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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 twosemester 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
PrenticeHall, 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 JouleThomson experiment, 33. 10. Application of
the First Law to ideal gases, 34. 1 1 . Examples of idealgas 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.
Freeenergy 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. Thirdlaw 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
ronClausius 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. Secondorder 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. Twocomponent
systems, 123. 8. Pressurecomposition diagrams, 723. 9. Temper
aturecomposition diagrams, 725. 10. Fractional distillation, 725.
II. Boilingpoint 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.
Boilingpoint diagrams, 736. 19. Partial miscibility, 737. 20. Con
densedliquid systems, 739. 21. Thermodynamics of nonideal solu
tions: the activity, 747. 22. Chemical equilibria in nonideal solutions,
143. 23. Gassolid equilibria, 144. 24. Equilibrium constant in solid
gas reactions, 145. 25. Solidliquid 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 solidsolid solubility, 757. 31. The iron
carbon diagram, 752. 32. Threecomponent 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. Xrays and atomic number, 213.
12. The radioactive disintegration series, 27 3. 13. Isotopes, 216. 14.
Positiveray 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.
Blackbody 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 Xray spectra, 268. 13. Particles and waves,
269. 14. Electron diffraction, 277. 15. The uncertainty principle,
272. 16. Waves and the uncertainty principle, 274. 17. Zeropoint
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 HH 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 farinfrared 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. Xray crystallography, 375. 7. The Bragg
treatment, 376. 8. The structures of NaCl and KC1, 377. 9. The
powder method, 382. 10. Rotatingcrystal 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 BornHaber 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. Xray diffraction of liquids, 4/5. 4. Results of liquidstructure
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 activitycoefficient
measurements, 457. IS^^rfie DebyeHtickel theory, 458. 1 9. Poisson's
equation, 458. 20. Tne PoissonBoltzmann equation, 460. 21. The
DebyeHiickel 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. Electrolyteconcentration 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. Surfacetension 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 reactionrate constant, 532. 6. Firstorder rate
equations, 533. 1. Secondorder rate equations, 534. 8. Thirdorder
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. Freeradical 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 transitionstate 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. Acidbase catalysis, 577.
34. General acidbase 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 highenergy 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 (16421727),
and usually begins with a statement of the wellknown 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 largescale 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 alcoholinglass "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 constantpressure 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
VLITERS
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 liquidinglass 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 Utube, 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 GayLussac in 1802,
extending earlier work by Charles (1787), showed that this value of a was a
constant for "permanent" gases. GayLussac found (1808) the value to
be 4 ^. By a much better experimental procedure, Regnault (1847) obtained
2^3. For every onedegree 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 GayLussac 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=CT (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: McGrawHill, 1933); also
J. H. Perry, ed., Chemical Engineers 9 Handbook (New York: McGrawHill, 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 mercuryinglass 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 NBUTANE
METHANE a ISOPENTANE
ETHANE
ETHYLENE
NHEPTANE
A CARBON DIOXIDE
WATER
23456
REDUCED PRESSURE, P R
Fig. 1.5. Compressibility factor as function of reduced state variables.
[From GouqJen 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 semiempirical 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 )(Vnb) = nRT
n*A'
P(Vnb')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 twophase 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 KammerlinghOnnes:
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, 229308 (1928).
J. A. Beattie, Chem. Rev., 44, 141192 (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 (rr 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 110volt 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 1liter
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 5.00g piece of Al is dropped into calorimeter A, and a
5.00g 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 10.0cmlong 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: PrenticeHall,
1950).
6. Lewis, G. N., and M. Randall, Thermodynamics and the Free Energy of
Chemical Substances (New York: McGrawHill, 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, 120149 (New Haven:
Yale Univ. Press, 1945), "The Mathematical Nature of Physical Theories."
2. Brescia, F.,/. Chem. Ed., 24, 123128 (1947), 'The Critical Temperature."
3. Reilly, D., /. Chem. Ed., 28, 178183 (1951), "Robert Boyle and His
Background."
4. Roseman, R., and S. KatzofT, J. Chem. Ed., 11, 350354 (1934), "The
Equation of State of a Perfect Gas."
5. Woolsey, G., J. Chem. Ed., 16, 6066 (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 = qw (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 massenergy, 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 117 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 JouleThomson 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 JouleThomson 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 JouleThomson experi
ment.
The JouleThomson 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
JOULETHOMSON 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: McGrawHill,
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 JouleThomson
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 JouleThomson experiment must be postponed until the Second Law
of Thermodynamics has been studied in the next chapter. It may be said,
however, that the porousplug 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 idealgas 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 :
^(r2732)=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
constantvolume 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 coworkers 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
rtButane
Isobutane
wPentane
2Methylbutane
Tetramethylmethane
Monolefines:
Ethylene
Propylene
1Butene
cis2Butene
trans2Butene
2Methylpropene
1Pentene
Diolefines:
Allene
1,3Butadiene
1,3Pentadiene
1,4Pentadiene
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 coworkers 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 heatcapacity data will be represented by
a power series:
CV a + bT\ cT 2 f . . . (2.31)
Examples of such heatcapacity equations are given in Table 2.4. These
threeterm equations fit the experimental data to within about 0.5 per cent
TABLE 2.4
HEAT CAPACITY OF GASES (2731 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
nHexane
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 heatcapacity 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 JouleThomson 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 heatcapacity 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
/iC 4 H 10  3 H 2  4 CH 4
isoC 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 whexane 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, 26266 (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, 311435, 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) (33)
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 (35)
<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. 33. 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 ATAS (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 GibbsHelmholtz 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 GibbsHelmholtz 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 freeenergy 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 paperand
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+PyTS
(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 JouleThomson experiment. Since
it follows from eq. (3.44) that
TQVfiT),.  V
(3.45)
It is apparent that the JouleThomson 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 JouleThomson coefficient vanishes when K  <xK o r. For an
ideal gas this is always true (Law of GayLussac) 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 JouleThomson 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 lowtemperature
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 PV
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 JouleThomson 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, 4146 (1949), "Principle of Cara
theodory."
2. Crawford, F. H., Am. J. Phys., 17, 15 (1949), "Jacobian Methods in
Thermodynamics."
3. Darrow, K. K., Am. J. Phys., 12, 18396 (1944), "Concept Of Entropy."
4. Dyson, F. J., Scientific American, 191, 5863 (1954), "What is Heat?"
5. LaMer, V. K., O. Foss, and H. Reiss, Ann. N. Y. Acad. Sci., 51, 60526
(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 seasalt 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 freeenergy function described
in Chapter 3 provides the true measure of chemical affinity under conditions
of constant temperature and pressure. The freeenergy change in a chemical
reaction can be defined as AF ^ F produt . t8 ^reactants When the freeenergy
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 freeenergy 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 freeenergy 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 freeenergy
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 platinumsponge 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. Freeenergy and cell reactions. Reactions occurring in electrochemical
cells with the production of electric energy are of especial interest in the
discussion of freeenergy 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 voltcoulomb 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 freeenergy 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 Tf >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 freeenergy 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 freeenergy 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 freeenergy 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
Freeenergy 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
FF = 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
PlP, = 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 freeenergy 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 hydrogeniodine 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 "watergas 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 watergas 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 pressuredependent. 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) 10 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 vanadiumpentoxide 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 freeenergy 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 1QT 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 freeenergy 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 perpetualmotion
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 JouleThomson 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 JouleThomson 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, KammerlinghOnnes,
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 JouleThomson 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 vaporpressure 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 lowtemperature
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. 1 120.
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 onecomponent 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 "ThirdLaw 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. Thirdlaw 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 heatcapacity data extrapolated to 0K to deter
mine socalled thirdlaw 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 HEATCAPACITY
MEASUREMENTS
Contribution
1. Extrapolation from 01 6K (Debye Theory, Sec. 1323)
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 188.07298.15K
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 thirdlawentropies
are collected in Table 4.6.
TABLE 4.6
THIRDLAW 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
rtHexane
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 lowtemperature
heatcapacity 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 highpressure 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^<**. (428)
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
VitfRTlnfi (4.31)
Now the treatment of equilibrium in Section 45 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 JPO 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 PRESSURFFfe
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 threetoone 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 cadmiumcalomel 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. Onehalf 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 + 1657. 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 ThirdLaw entropy of CH 3 NO 2
gas at 298. 1K and 1 atm pressure (assuming ideal gas behavior).
12. Using the ThirdLaw 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 rtpentane 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: McGrawHill, 1953).
4. Wenner, R. R., Thermochemical Calculations (New York: McGrawHill,
1941).
Also see Chapter 1, p. 25.
ARTICLES
1. Chem. Revs., 39, 357481 (1946), "Symposium on Low Temperature
Research."
2. Daniels, F., Scientific American, 191, 5863 (1954), "High Temperature
Chemistry,"
3. Huffman, H. M.,' Chem. Revs., 40, 114 (1947), "Low Temperature
Calorimetry."
4. Lemay, P., and R. Oesper, /. Chem. Ed., 23, 15865, 23036 (1946),
"Claude Berthollet "
5. Oesper, R., /. Chem. Ed., 21, 26364 (1944), "H. KammerlinghOnnes."
6. Urey, H. C., /. Chem. Soc., 56281 (1947), "Thermodynamic Properties
of Isotopic Substances."
7. Walden, P., /. Chem. Ed., 31, 2733 (1954), "Beginnings of the Doctrine
of Chemical Affinity."
8. Watson, R. G., Research, 7, 3440 (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 twophase system. The contents of a flask
of liquid benzene in contact with benzene vapor and air is a twophase
system; if we add a spoonful of sugar (practically insoluble in benzene) we
obtain a threephase 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
overall 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 overall 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 aggregationstate.) 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^cp + 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 (56)
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) + 2c(/> 1)
f=cp + 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 onecomponent 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 singlephase 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 vaporpressure 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 sublimationpressure 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 solidice region from the liquidwater 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 vaporpressure 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 meltingpoint 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 ClapeyronCIausius 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 ClapeyronClausius 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 straightline 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. 148). 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(10001)
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 vaporpressure measurements.
One of the most convenient static methods is the Smith Menzies isoteniscope
shown in Fig. 5.2. The bulb and short attached Utube 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 Utube. At each temperature the external
pressure is adjusted until the arms of the differential Utube manometer are
level, and the pressure and temperature are then recorded.
The gassaturation 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, 517550
(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. Solidsolid transformations the sulfur system. Sulfur provides the
classical example of a onecomponent system displaying a solidsolid 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 lowtemperature rhombic form and a hightemperature
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 lowpressure regions into prominence.
The curve AB is the vaporpressure curve of solid rhombic sulfur. At
point B it intersects the vaporpressure curve of monoclinic sulfur BE, and
also the transformation curve for rhombicmonoclinic 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 vaporpressure curve of liquid sulfur EF. Also from , there
extends the curve ED, the meltingpoint curve of monoclinic sulfur. The
density of liquid sulfur is less than that of the monoclinic solid, the usual
situation in a solidliquid 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 meltingpoint curve of rhombic sulfur
in this highpressure 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 meltingpoint 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 wellknown 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 onecomponent 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 solidstate 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
onecomponent system must be applied with caution.
12. Secondorder transitions. The usual change of state (solid to liquid,
liquid to vapor, etc.) is called & firstorder 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
secondorder transition.
13. Highpressure 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 highpressure 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
multiplechamber 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.
Highpressure 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 highpressure 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 (solidliquidvapor)
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 lowpressure 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 highpressure /? 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 liquidsolid 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 1000atm 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 secondorder 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: McGrawHill,
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, 10846 (New Haven: Yale
Univ. Press, 1942), "Recent Work in the Field of High Pressures."
2. Garner, W. E.,/. Chem. Soc., 19611973 (1952), "The Tammann Memor
ial Lecture."
3. Staveley, L. A. K., Quart. Rev., 3, 6581 (1949), "Transitions in Solids
and Liquids."
4. Swietoslawski, W., J. Chem. Ed., 23, 18385 (1946), "Phase Rule and the
Action of Gravity."
5. Ubbelohde, A. R., Quart. Rev., 4, 35681 (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 overall 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 GibbsDuhem 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:
PrenticeHall, 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 twocomponent 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 BX 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 GibbsDuhem 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 vaporpressure data. It
can be expressed as
PA = *A PA (69)
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 45, 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 gasliquid systems, but is followed by a
wide variety of fairly dilute solutions and by all solutions in the limit of
extreme dilution.
7. Twocomponent 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 twocomponent system therefore requires a
threedimensional diagram, with coordinates corresponding to pressure,
temperature, and composition. Since a threedimensional 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. Pressurecomposition diagrams. The example of a (PX) diagram in
Fig. 6.3a shows the system ethylene bromidepropylene 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 onephase 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. Pressurecomposition (mole
01 2345. 6 789 10
Fig. 6.3b. Temperaturecomposition
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 twophase 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 twophase region is called a tie line', it connects the liquid and vapor
compositions that are in equilibrium.
The overall 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 overall
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 onephase, allvapor region.
In the twophase 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. Temperaturecomposition diagrams. The temperaturecomposition dia
gram of the liquidvapor equilibrium is the boilingpoint diagram of the
solutions at the constant pressure chosen. If the pressure is one atmosphere,
the boiling points are the normal ones.
The boilingpoint 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 boilingpoint 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 vaporcomposition curve is therefore readily constructed from the
liquidus curve.
10. Fractional distillation. The application of the boilingpoint 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 "bubblecap" 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
bubblecap 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. Boilingpoint 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 bubblecap 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: McGrawHill,
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 boilingpoint 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
boilingpoint 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 boilingpoint 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 molecularweight determination
from the boilingpoint 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 molecularweight 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 "freezingpoint depression curve" and
"solubility curve" are merely two different names for the same thing that
is, a temperature vs. composition curve for a solidliquid equilibrium at
some constant pressure, usually chosen as one atmosphere. Such a diagram
is shown in Fig. 6.13 (p. 147) for the system benzenechloroform. 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 solidliquid 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 freezingpoint depression, or the solubility equation
for ideal solutions, is derived by essentially the same method used for the
boilingpoint 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 boilingpoint 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
molecularweight determination by freezingpoint 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 (622)
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 socalled "countercurrent distribution method."
14. Osmotic pressure. The classical trio of colligative properties, of which
boilingpoint elevation and freezingpoint 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 hydrogenrich to
the hydrogenpoor 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. 353414.
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 611, 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
bisulfidemethylal system, (b) Negative deviation from Raould's Law the PX
diagram of chloroformacetone 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 vaporpressure 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 vaporpressurecomposition 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 chloroformacetone 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 vaporpressure
curve, the vapor and the liquid must have the same composition.
18. Boilingpoint diagrams. The PX diagram in (a), Fig. 6.8, has its
counterpart in the boilingpoint 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 HC1H 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 boilingpoint 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 twophase 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 twophase 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 overall 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 anilinewater 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 anilinerich 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 wateraniline system is
drawn for the constant pressure of
one atmosphere (normalboilingpoint
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 phaseboundary 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. Condensedliquid 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 liquidliquid solubility curves. A classical example is the phenolwater
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) phenolwater system,
(b) tnethylaminewater system, (c) nicotinewater 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 phenolrich phase decreases and the amount of waterrich phase
increases.
Finally at Y the compositions of the two phases become identical,
the phenolrich 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 twophase 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 singlephase 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
triethylaminewater 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 nicotinewater 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 overall 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 vaporpressure 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 GibbsDuhem 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 GIBBSDUHEM 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: McGrawHill, 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. Gassolid 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 solidvapor and
solidliquid 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 twocomponent systems, with only a brief introduction to three
component phase diagrams.
A twocomponent gassolid system in which there is no appreciable
solidsolution 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 pressuretemperature diagram for such a system is therefore similar
to the vaporpressure 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 CaOCO 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 sulfatewater 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 solidgas 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 solidgas or liquidgas 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. Solidliquid equilibria: simple eutectic diagrams. For twocomponent
solidliquid equilibria in which the liquids are completely intersoluble in all
proportions and there is no appreciable solidssolid 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 twophase region, the relative amounts of pure A and residual
solution are given as usual by the ratio of the tieline 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
solidliquid systems. A twocompo
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 Pb20 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 twocomponent system, just as the melting point of one component, is
an invariant point at constant overall 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 constanttemperature 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 OHC 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
phenolcompound diagram, and the other the anilinecompound 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 OHC 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 phenolaniline.
by the coolingcurve 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 silicaalumina 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 coppernickel 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 SnBi 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 PbSnBi : threedimensional 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: McGrawHill, 1935); G. Mas
sing, Introduction to the Theory of Three ComponentSystems (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 twocomponent 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 AuTe 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: McGrawHill, 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. HumeRothery, 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, 1233 (1949), "Symposium on Thermodynamics of
Solutions."
2. Fleer, K. B., /. Chem. Ed., 22, 58892 (1945), "Azeotropism."
3. Hildebrand, J. H., J. Chem. Ed., 25, 7477 (1948), "Ammonia as a
Solvent."
4. Teller, A. J., Chem. Eng., 61, 16888 (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 (14931541), 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 (15961650)
helped to restore the idea of a corpuscular structure of matter. Gassendi
(15921655) 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, GayLussac 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. GayLussac, 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 (15611626) 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 kinetictheory 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 areplaced 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 idealgas 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 largescale 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 idealgas 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 kinetictheory
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 onesixth 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 kinetictheory 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 47T
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
\lnMJ
(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 effusiveflow 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 orificeeffusion 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 idealgas
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 surfacelayer
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 headon 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) Headon collision.
(b) Grazing collision, (c) Rightangle 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^viSj (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 crosssectional area, given by 2nr 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,
onethird 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 crosssectional 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 selfdiffusion coefficient is obtained as
D &c (7.24)
The results of the simple meanfreepath 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= 2.016 626.6
?70.93 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: McGrawHilJ,
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 kinetictheory 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 1 118.
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 rigidsphere
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 atommolecules 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 potentialenergy 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 potentialenergy 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 kinetictheory 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. Onedimensional 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/ Jc
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 onedimensional gas (one degree of freedom of trans
lation), a twodimensional 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 twodimensional 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'seye. In
the molecular case, each individual Fig . 1Mm Distribution of points in two
molecularvelocity 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 molecularvelocity 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 threedimensional 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
477cVc, 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 overall 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 socalled 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 MaxwellBoltzmann 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 socalled 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 dumbbelllike 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 potentialenergy 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 socalled 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 MaxwellBoltzmann 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 (IndoChina). 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 wellshuffled 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 onemillionth 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
freeenergy 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 lefthand 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: McGrawHill, 1938).
4. Knudsen, M., The Kinetic Theory of Gases (London: Methuen, 1950).
5. Loeb, L. B., Kinetic Theory of Gases (New York: McGrawHill, 1927).
ARTICLES
1. Furry, W. H., Am. J. Phys., 16, 6378 (1948), "Diffusion Phenomena in
Gases."
2. Pease, R. N., J. Chem. Ed., 16, 24247, 36673 (1939), "The Kinetic
Theory of Gases."
3. Rabi, I. L, Science in Progress, vol. IV (New Haven: Yale Univ. Press,
1945), 195204, "Streams of Atoms."
4. Rodebush, W. H., /. Chem. Ed., 27, 3943 (1950), "The Dynamics of
Gas Flow."
5. Wheeler, T. S., Endeavour, 11, 4752 (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 onefluid 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 staticelectrical 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 18061807, 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 gasdensity
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 twostep reaction
was postulated :
C 4 H 5 OOK 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 organicchemical 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 wellknown
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 (18941897), 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 onefluid 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, thirtytwo 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
////>.,
^XRAYS
Fig. 8.1. Production of Xrays.
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 earthconnected 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 welldefined 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 oildrop 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 darkroom
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 highvelocity 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 15inch
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 onehalf 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 Xray 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(Z6) (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* Xray 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 radioactivedecay 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 halflife period r, the time required for the activity to be
reduced to onehalf 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. Positiveray 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
(818)
^ '
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,
positiveray 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 Dempstertype 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 massspectrometer 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 positiveray 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 righthand 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 atomicweight 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 78 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 overall 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 pressurediffusion 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 socalled 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 singlestage 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. Radium226 decays by a particle emission with a half life of 1590
years, the product being radon222. 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 thoriumC is 60.5 minutes. How many disintegrations
would occur in 15 minutes from a sample containing initially 1 mg of ThC
(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 U238 is 4.56 x 10 9 years. The final decay product is
Pb206, the intermediate steps being fast compared with the uranium dis
integration. In Lower PreCambrian 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: McGrawHill, 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: McGrawHill, ,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: PrenticeHall, 1952).
ARTICLES
1. Birge, R. T., Am. J. Phys., 13, 6373 (1945), "Values of Atomic Con
stants."
2. Glasstone, S., /. Chem. Ed., 24, 47881 (1947), "William Prout."
3. Hooykaas, R., Chymia, 2, 6580 (1949), "Atomic and Molecular Theory
before Boyle."
4. Jauncey, G. E., Am. J. Phys., 14, 22641 (1946), "The Early Years of
Radioactivity."
5. Lemay, P., and R. E. Oesper, Chymia, 1, 171190 (1948), "Pierre Louis
Dulong."
6. Mayne, K. I., Rep. Prog. Phys., 15, 2448 (1952), "Mass Spectrometry."
7. Rayleigh, Lord, /. Chem. Soc., 46775 (1942), "Sir Joseph J. Thomson."
8. Urey, H. C, Science in Progress, vol. I (New Haven: Yale University
Press, 1939), 3577, "Separation of Isotopes."
9. Winderlich, R., J. Chem. Ed., 26, 35862 (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 massenergy. We can measure massenergy 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 cloudchamber 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
highvelocity 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 highvelocity 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
massspectrographic 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
cloudchamber 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 overall change
in mass and no overall change in energy; massenergy 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 massspectrometer data. The
TABLE 9.1
ATOMIC WEIGHTS (O 16  16.0000)
Atom
Mass Spectrometer
Value
MassEnergy
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
smallscale 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
largescale engineering.
One of the most generally useful of these atomsmashing 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 184in. 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. PrenticeHall, 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 billionvolt 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 Cshaped
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 CurieJoliot 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 hydrogenrich 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 cloudchamber 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 CurieJoliot 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 yray 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 yray photon, the wavelength would be
A  n,c ~ 9.U XIO
The y radiation obtained in 'the annihilation of electronpositron pairs has
either this wavelength or onehalf of it. The latter case corresponds to the
conversion of the masses of both e + and e~~ into a single yray photon. The
reverse process, the production of an electronpositron 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 7rmesons, discovered in 1947 by the Bristol cosmicray 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 massenergy 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 shortrange, 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 longerrange (inverse
square) force is outweighed by the shortrange 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 onetoone 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 packingfraction 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
oddodd nuclei are H 2 , Li 6 , B 10 , N 14 . Not only are the eveneven 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 1025 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 ENERGY6V
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 highvelocity 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 restmass 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 halflife.
Now the hypothetical 36 Kr 96 can get back to the protonneutron 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 liquiddrop 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 aradioactive
with 500 years halflife. 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,
AddisonWesley, 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 highvelocity
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
socalled 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 stellarenergy 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 4e'
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 protonproton 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 selfdiffusion 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 carbon14 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 , RC< 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 onehalf 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 onehalf. 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 (deuteriumsubstituted 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 10g sample of iodobenzene is shaken with 100 ml of a 1 M KI solu
tion containing 2500 counts per min radioiodine. 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 77meson has a mass about 285 times that of the electron; the
/tmeson has a mass about 215 times that of the electron. The 7rmeson
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 200in. 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 cosmicray 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 1ml 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 53day 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 100mg 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. Potassium40 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: AddisonWesley, 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: PrenticeHall, 1954).
9. Libby, W. F., Radiocarbon Dating (Chicago: Univ. of Chicago Press,
1952).
ARTICLES
1. Anderson, C. D., Science in Progress, 7, 236249 (1951), 'The Elementary
Particles of Physics."
2. Curtan, S. C., Quart. Rev., 7, 118 (1953), "Geological Age by Means of
Radioactivity/'
3. Dunning, J. R., Science in Progress, 7, 291355 (1951), "Atomic Structure
and Energy."
4. Hevesy, G. C., J. Chem. Soc., 16181639 (1951), "Radioactive Indicators
in Biochemistry."
5. Pryce, M. H. L., Rep. Prog. Phys., 17, 135 (1954), "Nuclear Shell
Structure."
6. Wilkinson, M. K., Am. J. Phys., 22, 26376 ( 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 MichelsonMorley
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
reexamination 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~(xct) (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 2nkx = 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,
onehalf a wavelength. Midway between the nodes are the positions of
maximum amplitude, or antinodes.
Solutions of the onedimensional 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 threedimensional 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 zeroorder 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 twodimensional 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 threedimensional diffraction
gratings for X rays. This prediction was immediately verified in the critical
experiment of Friedrich, Knipping, and Laue. A typical Xray diffraction
picture is shown in Fig. 13.7 on page 375. The farreaching 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. Blackbody radiation. The first definite failure of the old wave theory
of light was not found in the photoelectric effect, a particularly clearcut case,
but in the study of blackbody 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 socalled 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 blackbody 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 blackbody 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 (18771900) 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 lowfrequency oscillators and
80 per cent in the. highfrequency 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 highfrequency vibrations greatly outnumber the low
frequency ones. The onedimensional 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 highfrequency than lowfrequency 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^%7Tv*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 blackbody 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 blackbody 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
1027 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 blackbody 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
tO
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 energydistribution formula in
excellent agreement with the experimental data for blackbody 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.)
fcjJKftfVl^
MIBffP 8 ''' vCTP
Fig. 10.5b. Spectra of molecular nitrogen. (From Harrison, Lord, and
Loofbourow, Practical Spectroscopy, PrenticeHall, 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 heatcapacity 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\2n.
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\1n. 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 energylevel 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/rfold 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 Xray spectra. The origin of the characteristic Xray
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 Xray spectra are caused by transitions of the
inner electrons. X rays are generated when highvelocity 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 solarsystem 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 stationarystate 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 twodimensional 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 eectronic 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 onefifth 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 Xray 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 mansized objects, but
274 PARTICLES AND WAVES [Chap. 10
the disturbance of atomsized 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. Zeropoint 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 zeropoint 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 waveparticle 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 2nvt. On substitution in the original equation,
this yields
d*W 47T 2 V 2
T? + ^TV = (1034)
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 stationarystate 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 singlevalued, 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 highspeed 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 onedimensional 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 onedimensional 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 etaL, 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 potentialenergy 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 onedimensional result to a threedimensional 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
"onedimensional 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 potentialenergy 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 potentialenergy 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> = AeV* 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: McGrawHill, 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 threedimensional 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
^ ~
Vrr \<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 wavemechanical 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 gasvelocity 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 lightcolored 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 jtype 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/2rr, 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 higherlying
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 quantummechanical 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 rareearth 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* Xray 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, 41, +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 zeropoint 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 onefourth 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 onedimensional 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 Xray 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 Xray 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: McGrawHill, 1935).
6. Pitzer, K. S., Quantum Chemistry (New York: PrenticeHall, 1953).
7. Slater, J. C, Quantum Theory of Matter (New York: McGrawHill, 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, 8084 (1946), "Scattering of XRay
Photons."
2. de Vault, D., /. Chem. Ed., 21, 52634, 57581 (1944), "The Electronic
Structure of the Atom."
3. Glockler, G.,J. Chem. Ed., 18, 41823 (1941), "Teaching the Introduction
to Wave Mechanics."
4. Margenau, H., Am. J. Phys., 13, 7395 (1945); 12, 11930, 24768 (1944),
"Atomic and Molecular Theory Since Bohr."
5. Meggers, W. F., /. Opt. Soc. Am., 41, 1438 (1951), "Fundamental
Research in Atomic Spectra."
6. Zworykin, V. K., Science in Progress, vol. Ill (New Haven: Yale Univ.
Press, 1942), pp. 69107, "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 inertgas 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 threedimensional 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
(111)
This potentialenergy 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
potentialenergy 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 quantummechanical 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 twoelectron 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 twoelectron 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 HH 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 HH 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 lefthand 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 HH 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 HH 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, 2C2/4, 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 HeitlerLondon 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.
HeitlerLondon 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 HeitlerLondon theory is an example of the valencebond (V.B.)
approach to molecular structure.
5. Molecular orbitals. An alternative to the HeitlerLondon method of
applying quantum mechanics to molecular problems is the method of mole
cular orbitals, developed by Hund, Mulliken, and LennardJones. 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 outershell 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
waspwaisted 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 fy 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 (HeitlerLondon) 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 socalled 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 Vy<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 lowlying
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
sausageshaped 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 valencebond 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 valencebond 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 overall 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 hydrogenbonded 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 parallelplate 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 overall 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
J47r(erP) (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
DJ5+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
maoF (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
FB + *?* (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 ClausiusMossotti 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: McGrawHill, 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  2n 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 ClausiusMossotti 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 nonpolar
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 ClausiusMossotti 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 dipolemoment 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 symtrichlorobenzene 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 />diOH 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 highfrequency 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 lowerfrequency 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 dielectricconstant 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
LorenzLorentz 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 LorenzLorentz 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
temperaturedependence 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^ln, 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 lowpower radiofrequency 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 onetenth 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
electrondiffraction 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 4rr. Hence the required
average intensity becomes
, 4 ^o 2 r r jo r L^ 2 \] , M
lav ~ cos 2 2rr  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 carboncarbon 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
HgCl
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 so
.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 303311. 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 energylevel 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. Energylevel 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 vibrationrotation 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 farinfrared 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 hydrogenatom
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 rigidrotator 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: PrenticeHall, 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 nearinfrared 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
uncertaintyprinciple 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 potentialenergy 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 (1126)
In harmonic vibration the restoring force is directly proportional to the
displacement r. The potentialenergy curve is parabolic and dissociation can
never take place. Actual potentialenergy 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 potentialenergy 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 nearinfrared 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 farinfrared 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.
Potentialenergy 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 highpressure gaseousdischarge 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 cathoderay 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 FranckCondon 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 FranckCondon 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 midpoint 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 7rtype orbitais, that are readily raised
to excited orbitais.
In other cases, resonance gives rise to a series of lowlying 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 nearultraviolet 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 7relectron 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 2537A 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 vibrationrotational 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.
XYZ
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 highfrequency vibrations are those that stretch the bonds, and
the lower frequencies are bondbending 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 OO
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 fingerprinting of a suspect citizen. Some typical bond frequencies are
summarized in Table 1 1 .6.
TABLE 11.6
BONDFREQUENCY INTERVALS FOR INFRARED SPECTRA OF GASES*
Group
H <
H N<
H C=C<
I
H S
N^C
cc
Frequency Interval
35003700
33003500
33003400
30003100
25502650
22002300
21702270
Group
Frequency Interval
cm" 1
O=C<
17001850
>c=c<
15501650
s=c<
15001600
F C
11001300
Cl C
700800
Br C
500600
I C
400500
* 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 socalled 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 singlebond energies. 23 If the actual bond is markedly
polarized (partial ionic character), or if through resonance it acquires some
doublebond 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 .CHCH: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 zeropoint
energy of CO in cal per mole.
9. Sketch the potentialenergy 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 SchumannRunge 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
lowlying 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 40kv
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 477/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: PrenticeHall, 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, 36574 (1947), "The FranckCondon
Principle and Related Topics."
2. Klotz, I. M., /. Chem. Ed., 22, 32836 (1945), "Ultraviolet Absorption
Spectroscopy."
3. Mills, W. H., J. Chem. Soc., 1942, 45766 (1942), "The Basis of Stereo
chemistry."
4. Pake, G. E., Am. J. Phys., 18, 43873 (1950), "Nuclear Magnetic
Resonance."
5. Pauling, L., /. Chem. Soc., 146167 (1948), "The Modern Theory of
Valency."
6. Purcell, E. M., Science, 118, 43136 (1953), "Nuclear Magnetic
Resonance."
346 THE STRUCTURE OF MOLECULES [Chap. 11
7. Selwood, P. W., /. Chem. Ed., 79, 18188 (1942), "Magnetism and
Molecular Structure."
8. Spurr, R., and L. Pauling, J. Chem. Ed., 18, 45865 (1941), "Electron
Diffraction of Gases."
9. Sugden, S., J. Chem. Soc., 32833 (1943), "Magnetochemistry."
10. Thompson, H. W., J. Chem. Soc. 9 18392 (1944), "Infrared Measure
ments in Chemistry."
11. Wilson, E. B., Ann. Rev. Phys. Chem., 2, 15176 (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
fiftytwo. 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 10minute 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 largescale 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 molecularkinetic 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: PrenticeHall, 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
*** (1212)
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 largescale
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 ZS&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 sumoverstates
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 temperatureindependent
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 statisticalmechanical 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 nuclearspin orientations, there will be a residual
entropy at 0K. In chemical problems such effects are of no importance,
since in any chemical reaction the nuclearspin 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 ThirdLaw values based on heatcapacity data.
Examples are given in Table 12.1.
TABLE 12.1
COMPARISON OF STATISTICAL (SPECTROSCOPIC) AND THIRDLAW (HEATCAPACITY)
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, heatcapacity 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 ThirdLaw 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 = eW* 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 1020 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 "doubleup" and hence the assumption of single occupancy is a good one. For
a further discussion, see Tolman, he. cit., pp. 569572.
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 m6.63 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 rigidrotator 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 ,
a3;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 (1237)
(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)
(FE )
hv
(E  )
(FEj
* 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 lowtemperature
energies, entropies, and specific heats. The data in Table 7.6 (page 192)
confirm this conclusion. In Fig. 12.1, the heatcapacity 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 potentialenergy 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
nuclearspin 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 nuclearspin weights are g NS  3 for ortho, corresponding
to allowed directions 41,0, 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, liquidair 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 liquidair temperatures, and then warming the
gas in the absence of catalyst.
The calculated heat capacities of pure />H 2 , pure oH 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 hightempera
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 parahydrogen, pure orthohydrogen,
and 3o 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 nuclearspin 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) FermiDirac
(2) BoseEinstein
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 BoseEinstein statistics,
indicating that they are complex particles and recalling the formation of
electronpositron pairs from Xray 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 FermiDirac 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 C1C1 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, heatcapacity data were listed for a calculation of the
ThirdLaw 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:
PrenticeHall, 1954).
3. Gurney, R. W., Introduction to Statistical Mechanics (New York:
McGrawHill, 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, 8498 (1946), "Practical Statistics for
Practical Physicists/'
2. Eyring, H., and J. Walter, /. Chem. Ed., 18, 7378 (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 Xray 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
OLft=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. Twodimensional 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 threedimensional 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 threedimensional array. The choice
of the 14 lattices is somewhat arbitrary, since in certain cases alternative
descriptions are possible.
TRICLINIC
A
2 SIMPLE 3 SIDECENTERED
MONOCLINIC MONOCLINIC
\
4. SIMPLE 5 ENDCENTERED
ORTHORHOMBIC ORTHORHOMBIC
6. FACECENTERED 7 BODY
ORTHORHOMBIC CENTERED
ORTHORHOMBIC
9 RHOMBOHEDRAL 10 SIMPLE
II BODYCENTERED
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 endcentered points can always be reduced to
one that does not (primitive lattice). Thus the facecentered 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 4fold axes, four 3fold axes, six 2fold axes, three mirror planes
perpendicular to the 4fold axes, six mirror planes perpendicular to the 2fold
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 2fold axes with the six planes normal to
them, and the 4fold axes have been reduced to 2fold 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 Xray diffraction.
6. Xray 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 Xrays. (From Lapp and Andrews,
Nuclear Radiation Physics, 2nd Ed., PrenticeHall, 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 Xray 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 spjicegroup 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 Xray 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 threedimensional 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 socalled 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 Lauetype 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 Xray beam by the amount of ioniza
tion it produced, and he found that the characteristic Xray 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 Xray 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
/7A2</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 Xray
IXRAY BEAM
LV/V//J
SLIT SYSTEM
IONIZATION
CHAMBER
**" ^^J^^
DIVIDED
SCALE
/TO ELECTROMETER
Fig. 13.9. Bragg Xray 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, bodycentered, or facecentered.
By comparing the spacings calculated from Xray 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 Xray 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 Xray 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 "firstorder 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) bodycentered cubic;
(c) facecentered 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 bodycentered cubic lattice.
The (110) planes, as in the simplecubic case, pass through all the lattice
points, and a strong firstorder (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
firstorder (100) reflection will appear. If, however, the atoms are different,
the firstorder (100) will still appear, but with a reduced intensity given
by the difference between the scatterings from the two interleaved sets of
planes.
The secondorder 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 bodycentered 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 firstorder (111) will be weak or extinguished, but the secondorder (111),
i.e. the (222) planes, will give strong scattering. If we examine successive
planes (hkl) in this way, we find for the bodycentered 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 .         
bodycentered cubic :    III
facecentered 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 facecentered 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 Xray 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 facecentered
cubic array of Na+ ions and an interpenetrating facecentered 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 firstorder (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 firstorder reflections are
altogether extinguished by interference. Thus the postulated structure is in
complete agreement with the experimental Xray 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 Xray 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 Geigercounter tube replaces the electrom
eter and ionization chamber.
9. The powder method. The simplest technique for obtaining Xray 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 Xray beam, a
CYLINDRICAL
" CAMERA
POWDER
SPECIMEN
XRAY
BEAM
FILM
Fig. 13.13. The powder method. Powder picture of sodium chloride, CuK 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 thinwalled 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 Xray 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 Xray powder picture is shown in (c), Fig. 13.13.
It may be compared with the electrondiffraction 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 unitcell 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 unitcell dimensions can be calculated from
highindex spacings. The general formulae giving the interplanar spacings
are straightforward derivations from analytical geometry. 5
10. Rotatingcrystal method. The rotatingsinglecrystal 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 halfmillimeter 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 crystalstructure
determinations.
11. Crystalstructure determinations: the structure factor. The problem of
reconstructing a crystal structure from the intensities of the various Xray
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 Xray 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 Xray 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 XRay 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. Xray 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 (twodimensional) 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,
^' (134)
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. Structurefactor 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 facecentered 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 facecentered 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 threedimensional 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 Xray 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 Xray 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 twodimensional 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 Xray
crystallography, since it was achieved before the organic chemists knew the
structural formula.
13. Neutron diffraction. Not only Xray 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 rocksalt 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 Xray 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 neutrondiffraction analysis. 13
14. Closest packing of spheres. Quite a while before the first Xray 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 hexagonalclosestpacked (hep) and the
13 S. W. Peterson and H. A. Levy, /. Chem. Phys., 20, 704 (1952).
Sec. 14]
CRYSTALS
391
cubicclosestpacked (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 facecentered
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 hightemperature 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 bodycenteredcubic structure. Some examples are collected in Table
13.3. Other structures include the following: 14 the diamondtype cubic of
TABLE 13.3
STRUCTURES OF THE METALS
Cubic Closest Packed
(fee) or (ccp)
Hexagonal Closest Packed
(hep)
BodyCentered 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 facecentered tetragonal, a distorted fee, of
ymanganese and indium; the rhombohedral layered structures of bismuth,
arsenic, and antimony; and the bodycentered 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 socalled 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 atomicorbital 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 dipoledipole and dipolepolarization 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 threedimensional 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 sodiumchloride 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 overall 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 facecentered cubic lattices. Each point
in one lattice is surrounded tetrahedrally by four equidistant points in the
other lattice. This arrangement constitutes a threedimensional 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 outershell 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 localizedpair (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 threedimen
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,
eightmembered 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?^ intermediatetype 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: raregas 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 therocksalt 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 zincblende 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 electronpair 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 onedimensional 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 potentialenergy 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^/^Ss 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
potentialenergy 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 potentialenergy 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 Xray emission spectroscopy. 16 For example, if an electron is driven out of
the \s level in sodium metal (Fig. 13.23b) the K a Xray 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, 761 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 bandpass 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 CuAu and AgAu.
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 facecenteredcubic structure and dissolves up to about
38 per cent zinc in this a phase. Then the bodycenteredcubic 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. HumeRothery 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 HUMEROTHERY 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 42
2
Cu 3 Al
343
4
Cu 6 Sn
544
6
CoAl
043
2
FeAl
043
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
542x8
13
21 :13
542x8
13
21 :13
+ 2 x 21
26
42 : 26
943x4
13
21 : 13
913x4
13
21 :13
31 44 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,
142x3
142x3
344
344
543x3
543x3
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 bodycentered /? brass structure illustrates another interesting
property of some alloy systems, the orderdisorder 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
bodycentered 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 ionicradius ratios of a number of CA compounds
are summarized in Table 13.6. The radiusratio 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
0.7320.414
0.4140.225
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 cadmiumiodide 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., onehalf 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) Threedimensional 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) hexagonaltype sheets;
(c) extended chains; (d) threedimensional 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 BornHaber 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 inversesquare 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 socalled 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 BornHaber 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 alkalihalide 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 BornHaber 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 BORNHABER 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 hightemperature 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 heatcapacity
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 blackbody 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, McGrawHill, 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
heatcapacity data to absolute zero in connection with studies based on the
Third Law of Thermodynamics (cf. page 90).
PROBLEMS
1. Show that a facecenteredcubic 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 DebyeScherrer 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: McGrawHill, 1952).
2. Bragg, W. H., and W. L. Bragg, The Crystalline State, vol. I (London:
Bell, 1934).
3. Buerger, M. J., XRay 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. HumeRothery, W., Atomic Theory for Students of Metallurgy (London:
Institute of Metals, 1947).
7. Kittel, C., Introduction to SolidState Physics (New York: Wiley, 1953).
8. Lonsdale, K., Crystals and XRays (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., 64366 (1946), "The Past and Future of
XRay Crystallography."
2. DuBridge, L. A., Am. J. Phys., 16, 19198 (1948), "Electron Emission
from Metal Surfaces."
3. Frank, F. C., Adv. Phys., /, 91109 (1952), "Crystal Growth and Disloca
tions."
4. Fuoss, R. M., J. Chem. Ed., 19, 19093, 23135 (1942), "Electrical Pro
perties of Solids."
5. Lonsdale, K., Endeavour, 6, 13946 (1947), "XRays and the Carbon
Atom."
6. Robertson, J. M., /. Chem. Soc., 24957 (1945), "Diffraction Methods in
Modern Structural Chemistry."
7. Sidhu, S. S., Am. J. Phys. 9 16, 199205 (1948), "Structure of Cubic
Crystals."
8. Smoluchowski, R., and J. S. Koehler, Ann. Rev. Phys. Chem., 2, 187216
(1951), "Band Theory and Crystal Structure."
9. Weisskopf, V. F., Am. J. Phys., 11, 11112 (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 freeenergy
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. Twodimensional 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. Twodimensional 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 closepacked 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 longrange
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 socalled "fluid state" of
matter exists. On the other hand, the liquid may be considered as similar to
a crystal, except that the wellordered arrangement of units extends over
a short range only, five or six molecular diameters, instead of over the whole
specimen. This is sometimes called "shortrange order and longrange 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 imperfectgas theory of liquids would be suitable close to the critical
point; the disorderedcrystal 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 Xray diffraction.
3. Xray diffraction of liquids. The study of the Xray 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 Xray 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 closepacked 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 FILMCm
5
10 II T2
Fig. 14.2.
4 5 6 7 8 9
(b) f ANGSTROM UNITS
(a) Photometric tracing of liquidmercury picture; (b) radial
distribution function for liquid mercury.
is transformed on melting into a closepacked 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 shortrange order and increasing
disorder at longer range. In order to obtain the maxima corresponding to
smaller interplanar spacings or higher orders of diffraction, the longrange
order of the crystal must be present.
The diffraction maxima observed with crystals or liquids should be dis
tinguished from those obtained by the Xray 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 threedimensional 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 Xray 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 radialdistribution
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 liquidmercury 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 liquidstructure investigations. Xray 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 closepacked 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 closepacked 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 Xray 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 radialdistribution 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 closestpacked 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 Xray 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 /7anisalaminocinnamate,
: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 liquidcrystalline form.
Liquid crystals tend to occur in compounds whose molecules are
markedly unsymmetrical in shape. For example, in the crystalline state
longchain 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 welldefined 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 liquidcrystalline 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 twodimensional 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. Twodimensional 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 closepacked 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 disorderedcrystai
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 socalled 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
/iOctane
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. Intelmolecular 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 longrange forces and shortrange 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 longrange 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. LennardJones, 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 snapshotdipole 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 quantummechanical 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 socalled 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 culdesac.
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: McGrawHill, 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 longcontinued 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 kineticmolecular 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 viscousflow process requires a free space about onethird 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 onequarter 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, 31017 (1954), 'The Glassy State."
2. Hildebrand, J. H., Proc. Phys. Soc. (London), 56, 22139 (1944), 'The
Liquid State."
434 LIQUIDS [Chap. 14
3. Hirschfelder, J. O., J. Chem. Ed., 16, 54044 (1939), 'The Structure of
Liquids."
4. Kirkwood, J. G., Science in Progress, vol. Ill (New Haven: Yale Uni
versity Press, 1942), 208221, 'The Structure of Liquids."
5. Rowlinson, J. S., Quart. Rev., 8, 16891 (1954), "Tntermolecular Forces
and Properties of Matter."
6. Ubbelohde, A. R., Quart. Rev., 4, 35681 (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. 591598.
435
436
ELECTROCHEMISTRY
[Chap. 15
Here / is the length and A the crosssectional 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 alternatingcurrent
bridge, such as that shown in (a), Fig. 15.1. With ac frequencies in the audio
range (10004000 cycles per sec) the direction of the current changes so
rapidly that polarization effects are eliminated. One difficulty with the ac
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 cathoderay oscilloscope. The voltage from the bridge midpoint 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 phaseshifting 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 osmoticpressure 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 massaction 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 iondipole attractions.
Thus in many cases the solution of ionic salts is an exothermic reaction.
The equilibrium position is of course determined by the freeenergy 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 onesquarecentimeter 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 overall 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. Movingboundary 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 crosssectional 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.
Anion
>.
H+
349.82
OH
198.0
Li+
38.69
ci
76.34
Na+
50.11
Br~
78.4
K+
73.52
I
76.8
NH 4 +
Ag+
iCa++
73.4
61.92
59.50
63.64
NO S 
CH,COO
C10 4 ~
71.44
40.9
68.0
79.8
i Sr+ +
59.46
i Mg^ +
53.06
* u+++
69.6
D. Maclnnes, Principles of Electrochemistry (New York: Reinhold, 1939).
Q
446
ELECTROCHEMISTRY
[Chap. 15
be combined to yield values for the equivalent conductivities A of a wide
variety of electrolytes.
It has been mentioned that ions in solution are undoubtedly hydrated,
so that the observed transport numbers are actually not those of the "bare
ions" but of solvated ions. It was pointed out by Nernst that if a nonionized
substance, such as urea or a sugar, is added to the solution in a Hittorf
transference apparatus, its concentration in the end compartments should
change as a result of the transport of water by the hydrated ions. This
expectation was experimentally realized. From the equivalent ionic conduc
tivities it is possible to calculate the mobilities of the ions by use of eq.
(15.13). Some results are given in Table 15.4. The effect of hydration is
shown in the set of values for Li+, Na+, K+. Although Li+ is undoubtedly
the smallest ion, it has the lowest mobility; i.e., the resistance to its motion
through the solution is highest. This resistance must be partly due to a
tightly held sheath of water molecules, bound by the intense electric field of
the small ion.
TABLE 15.4
MOBILITIES OF IONS IN WATER SOLUTIONS AT 25C
(cm 2 sec" 1 volt" 1 )
Mobility
Anions
36.30 x 10~ 4
OH
7.62 x 10~ 4
6.59 x 10" 4
S0 4 ~
Cl
5.19 x 10~ 4
4.01 x 10" 4
N0 3
HC0 3 
Mobility
20.50 x 10~ 4
8.27 x 10 4
7.91 x 10 4
7.40 x 10~ 4
4.61 X 10*
Cations
H+
K+
Ba 4
Na+
The passage of an ion t