be thought that this result is in disagree- 
ment with the fact that there is a contraction in volume of about 9 per cent 
when ice melts. Further analysis shows, however, that the coordination in 
liquid water is not exactly the same as the tetrahedral coordination of four 
nearest neighbors in ice. The number of nearest neighbors can be estimated 
from the area under the peaks in the radial-distribution curve, with the 
following results: 



Temperature, C: 
Number nearest neighbors: 



1.5 
4.4 



13 

4.4 



30 
4.6 



62 
4.9 



83 
4.9 



Thus the tetrahedral arrangement in ice is partially broken down in water, 
to an extent that increases with temperature. This breakdown permits closer 
packing, although water is of course far from being a closest-packed struc- 
ture. The combination of this effect with the usual increase of intermolecular 
separation with temperature explains the occurrence of the maximum in the 
density of water at 4C. 

Among other structures that have been investigated, those of the long- 
chain hydrocarbons may be mentioned. These molecules tend to pack with 
parallel orientations of the chains, sometimes suggesting an approach to 
Stewart's cybotactic models. 

5. Liquid crystals. In some substances the tendency toward an ordered 
arrangement is so great that the crystalline form does not melt directly to a 

2 J. Chem. Phys., 6, 666 (1938). This paper is recommended as a clear and excellent 
example of the X-ray method as applied to liquids. 



Sec. 5] 



LIQUIDS 



419 



liquid phase at all, but first passes through an intermediate stage (the meso- 
morphic or paracrystalline state), which at a higher temperature undergoes 
a transition to the liquid state. These intermediate states have been called 
liquid crystals, since they display some of the properties of each of the 
adjacent states. Thus some paracrystalline substances flow quite freely but 




(a) 



I I Hill 1 1111 II II III! 



(b) 





(c) (d) 

Fig. 14.3. Degrees of order: (a) crystalline orientation and periodicity; 
(b) smectic orientation and arrangement in equispaced planes, but no periodicity 
within planes; (c) nematic orientation without periodicity; (d) isotropic fluid 
neither orientation nor periodicity. 

are not isotropic, exhibiting interference figures when examined with polar- 
ized light ; other varieties flow in a gliding stepwise fashion and form "graded 
droplets" having terracelike surfaces. 

A compound frequently studied in its paracrystalline state is />-azoxy- 
anisole, 

O 





OCH a 



CH 



The solid form melts at 84 to the liquid crystal, which is stable to 150 at 
which point it undergoes a transition to an isotropic liquid. The compound 
ethyl /7-anisalaminocinnamate, 



:H=N/ \ CH=C 



P/ \_C H=N / \_ CH=CH COOC 2 H 5 



420 LIQUIDS [Chap. 14 

passes through three distinct paracrystalline phases between 83 and 139. 
Cholesteryl bromide behaves rather differently. 3 The solid melts at 94 to 
an isotropic liquid, but this liquid can be supercooled to 67 where it passes 
over into a metastable liquid-crystalline form. 

Liquid crystals tend to occur in compounds whose molecules are 
markedly unsymmetrical in shape. For example, in the crystalline state 
long-chain molecules may be lined up as shown in (a), Fig. 14.3. On raising 
the temperature, the kinetic energy may become sufficient to disrupt the 
binding between the ends of the molecules but insufficient to overcome the 
strong lateral attractions between the long chains. Two types of anisotropic 
melt might then be obtained, shown in (b) and (c), Fig. 14.3. In the smectic 
(ojurjypQL, "soap") state the molecules are oriented in well-defined planes. 
When a stress is applied, one plane glides over another. In the nematic 
(*>/7//a, "thread") state the planar structure is lost, but the orientation is pre- 
served. With some substances, notably the soaps, several different phases, 
differentiated by optical and flow properties, can be distinguished between 
typical crystal and typical liquid. 

It has been suggested that many of the secrets of living substances may 
be elucidated when we know more about the liquid-crystalline state. Joseph 
Needham 4 has written : 

Liquid crystals, it is to be noted, are not important for biology and embryology 
because they manifest certain properties which can be regarded as analogous to 
those which living systems manifest (models), but because living systems actually 
are liquid crystals, or, it would be more correct to say, the paracrystalline state 
undoubtedly exists in living cells. The doubly refracting portions of the striated 
muscle fibre are, of course, the classical instance of this arrangement, but there are 
many other equally striking instances, such as cephalopod spermatozoa, or the 
axons of nerve cells, or cilia, or birefringent phases in molluscan eggs, or in nucleus 
and cytoplasm of echinoderm eggs. . . . 

The paracrystalline state seems the most suited to biological functions, as it 
combines the fluidity and diffusibility of liquids while preserving the possibilities of 
internal structure characteristic of crystalline solids. 

6. Rubbers. Natural rubber is a polymerized isoprene,' with long hydro- 
carbon chains of the following structure: 

( CH 2 CH CH 2 CH 2 ) n 

I 
CH 3 

The various synthetic rubbers are al$o long, linear polymers, with similar 
structures. The elasticity of rubber is a consequence of the different degrees 
of ordering of these chains in the stretched and unstretched states. An 
idealized model of the rubber chains when stretched and when contracted 

3 J. Fischer, Zeit. physik. Chern., 160A, 110 (1932). 

4 Joseph Needham, Biochemistry and Morphogenesis (London: Cambridge, 1942), p. 661. 



Sec. 6] 



LIQUIDS 



421 



is shown in Fig. 14.4. Stretching forces the randomly oriented chains into a 
much more ordered alignment. The unstretched, disordered configuration is 
a state of greater entropy, and if the tension is released, the stretched rubber 
spontaneously reverts to the unstretched condition. 

Robert Boyle and his contemporaries talked about the "elasticity of a 
gas," and although we hear this term infrequently today, it is interesting to 
noie that the thermodynamic interpretations of the elasticity of a gas and of 
the elasticity of a rubber band are in fact the same. If the pressure is released 
on a piston that holds gas in a cylinder, the piston springs back as the gas 
expands. The expanded gas is in a state of higher entropy than the com- 




(o) 



(b) 



Fig. 14.4. Idealized models of chains in rubber: 
(a) stretched; (b) contracted. 

pressed gas: it is in a more disordered state since each molecule has a larger 
volume in which to move. Hence the compressed gas spontaneously expands 
for the same reason that the stretched rubber band spontaneously contracts. 
From eq. (6) on page 65, the pressure is 



(14.3) 



For a gas, the (dE/3V) T term is small, so that effectively P = T(dS/dV) T , 
and the pressure varies directly with r, and is determined by the change in 
entropy with the volume. The analog of eq. (14.3) for a rubber band of 
length L in which the tension is K is 



(14.4) 



It was found experimentally that K varies directly with T, so that, just as in 
the case of an ideal gas, the term involving the energy must be negligible. 
It was this observation that first led to the interpretation of rubber elasticity 
as an entropy effect. 



422 LIQUIDS [Chap. 14 

7. Glasses. The glassy or vitreous state of matter is another example of 
a compromise between crystalline and liquid properties. The structure of a 
glass is essentially similar to that of an associated liquid such as water, so 
that there is a good deal of truth in the old description of glasses as super- 
cooled liquids. The two-dimensional models in Fig. 14.5, given by W. H. 
Zachariasen, illustrate the differences between a glass and a crystal. 

The bonds are the same in both cases, e.g., in silica the strong electro- 
static Si O bonds. Thus both quartz crystals and vitreous silica are hard 
and mechanically strong. The bonds in the glass differ considerably in length 
and therefore in strength. Thus a glass on heating softens gradually rather 




(o) (b) 

Fig. 14.5. Two-dimensional models for (a) crystal and (b) glass. 

than melts sharply, since there is no one temperature at which all the bonds 
become loosened simultaneously. 

The extremely low coefficient of thermal expansion of some glasses, 
notably vitreous silica, is explicable in terms of a structure such as that in 
Fig. 14.5. The structure is a very loose one, and just as in the previously 
discussed case of liquid water, increasing the temperature may allow a closer 
coordination. To a certain extent, therefore, the structure may "expand into 
itself." This effect counteracts the normal expansion in interatomic distance 
with temperature. 

8. Melting. In Table 14.1 are collected some data on the melting point, 
latent heat of fusion, latent heat of vaporization, and entropies of fusion 
and vaporization of a number of substances. 

It will be noted that the heats of fusion are much less than the heats of 
vaporization. It requires much less energy to convert a crystal to liquid than 
to vaporize a liquid. 

The entropies of fusion are also considerably lower than the entropies of 
vaporization. The latter are quite constant, around 21.6 eu (Trouton's rule). 
The constancy of the former is not so marked. For some classes of sub- 
stances, however, notably the close-packed metals, the entropies of fusion 
are seen to be remarkably constant. 

9. Cohesion of liquids the internal pressure. We have so far been dis- 
cussing the properties of liquids principally from the disordered-crystai 
point of view. Whatever the model chosen for the liquid state, the cohesive 



Sec. 9] 



LIQUIDS 



423 



TABLE 14.1 
DATA ON MELTING AND VAPORIZATION 



ince 


Heat of 
Fusion 
(kcal/mole) 


Heat of 
Vaporization 
(kcal/mole) 


Melting 
Point 
(K) 


Entropy of 
Fusion 
(cal/) 


Entropy of 
Vaporiza- 
tion 
(cain 



Metals 



Na 
Al 
K 
Fe 

Ag 
Pt 

Hg 





0.63 


24.6 


371 


1.70 


21.1 




2.55 


67.6 


932 


2.73 


29.0 




0.58 


21.9 


336 


1.72 


21.0 




3.56 


96.5 


1802 


1.97 


29.5 




2.70 


69.4 


1234 


2.19 


27.8 




5.33 


125 


2028 


2.63 


26.7 




0.58 


15.5 


234 


2.48 


24.5 



Ionic Crystals 



NaCl 


7.22 


183 


1073 


KC1 


6.41 


165 


1043 


AgCl 


3.15 





728 


KN0 3 


2.57 





581 


BaCl 2 


5.75 





1232 


K 2 Cr 2 O 7 


.77 





671 



6.72 
6.15 
4.33 
4.42 
4.65 
13.07 



109 
93 



Molecular Crystals 



H 2 

H 2 

A 

NH 3 

C 2 H 5 OH 



0.028 

1.43 

0.280 

1.84 

1.10 

2.35 



0.22 
11.3 

1.88 

7.14 
10.4 

8.3 



14 


2.0 


15.8 


273 


5.25 


30.1 


83 


3.38 


21.6 


198 


9.30 


29.7 


156 


7.10 


29.6 


278 


8.45 


23.5 



forces are of primary importance. Ignoring, for the time being, the origin of 
these forces, we can obtain an estimate of their magnitude from thermo- 
dynamic considerations. This estimate is provided by the so-called internal 
pressure. 

We recall from eq. (3.43) that 



v- = T 



- 

dT 



p 



(14.5) 



In the case of an ideal gas, the internal pressure term />, = (5Ej^V) T is 
zero since intermolecular forces are absent. In the case of an imperfect gas, 
the (dE/dV) T term becomes appreciable, and in the case of a liquid it may 
become much greater than the external pressure. 

The internal pressure is the resultant of the forces of attraction and the 
forces of repulsion between the molecules in a liquid. It therefore depends 



424 



LIQUIDS 



[Chap. 14 



markedly on the volume K, and thus on the external pressure P. This effect 
is shown in the following data for diethyl ether at 25C. 



P(atm): 200 
P<(atm): 2790 



800 
2840 



2000 
2530 



5300 
2020 



7260 
40 



9200 
-1590 



11,100 
-4380 



For moderate increases in P, the P t decreases only slightly, but as P exceeds 
5000 atm, the P t begins to decrease rapidly, and goes to large negative values 
as the liquid is further compressed. This behavior is a reflection, on a larger 
scale, of the law of force between individual molecules that was illustrated 
in Fig. 7.8. on page 181. 

Internal pressures at 1 atm and 25C are summarized in Table 14.2, 
taken from a compilation by J. H. Hildebrand. With normal aliphatic hydro- 
carbons there appears to be a gradual increase in P t with the length of the 
chain. Dipolar liquids tend to have somewhat larger values than nonpolar 
liquids. The effect of dipole interaction is nevertheless not predominant. As 
might be expected, water with its strong hydrogen bonds has an exceptionally 
high internal pressure. 

TABLE 14.2 

INTERNAL PRESSURES OF LIQUIDS 
(25C and 1 atm) 



Compound 



Diethyl ether 
/^-Heptane 
/i-Octane 










Tin tetrachloride 










Carbon tetrachlork 


e 








Benzene 










Chloroform . 










Carbon bisulfide 










Mercury 
Water . 











Pi atm 

2,370 
2,510 
2,970 
3,240 
3,310 
3,640 
3,660 
3,670 
13,200 
20,000 



Hildebrand was the first to point out the significance of the internal 
pressures of liquids in determining solubility relationships. If two liquids 
have about the same P t , their solution has little tendency toward positive 
deviations from Raoult's Law. The solution of two liquids differing con- 
siderably in P t will usually exhibit considerable positive deviation from 
ideality, i.e., a tendency toward lowered mutual solubility. Negative devia- 
tions from ideality are still ascribed to incipient compound or complex 
formation, 

10. Intel-molecular forces. It should be clearly understood from earlier 
discussions (cf. Chapter 11) that all the forces between atoms and molecules 
are electrostatic in origin. They are ultimately based on Coulomb's Law of 
the attraction between unlike, and the repulsion between like charges. One 
often speaks of long-range forces and short-range forces. Thus a force that 



Sec. 10] LIQUIDS 425 

depends on 1/r 2 will be effective over a longer range than one dependent on 
1/r 7 . All these forces may be represented as the gradient of a potential func- 
tion,/ = 3//3r, and it is often convenient to describe the potential energies 
rather than the forces themselves (See Fig. 7.8, page 181.) The following 
varieties of intermolecular and interionic potential energies may then be 
distinguished: 

(1) The coulombic energy of interaction between ions with net charges, 
leading to a long-range attraction, with U ~ r^ 1 . 

(2) The energy of interaction between permanent dipoles, with U ~ r~ 6 . 

(3) The energy of interaction between an ion and a dipole induced by it 
in another molecule, with U ~ r~ 4 . 

(4) The energy of interaction between a permanent dipole and a dipole 
induced by it in another molecule, with U ~ r~ 6 . 

(5) The forces between neutral atoms or molecules, such as the inert 
gases, with U ~ r~ B . 

(6) The overlap energy arising from the interaction of the positive nuclei 
and electron cloud of one molecule with those of another. The overlap leads 
to repulsion at very close intermolecular separations, with an r~ 9 to r~ 12 
potential. 

The van der Waais attractions between molecules must arise from inter- 
actions belonging to classes (2), (4), and (5). 

The first attempt to explain them theoretically was that of W. H. Keesom 
(1912), based on the interaction between permanent dipoles. Two dipoles in 
rapid thermal motion may sometimes be oriented so as to attract each other, 
sometimes so as to repel each other. On the average they are somewhat closer 
together in attractive configurations, and there is a net attractive energy. 
This energy was calculated 5 to be 

,-- 

where ^ is the dipole moment. The observed r~ 6 dependence of the interaction 
energy, or r~ 7 dependence of the forces, is in agreement with deductions from 
experiment. This theory is of course not an adequate general explanation of 
van der Waals' forces, since there are considerable attractive forces between 
molecules, such as the inert gases, with no vestige of a permanent dipole 
moment. 

Debye, in 1920, extended the dipole theory to take into account the 
induction effect. A permanent dipole induces a dipole in another molecule 
and a mutual attraction results. This interaction depends on the polariza- 
bility a of the molecules, and leads to a formula, 



U U = - (14.7) 

5 J. E. Lennard-Jones, Proc. Phys. Soc. (London), 43, 461 (1931). 



426 LIQUIDS [Chap. 14 

This effect is quite small and does not help us to explain the case of the inert 
gases. 

In 1930, F. London solved this problem by a brilliant application of 
quantum mechanics. Let us consider a neutral molecule, such as argon. The 
positive nucleus is surrounded by a cloud 6 of negative charge. Although the 
time average of this charge distribution is spherically symmetrical, at any 
instant the distribution will be somewhat distorted. (This may be visualized 
very clearly in the case of the neutral hydrogen atom, in which the electron 
is sometimes on one side of the proton, sometimes on the other.) Thus a 
"snapshot" taken of an argon atom would reveal a little dipole with a certain 
orientation. An instant later the orientation would be different, and so on, 
so that over any macroscopic period of time the instantaneous dipole 
moments would average to zero. 

Now it should not be thought that these little snapshot dipoles interact 
with those of other molecules to produce an attractive potential. This cannot 
happen since there will be repulsion just as often as attraction; there is no 
time for the instantaneous dipoles to line up with one another. There is, 
however, a snapshot-dipole polarization interaction. Each instantaneous 
argon dipole induces an appropriately oriented dipole moment in neighboring 
atoms, and these moments interact with the original to produce an in- 
stantaneous attraction. The polarizing field, traveling with the speed of light, 
does not take long to traverse the short distances between molecules. Cal- 
culations show that this dispersion interaction leads to a potential, 

U m -- -IK^ (14.8) 

where V Q is the characteristic frequency of oscillation of the charge dis- 
tribution. 7 

The magnitudes of the contributions from the orientation, induction, and 
dispersion effects are shown in Table 14.3 for a number of simple molecules. 

It is noteworthy that all the contributions to the potential energy of inter- 
molecular attraction display an r~ 6 dependence. The complete expression for 
the inter molecular energy must include also a repulsive term, the overlap 
energy, which becomes appreciable at very close distances. Thus we may 
write 

U=- ~Ar - + Br~ n (14.9) 

The value of the exponent n is from 9 to 12. 

11. Equation of state and intermolecular forces. The calculation of the 
equation of state of a substance from a knowledge of the intermolecular 

6 At least a "probability cloud" see p. 276. 

7 The r~* dependence of the potential can be readily derived in this case from electro- 
static theory. The field due to a dipole varies as 1/r 3 , and the potential energy of an induced 
dipole in a field F is fcaF 2 . See Harnwell, Electricity and Magnetism, p. 59. 

For a simple quantum-mechanical derivation of eq. (14.8), see R. H. Fowler, Statistical 
Mechanics (London: Cambridge, 1936), p. 296. 



Sec. ii] LIQUIDS 

TABLE 14.3 
RELATIVE MAGNITUDES OF INTERMOLECULAR INTERACTIONS* 



427 



Molecule 


Dipole 
Moment 
/ x 10 18 


Polariz- 
ability 
a x 10 24 


Energy 
hv (ev) 


Orienta- 
tionf 


Induo 
tionf 


Disper- 
sionf 




(esu cm) 


(cc) 




/ 


;x 


|/IVo 


CO 


0.12 


1.99 


14.3 


0.0034 


0.057 


67.5 


HI 


0.38 


5.4 


12 


0.35 


1.68 


382 


HBr 


0.78 


3.58 


13.3 


6.2 


4.05 


176 


HC1 


1.03 


2.63 


13.7 


18.6 


5.4 


105 


NH 3 
HoO 
He 


1.5 
1.84 



2.21 
1.48 
0.20 


16 
18 

24.5 


84 
190 



10 
10 



93 
47 
1.2 


A 





1.63 


15.4 








52 


Xe 





4.00 


11.5 








217 



* J. A. V. Butler, Ann. Rep. Chem. Soc. (London), 34, 75 (1937). 
| Units of erg cm x 10 60 . 

forces is in general a problem of great complexity. The method of attack 
may be outlined in principle, but so far the mathematical difficulties have 
proved so formidable that in practice a solution has been obtained only for 
a few drastically simplified cases. 

We recall that the calculation of the equation of state reduces to cal- 
culating the partition function Z for the system. From Z the Helmholtz free 
energy A is immediately derivable, and hence the pressure, P --- ~(dA/dV) T . 

To determine the partition function, Z = S e ~E i /kT^ ^ Q energy levels 
of the system must be known. In the cases of ideal gases and crystals it is 
possible to use energy levels for individual constituents of the system, such 
as molecules or oscillators, ignoring interactions between them. In the case 
of liquids, this is not possible since it is precisely the interaction between 
different molecules that is responsible for the characteristic properties of a 
liquid. It would therefore be necessary to know the energy levels of the 
system as a whole, for example, one mole of liquid. So far this problem has 
not been solved. 

An indication of the difficulties of a more general theory may be obtained 
by a consideration of the theory of imperfect gases. In this case we consider 
that the total energy of the system H can be divided into two terms, the 
kinetic energy E K , and the intermolecular potential energy U: H = E K + U. 

For a mole of gas, U is a function of the positions of all the molecules. 
For the N molecules in a mole there are 37V positional coordinates, q, ft, 
ft . . . ft^ Therefore, U = U(q l9 ft, ft ft,v)- 

The partition function may now be written 

Z = S e - (E *+ U)lkT = 2 e- s * lkT S e~ ulkT (14.10) 

It is not necessary to consider quantized energy levels, and Z may be 



428 LIQUIDS [Chap. 14 

written in terms of an integration, rather than a summation over discrete 
levels. 

Z=Xe-** l * T S. . .Se- u >" '**** dq l9 dqt. . . dq^ (14.11) 

The theoretical treatment of the imperfect gas reduces to the evaluation 
of the so-called configuration integral, 

f)(T) = /.../ <?-"<' ' '" dq lt dq z . . . dq w (14.12) 

Since this is a repeated integral over 3 N coordinates, </ it will be easily 
appreciated that its general evaluation is a matter of unconscionable diffi- 
culty, so that the general theory has ended in a mathematical cul-de-sac. 

Physically, however, it is evident that the potential energy of interaction, 
even in a moderately dense gas, does not extend much beyond the nearest 
neighbors of any given molecule. This simplification still leaves a problem of 
great difficulty, which is at present the subject of active research. 

The only simple approach is to consider interactions between pairs of 
molecules only. This would be a suitable approximation for a slightly im- 
perfect gas. One lets <f>(r l9 ) be the potential energy of interaction between two 
molecules / and j separated by a distance /, and assumes that the total 
potential energy is the sum of such terms. 



over pairs 

When this potential is substituted in eq. (14.12), the configuration integral 
can be evaluated. The details of this very interesting, but rather long, 
calculation will not be given here. 8 

Efforts have been made to solve the configuration integral for more 
exact assumptions than the interaction between pairs. These important 
advances toward a comprehensive theory for dense fluids are to be found 
in the works of J. E. Lennard- Jones, J. E. Mayer, J. G. Kirkwood, and 
Max Born. 

12. The free volume and holes in liquids. There have been many attempts 
to devise a workable theory for liquids that would avoid entanglement with 
the terrible intricacies of the configuration integral. One of the most success- 
ful efforts has been that of Henry Eyring, based on the concept of a free 
volume. The liquid is supposed to be in many respects similar to a gas. In a 
gas, the molecules are free to move throughout virtually the whole container, 
the excluded volume (four times van der Waals' b) being almost negligible 
at low densities. In a liquid, however, most of the volume is excluded volume, 
and only a relatively small proportion is a void space or free volume in which 
the centers of the molecules can manoeuvre. 

Eyring then assumes that the partition function for a liquid differs from 
that for a gas in two respects: (I) the free volume V 1 is substituted for the 

8 See J. C. Slater, Introduction to Chemical Physics (New York: McGraw-Hill, 1 939), 
p. I9l. 



Sec. 12] 



LIQUIDS 



429 



total volume F; (2) the zero point of energy is changed by the subtraction 
of the latent heat of vaporization from the energy levels of the gas. Thus, 
instead of eq. (12.28), one obtains 



( ' 



__ / 

11(1 ~ Nl[ If \ 



The idea of a free volume in liquids is supported experimentally by 
Bridgman's studies of liquid compressibilities. These are high at low pressures, 




07 



08 

T Ac 

REDUCED TEMPERATURE 



0.9 



1.0 



Fig. 14.6. Law of rectilinear diameters. 

but after a compression in volume of about 3 per cent, the compressibility 
coefficient decreases markedly. The initial high compressibility corresponds 
to "taking up the slack" in the liquid structure or using up the free volume. 

A useful model is sometimes provided by considering that the free 
volume is distributed throughout the liquid in the form of definite holes 
in a more closely packed structure. We should not think of these holes as 
being of molecular size, since there is probably a distribution of smaller 
holes of various sizes. The vapor is mostly void space with a few molecules 
moving at random. The liquid is a sort of inverse of this picture, being 
mostly material substance with a few holes moving at random. 

As the temperature of a liquid is raised, the concentration of molecules 
in its vapor increases and the concentration of holes in the liquid alsp in- 
creases. Thus as the vapor density increases the liquid density decreases, 
until they become equal at the critical point. We might therefore expect the 



430 LIQUIDS [Chap. 14 

average density of liquid and vapor to be constant. Actually, there is a slight 
linear decrease with temperature. This behavior was discovered by L. 
Cailletet and E. Mathias (1886), and has been called the law of rectilinear 
diameters. It may be expressed as p av = p aT, where p av is the arith- 
metical mean of the densities of the liquid and the vapor in equilibrium with 
it, and p and a are characteristic constants for each substance. The relation- 
ship is illustrated in Fig. 14.6 where the data for helium, argon, and ether 
are plotted in terms of reduced variables to bring them onto the same scale. 

13. The flow of liquids. Perhaps most typical of all the properties of 
fluids is the fact that they begin to flow appreciably as soon as a shearing 
stress is applied. A solid, on the other hand, apparently supports a very con- 
siderable shear stress, opposing to it an elastic restoring force proportional 
to the strain, and given by Hooke's Law,/ KX. 

Even a solid flows somewhat, but usually the stress must be maintained 
for a long time before the flow is noticeable. This slow flow of solids is called 
creep, and it can become a serious concern to designers of metal structural 
parts. Under high stresses, creep passes over into the plastic deformation of 
solids, for example, in the rolling, drawing, or forging of metals. These 
operations proceed by a mechanism involving the gliding of slip planes 
(page 391). Although creep is usually small, it must be admitted that the 
flow properties of liquids and solids differ in degree and not in kind. 

The fact that liquids flow immediately under even a very small shear 
force does not necessarily mean that there are no elastic restoring forces 
within the liquid structure. These forces may exist without having a chance 
to be effective, owing to the rapidity of the flow process. The skipping of a 
thin stone on the surface of a pond demonstrates the elasticity of a liquid 
very well. An interesting substance, allied to the silicone rubbers, has been 
widely exhibited under the name of "bouncing putty." This curious material 
is truly a hybrid of solid and liquid in regard to its flow properties. Rolled 
into a sphere and thrown at a wall, it bounces back as well as any rubber 
ball. Set the ball on a table and it gradually collapses into a puddle of viscous 
putty. Thus under long-continued stress it flows slowly like a liquid, but 
under a sudden sharp blow it reacts like a rubber. 

Some of the hydrodynamic theory of fluid flow was discussed in Chapter 
7 (page 173) in connection with the viscosity of gases. It was shown how the 
viscosity coefficient could be measured from the rate of flow through cylin- 
drical tubes. This is one of the most convenient methods for use with liquids 
as well as gases, the viscosity being calculated from the Poiseuille equation, 



Note that the equation for an incompressible fluid is suitable for liquids, 
whereas that for a compressible fluid is used for gases. 

In the Ostwald type of viscometer, one measures the time required for 



Sec. 14] LIQUIDS 431 

a bulb of liquid to discharge through a capillary under the force of its own 
weight. It is usual to make relative rather than absolute measurements with 
these instruments, so that the dimensions of the capillary tube and volume 
of the bulb need not be known. The time 7 required for a liquid of known 
viscosity ?? , usually water, to flow out of the bulb is noted. The time t 9 for 
the unknown liquid is similarly measured. The viscosity of the unknown is 



where p and p x are the densities of water and unknown. 

Another useful viscometer is the Happier type, based on Stokes' formula 
[eq.(S.ll)]: 

\^ JL. = ( m 



By measuring the rate of fall in the liquid (terminal velocity v) of metal 
spheres of known radius r and mass w, the viscosity may be calculated, 
since the force /is equal to (m m Q )g, where m is the mass of liquid dis- 
placed by the ball. 

14. Theory of viscosity. The hydrodynamic theories for the flow of liquids 
and gases are very similar. The kinetic-molecular mechanisms differ widely, 
as might be immediately suspected from the difference in the dependence of 
gas and liquid viscosities on temperature and pressure. In a gas, the viscosity 
increases with the temperature and is practically independent of the pressure. 
In a liquid, the viscosity increases with the pressure and decreases exponen- 
tially with increasing temperature. 

The exponential dependence of liquid viscosity on temperature was first 
pointed out by J. deGuzman Carrancio in 1913. Thus the viscosity coefficient 
may be written 

77=-- Ae* K ^ IRT (14.14) 



The quantity A vl8 is a measure of the energy barrier that must be overcome 
before the elementary flow process can occur. It is expressed per mole of 
liquid. The term e ~^ E ^ RT can then be explained as a Boltzmann factor 
giving the fraction of the molecules having the requisite energy to surmount 
the barrier. 

In Table 14.4 are collected the values A vig for a number of liquids, 
together with values of AZ^p for purposes of comparison. 9 The energy 
required to create a hole of molecular size in a liquid is A vap . The fact 
that the ratio of AE vl8 to A is vap about \ to \ for many liquids suggests 
that the viscous-flow process requires a free space about one-third to one- 
fourth the volume of a molecule. A noteworthy exception to the constancy 
of the A vig : A vftp ratio is provided by the liquid metals, for which the 

9 R. H. Ewell and Henry Eyring, J. Chem. Phys., 5, 726 (1937). 



432 



LIQUIDS 



[Chap. 14 



TABLE 14.4 
VALUES OF AVis 



Liquid 


A V 18 
(cal/mole) 


AEyap 
(cal/mole) 


AfVap/AEvis 


CC14 


2500 


6600 


2.66 


QH. 


2540 


6660 


2.62 


CH 4 


719 


1820 


2.53 


A 


516 


1420 


2.75 


N 2 


449 


1210 


2.70 


2 


398 


1470 


3.69 


CHC1 3 


1760 


6630 


3.76 


C 2 H 5 Br 


1585 


6080 


3.84 


CS 2 


1280 


5920 


4.63 


Na 


1450 


23,400 


16.1 


K 


1130 


19,000 


16.7 


Ag 


4820 


60,700 


12.5 


Hg 


650 


13,600 


20.8 



values range from & to 7 V- This low ratio has been interpreted as indicating 
that the units that flow in liquid metals are ions, whereas the units that 
vaporize are the much larger atoms. 

The pressure dependence of the viscosity follows an equation similar in 
form to eq. (14.14), 



The AF* is the volume of the hole that must be created for the flow process 
to occur. As we have seen, AF* for most liquids is about one-quarter of 
the molar volume, but for liquid metahs AF* is exceptionally small. Thus 
the viscosity of liquid metals increases only slowly with the pressure. Some 
properties of the earth provide an interesting confirmation of these ideas. 
To a depth of about 3500 km the earth consists of silicates. Except for a 
relatively thin solid crust, these must be molten, since the temperature is 
about 3000C. The pressure is so high, however, that the molten silicates 
have the flow properties of a solid; this result is shown by the fact that 
seismic waves can pass through the material with little damping. The core 
of the earth, with a radius of about 3000 km, is believed to consist of molten 
metal. Although the pressures are even higher than in the silicate layer, the 
metallic core behaves as a typical liquid, and does not transmit seismic 
waves. In other words, the AF* term for a molten metal is so small that 
the viscosity is not greatly increased even by very high pressures. 

PROBLEMS 

1. For CC1 4 the thermal pressure coefficient @P/dT) y at 20.4C and 
1 atm is 11.63 atm per deg. .Calculate the internal pressure in atmospheres. 



Chap. 14] LIQUIDS 433 

2. The following values were found for the viscosity of liquid CC1 4 : 



r,C 





20 


40 


60 


80 


ry millipoise 


13.47 


9,09 


7.38 


5.84 


4.68 



Plot these data and calculate AE vis . 

3. The liquid and vapor densities of ethanol in equilibrium at various 
temperatures are: 



f, C 


100 


150 


200 


220 


240 


<w g/ 00 

Pvap* g/CC 


0.7157 
0.00351 


0.6489 
0.0193 


0.5568 
0.0508 


0.4958 
0.0854 


0.3825 
0.1716 



The critical temperature is 243C. What is the critical volume? 

4. Suppose that the holes in liquid benzene are of the same order of size 
as the molecules. Estimate the number of holes per cc at 20C where the 
vapor pressure is 77 mm. Hence estimate the free volume in liquid benzene 
in cc per mole. Van der Waals' b for benzene is 1 1 5 cc per mole. If b is four 
times the volume of the molecules and molecules are closely packed in liquid 
benzene, estimate the free volume on this basis. The density of benzene at 
20C is 0.879 g per cc. 

5. The equation of state of a rubber band is K = CT(L/L Q L 2 /L 2 ), 
where K is the tension and L is the length at zero tension. In a case with 
L = 20cm and C = 1.33 x 10 3 dynes deg- 1 , a band is stretched at 25C 
to 40cm. What is the decrease in entropy of the rubber band? 

6. A certain glass at 800C has a viscosity of 10 6 poise and a density of 
3.5 g crrr 3 . How long would a 5 mm diameter platinum ball require to fall 
1 .0 cm through the hot glass ? 

REFERENCES 

BOOKS 

1. Frenkel, J., The Kinetic Theory of Liquids (New York: Oxford, 1946). 

2. Green, H. S., The Molecular Theory of Fluids (New York: Interscience, 
1952). 

3. Kimball, G. E., 'The Liquid State," in Treatise on Physical Chemistry, 
vol. II, ed. by H. S. Taylor and S. Glasstone (New York: Van Nostrand, 
1951). 

ARTICLES 

1. Condon, E. U., Am. J. Phys., 22, 310-17 (1954), 'The Glassy State." 

2. Hildebrand, J. H., Proc. Phys. Soc. (London), 56, 221-39 (1944), 'The 
Liquid State." 



434 LIQUIDS [Chap. 14 

3. Hirschfelder, J. O., J. Chem. Ed., 16, 540-44 (1939), 'The Structure of 
Liquids." 

4. Kirkwood, J. G., Science in Progress, vol. Ill (New Haven: Yale Uni- 
versity Press, 1942), 208-221, 'The Structure of Liquids." 

5. Rowlinson, J. S., Quart. Rev., 8, 168-91 (1954), "Tntermolecular Forces 
and Properties of Matter." 

6. Ubbelohde, A. R., Quart. Rev., 4, 356-81 (1950), "Melting and Crystal 
Structure." 



CHAPTER 15 

Electrochemistry 



1. Electrochemistry: coulometers. The subject of electrochemistry, the 
interrelations of electrical and chemical phenomena, is an exceedingly broad 
one since, as we have seen, all chemical interactions are fundamentally 
electrical in nature. In a more restricted sense, however, electrochemistry 
has come to mean the study of solutions of electrolytes and the phenomena 
occurring at electrodes immersed in such solutions. The electrochemistry of 
solutions may claim a special interest from physical chemists since it was in 
this field that physical chemistry first emerged as a distinct and characteristic 
science. Its first journal, Die Zeitschrift fur physikalische Chemie, was founded 
in 1887 by Wilhelm Ostwald, and the early volumes are devoted mainly to 
the researches in electrochemistry of Ostwald, van't Hoff, Kohlrausch, 
Arrhenius, and others of their "school." 

The early history of electrical science has already been discussed as an 
introduction to Chapter 8. Its culmination was Faraday's discovery of the 
quantitative laws of electrolysis. 

These laws became the basis for the construction of coulometers for 
measuring quantity of electricity. A standard instrument is the silver coulo- 
meter, based on the mass of silver deposited at a platinum cathode by the 
passage of the electric current through an aqueous silver nitrate solution. 
One coulomb of electricity is equivalent to 0.001 11 800 g of silver. The 
iodine coulometer depends on the volumetric estimation of the iodine 
liberated by electrolysis of a potassium iodide solution. The experimental 
precautions needed for precise coulometry have been extensively studied. 1 

2. Conductivity measurements. From the very beginning one of the 
fundamental theoretical problems in electrochemistry was how the solutions 
of electrolytes conducted an electric current. 

Metallic conductors were known to obey Ohm's Law, 

,- , 

where / is the current (amperes), ff is the electromotive force, emf (volts), 
and the proportionality constant R is called the resistance (ohms). The 
resistance depends on the dimensions of the conductor: 

R = p l (15.2) 

A 

1 See, for example, H. S. Taylor, Treatise on Physical Chemistry, 2nd ed., pp. 591-598. 

435 



436 



ELECTROCHEMISTRY 



[Chap. 15 



Here / is the length and A the cross-sectional area, and the specific resistance 
p (ohm cm) is called the resistivity. The reciprocal of the resistance is called 
the conductance (chirr 1 ) and the reciprocal of the resistivity, the specific 
conductance or conductivity K (ohm" 1 cm" 1 ). 

The earliest studies of the conductivity of solutions were made with 
rather large direct currents. The resulting electrochemical action was so great 
that erratic results were obtained, and it appeared that Ohm's Law was not 
obeyed; i.e., the conductivity seemed to depend on the emf. This result was 
largely due to polarization at the electrodes of the conductivity cell, i.e., a 
departure from equilibrium conditions in the surrounding electrolyte. 





AC 

-1000 CYCLES 

(a) 



LABORATORY 
CELL 



^| DIPPING 
PJCELL 



(b) 
Fig. 15.1. Conductivity measurement. 



These difficulties were overcome by the use of an alternating-current 
bridge, such as that shown in (a), Fig. 15.1. With a-c frequencies in the audio 
range (1000-4000 cycles per sec) the direction of the current changes so 
rapidly that polarization effects are eliminated. One difficulty with the a-c 
bridge is that the cell acts as a capacitance in parallel with a resistance, so 
that even when the resistance arms are balanced there is a residual unbalance 
due to the capacitances. This effect can be partially overcome by inserting 
a variable capacitance in the other arm of the bridge, but for very precise 
work further refinements are necessary. 2 Microphones formerly were used 
to indicate the balance point of the bridge, but the preferred indicator is 
now the cathode-ray oscilloscope. The voltage from the bridge mid-point is 
filtered, amplified, and fed to the vertical plates of the oscilloscope. A small 
portion of the bridge input signal is fed to the horizontal plates of the scope 
through a suitable phase-shifting network. When the two signals are properly 
phased, the balance of capacitance is indicated by the closing of the loop 
on the oscilloscope screen, and the balance of resistance is indicated by 
the tilt of the loop from horizontal. 

Typical conductivity cells are also shown in Fig. 15.1. Instead of measur- 
ing their dimensions, we now usually calibrate these cells before use with a 
solution of known conductivity, such as normal potassium chloride. The 
cell must be well thermostated since the conductivity increases with the 
temperature. 

1 T. Shedlovsky, /. Am. Chenr. Soc., 54, 141 1 (1932); W. F. Luder, ibid.. 62, 89 (1940). 



Sec. 3] ELECTROCHEMISTRY 437 

As soon as reliable conductivity data were available it became apparent 
that solutions of electrolytes followed Ohm's Law. The resistance was in- 
dependent of the emf, and the smallest applied voltage sufficed to produce 
a current of electricity. Any conductivity theory would have to explain this 
fact: the electrolyte is always ready to conduct electricity and this capability 
is not something produced or influenced by the applied emf. 

On this score, the ingenious theory proposed in 1805 by Baron C. J. 
von Grotthuss must be adjudged inadequate. According to his theory, the 
molecules of electrolyte were supposed to be very polar, with positive and 
negative ends. An applied field lined them up in a chain. Then the field 
caused the molecules at the end of the chain to dissociate, the free ions thus 
formed being discharged at the electrodes. Thereupon, there was an exchange 
of partners along the chain. Before further conduction could occur, each 
molecule had to rotate under the influence of the field to reform the original 
oriented chain. 

Despite its shortcomings, the Grotthuss theory was valuable in empha- 
sizing the necessity of having free ions in the solution to explain the observed 
conductivity. We shall see later that there are some cases in which a mechan- 
ism similar to that of Grotthuss may actually be followed. 

In 1857, Clausius proposed that especially energetic collisions between 
undissociated molecules in electrolytes maintained at equilibrium a small 
number of charged particles. These particles were believed to be responsible 
for the observed conductivity. 

3. Equivalent conductivities. From 1869 to 1880, Friedrich Kohlrausch 
and his coworkers published a long series of careful conductivity investiga- 
tions. The measurements were made over a range of temperatures, pressures, 
and concentrations. 

Typical of the painstaking work of Kohlrausch was his extensive purifica- 
tion of the water used as a solvent. After 42 successive distillations in vacuo, 
he obtained a conductivity water having a K of 0.043 x 10~ 6 ohm" 1 cm" 1 at 
18C. Ordinary distilled water in equilibrium with the carbon dioxide of the 
air has a conductivity of about 0.7 x 10~ 6 . 

To reduce his results to a common concentration basis, Kohlrausch 
defined a function called the equivalent conductivity, 

A =4 (15.3) 

c* 

The concentration c* is in units of equivalents per cc; the reciprocal (f> -- \/c* 
is called the dilution, in cc per equivalent. The equivalent conductivity would 
be the conductance of a cube of solution having one square centimeter cross 
section and containing one equivalent of dissolved electrolyte. 

Some values for A are plotted in Fig. 15.2. On the basis of their con- 
ductivity behavior two classes of electrolytes can be distinguished. Strong 
electrolytes, such as most salts and acids like hydrochloric, nitric, and 



438 



ELECTROCHEMISTRY 



[Chap. 15 



sulfuric, have high equivalent conductivities which increase only moderately 
with increasing dilution. Weak electrolytes, such as acetic and other organic 
acids and aqueous ammonia, have much lower equivalent conductivities at 
high concentrations, but the values increase greatly with increasing dilution. 
The value of A extrapolated to zero concentration is called the equivalent 
conductivity at infinite dilution, A . The extrapolation is made readily for 
strong electrolytes but is impossible to make accurately for weak electrolytes 

175 




4 6 .8 1.0 

Vc"- C IN EQUIVALENTS/LITER 
Fig. 15.2. Equivalent conductivities vs. square roots of concentration. 

because of their tremendous increase in A at high dilutions, where the 
experimental measurements become very uncertain. It was found that the 
data were fairly well represented by the empirical equation 

A = AO - k e c 1 ' 2 (15.4) 

where k c is an experimental constant. 

Kohlrausch observed certain interesting relations between the values of 
AD for different electrolytes : the difference in A for pairs of salts having a 
common ion was always nearly constant. For example (at 18C): 

NaCl 108.99 NaNO 3 105.33 NaBr 111.10 

KC1 130.10 KNO 3 126.50 KBr 132.30 



21.11 



21.17 



21.20 



Sec. 4] ELECTROCHEMISTRY 439 

Thus no matter what the anion might be, there was a constant difference 
between the conductivities of potassium and sodium salts. This behavior 
could be readily explained if A is the sum of two independent terms, one 
characteristic of the anion and one of the cation. Thus 

A - V + V (15.5) 

where A + and A ~ are the equivalent ionic conductivities at infinite dilution. 
This is Kohlrausch's law of the independent migration of ions. 

This rule made it possible to calculate the A for weak electrolytes like 
organic acids from values for their salts, which are strong electrolytes. For 
example (at 18C): 

A (HAc) - A (NaAc) + A (HC1) - A (NaCl) 
- 87.4 + 379.4 109.0 =-- 357.8 

4. The Arrhenius ionization theory. From 1882 to 1886, Julius Thomsen 
published data on the heats of neutralization of acids and bases. He found 
that the heat of neutralization of a strong acid by a strong base in dilute 
solution was always very nearly constant, being about 13,800 calories per 
equivalent at 25C. The neutralization heats of weak acids and bases were 
lower, and indeed the "strength" of an acid appeared to be proportional to 
its heat of neutralization by a strong base such as NaOH. 

These results and the available conductivity data' led Svante Arrhenius 
in 1887 to propose a new theory for the behavior of electrolytic solutions. 
He suggested that an equilibrium exists in the solution between undissociated 
solute molecules and ions which arise from these by electrolytic dissociation. 
Strong acids and bases being almost completely dissociated, their interaction 
was in every case simply H+ ~f OH H 2 O, thus explaining the constant 
heat of neutralization. 

While Arrhenius was working on this theory, the osmotic-pressure studies 
of van't Hoff appeared, which provided a striking confirmation of the new 
ideas. It will be recalled (page 132) that van't Hoff found that the osmotic 
pressures of dilute solutions of nonelectrolytes often followed the equation 
II = cRT. The osmotic pressures of electrolytes were always higher than 
predicted from this equation, often by a factor of two, three, or more, so 
that a modified equation was written as 

II - icRT (15.6) 

Now it was noted that the van't Hoff u / factor" for strong electrolytes was 
very closely equal to the number of ions that would be formed if a solute 
molecule dissociated according to the Arrhenius theory. Thus for NaCl, 
KC1, and other uniunivalent electrolytes, i = 2; for BaCl 2 , K 2 SO 4 , and other 
unibivalent species, i = 3 ; for LaCl 3 , / = 4. 

On April 13, 1887, Arrhenius wrote to van't Hoff as follows: 

It is true that Clausius had assumed that only a minute quantity of dissolved 
electrolyte is dissociated, and that all other physicists and chemists had followed 



440 ELECTROCHEMISTRY [Chap. 15 

him; but the only reason for this assumption, as far as I can understand, is a strong 
feeling of aversion to a dissociation at so low a temperature, without any actual 
facts against it being brought forward. ... At extreme dilution all salt molecules 
are completely dissociated. The degree of dissociation can be simply found on this 
assumption by taking the ratio of the equivalent conductivity of the solution in 
question to the equivalent conductivity at the most extreme dilution. 

Thus Arrhenius would write the degree of dissociation a as 

a = A (15.7) 

A o 

The van't Hoff / factor can also be related to a. If one molecule of solute 
capable of dissociating into n ions per molecule is dissolved, the total number 
of particles present will be i = 1 a + nan. Therefore 

a = i^-i (15.8) 

n 1 

Values of a for weak electrolytes calculated from eqs. (15.7) and (15.8) were 
found to be in good agreement. 

Applying the mass-action principle to ionization, Ostwald obtained a 
dilution law, governing the variation of equivalent conductivity A with con- 
centration. For a binary electrolyte AB with degree of dissociation a, whose 
concentration is c moles per liter: 

AB ^ A+ + B~ 

c(l a) etc <xc 



(1 -a) 
From eq. (15.7), therefore, 



- 
AO(A O - A) 

This equation was closely obeyed by weak electrolytes in dilute solutions. 
An example is shown in Table 15.1. In this case, the "law" is obeyed at 
concentrations below about 0. 1 molar, but discrepancies begin to appear at 
higher concentrations. 

The accumulated evidence gradually won general acceptance for the 
Arrhenius theory, although to the chemists at the time it still seemed most 
unnatural that a stable molecule when placed in water should spontaneously 
dissociate into ions. This criticism was in fact justified and it soon became 
evident that the solvent must play more than a purely passive role in the 
formation of an ionic solution. 

We now know that the crystalline salts are themselves formed of ions in 
regular array, so that there is no question of "ionic dissociation" when they 
are dissolved. The process of solution simply allows the ions to be separated 
from one another. The separation is particularly easy in aqueous solutions 



Sec. 4] ELECTROCHEMISTRY 441 

TABLE 15.1 

TEST OF OSTWALD'S DILUTION LAW 
Acetic Acid at 25C, A = 387.9* 



c 
(moles/liter) 


A 


Per Cent 
Dissociation 
lOOa - lOOlA/A^) K 


v? 


1.011 


1.443 


0.372 


.405 


0.2529 


3.221 


0.838 


.759 


0.06323 


6.561 


1.694 


.841 


0.03162 


9.260 


2.389 


.846 


0.01581 


13.03 


3.360 


.846 


0.003952 


25.60 


6.605 


.843 


0.001976 


35.67 


9.20 


.841 


0.000988 


49.50 


12.77 


.844 


0.000494 


68.22 


17.60 


.853 



w 



* D. A. Maclnnes and T. Shedlovsky, J. Am. Chem. Soc., 54, 1429 (1932). 

owing to the high dielectric constant of water, e =r 82.0. If we compare, for 
water and a vacuum, the work necessary to separate two ions, say Na+ and 
Cl~, from a distance of 2 A to infinity, we find: 3 

Vacuum Water 

f 00 ,, f^2 , [ 

fdr = \ r dr w - 

J2A h r 2 J 2 

_ (4.80 X IP" 10 ) 2 _ __ (4.80 X 1Q- 10 ) 2 _ 

2 x 10- ~~ ""82 x "2 X 10- 8 ~~ 

1.15 x 10- n erg 1.40 x 10~ 13 erg 

Counteracting the energy necessary to separate the ions is the energy of 
hydration of the ions, which arises from the strong ion-dipole attractions. 
Thus in many cases the solution of ionic salts is an exothermic reaction. 
The equilibrium position is of course determined by the free-energy change. 
The increased randomness of the ions in solution, compared with the ionic 
crystal, leads to an increase in entropy, but this is sometimes outweighed 
by an entropy decrease due to the ordering effect of the ions on the water 
molecules. 

In the case of acids such as HC1, the solution process probably occurs as 
follows: HC1 + H 2 O - OH 3 + + Cl~. In both HC1 and H 2 O the bonds are 
predominantly covalent in character. The ionization that occurs in solution 
is promoted by the high energy of hydration of the proton to form the 
hydronium ion, OH 3 + . 

Whatever the detailed mechanisms may be, it has been clear since the 
work of Arrhenius that in electrolytic solutions the solute is ionized, and the 

8 This assumes that e in the neighborhood of an ion is the same as c for bulk water, 
which is an approximation. 



442 ELECTROCHEMISTRY [Chap. 15 

transport of the ions in an electric field is responsible for the conductivity of 
the solutions. 

5. Transport numbers and mobilities. The fraction of the current carried 
by a giverriomc speci&~l7Tsolution is called the transport number or trans- 
ference number of that ion. 

From Kohlrausch's law, eq. (15.5), the transference numbers / + and / ~ 
of cation and anion at infinite dilution may be written 

V - V = (15.10) 

A o yx o 

The mobility I of an ion is defined as its velocity in an electric field of 
unit strength. The usual units are cm sec * per volt cm" 1 (cm 2 sec" 1 volt" 1 ). 

Consider a one-square-centimeter cross section taken normal to the 
direction of the current in an electrolyte. The total current / passing through 
this area is the sum of that carried by the positive and that carried by the 
negative ions. Thus / ^ n\v\z+e + n_v z^e, where n+ and /?_ are the con- 
centrations in ions per cc, z + and z_ are the number of charges on positive 
and negative ions, v { and v_ are the velocities of the ions, and e is the elec- 
tronic charge. The requirement for over-all electrical neutrality is n+z + = 
/i._z_, so that 

/ n+z + e(v, \ v_) (15.11) 

For a unit cube of electrolyte, from eqs. (15.2) and (15.3): / = &\p ---- K$ 
(2f Ac*. Since n\z^e is the charge in one cc and J*", the faraday, is the charge 
in one equivalent, 



From eqs. (15.1 1) and (15.12), 

"^ "= "+z+*(+ 



Now v + / and vj are the mobilities, l + and /_. Therefore 
JiJP = /+ + /_, and 



It follows also from eq. (15.10) that 

/ = i__, 



6. Measurement of transport numbers Hittorf method. The method of 
Hittorf is based on concentration changes in the neighborhood of the elec- 
trodes caused by the passage of current through the electrolyte. The principle 




Sec. 6] ELECTROCHEMISTRY 443 

of the method may be illustrated by reference to Fig. 15.3. Imagine a cell 
divided into three compartments as shown. The situation of the ions before 
the passage of any current is represented schematically as (a), each -| or 
sign indicating one equivalent of the corresponding ion. 

Now let us assume that the mobility of the positive ion is three times 
that of the negative ion, / f 3/_. Let 4 faradays of electricity be passed 
through the cell. At the anode, therefore, four equivalents of negative ions 
are discharged, and at the cathode, four equivalents of positive ions. Four 
faradays must pass across any boundary 
plane drawn through the electrolyte par- 
allel to the electrodes. Since the posi- 
tive ions travel three times faster than 
the negative ions, 3 faradays are carried 
across the plane from left to right by the +3=lrj 
positive ions while one faraday is being 
carried from right to left by the negative 
ions. This transfer is depicted in panel (b) 
of the picture. The final situation is 
shown in (c). The change in number of 
equivalents around the anode, A* a = 6 p . g 15 3 T rt numbers 
- 3 = 3; around the cathode, A* c =.- (Hittorf method). 

6 5 -- 1. The ratio of these concentra- 
tion changes is necessarily identical with the ratio of the ionic mobilities: 
A* a /A c = IJL - 3. 

Suppose the amount of electricity passed through the cell has been 
measured by a coulometer in series, and found to be q faradays. Provided 
the electrodes are inert, q equivalents of cations have therefore been dis- 
charged at the cathode, and q equivalents of anions at the anode. The net 
loss of solute from the cathode compartment is 

A/? c - q - t+ q - q(\ - /+) = qt_ 
Thus /_ = * /+ = (15.15) 

where A/I O is the net loss of solute from the anode compartment. Since 
/+ + *- = 1, both transport numbers can be determined from measurements 
on either compartment, but it is useful to have both analyses as a check. 

In the experiment just described, it is assumed that the electrodes are 
inert. In other cases ions may pass into the solution from the electrodes. 
Consider, for example, a silver anode in a silver nitrate solution. When 
electricity passes through the cell, there will be a net increase in the amount 
of electrolyte in the anode compartment, equal to the number of equivalents 
of silver entering the solution at the anode minus the number of equivalents 
of silver crossing the boundary of the anode compartment. 



444 



ELECTROCHEMISTRY 



[Chap. 15 



Fig. 15.4. Hittorf transport 
apparatus. 



An experimental apparatus for carrying out these determinations is shown 
in Fig. 15.4. The apparatus is filled with a standardized electrolyte solution 
and a current, kept low to minimize thermal effects, is passed through the 

solution for some time. The total amount of 
electricity is measured with a coulometer. 
The solutions are drawn separately from 
the three sections of the cell and analyzed. 
Analysis gives the mass of solute and the 
mass of solvent in the solutions from the 
electrode compartments. Since the mass of 
solute originally associated with this mass 
of solvent is known, A a and A c can be 
found by difference. Ideally there should be 
no change in concentration in the middle 
compartment, but small changes arising from diffusion may detract from 
the accuracy of the determination. 

7. Transport numbers moving boundary method. This method is based 
on the early work of Sir Oliver Lodge (1886) who used an indicator to follow 
the migration of ions in a conducting gel. For example, a solution of barium 
chloride may be placed around platinum electrodes serving as anode and 
cathode. The two sides of the cell are then connected by means of a tube 
filled with gelatin acidified with acetic acid to make it conducting and con- 
taining a small amount of dissolved silver sulfate as indicator. As the current 
is passed, the Ba++ and Cl~ ions migrate into 
the gel from opposite ends, forming pre- 
cipitates of BaSO 4 and AgCl, respectively. 
From the rate of progression of the white 
precipitate boundaries, the relative velocities 
of the ions may be estimated. 

The more recent applications of this 
method discard the gel and indicator and use 
an apparatus such as that in Fig. 15.5, to 
follow the moving boundary between two 
liquid solutions. For example, the electrolyte 
to be studied, CA, is introduced into the 
apparatus in a layer above a solution of a 
salt with a common anion, C'A, and a cation 
whose mobility is considerably less than that 



dfr 



CA 



C'A 



Fig. 153. Moving-boundary cell. 



of the ion C+. When a current is passed through the cell, A~ ions move 
downwards toward the anode, while C+ and C"+ ions move upwards toward 
the cathode. A sharp boundary is preserved between the two solutions since 
the more slowly moving C"+ ions never overtake the C+ ions; nor do the 
following ions, C"+, fall far behind, because if they began to lag, the solution 
behind the boundary would become more dilute, and its higher resistance 



Sec. 8] 



ELECTROCHEMISTRY 



445 



and therefore steeper potential drop would increase the ionic velocity. Even 
with colorless solutions, the sharp boundary is visible owing to the different 
refractive indices of the two solutions. 

Suppose the boundary moves a distance x for the passage of q coulombs. 
The number of equivalents transported is then qj^, of which t+ql^ are 
carried by the positive ion. Recalling that c* is the concentration in equi- 
valents per cc, the volume of solution swept out by the boundary during the 
passage of q coulombs is t+qf&c*. If a is the cross-sectional area of the tube, 
xa = t+q/^c*, or 

(.5.16) 



8. Results of transference experiments. Some of the measured transport 
numbers are summarized in Table 15.2. With these values it is possible to 

TABLE 15.2 
TRANSPORT NUMBERS OF CATIONS IN WATER SOLUTIONS AT 25C* 



Normality 
Solution 


AgNO, 


BaCl 2 


LiCl 


NaCl 


KC1 


KN0 8 


LaCl s 


HCI 


0.01 


0.4648 


0.440 


0.3289 


0.3918 


0.4902 


0.5084 


0.4625 


0.8251 


0.05 ! 0.4664 


0.4317 


0.3211 


0.3876 


0.4899 


0.5093 


0.4482 


0.8292 


0.10 


0.4682 


0.4253 


0.3168 


0.3854 


0.4898 


0.5103 


0.4375 


0.8314 


0.50 





0.3986 


0.300 





0.4888 





0.3958 





1.0 





0.3792 


0.287 





0.4882 












* L. G. Longsworth, J. Am. Chem. Soc. t 57, 1185 (1935); 60, 3070 (1938). 

calculate from eq. (15.10) the equivalent ionic conductivities A, some of 
which are given in Table 15.3. By the use of Kohlrausch's rule, they may 

TABLE 15.3 
EQUIVALENT IONIC CONDUCTIVITIES AT INFINITE DILUTION, ^, AT 25C* 



Cation 


A.