TIGHT BINGING BOOK a L< OU 166588 >m > Q^ 73 7) ^ CD :< co Preface to the First Edition This book is an account of physical chemistry designed for stude.its in the sciences and in engineering. It should also prove useful to chemists in industry who desire a review of the subject. The treatment is somewhat more precise than is customary in elementary books, and most of the important relationships have been given at least a heuristic derivation from fundamental principles. A prerequisite knowledge of calculus, college physics, and two years of college chemistry is assumed. The difficulty in elementary physical chemistry lies not in the mathematics itself, but in the application of simple mathematics to complex physical situations. This statement is apt to be small comfort to the beginner, who finds in physical chemistry his 'ftrst experience with such applied mathematics. The familiar x's and /s of the calculus course are replaced by a bewildering array of electrons, energy levels, and probability functions. By the time these ingredients are mixed well with a few integration signs, it is not difficult to become convinced that one is dealing with an extremely abstruse subject. Yet the alternative is to avoid the integration signs and to present a series of final equations with little indication of their origins, and such a procedure is likely to make physical chemistry not only abstruse but also permanently mysterious. The derivations are important because the essence of the subject is not in the answers we have today, but in the procedure that must be followed to obtain these and tomorrow's answers. The student should try not only to remember facts but also to learn methods. There is more material included in this book than can profitably be dis- cussed in the usual two-semester course. There has been a growing tendency to extend the course in basic physical chemistry to three semesters. In our own course we do not attempt to cover the material on atomic and nuclear physics in formal lectures. These subjects are included in the text because many students in chemistry, and most in chemical engineering, do not acquire sufficient familiarity with them in their physics courses. Since the treatment in these sections is fairly descriptive, they may conveniently be used for independent reading. In writing a book on as broad a subject as this, the author incurs an indebtedness to so many previous workers in the field that proper acknow- ledgement becomes impossible. Great assistance was obtained from many excellent standard reference works and monographs. To my colleagues Hugh M. Hulburt, Keith J. Laidler, and Francis O. Rice, I am indebted for many helpful suggestions and comments. The skillful work of Lorraine Lawrence, R.S.C.J., in reading both galley and page proofs, was an invaluable assistance. I wish to thank the staff of Prentice-Hall, Inc. for their understanding cooperation in bringing thfc vi PREFACE book to press. Last, but by no means least, are the thanks due to my wife, Patricia Moore, who undertook many difficult tasks in the preparation of the manuscript. PREFACE TO THE SECOND EDITION In preparing the second edition of this book, numerous corrections of details and improvements in presentation have been made in every chapter, but the general plan of the book has not been altered. My fellow physical chemists have contributed generously of their time and experience, suggesting many desirable changes. Special thanks in this regard are due to R. M. Noyes, R. E. Powell, A. V. Tobolsky, A. A. Frost, and C. O'Briain. A new chapter on photochemistry has been added, and recent advances in nuclear, atomic, and molecular structure have been described. W. J. MOORE Btoomington, Indiana Contents 1. The Description of Physicochemical Systems . . 1 1. The description of our universe, /. 2. Physical chemistry, /. 3. Mechanics: force, 2. 4. Work and energy, .?. 5. Equilibrium, 5. 6. The thermal properties of matter, 6. 1. Definition of temperature, 8. 8. The equation of state, 8. 9. Gas thermometry: the ideal gas, JO. 10. Relationships of pressure, volume, and temperature, 12. 11. Law of corresponding states, 14. 12. Equations of state for gases, 75, 13. The critical region, 16. 14. The van der Waals equation and lique- faction of gases, 18. 15. Other equations of state,' 19. 16. Heat, 19. 17. Work in thermodynamic systems, 21. 18. Reversible processes, 22. 19. Mciximum work, 23. 20. Thermodynamics and thermostatics, 23. 2. The First Law of Thermodynamics 27 *1. The history of the First Law, 27. 2. Formulation of the First Law, 28. 3? The nature of internal energy, 28. 4. Properties of exact differ- entials, 29. 5? Adiabatic and isothermal processes, 30. 6? The heat content or enthalpy, 30. 1? Heat capacities, 31. 8. The Joule experi- ment, 32. 9. The Joule-Thomson experiment, 33. 10. Application of the First Law to ideal gases, 34. 1 1 . Examples of ideal-gas calcula- tions, 36. 12. Thermochemistry heats of reaction, 38. 13. Heats of formation, 39. 14. Experimental measurements of reaction heats, 40. 15. Heats of solution, 41. 16. Temperature dependence of reaction heats, 43. 17. Chemical affinity, 45. 3. The Second Law of Thermodynamics .... 48 1. The efficiency of heat engines, 48. 2. The Carnot cycle, 48. 3. The Second Law of Thermodynamics, 51. 4. The thermodynamic temper- ature scale, 51. 5. Application to ideal gases, 53. 6. Entropy, 53. 1. The inequality of Clausius, 55. 8. Entropy changes in an ideal gas, 55. 9. Entropy changes in isolated systems, 56- 10. Change of entropy in changes of state of aggregation, 58. 1 r? Entropy and equilibrium, 58. 12. The free energy and work functions, 59. 13. Free energy and equilibrium, 61. 14. Pressure dependence of the free energy, 61. 15. Temperature dependence of free energy, 62. 1 6. Variation of entropy with temperature and pressure, 63. 17. The entropy of mixing, 64. 18. The calculation of thermodynamic relations, 64. 4. Thermodynamics and Chemical Equilibrium . . 69 1. Chemical affinity, 69. 2. Free energy and chemical affinity, 71. 3. Free-energy and cell reactions, 72. 4. Standard free energies, 74. 5. Free energy and equilibrium constant of ideal gas reactions, 75. 6. The measurement of homogeneous gas equilibria, 7? 7. The principle viii CONTENTS of Le Chatelier, 79. 8. Pressure dependence of equilibrium constant, 80. 9. Effect of an inert gas on equilibrium, 81. 10. Temperature dependence of the equilibrium constant, 83. 11. Equilibrium constants from thermal data, 85. 12. The approach to absolute zero, 85. 13. The Third Law of Thermodynamics, 87. 14. Third-law entropies, 89. 15. General theory of chemical equilibrium: the chemical potential, 91. 16. The fugacity, 93. 17. Use of fugacity in equilibrium calcula- tions, 95. 5. Changes of State 99 1. Phase equilibria, 99. 2. Components, 99. 3. Degrees of freedom, 100. 4. Conditions for equilibrium between phases, 101. 5. The phase rule, 702. 6. Systems of one component water, 104. 1. The Clapey- ron-Clausius equation, 105. 8. Vapor pressure and external pressure, 107. 9. Experimental measurement of vapor pressure, 108. 10. Solid- solid transformations the sulfur system, 109. 11. Enantiotropism and monotropism, 111. 12. Second-order transitions, 112. 13. High- pressure studies, 112. 6. Solutions and Phase Equilibria 116 I. The description of solutions, 116. 2. Partial molar quantities: partial molar volume, 116. 3. The determination of partial molar quantities, 118. 4. The ideal solution Raoult's Law, 120. 5. Equil- ibria in ideal solutions, 722. 6. Henry's Law, 722. 7. Two-component systems, 123. 8. Pressure-composition diagrams, 723. 9. Temper- ature-composition diagrams, 725. 10. Fractional distillation, 725. II. Boiling-point elevation, 126. 12. Solid and liquid phases in equil- ibrium, 128. 13. The Distribution Law, 130. 14. Osmotic pressure, 757. 1 5. Measurement of osmotic pressure, 133. 16. Osmotic pressure and vapor pressure, 134\+ 17. Deviations from Raoult's Law, 135. 18. Boiling-point diagrams, 736. 19. Partial miscibility, 737. 20. Con- densed-liquid systems, 739. 21. Thermodynamics of nonideal solu- tions: the activity, 747. 22. Chemical equilibria in nonideal solutions, 143. 23. Gas-solid equilibria, 144. 24. Equilibrium constant in solid- gas reactions, 145. 25. Solid-liquid equilibria: simple eutectic dia- grams, 145, 26. Cooling curves, 147. 27. Compound formation, 148. 28. Solid compounds with incongruent melting points, 149. 29. Solid solutions, 750. 30. Limited solid-solid solubility, 757. 31. The iron- carbon diagram, 752. 32. Three-component systems, 753. 33. System with ternary eutectic, 154. 1. The Kinetic Theory 160 1 . The beginning of the atom, 160. 2. The renascence of the atom, 767. 3, Atoms and molecules, 762. 4. The kinetic theory of heat, 763. 5. The pressure of a gas, 764. 6. Kinetic energy and temperature, 765. 7. Molecular speeds, 766. 8. Molecular effusion, 766. 9. Imperfect gases van \der Waal's equation, 769. 10. Collisions between mole- cules, 777. IK Mean free paths, 772. 12. The viscosity of a gas, 773. 13. Kinetic thec 4 *y of gas viscosity, 775. 14. Thermal conductivity CONTENTS ix and diffusion, 777, 15. Avogadro's Number and molecular dimen- sions, 178. 16. The softening of the atom, 180. 17. The distribution of molecular velocities, 181. 18. The barometric formula, 182. 19. The distribution of kinetic energies, 183. 20. Consequences of the distribu- tion law, 183. 21. Distribution law in three dimensions, 186. 22. The average speed, 187. 23. The equipartition of energy, J88. 24. Rota- tion and vibration of diatomic molecules, 189. 26. The equipartjtion principle and the heat capacity of gases, 792. 27. Brownian motion, 193. 28. Thermodynamics and Brownian motion, 194. 29. Entropy and probability, 795. 8. The Structure of the Atom 200 1. Electricity, 200. 2. Faraday's Laws and electrochemical equiva- lents, 201. 3. The development of valence theory, 202. 4. The Periodic Law, 204. 5. The discharge of electricity through gases, 205. 6. The electron, 205. 7. The ratio of charge to mass of the cathode particles, 206. 8. The charge of the electron, 209. 9 Radioactivity, 277. 10. The nuclear atom, 272. 1 1. X-rays and atomic number, 213. 12. The radioactive disintegration series, 27 3. 13. Isotopes, 216. 14. Positive-ray analysis, 216. 15. Mass spectra the Dempster method, 218. 16. Mass spectra Aston's mass spectrograph, 279. 17. Atomic weights and isotopes, 227. 18. Separation of isotopes, 223. 19. Heavy hydrogen, 225. 9. Nuclear Chemistry and Physics 228 1. Mass and energy, 228. 2. Artificial disintegration of atomic nuclei, 229. 3. Methods for obtaining nuclear projectiles, 237. 4. The photon, 232. 5. The neutron, 234. 6. Positron, meson, neutrino, 2J5. 7. The structure of the nucleus, 236. 8. Neutrons and nuclei, 238. 9. Nuclear reactions, 240. 10. Nuclear fission, 241. 11. The trans- uranimTLj^meB4^^2<O. 12, Nuclear chain reactions., 243. I IJEnergy production by the stars, 244. 14. Tracers, 245. 15. Nuclear spin, 247. 10. Particles and Waves 251 1. The dual nature of light, 257. 2. Periodic and wave motion, 257. 3. Stationary waves, 253. 4. Interference and diffraction, 255. 5. Black-body radiation, 257. 6. Plank's distribution law, 259. 7. Atomic spectra, 261. 8. The Bohr theory, 252. 9. Spectra of the alkali metals, 265. 10. Space quantization, 267. 11. Dissociation as series limit, 268. 12. The origin of X-ray spectra, 268. 13. Particles and waves, 269. 14. Electron diffraction, 277. 15. The uncertainty principle, 272. 16. Waves and the uncertainty principle, 274. 17. Zero-point energy, 275. 18. Wave mechanics the Schrodinger equation, 275. 19. Interpretation of the y) functions, 276. 20. Solution of wave equation the particle in a box, 277. 21. The tunnel effect, 279. 22. The hydrogen atom, 280. 23. The radial wave functions, 282. 24. The spinning electron, 284. 25. The PauJi Exclusion Principle, 285. 26, Structure of the periodic table, 285. 27. Atomic energy levels, 287. x CONTENTS 11. The Structure of Molecules ....... 295 1. The development of valence theory, 295. 2. The ionic bond, 295. 3. The covalent bond, 297. 4. Calculation of the energy in H-H mole- cule, 301. 5. Molecular orbitals, 303. 6. Homonuclear diatomic mole- cules, 303. 1. Heteronuclear diatomic molecules, 307. 8. Comparison of M.O. and V.B. methods, 307. 9. Directed valence, 308. 10. Non- localized molecular orbitals, 310. 11. Resonance between valence- bond structures, 311. 12. The hydrogen bond, 313. 13. Pipole_ momejj*s7 314. 14. Polarization of dielectrics, 314. 15. The induced -^polarization, 316. 16. Determination of the dipole moment, 316. 17. Dipole moments and molecular structure, 319. 18. Polarization and refractivity, 320. 19. Dipole moments by combining dielectric con- stant and refractive index measurements, 321. 20. Magnetism and molecular structure, 322. 21. Nuclear paramagnetism, 324. 23. Application of Wierl equation to experimental data, 329. 24. Mole- cular spectra, 331. 25. Rotational levels far-infrared spectra, 333. 26. Internuclear distances from rotation spectra, 334. 21. Vibrational energy levels, 334. 28. Microwave spectroscopy, 336. 29. Electronic band spectra, 337. 30. Color and resonance, 339. 31. Raman spectra, 340. 32. Molecular data from spectroscopy, 341. 33. Bond energies, 342. 12. Chemical Statistics v ......... 347 1. The statistical method, 347. 2. Probability of a distribution, 348. 3. The Boltzmann distribution, 349. 4. Internal energy and heat capacity, 552. 5. Entropy and the Third Law, 352. 6. Free energy and pressure, 354. 1. Evaluation of molar partition functions, 354. 8. Monatomic gases translational partition function, 356. 9. Diatomic molecules rotational partition function, 358. 10. Polyatomic mole- cules rotational partition 'function, 359. 1 1 . Vibrational partition function, 359. 12. Equilibrium constant for ideal gas reactions, 361. \ 3. The heat capacity of gases, 361. 14. The electronic partition func- tion, 363. 1 5. Internal rotation, 363. 1 6. The hydrogen molecules, 363. 17. Quantum statistics, 365. 13. Crystals <2 .......... 369 1 . The growth and form of crystals, 369. 2. The crystal systems, 370. 3. Lattices and crystal structures, 371. 4. Symmetry properties, 372. 5. Space groups, 374. 6. X-ray crystallography, 375. 7. The Bragg treatment, 376. 8. The structures of NaCl and KC1, 377. 9. The powder method, 382. 10. Rotating-crystal method, 383. 11. Crystal- structure determinations: the structure factor, 384. 12. Fourier syntheses, 387. 13. Neutron diffraction, 389. 14. Closest packing of spheres, 390. 15. Binding in crystals, 392. 16. The bond model, 392. 17. The band model, 395. 18. Semiconductors, 398. 19. Brillouin zones, 399. 20. Alloy systems electron compounds, 399. 21. Ionic crystals, 401. 22. Coordination polyhedra and Pauling's Rule, 403. 23. Crystal energy the Born-Haber cycle, 405. 24. Statistical thermo- dynamics of crystals: the Einstein model, 406. 25. The Debye model, 408. CONTENTS xi 14. Liquids 413 1. The liquid state, 413. 2. Approaches to a theory for liquids, 415. 3. X-ray diffraction of liquids, 4/5. 4. Results of liquid-structure investigations, 417. 5. Liquid crystals, 418. 6. Rubbers, 420. 7. Glasses, 422. 8. Melting, 422. 9. Cohesion of liquids the internal pressure, 422. 10. Intermolecular forces, 424. 11. Equation of state and intermolecular forces, 426. 12. The free volume and holes in liquids, 428. 13. The flow of liquids, 430. 14. Theory of viscosity, 431. 15. Electrochemistry 435 1. Electrochemistry: coulometers, 435. 2. Conductivity measure- ments, 435. 3. Equivalent conductivities, 437. 4. The Arrhenius ionization theory, 439. 5. Transport numbers and mobilities, 442. 6. Measurement of transport numbers Hittorf method, 442. 7. Transport numbers moving boundary method, 444. 8. Results of transference experiments, 445. 9. Mobilities of hydrogen and hydroxyl ions, 447. 10. Diffusion and ionic mobility, 447. 1 1 . A solu- tion of the diffusion equation, 448. 12. Failures of the Arrhenius theory, 450. 1 3. Activities and standard states, 451. 14. Ion activities, 454. 15. Activity coefficients from freezing points, 455. 16. Activity coefficients from solubilities, 456. 17. Results of activity-coefficient measurements, 457. IS^^rfie Debye-Htickel theory, 458. 1 9. Poisson's equation, 458. 20. Tne Poisson-Boltzmann equation, 460. 21. The Debye-Hiickel limiting law, 462. 22. Advances beyond the Debye- Hiickel theory, 465. 23. Theory of conductivity, 466. 21. Acids and bases, 469. 25. Dissociation constants of acids and bases, 471. 26. Electrode processes: reversible cells, 473. 21. Types of half cells, 474. 28. Electrochemical cells, 475. 29. The standard emf of cells, 476. 30. Standard electrode potentials, 478. 31. Standard free energies and entropies of aqueous ions, 481. 32. Measurement of solubility pro- ducts, 482. 33. Electrolyte-concentration cells, 482. 34. Electrode- concentration cells, 483. 16. Surface Chemistry 498 1. Surfaces and colloids, 498. 2. Pressure difference across curved surfaces, 500. 4. Maximum bubble pressure, 502. 5. The Du Notiy tensiometer, 502. 6. Surface-tension data, 502. 1. The Kelvin equa- tion, 504. 8. Thermodynamics of surfaces, 506. 9. The Gibbs adsorp- tion isotherm, 507. 10. Insoluble surface films the surface balance, 508. 11. Equations of state of monolayers, 577. 12. Surface films of soluble substances, 572. 13. Adsorption of gases on solids, 572. 14. The Langmuir adsorption isotherm, 575. 15. Thermodynamics of the adsorption isotherm, 576. 16. Adsorption from solution, 577. 17. Ion exchange, 518. 18. Electrical phenomena at interfaces, 579. 19. Electrokinetic phenomena, 520. 20. The stability of sols, 522. 17. Chemical Kinetics 528 1 . The rate of chemical change, 525. 2. Experimental methods ii| kinetics, 529. 3. Order of a reaction, 530. 4. Molecularity of a rcac- xii CONTENTS tion, 531. 5. The reaction-rate constant, 532. 6. First-order rate equations, 533. 1. Second-order rate equations, 534. 8. Third-order rate equation, 536. 9. Opposing reactions, 537. 10. Consecutive reactions, 559. II. Parallel reactions, 541. 12. Determination of the reaction order, 541. 13. Reactions in flow systems, 543. 14. Effect of temperature on reaction rate, 546. \ 5. Collision theory of gas reac- tions, 547. 16. Collision theory and activation energy, 551. 17. First- order reactions and collision theory, 557. 18. Activation in many degrees of freedom, 554. 19. Chain reactions: formation of hydrogen bromide, 555. 20. Free-radical chains, 557. 21. Branching chains explosive reactions, 559. 22. Trimolecular reactions, 562. 23. The path of a reaction, and the activated complex, 56 3. 24. The transition- state theory, 566. 25. Collision theory and transition-state theory, 568. 26. The entropy of activation, 569. 27. Theory of unimolecular reactions, 570. 28. Reactions in solution, 577. 29. Ionic reactions salt effects, 572. 30. Ionic reaction mechanisms, 574. 31. Catalysis, 575. 32. Homogeneous catalysis, 576. 33. Acid-base catalysis, 577. 34. General acid-base catalysis, 579. 35. Heterogeneous reactions, 580. 36. Gas reactions at solid surfaces, 582. 37. Inhibition by pro- ducts, 583. 38. Two reactants on a surface, 583. 39. Effect of temper- ature on surface reactions, 585. 40. Activated adsorption, 586. 41. Poisoning of catalysts, 587. 42. The nature of the catalytic surface, 588. 43. Enzyme reactions, 589. 18. Photochemistry and Radiation Chemistry ... 595 1. Radiation and chemical reactions, 595. 2. Light absorption and quantum yield, 595. 3. Primary processes in photochemistry, 597. 4. Secondary processes in photochemistry: fluorescence, 598. 5. Luminescence in solids, 601. 6. Thermoluminescence, 60 3. 7. Secondary photochemical processes: initiation of chain reactions, 604. 8. Flash photolysis, 606. 9. Effects of intermittent light, 607. 10. Photosynthesis in green plants, 609. 11. The photographic pro- cess, 6/7. 12. Primary processes with high-energy radiation, 672. 13. Secondary processes in radiation chemistry, 614. 14. Chemical effects of nuclear recoil, 6/5. Physical Constants and Conversion Factors . . . 618 Name Index 619 Subject Index 623 CHAPTER 1 The Description of Physicochemical Systems J/Thc The description of our universe. Since man is a rational being, he has always tried to increase his understanding of the world in which he lives. This endeavor has taken many forms. The fundamental questions of the end and purpose of man's life have been illumined by philosophy and religion. The form and structure of life have found expression in art. The nature of the physical world as perceived through man's senses has been investigated by science. The essential components of the scientific method are experiment and theory. Experiments are planned observations of the physical world. A theory seeks to correlate observables with ideals. These ideals have often taken the form of simplified models, based again on everyday experience. We have, for example, the little billiard balls of the kinetic theory of gases, the miniature hooks and springs of chemical bonds, and the microcosmic solar systems of atomic theory. As man's investigation of the universe progressed to the almost infinitely large distances of interstellar space or to the almost infinitesimal magnitudes of atomic structures, it began to be realized that these other worlds could not be adequately described in terms of the bricks and mortar and plumbing of terrestrial architecture. Thus a straight line might be the shortest distance between two points on a blackboard, but not between Sirius and Aldebaran. We can ask whether John Doe is in Chicago, but we cannot ask whether electron A is at point B. Intensive research into the ultimate nature of our universe is thus gradu- ally changing the meaning we attach to such words as "explanation" or "understanding." Originally they signified a representation of the strange in terms of the commonplace; nowadays, scientific explanation tends more to be a description of the relatively familiar in terms of the unfamiliar, light in terms of photons, matter in terms of waves. Yet, in our search for under- standing, we still consider it important to "get a physical picture" of the process behind the mathematical treatment of a theory. It is because physical science is at a transitional stage in its development that there is an inevitable question- as to what sorts of concepts provide the clearest picture. v^J/Thysical chemistry. There are therefore probably two equally logical approaches to the study of a branch of scientific knowledge such as physical chemistry. We may adopt a synthetic approach and, beginning with the structure and behavior of matter in its finest known states of subdivision, gradually progress from electrons to atoms to molecules to states of i 2 PHYSICOCHEMICAL SYSTEMS [Chap. 1 aggregation and chemical reactions. Alternatively, we may adopt an analyti- cal treatment and, starting with matter or chemicals as we find them in the laboratory, gradually work our way back to finer states of subdivision as we require them to explain our experimental results. This latter method follows more closely the historical development, although a strict adherence to his- tory is impossible in a broad subject whose different branches have progressed at very different rates. Two main problems have occupied most of the efforts of physical chem- ists: the question of the position of chemical equilibrium, which is the principal problem of chemical thermodynamics; and the question of the rate of chemical reactions, which is the field of chemical kinetics. Since these problems are ultimately concerned with the interaction of molecules, their final solution should be implicit in the mechanics of molecules and molecular aggregates. Therefore molecular structure is an important part of physical chemistry. The discipline that allows us to bring our knowledge of molecular structure to bear on the problems of equilibrium and kinetics is found in the study of statistical mechanics. We shall begin our introduction to physical chemistry with thermo- dynamics, which is based on concepts common to the everyday world of sticks and stones. Instead of trying to achieve a completely logical presenta- tion, we shall follow quite closely the historical development of the subject, since more knowledge can be gained by watching the construction of some- thing than by inspecting the polished final product. j Mechanics: force. The first thing that may be said of thermodynamics is that the word itself is evidently derived from "dynamics,'* which is a branch of mechanics dealing with matter in motion. Mechanics is still founded on the work of Sir Isaac Newton (1642-1727), and usually begins with a statement of the well-known equation WIth /= ma dv The equation states the proportionality between a vector quantity f, called the force applied to a particle of matter, and the acceleration a of the particle, a vector in the same direction, with a proportionality factor w, called the mass. A vector is a quantity that has a definite direction as well as a definite magnitude. Equation (1.1) may also be written f. where the product of mass and velocity is called the momentum. With the mass in grams, time in seconds, and displacement r in centi- meters (COS system), the unit force is the dyne. With mass in kilograms, time Sec. 4] PHYSICOCHEMICAL SYSTEMS 3 in seconds, and displacement in meters (MKS system), the unit force is the newton. Mass might also be introduced by Newton's "Law of Universal Gravi- tation," f-'~rf which states that there is an attractive force between two masses propor- tional to their product and inversely proportional to the square of their separation. If this gravitational mass is to be the same as the inertia! mass of eq. (1.1), the proportionality constant ft 6.66 x 10~ 8 cm 3 sec"" 2 g" 1 . The weight of a body, W, is the force with which it is attracted towards the earth, and naturally may vary slightly at various points on the earth's surface, owing to the slight variation of r 12 with latitude and elevation, and of the effective mass of the earth with subterranean density. Thus At New York City, g = 980.267 cm per sec 2 ; at Spitzbergen, g = 982.899; at Panama, g = 978.243. In practice, the mass of a body is measured by comparing its weight by meanspf a balance with that of known standards (mjm 2 = W^W^). Jltwo and energy. The differential element of work dw done by a force / that moves a particle a distance dr in the direction of the force is defined as the product of force and displacement, dw-^fdr (1.3) For a finite displacement from r Q to r t , and a force that depends only on the position r, *' = P/(r)dr (1.4) Jr The integral over distance can be transformed to an integral over time: f l dr M* Jt/ at t Introducing Newton's Law of Force, eq. (1.1), we obtain f'i d*rdr , w = I m~~~-dt Jt dt 2 dt Since (d/dt)(dr/dt)* = 2(dr/dt)d*r/dt 2 , the integral becomes w = \rnvf - \rnvf (1.5) The kinetic energy is defined by E K = JlMP 2 4 PHYSICOCHEMICAL SYSTEMS [Chap, l It is evident from eq. (1.5), therefore, that the work expended equals the difference in kinetic energy between the initial and the final states, \r)dr = E Kl -E KQ (1.6) An example of a force that depends only on position r is the force of gravity acting on a body falling in a vacuum; as the body falls from a higher to a lower level it gains kinetic energy according to eq. (1.6). Since the force is a function only of r, the integral in eq. (1.6) defines another function of r, which we may write J/(r) dr -= - U(r) Or /(/)- -dU/dr (1.7) This new function U(r) is called the potential energy. It may be noted that, whereas the kinetic energy E K is zero for a body at rest, there is no naturally defined zero of potential energy; only differences in potential energy can be measured. Sometimes, however, a zero of potential energy is chosen by convention', an example is the choice U(r) for the gravitational potential energy when two bodies are infinitely far apart. Equation (1.6) can now be written \ dr - Ufa) - Ufa) - E Kl -- E KQ The sum of the potential and the kinetic energies, U + E K , is the total mechanical energy of the body, and this sum evidently remains constant during the motion. Equation (1.8) has the typical form of an equation of conservation. It is a statement of the mechanical principle of the conservation of energy. For example, the gain in kinetic energy of a body falling in a vacuum is exactly balanced by an equal loss in potential energy. A force that can be represented by eq. (1.7) is called a conservative force. If a force depends on velocity as well as position, the situation is more complex. This would be the case if a body is falling, not in a vacuum, but in a viscous fluid like air or water. The higher the velocity, the greater is the frictional or viscous resistance opposed to the gravitational force. We can no longer write /(r) = dU/dr, and we can no longer obtain an equation such as (1.8). The mechanical energy is no longer conserved. From the dawn of history it has been known that the frictional dissipation of energy is attended by the evolution of something called heat. We shall see later how the quantitative study of such processes finally led to the inclusion of heat as a form of energy, and hence to a new and broader principle of the conservation of energy. The unit of work and of energy in the COS system is the erg, which is the work done by a force of one dyne acting through a distance of one centimeter. Since the erg is a very small unit for large-scale processes, it is Sec. 5] PHYSICOCHEMICAL SYSTEMS often convenient to use a larger unit, the joule, which is the unit of work in the MKS system. Thus, 1 joule = 1 newton meter 10 7 ergs The joule is related to the absolute practical electrical units since 1 joule = 1 volt coulomb The unit of power is the watt. 1 watt = 1 joule per sec = 1 volt coulomb per sec = 1 volt ampere <-& Equilibrium. The ordinary subjects for chemical experimentation are not individual particles of any sort but more complex systems, which may contain solids, liquids, and gases. A system is a part of the world isolated from the rest of the world by definite boundaries. The experiments that we perform on a system are said to measure its properties, these being the attri- butes that enable us to describe it with all requisite completeness. This complete description is said to define the state of the system. A B c Fig. l.la. Illustration of equilibrium. The idea of predictability enters here; having once measured the prop- erties of a system, we expect to be able to predict the behavior of a second system with the same set of properties from our knowledge of the behavior of the original. This is, in general, possible only when the system has attained a state called equilibrium. A system is said to have attained a state of equi- librium when it shows no further tendency to change its properties with time. A simple mechanical illustration will clarify the concept of equilibrium. Fig. l.la shows three different equilibrium positions of a box resting on a table. In both positions A and C the center of gravity of the box is lower than in any slightly displaced position, and if the box is tilted slightly it will tend to return spontaneously to its original equilibrium position. The gravi- tational potential energy of the box in positions A or C is at a minimum, and both positions represent stable equilibrium states. Yet it is apparent that position C is more stable than position A, and a certain large tilt of A will suffice to push it over into C. The position A is therefore said to be in meta- stable equilibrium. Position B is also an equilibrium position, but it is a state of unstable equilibrium, as anyone who has tried to balance a chair on two legs will PHYSICOCHEMICAL SYSTEMS [Chap. 1 agree. The center of gravity of the box in B is higher than in any slightly dis- placed position, and the tiniest tilt will send the box into either position A or C. The potential energy at a position of unstable equilibrium is a maximum, and such a position could be realized only in the absence of any disturbing forces. These relations may be presented in more mathematical form by plotting in Fig. l.lb the potential energy of the system as a function of height r of the center of gravity. Positions of stable equilibrium are seen to be minima in the curve, and the posi- tion of unstable equilibrium is represented by a maximum. Posi- tions of stable and unstable equi- librium thus alternate in any system. For an equilibrium position, the slope of the U vs. r curve, dU/dr, equals zero and one may write the equilibrium condition as at constant r (= r ), dU ABC POSITION OF CENTER OF GRAVITY Fig. l.lb. Potential energy diagram. Although these considerations have been presented in terms of a simple mechanical model, the same kind of principles will be found to apply in the more complex physicochemical systems that we shall study. In addition to purely mechanical changes, such systems may undergo temperature changes, changes of state of aggregation, and chemical reactions. The problem of thermodynamics is to discover or invent new functions that will play the role in these more general systems that the potential energy plays in mechanics. ^f. The thermal properties of matter. What variables are necessary in order to describe the state of a pure substance ? For simplicity, let us assume that the substance is at rest in the absence of gravitational and electromagnetic forces. These forces are indeed always present, but their effect is most often negligible in systems of purely chemical interest. Furthermore let us assume that we are dealing with a fluid or an isotropic solid, and that shear forces are absent. To make the problem more concrete, let us suppose our substance is a flask of water. Now to specify the state of this water we have to describe it in unequivocal terms so that, for example, we could write to a fellow scientist in Pasadena or Cambridge and say, "I have some water with the following properties. . . . You can repeat my experiments exactly if you bring a sample of water to these same conditions." First of all we might specify how much water we have by naming the mass m of our substance; alternatively we could measure the volume K, and the density p. Another useful property, the pressure, is defined as the force normal to unit area of the boundary of a body (e.g., dynes per square centimeter). In Sec. 6] PHYSICOCHEMICAL SYSTEMS 7 a state of equilibrium the pressure exerted by a body is equal to the pressure exerted upon the body by its surroundings. If this external pressure is denoted by P ex and the pressure of the substance by P, at equilibrium P = P ex . We have now enumerated the following properties: mass, volume, den- sity, and pressure (m, K, p, P). These properties are all mechanical in nature; they do not take us beyond the realm of ordinary dynamics. How many of these properties are really necessary for a complete description? We ob- viously must state how much water we are dealing with, so let us choose the mass m as our first property. Then if we choose the volume F, we do not need the density p, since p ml V. We are left with m, V, and P. Then we find experimentally that, as far as mechanics is concerned, if any two of these properties are fixed in value, the value of the third is always fixed. For a given mass of water at a given pressure, the volume is always the same; or if the volume and mass are fixed, we can no longer arbitrarily choose the pressure. Only two of the three variables of state are independent variables. In what follows we shall assume that a definite mass has been taken say one kilogram. Then the pressure and the volume are not independently variable in mechanics. The value of the volume is determined by the value of the pressure, or vice versa. This dependence can be expressed by saying that V is a function of P, which is written V=-f(P) or F(P 9 K) = (1.9) According to this equation, if the pressure is held constant, the volume of our kilogram of water should also remain constant. Our specification of the properties of the water has so far been restricted to mechanical variables. When we try to verify eq. (1.9), we shall find that on some days it appears to hold, but on other days it fails badly. The equation fails, for example, when somebody opens a window and lets in a blast of cold air, or when somebody lights a hot flame near our equipment. A new variable, a thermal variable, has been added to the mechanical ones. If the pressure is held constant, the volume of our kilogram of water is greater on the hot days than on the cold days. The earliest devices for measuring "degrees of hotness" were based on exactly this sort of observation of the changes in volume of a liquid. 1 In 1631, the French physician Jean Rey used a glass bulb and stem partly filled with water to follow the progress of fevers in his patients. In 1641, Ferdi- nand II, Grand Duke of Tuscany, invented an alcohol-in-glass "thermo- scope." Scales were added by marking equal divisions between the volumes at "coldest winter cold" and "hottest summer heat." A calibration based on two fixed points was introduced in 1688 by Dalence, who chose the melting point of snow as 10, and the melting point of butter as +10. In 1694 1 A detailed historical account is given by D. Roller in No. 3 of the Harvard Case Histories in Experimental Science, The Early Development of the Concepts of Temperature and Heat (Cambridge, Mass.: Harvard Univ. Press, 1950). 8 PHYSICOCHEMICAL SYSTEMS [Chap. 1 Rinaldi took the boiling point of water as the upper fixed point. If one adds the requirement that both the melting point of ice and the boiling point of water are to be taken at a constant pressure of one atmosphere, the fixed points^are precisely defined. ^Definition of temperature. We have seen how our sensory perception of relative "degrees of hotness" came to be roughly correlated with volume readings on constant-pressure thermometers. We have not yet demonstrated, however, that these readings in fact measure one of the variables that define the state of a thermodynamic system. Let us consider, for example, two blocks of lead with known masses. At equilibrium the state of block I can be specified by the independent variables P l and F x . Similarly P 2 and K 2 specify the state of block II. If we bring the two blocks together and wait until equilibrium is again attained, i.e., until P 19 V l9 P& and K 2 have reached constant values, we shall discover as an experimental fact that P l9 V 19 P 2 , and K 2 are no longer all independent. They are now connected by a relation, the equilibrium condition, which may be written Furthermore, it is found experimentally that two bodies separately in equilibrium with the same third are also in equilibrium with each other. That is, if and F(P 29 V 29 P 39 K a ) = it necessarily follows that It is apparent that these equations can be satisfied if the function F has the special form K 2 ) - o (i. 10) Thus F is the difference of two functions each containing properties pertain- ing to one body only. The function /(P, V) defined in this way is called the empirical temperature t. This definition of / is sometimes called the Zeroth Law of Thermodynamics. From eq. (1.10) the condition for thermal equi- librium between two systems is therefore It may be noted that, strictly speaking, the temperature is defined only for a state of equilibrium. The state of our one kilogram of water, or lead, is now specified in terms of three thermodynamic variables, P 9 V, and /, of which only two are independent. 8. The equation of state. The properties of a system may be classified as extensive or Intensive. Extensive properties are additive; their value for the whole system is equal- to the sum of their values for the individual parts. Sec. 8] PHYSICOCHEMICAL SYSTEMS Sometimes they are called capacity factors. Examples are the volume and the mass. Intensive properties, or intensity factors, are not additive. Examples are temperature and pressure. The temperature of any small part of a system in equilibrium is the same as the temperature of the whole. If P and V are chosen as independent variables, the temperature is some function of P and V. Thus ) (1.12) For any fixed value of t, this equation defines an isotherm of the body under consideration. The state of a body in thermal equilibrium can be fixed by specifying any two of the three variables, pressure, volume, and temperature. 2 4 6 8 10 12 14 16 16 20 V-LITERS 400 800 1200 07 06 05 0.4 03 02 01 00 ISOCHORES 200 400 600 Fig. 1.2. Isotherms, isobars, and isochores for one gram of hydrogen. The third variable can then be found by solving the equation. Thus, by analogy with eq. (1.12) we may have: V=f(t,P) (1.13) P=--f(t,V) (1.14) Equations such as (1.12), (1.13), (1.14) are called equations of state. Geometrically considered, the state of a body in equilibrium can be represented by a point in the PV plane, and its isotherm by a curve in the PV plane connecting points at constant temperature. Alternatively, the state can be represented by a point in the Vt plane or the Pt plane, the curves connecting equilibrium points in these planes being called the isobars (con- stant pressure) and isochores or isometrics (constant volume) respectively. Examples of these curves for one gram of hydrogen gas are shown in Fig. 1.2. We have already seen how eq. (1.12) can be the basis for a quantitative measure of temperature. For a liquid-in-glass thermometer, P is constant, and the change in volume measures the change in temperature. The Celsius (centigrade) calibration calls the melting point of ice at 1 atm pressure 0C, 10 PHYSICOCHEMICAL SYSTEMS [Chap. 1 and the boiling point of water at I atm pressure 100C. The reading at other temperatures depends on the coefficient of thermal expansion a of the thermo- metric fluid, 1 /9K\ (1.15) where K is the volume at 0C and at the pressure of the measurements. If a is a constant over the temperature range in question, the volume increases linearly with temperature: K t = K + a/K (1.16) This is approximately true for mercury, but may be quite far from true for other substances. Thus, although many substances could theoretically be used as thermometers, the readings of these various thermometers would in general agree only at the two fixed points chosen by convention. 9. Gas thermometry: the ideal gas. Gases such as hydrogen, nitrogen, oxygen, and helium, which are rather difficult to condense to liquids, have been found to obey approximately certain simple laws which make them especially useful as thermometric fluids. In his book, On the Spring of the Air, Robert Boyle 2 reported in 1660 experiments confirming Torricelli's idea that the barometer was supported by the pressure of the air. An alternative theory proposed that the mercury column was held up by an invisible rigid thread in its interior. In answering this, Boyle placed air in the closed arm of a U-tube, compressed it by adding mercury to the other arm, and observed that the volume of gas varied in- versely as the pressure. He worked under conditions of practically constant temperature. Thus, at any constant temperature, he found PV = constant (1.17) If the gas at constant pressure is used as a thermometer, the volume of the gas will be a function of the temperature alone. By measuring the volume at 0C and at 100C a mean value of a can be calculated from eq. (1.16), ^100 - W + 1005) or a = The measurements on gases published by Joseph Gay-Lussac in 1802, extending earlier work by Charles (1787), showed that this value of a was a constant for "permanent" gases. Gay-Lussac found (1808) the value to be 4 ^. By a much better experimental procedure, Regnault (1847) obtained 2^3. For every one-degree rise in temperature the fractional increase in the gas volume is 3 of the volume at 0C. 2 Robert Boyle's Experiments in Pneumatics, Harvard Case Histories in Experimental Science No. 1 (Cambridge, Mass.: Harvard Univ. Press, 1950) is a delightful account of this work. Sec. 9] PHYSICOCHEMICAL SYSTEMS 11 Later and more refined experiments revealed that the closeness with which the laws of Boyle and Gay-Lussac are obeyed varies from gas to gas. Helium obeys most closely, whereas carbon dioxide, for example, is rela- tively disobedient. It has been found that the laws are more nearly obeyed the lower the pressure of the gas. It is very useful to introduce the concept of an ideal gas, one that follows the laws perfectly. The properties of such a gas usually can be obtained by extrapolation of values measured with real gases to zero pressure. Examples 36.75i 36.70 36.65 36.60 200 1200 400 600 800 1000 PRESSURE - mm of Hg Fig. 1.3. Extrapolation of thermal expansion coefficients to zero pressure. are found in some modern redetermi nations of the coefficient a shown plotted in Fig. 1.3. The extrapolated value at zero pressure is oc () - 36.608 x 10~ 4 , or l/a --- 273.16 We may use such carefully measured values to define an ideal gas tem- perature scale, by introducing a new temperature, =t + = t + (273.16 0.01) "o (1.18) The new temperature T is called the absolute temperature (K); the zero on this scale represents the limit of the thermal contraction of an ideal gas. From eq. (1.16), V T V T F- ('-I') 273.16 where V is now the volume of gas at 0C and standard atmospheric pressure P Q9 and V T j> o is the volume at P Q and any other temperature T. The tempera- ture of the ice point on the absolute scale is written as T (273.16). Boyle's Law eq. (1.17) states that for a gas at temperature T 12 PHYSICOCHEMICAL SYSTEMS [Chap. 1 Combining with eq. (1.19), we obtain PV^ f ^T=C-T (1.20) M) The value of the constant C depends on the amount of gas taken, but for a given volume of gas, it is the same for ail ideal gases. Thus for 1 cc of gas at 1 atm pressure, PV = 7)273. For chemical purposes, the most significant volume is that of a mole of gas, a molecular weight in grams. In conformity with the hypothesis of Avogadro, this volume is the same for all ideal gases, being 22,414 cc at 0C and 1 atm. Per mole, therefore, PV=RT (1.21) where R = 22,414/273.16 - 82.057 cc atm per C. For n moles, PY=nRT^^-RT (1.22) M where m is the mass of gas of molecular weight M. In all future discussions the volume V will be taken as the molar volume unless otherwise specified. It is often useful to have the gas constant in other units. A pressure of 1 atm corresponds to 76.00 cm of mercury. A pressure of 1 atm in units of dynes cm~ 2 is 76.00 /3 H go where /> Hg is the density of mercury at 0C and 1 atm, and g Q is the standard gravitational acceleration. Thus 1 atm = 76.00 x 13.595 x 980.665 =1.0130 X 10 6 dyne cm- 2 . The gas constant R - 82.057 x 1.0130 x 10 6 - 8.3144 x 10 7 ergs deg~ l mole- 1 == 8.3144 joules deg" 1 mole* 1 . 10. Relationships of pressure, volume, and temperature. The pressure, volume, temperature (PVT) relationships for gases, liquids, and solids would preferably all be succinctly summarized in the form of equations of state of the general form of eqs. (1.12), (1.13), and (1.14). Only in the case of gases has there been much progress in the development of these state equations. They are obtained not only by correlation of empirical PVT data, but also from theoretical considerations based on atomic and molecular structure. These theories are farthest advanced for gases, but more recent developments in the theory of liquids and solids give promise that suitable state equations may eventually be available in these fields also. The ideal gas equation PV = RT describes the PVT behavior of real gases only to a first approximation. A convenient way of showing the devia- tions from ideality is to write for the real gas : PV=zRT (1.23) The factor z is called the compressibility factor. It is equal to PV/RT. For an ideal gas z = 1, and departure from ideality will be measured by the deviation of the compressibility factor from unity. The extent of deviations from Sec. 10] PHYSICOCHEMICAL SYSTEMS 13 ideality depends on the temperature and pressure, so z is a function of T and P. Some compressibility factor curves are shown in Fig. 1.4; these are determined from experimental measurements of the volumes of the gases at different pressures. Useful PVT data for many substances are contained in the tabulated values at different pressures and temperatures of thermal expansion co- efficients a [eq. (1.15)] and compressibilities /?. 3 The compressibility* is defined by 1 IAV\ (1.24) The minus sign is introduced because (3V/dP) T is itself negative, the volume decreasing with increasing pressure. Z2{ C 2 H4/-N2 ^>CH4 200 400 600 800 1000 PRESSURE -ATM Fig. 1.4. Compressibility factors at 0C. 1200 Since V /(P, T), a differential change in volume can be written 5 : For a condition of constant volume, V = constant, dV =- 0, and - ,,,6, , v (3K/aP) r - 3 See, for example, International Critical Tables (New York: McGraw-Hill, 1933); also J. H. Perry, ed., Chemical Engineers 9 Handbook (New York: McGraw-Hill, 1950), pp. 200, 205. 4 Be careful not to confuse compressibility with compressibility factor. They are two distinctly different quantities. 6 Granville, Smith, Longley, Calculus (Boston: Ginn, 1934), p. 412. 14 PHYSICOCHEMICAL SYSTEMS [Chap. 1 Or, from eqs. (1.15) and (1.24), (3Pl3T) r =--- a/0. The variation of P with T can therefore readily be calculated if we know a and ft. An interesting example is suggested by a common laboratory accident, the breaking of a mercury-in-glass thermometer by overheating. If a thermo- meter is exactly filled with mercury at 50C, what pressure will be developed within the thermometer if it is heated to 52C? For mercury, a 1.8 X 10~ 4 deg- 1 , p - 3.9 x 10- 6 atm- 1 . Therefore (2P/dT) v -- <x/ft =-- 46 atm per deg. For AT 2, A/> =-- 92 atm. It is apparent why even a little overheating will break the usual thermometer. 11. Law of corresponding states. If a gas is cooled to a low enough tem- perature and then compressed, it can be liquefied. For each gas there is a characteristic temperature above which it cannot be liquefied, no matter how great the applied pressure. This temperature is called the critical temperature T fy and the pressure that just suffices to liquefy the gas at T c is called the critical pressure P c . The volume occupied at T c and P c is the critical volume V c . A gas below the critical temperature is often called a vapor. The critical constants for various gases are collected in Table 1.1. TABLE 1.1 CRITICAL POINT DATA AND VAN DER WAALS CONSTANTS Formula T c (K) P c (atm) V t (cc/mole) a (I 2 atm/mole 2 ) b (cc/mole) He 5.3 2.26 57.6 0.0341 23.7 H 2 | i 33.3 12.8 65.0 0.244 26.6 N 2 126.1 33.5 90.0 1.39 39.1 CO 134.0 35.0 90.0 1.49 39.9 2 153.4 49.7 74.4 1.36 31.8 C 2 H 4 282.9 50.9 127.5 4.47 57.1 CO 2 304.2 73.0 95.7 3.59 42.7 NH 3 405.6 111.5 72.4 4.17 37.1 H 2 O 647.2 217.7 45.0 5.46 30.5 Hg 1823.0 200.0 45.0 8.09 17.0 The ratios of P, K, and T to the critical values P c , K c , and T c are called the reduced pressure, volume, and temperature. These reduced variables may be written P V T p r V T (\ ">7\ r n ~ p ' ' it ~ is 9 1 R ~ T \ 1 '^') * c Y c 2 c To a fairly good approximation, especially at moderate pressures, all gases obey the same equation of state when described in terms of the reduced variables, P n , V w T R , instead of P, K, T. If two different gases have identical values for two reduced variables, they therefore have approximately identical values for the third: They are then said to be in corresponding states, and Sec. 12] PHYSICOCHEMICAL SYSTEMS 15 LEGEND NITROGEN a N-BUTANE METHANE a ISOPENTANE ETHANE ETHYLENE N-HEPTANE A CARBON DIOXIDE WATER 23456 REDUCED PRESSURE, P R Fig. 1.5. Compressibility factor as function of reduced state variables. [From Gouq-Jen Su, Ind. Eng. Chem., 38, 803 (1946).] this approximation is called the Law of Corresponding States. This is equiva- lent to Sciying that the compressibility facror z is the same function of the reduced variables for all gases. This rule is illustrated in Fig. 1.5 for a number of different gases, where z PV/RT is plotted at various reduced tempera- tures, against the reduced pressure. 12. Equations of state for gases. If the equation of state is written in terms of reduced variables as F(P& V E } ^= T R , it is evident that it contains at least two independent constants, characteristic of the gas in question, for example P c and K r . Many equations of state, proposed on semi-empirical grounds, serve to represent the PVT data more accurately than does the ideal gas equation. Several of the best known of these also contain two added con- stants. For example: 16 PHYSICOCHEMICAL SYSTEMS [Chap. 1 Equation of van der Waals: Equation of Berthelot: ( P Equation of Dieterici: P(V - b')e a ' IRTV = RT (1.30) Van der Waals' equation provides a reasonably good representation of the PVT data of gases in the range of moderate deviations from ideality. For example, consider the following values in liter atm of the PV product for carbon dioxide at 40C, as observed experimentally and as calculated from the van der Waals equation: P, atm 1 10 50 100 200 500 1100 PF, obs. 25.57 24.49 19.00 6.93 10.50 22.00 40.00 PK,calc. 25.60 24.71 19.75 8.89 14.10 29.70 54.20 The constants a and b are evaluated by fitting the equation to experimental PVT measurements, or more usually from the critical constants of the gas. Some values for van der Waals' a and b are included in Table 1.1. Berthelot's equation is somewhat better than van der Waals' at pressures not much above one atmosphere, and is preferred for general use in this range. Equations (1.28), (1.29), and (1.30) are all written for one mole of gas. For n moles they become: f )(V-nb) = nRT n*A' P(V-nb')e na ' IRTV ^ nRT The way in which the constants in these equations are evaluated from critical data will now be described, using the van der Waals equation as an example. 13. The critical region. The behavior of a gas in the neighborhood of its critical region was first studied by Thomas Andrews in 1869, in a classic series of measurements on carbon dioxide. Results of recent determinations of these PV isotherms around the critical temperature of 31.01C are shown in Fig. 1.6. Consider the isotherm at 30.4, which is below T c . As the vapor is com- pressed the PV curve first follows AB, which is approximately a Boyle's law isotherm. When the point B is reached, liquid is observed to form by the appearance of a meniscus between vapor and liquid. Further compression Sec. 13] 75 74 <73 CO UJ 72 71 PHYSICOCHEMICAL SYSTEMS 17 31.523' 31.013' VAN OER WAALS B \ \ 30.409 32 36 40 44 48 52 56 60 VOLUME Fig. 1.6. Isotherms of carbon dioxide near the critical point. 64 then occurs at constant pressure until the point C is reached, at which all the vapor has been converted into liquid. The curve CD is the isotherm of liquid carbon dioxide, its steepness indicating the low compressibility of the liquid. As isotherms are taken at successively higher temperatures the points of discontinuity B and C are observed to approach each other gradually, until at 31.0lC they coalesce, and no gradual formation of a liquid is observable. This isotherm corresponds to the critical temperature of carbon dioxide. Isotherms above this temperature exhibit no formation of a liquid no matter how great the applied pressure. Above the critical temperature there is no reason to draw any distinction 18 PHYSICOCHEMICAL SYSTEMS [Chap. 1 between liquid and vapor, since there is a complete continuity of states. This may be demonstrated by following the path EFGH. The vapor at point E, at a temperature below T c , is warmed at constant volume to point /% above T c . It is then compressed along the isotherm FG, and finally cooled at constant volume along GH. At the point //, below T c , the carbon dioxide exists as a liquid, but at no point along this path are two phases, liquid and vapor, simultaneously present. One must conclude that the transformation from vapor to liquid occurs smoothly and continuously. 14. The van der Waals equation and liquefaction of gases. The van der Waals equation provides a reasonably accurate representation of the PVT data of gases under conditions that deviate only moderately from ideality. When an attempt is made to apply the equation to gases in states departing greatly from ideality, it is found that, although a quantitative representation of the data is not obtained, an interesting qualitative picture is still provided. Typical of such applications is the example shown in Fig. 1.6, where the van der Waals isotherms, drawn as dashed lines, are compared with the experimental isotherms for carbon dioxide in the neighborhood of the critical point. The van der Waals equation provides an adequate representation of the isotherms for the homogeneous vapor and even for the homogeneous liquid. As might be expected, the equation cannot represent the discontinuities arising during liquefaction. Instead of the experimental straight line, it exhibits a maximum and a minimum within the two-phase region. We note that as the temperature gradually approaches the critical temperature, the maximum and the minimum gradually approach each other. At the critical point itself they have merged to become a point of inflection in the PKcurve. The analytical condition for a maximum is that (OP/OK) and (d 2 P/dV 2 ) < 0; for a minimum, (ZPfiV) = and (D 2 />/3K 2 ) > 0. At the point of in- flection, both the first and the second derivatives vanish, (DP/3K) According to van der Waals' equation, therefore, the following three equations must be satisfied simultaneously at the critical point (T = T c , V= V n P=-.P t ): RT r *r ' V, - b V* RT f la PP\ w) - =-- (v t -bp y* When these equations are solved for the critical constants we find (L31) Sec. 15] PHYSICOCHEMICAL SYSTEMS 19 The values for the van der Waals constants are usually calculated from these equations. In terms of the reduced variables of state, P Jf , V R , and T ]{ , one obtains from eq. (1.31): The van der Waals equation then reduces to As was pointed out previously, it is evident that a reduced equation of state similar to (1.32) can be obtained from any equation of state containing no more than two arbitrary constants, such as a and b. The Berthelot equa- tion is usually used in the following form, applicable at pressures of the order of one atmosphere: + 15 (-)] 15. Other equations of state. In order to represent the behavior of gases with greater accuracy, especially at high pressures or near their condensation temperatures, it is necessary to use expressions having more than two adjust- able parameters. Typical of such expressions is the very general virial equation of Kammerlingh-Onnes: 4 I f t ^ 2 i 3 -t- . . The factors B(T) 9 C(T) 9 D(T), etc., are functions of the temperature, called the second, third, fourth, etc., virial coefficients. An equation like this, though difficult to use, can be extended to as many terms as are needed to reproduce the experimental PKTdata with any desired accuracy. One of the best of the empirical equations is that proposed by Beattie and Bridgeman in 1928. 6 This equation contains five constants in addition to R, and fits the PKTdata over a wide range of pressures and temperatures, even near the critical point, to within 0.5 per cent. 16. Heat. The experimental observations that led to the concept of tem- perature led also to the concept of heat. Temperature, we recall, has been defined only in terms of the equilibrium condition that is reached when two bodies are placed in contact. A typical experiment might be the introduction of a piece of metal at temperature T 2 into a vessel of water at temperature 7\. To simplify the problem, let us assume that: (1) the system is isolated com- pletely from its surroundings; (2) the change in temperature of the container itself may be neglected; (3) there is no change in the state of aggregation of either body, i.e., no melting, vaporization, or the like. The end result is that ' J. A. Beattie and O. C. Bridgeman, Proc. Am. Acad. Arts Sci., 63, 229-308 (1928). J. A. Beattie, Chem. Rev., 44, 141-192 (1949). 20 PHYSICOCHEMICAL SYSTEMS [Chap. 1 the entire system finally reaches a new temperature T, somewhere between 7^ and T 2 . This final temperature depends on certain properties of the water and of the metal. It is found experimentally that the temperatures can always be related by an equation having the form C 2 (T 2 ~ 7')=C 1 (r-r 1 ) (1.34) Here C\ and C 2 are functions of the mass and constitution of the metal and of the water respectively. Thus, a gram of lead would cause a smaller tem- perature change than a gram of copper; 10 grams of lead would produce 10 times the temperature change caused by one gram. Equation (1.34) has the form of an equation of conservation, such as eq. (1.8). Very early in the development of the subject it was postulated that when two bodies at different temperatures are placed in contact, something flows from the hotter to the colder. This was originally supposed to be a weightless material substance, called caloric. Lavoisier, for example, in his Traite elementaire de Chimle (1789), included both caloric and light among the chemical elements. We now speak of a flow of heat q, given by q - C 2 (T 2 - r) - CAT - T,) (1.35) The coefficients C are called the heat capacities of the bodies. If the heat capacity is reckoned for one gram of material, it is called the specific heat; for one mole of material, the molar heat capacity. The unit of heat was originally defined in terms of just such an experiment in calorimetry as has been described. The gram calorie was the heat that must be absorbed by one gram of water to raise its temperature 1C. It followed that the specific heat of water was 1 cal per C. More careful experiments showed that the specific heat was itself a func- tion of the temperature. It therefore became necessary to redefine the calorie by specifying the range over which it was measured. The standard was taken to be the 75 calorie, probably because of the lack of central heating in European laboratories. This is the heat required to raise the temperature of a gram of water from 14.5 to 15.5C. Finally another change in the definition of the calorie was found to be desirable. Electrical measurements are capable of greater precision than calorimetric measurements. The Ninth International Conference on Weights and Measures (1948) therefore recommended that the joule (volt coulomb) be used as the unit of heat. The calorie, however, is still popular among chemists, and the National Bureau of Standards uses a defined calorie equal to exactly 4.1840 joules. The specific heat, being a function of temperature, should be defined precisely only in terms of a differential heat flow dq and temperature change dT. Thus, in the limit, eq. (1.35) becomes or C = (1.36) Sec. 17] PHYSICOCHEMICAL SYSTEMS 21 The heat added to a body in raising its temperature from 7\ to T 2 is therefore (1.37) *=\ T \ CdT Since C depends on the exact process by which the heat is transferred, this integral can be evaluated only when the process is specified. If our calorimeter had contained ice at 0C instead of water, the heat added to it would not have raised its temperature until all the ice had melted. Such heat absorption or evolution accompanying a change in state of aggre- gation was first studied quantitatively by Joseph Black (1761), who called it latent heat. It may be thought of as somewhat analogous to potential energy. Thus we have latent heat of fusion, latent heat of vaporization, or latent heat accompanying a change of one crystalline form to another, for example rhombic to monoclinic sulfur. 17. Work in thermodynamic systems. In our discussion of the transfer of heat we have so far carefully restricted our attention to the simple case in which the system is completely isolated and is not allowed to interact mechanically with its surroundings. If this restriction does not apply, the system may either do work on its surroundings or have work done on itself. Thus, in certain cases, only a part of the heat added to a substance causes its tem- perature to rise, the remainder being used in the work of expanding the substance. The amount of heat that must be added to produce a certain temperature change depends on the exact process by which the change is effected. A differential element of work may be defined by reference to eq. (1.3) as dw / 'dr, the product of a displacement and the component of force in the same direction. In the case of a simple thermodynamic system, a fluid confined in a cylinder with a movable piston (assumed frictionless), the work done by the fluid against the external force on the piston (see Fig. 1.7) in a differential expansion dV would be Fig. 1.7. Work in expansion. dw - J - A dr - /> ex dV Note that the work is done against the external pressure P ex . If the pressure is kept constant during a finite expansion from w dV = AK (1.38) to V* (1.39) If a finite expansion is carried out in such a way that each successive state is an equilibrium state, it can be represented by a curve on the PV diagram, 22 PHYSICOCHEMICAL SYSTEMS [Chap. 1 since then we always have P ex = P. This is shown in (a), Fig. 1.8. In this case, dw = P dV (1.40) On integration, \v^j*PdY (1.41) The value of the integral is given by the area under the PV curve. Only when equilibrium is always maintained can the work be evaluated from functions of the state of the substance itself, P and Y 9 for only in this case does P -=- P ex . It is evident that the work done in going from point I to point 2 in the PV diagram, or from one state to another, depends upon the particular path that is traversed. Consider, for example, two alternate paths from A to B in (b), Fig. 1.8. More work will be done in going by the path ADB than by the path ACB, as'is evident from the greater area under curve ADB. If we proceed (a) Fig. 1.8. Indicator diagrams for work. from state A to state B by path ADB and return to A along BCA, we shall have completed a cyclic process. The net work done by the system during this cycle is seen to be equal to the difference between the areas under the two paths, which is the shaded area in (b), Fig. 1.8. It is evident, therefore, that in going from one state to another both the work done by a system and the heat added to a system depend on the par- ticular path that is followed. The reason why alternate paths are possible in (b), Fig. 1.8 is that for any given volume, the fluid may exert different pres- sures depending on the temperature that is chosen. 18. Reversible processes. The paths followed in the PV diagrams of Fig. 1.8 belong to a special class, of great importance in thermodynamic arguments. They are called reversible paths. A reversible path is one connect- ing intermediate states all of which are equilibrium states. A process carried out along such an equilibrium path will be called a reversible process. In order, for example, to expand a gas reversibly, the pressure on the piston must be released so slowly, in the limit infinitely slowly, that at every instant the pressure everywhere within the gas volume is exactly the same and is just equal to the opposing pressure on the piston. Only in this case can the state of the gas- be represented by the variables of state, P and V Sec. 19] PHYSICOCHEMICAL SYSTEMS 23 Geometrically speaking the state is represented by a point in the PV plane. The line joining such points is a line joining points of equilibrium. Consider the situation if the piston were drawn back suddenly. Gas would rush in to fill the space, pressure differences would be set up throughout the gas volume, and even a condition of turbulence might ensue. The state of the gas under such conditions could no longer be represented by the two variables, P and V. Indeed a tremendous number of variables would be required, corre- sponding to the many different pressures at different points throughout the gas volume. Such a rapid expansion is a typical irreversible process; the inter- mediate states are no longer equilibrium states. It will be recognized immediately that reversible processes are never realizable in actuality since they must be carried out infinitely slowly. All naturally occurring processes are therefore irreversible. The reversible path is the limiting path that is reached as we carry out an irreversible process under conditions that approach more and more closely to equilibrium con- ditions. We can define a reversible path exactly and calculate the work done in moving along it, even though we can never carry out an actual change reversibly. It will be seen later that the conditions for reversibility can be closely approximated in certain experiments. 19. Maximum work. In (b), Fig. 1.8, the change from A to B can be carried out along different reversible paths, of which two (ACB and ADB) are drawn. These different paths are possible because the volume Kis a func- tion of the temperature 7, as well as of the pressure P. If one particular tem- perature is chosen and held constant throughout the process, only one rever- sible path is possible. Under such an isothermal condition the work obtained in going from A to B via a path that is reversible is the maximum work possible for the particular temperature in question. This is true because in the rever- sible case the expansion takes place against the maximum possible opposing force, which is one exactly in equilibrium with the driving force. If the opposing force, e.g., pressure on a piston, were any greater, the process would occur in the reverse direction ; instead of expanding and doing work the gas in the cylinder would have work done upon it and would be com- pressed. 20. Thermodynamics and thermostatics. From the way in which the variables of state have been defined, it would appear that thermodynamics might justly be called the study of equilibrium conditions. The very nature of the concepts and operations that have been outlined requires this restric- tion. Nowhere does time enter as a variable, and therefore the question of the rate of physicochemical processes is completely outside the scope of this kind of thermodynamic discussion. It would seem to be an unfortunate accident of language that this equilibrium study is called thermodynamics', a better term would be thermostatics. This would leave the term thermo- dynamics to cover the problems in which time occurs as a variable, e.g., thermal conductivity, chemical reaction rates, and the like. The analogy with 24 PHYSICOCHEMICAL SYSTEMS [Chap. 1 dynamics and statics as the two subdivisions of mechanics would then be preserved. Although the thermodynamics we shall employ will be really a thermo- statics, i.e., a thermodynamics of reversible (equilibrium) processes, it should be possible to develop a much broader study that would include irreversible processes as well. Some progress along these lines has been made and the field should be a fruitful one for future investigation. 7 PROBLEMS 1. The coefficient of thermal expansion of ethanol is given by a 1.0414 x 10~ 3 t- 1.5672 x 10~ 6 / + 5.148 x 10~ 8 / 2 , where t is the centigrade tem- perature. If and 50 are taken as fixed points on a centigrade scale, what will be the reading of the alcohol thermometer when an ideal gas thermo- meter reads 30C? 2. In a series of measurements by J. A. Beattie, the following values were found for a of nitrogen : P (cm) . . . 99.828 74.966 59.959 44.942 33.311 axlOVK- 1 . . 3.6740 3.6707 3.6686 3.6667 3.6652 Calculate from these data the melting point of ice on the absolute ideal gas scale. 3. An evacuated glass bulb weighs 37.9365 g. Filled with dry air at 1 atm pressure and 25C, it weighs 38.0739 g. Filled with a mixture of methane and ethane it weighs 38.0347 g. Calculate the percentage of methane in the gas mixture. 4. An oil bath maintained at 50C loses heat to its surroundings at the rate of 1000 calories per minute. Its temperature is maintained by an electri- cally heated coil with a resistance of 50 ohms operated on a 110-volt line, A thermoregulator switches the current on and off. What percentage of the time will the current be turned on? 5. Calculate the work done in accelerating a 2000 kg car from rest to a speed of 50 km per hr, neglecting friction. 6. A lead bullet is fired at a wooden plank. At what speed must it be traveling to melt on impact, if its initial temperature is 25 and heating of the plank is neglected? The melting point of lead is 327 and its specific heat is 0.030 cal deg~ ' l g- 1 . 7. What is the average power production in watts of a man who burns 2500 kcal of food in a day? 8. Show that 7 See, for example, P. W. Bridgman, The Nature of Thermodynamics (New Haven: Yale Univ. Press, 1941); K. G. Denbigh, The Thermodynamics of the Steady State (London: Methuen, 1951). Chap. 1] PHYSICOCHEMICAL SYSTEMS 25 9. Calculate the pressure exerted by 10 g of nitrogen in a closed 1-liter vessel at 25C using (a) the ideal gas equation, (b) van der Waals' equation. 10. Use Berthelot's equation to calculate the pressure exerted by 0.1 g of ammonia, NH 3 , in a volume of 1 liter at 20C 11. Evaluate the constants a and b' in Dieterici's equation in terms of the critical constants P c , V c , T c of a gas. 12. Derive an expression for the coefficient of thermal expansion a for a gas that follows (a) the ideal gas law, (b) the van der Waals equation. 13. The gas densities (g per liter) at 0C and 1 atm of (a) CO 2 and (b) SO 2 are (a) 1.9769 and (b) 2.9269. Calculate the molar volumes of the gases and compare with the values given by Berthelot's equation. 14. The density of solid aluminum at 20C is 2.70 g per cc; of the liquid at 660C, 2.38 g per cc. Calculate the work done on the surroundings when 10 kg of Al are heated under atmospheric pressure from 20 to 660C. 15. One mole of an ideal gas at 25C is held in a cylinder by a piston at a pressure of 100 atm. The piston pressure is released in three stages: first to 50 atm, then to 20 atm, and finally to 10 atm. Calculate the work done by the gas during these irreversible isothermal expansions and compare it with that done in an isothermal reversible expansion from 100 to 10 atm at 25C. 16. Two identical calorimeters are prepared, containing equal volumes of water at 20.0. A 5.00-g piece of Al is dropped into calorimeter A, and a 5.00-g piece of alloy into calorimeter B. The equilibrium temperature in A is 22.0, that in B is 21.5. Take the specific heat of water to be independent of temperature and equal to 4.18 joule deg~ l . If the specific heat of Al is 0.887 joule deg" 1 , estimate the specific heat of the alloy. 17. According to Hooke's Law the restoring force/ on a stretched spring is proportional to the displacement r (/-=- /cr). How much work must be expended to stretch a 10.0-cm-long spring by 10 per cent, if its force constant AC 10 5 dynes cm" 1 ? 18. A kilogram of ammonia is compressed from 1000 liters to 100 liters at 50. Calculate the minimum work that must be expended assuming (a) ideal gas, (b) van der Waals' equation. REFERENCES BOOKS 1. Berry, A. J., Modern Chemistry (Historical Development) (London: Cambridge, 1948). 2. Epstein, P. S., Textbook of Thermodynamics (New York: Wiley, 1937). 3. Guggenheim, E, A., Modern Thermodynamics by the Methods of Willard Gibbs (London: Methuen, 1933). 4. Keenan, J. G., Thermodynamics (New York: Wiley, 1941). 26 PHYSICOCHEMICAL SYSTEMS [Chap. 1 5. Klotz, I. M., Chemical Thermodynamics (New York: Prentice-Hall, 1950). 6. Lewis, G. N., and M. Randall, Thermodynamics and the Free Energy of Chemical Substances (New York: McGraw-Hill, 1923). 7. MacDougall, F. H., Thermodynamics and Chemistry (New York: Wiley, 1939). 8. Planck, M., Treatise on Thermodynamics (New York: Dover, 1945). 9. Roberts, J. K., Heat and Thermodynamics (London: Blackie, 1951). 10. Rossini, F. D., Chemical Thermodynamics (New York: Wiley, 1950). 11. Sears, F. W., An Introduction to Thermodynamics, The Kinetic Theory of Gases, and Statistical Mechanics (Boston: Addison- Wesley, 1950). 12. Zemansky, M. W., Heat and Thermodynamics (New York: McGraw- Hill, 1951). ARTICLES 1. Birkhoff, G. D., Science in Progress, vol. IV, 120-149 (New Haven: Yale Univ. Press, 1945), "The Mathematical Nature of Physical Theories." 2. Brescia, F.,/. Chem. Ed., 24, 123-128 (1947), 'The Critical Temperature." 3. Reilly, D., /. Chem. Ed., 28, 178-183 (1951), "Robert Boyle and His Background." 4. Roseman, R., and S. KatzofT, J. Chem. Ed., 11, 350-354 (1934), "The Equation of State of a Perfect Gas." 5. Woolsey, G., J. Chem. Ed., 16, 60-66 (1939), "Equations of State." CHAPTER 2 The First Law of Thermodynamics 1. The history of the First Law. The First Law of Thermodynamics is an extension of the principle of the conservation of mechanical energy. This extension became natural when it was realized that work could be converted into heat, the expenditure of a fixed amount of work always giving rise to the production of the same amount of heat. To give the law an analytical formulation, it was only necessary to define a new energy function that included the heat. The first quantitative experiments on this subject were carried out by Benjamin Thompson, a native of Woburn, Massachusetts, who became Count Rumford of The Holy Roman Empire. Commissioned by the King of Bavaria to supervise the boring of cannon at the Munich Arsenal, he became impressed by the tremendous generation of heat during this opera- tion. He suggested (1798) that the heat arose from the mechanical energy expended, and was able to estimate the amount of heat produced by a horse working for an hour; in modern units his value would be 0.183 calorie per joule. The reaction at the time to these experiments was that the heat was produced owing to a lower specific heat of the metal in the form of fine turnings. Thus when bulk metal was reduced to turnings it had to release heat. Rumford then substituted a blunt borer, producing just as much heat with very few turnings. The adherents of the caloric hypothesis thereupon shifted their ground and claimed that the heat arose from the action of air on the metallic surfaces. Then, in 1799, Sir Humphry Davy provided further support for Rumford's theory by rubbing together two pieces of ice by clock- work in a vacuum and noting their rapid melting, showing that, even in the absence of air, this latent heat could be provided by mechanical work. Nevertheless, the time did not become scientifically ripe for a mechanical theory of heat until the work of Dalton and others provided an atomic theory of matter, and gradually an understanding of heat in terms of molecular motion. This development will be considered in some detail in Chapter 7. James Joule, at the age of twenty, began his studies in 1840 in a labora- tory provided by his father in a Manchester brewery. In 1843, he published his results on the heating effect of the electric current. In 1849, he carefully determined the mechanical equivalent of heat by measuring the work input and the temperature rise in a vessel of water vigorously stirred with paddle wheels. His value, converted into our units, was 0.241 calorie per joule; the accepted modern figure is 0.239. Joule converted electric energy and mechanical 28 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 energy into heat in a variety of ways: electric heating, mechanical stirring, compression of gases. By every method he found very nearly the same value for the conversion factor, thus clearly demonstrating that a given amount of work always produced the same amount of heat, to within the experimental error of his measurements. 2. Formulation of the First Law. The interconversion of heat and work having been demonstrated, it is possible to define a new function called the internal energy E. In any process the change in internal energy A, in passing from one state A to another 5, is equal to the sum of the heat added to the system q and the work done on the system w. (Note that by convention work done by the system is called positive, 4 H>.) Thus, A - q vv. Now the first law of thermodynamics states that this difference in energy A depends only on the final state B and the initial state A 9 and not on the path between A and B. &E=E B -E A = q-w (2.1) Both q and w depend upon the path, but their difference^ w is independent of the path. Equation (2.1) therefore defines a new state function E. Robert Mayer (1842) was probably the first to generalize the energy in this way. For a differential change eq. (2.1) becomes dE = dq - dw (2.2) The energy function is undetermined to the extent of an arbitrary addi- tive constant; it has been defined only in terms of the difference in energy between one state and another. Sometimes, as a matter of convenience, we may adopt a conventional standard state for a system, and set its energy in this state equal to zero. For example, we might choose the state of the system at 0K and 1 atm pressure as our standard. Then the energy E in any other state would be the change in energy in going from the standard state to the state in question. The First Law has often been stated in terms of the universal human experience that it is impossible to construct a perpetual motion machine, that is, a machine that will continuously produce useful work or energy from nothing. To see how this experience is embodied in the First Law, consider a cyclic process from state A to B and back to A again. If perpetual motion were ever possible, it would sometimes be possible to obtain a net increase in energy A > by such a cycle. That this is impossible can be ascertained from eq. (2.1), which indicates that for any such cycle &E = (E n E A ) + (E A E B ) = 0. A more general way of expressing this fact is to say that for any cyclic process the integral of dE vanishes: dE=Q (2.3) 3. The nature of internal energy. On page 6 we restricted the systems under consideration to those in a state of rest in the absence of gravitational or electromagnetic fields. With these restrictions, changes in the internal Sec. 4] THE FIRST LAW OF THERMODYNAMICS 29 energy E include changes in the potential energy of the system, and energy associated with the addition or subtraction of heat. The potential energy changes may be considered in a broad sense to include also the energy changes caused by the rearrangements of molecular configurations that take place during changes in state of aggregation, or in chemical reactions. If the system were moving, the kinetic energy would have to be added to E. If the restriction on electromagnetic fields were removed, the definition of E would have to be expanded to include the electromagnetic energy. Simi- larly, if gravitational effects were of interest, as in centrifugal operations, the energy of the gravitational field would have to be included in or added to E before applying the First Law. In view of these facts, it has been remarked that even if somebody did invent a perpetual motion machine, we should simply invent a new variety of energy to explain it, and so preserve the validity of the First Law. From this point of view, the First Law is essentially a definition of a function called the energy. What gives the Law real meaning and usefulness is the practical fact that a very small number of different kinds of energy suffice to describe the physical world. In anticipation of future discussions, it may be mentioned that experi- mental proof of the interconversion of mass and energy has been provided by the nuclear physicists. The First Law should therefore become a law of the conservation of mass-energy, and the extension of thermodynamics along these lines is beginning to be studied. The changes in mass theoretically associated with the energy changes in chemical reactions are so small that they lie just outside the range of our present methods of measurement. Thus they need not be considered in ordinary chemical thermodynamics. 4. Properties of exact differentials. We have seen in Section 1-17 that the work done by a system in going from one state to another is a function of the path between the states, and that dw is not in general equal to zero. The reason was readily apparent when the reversible process was considered. Then, dw \ P dV. The differential expression P dV cannot be inte- J A J A grated when only the initial and final states are known, since P is a function not only of the volume Kbut also of the temperature 7, and this temperature C B may also change along the path of integration. On the other hand, I dE can always be carried out, giving E n E A , since is a function of the state of the system alone, and is not dependent on the path by which that state is reached or on the previous history of the system. Mathematically, therefore, we distinguish two classes of differential ex- pressions. Those such as dE are called exact differentials since they are obtained by differentiation of some state function such as E. Those such as dq or dw are inexact differentials, since they cannot be obtained by differen- tiation of a function of the state of the system alone. Conversely, dq or dw cannot be integrated to yield a q or w. The First Law states that although 30 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 dq and dw are not exact differentials, their difference dE = dq dw is an exact differential. The following statements are mathematically completely equivalent : (1) The function E is a function of the state of a system. (2) The differential dE is an exact differential. (3) The integral of dE about a closed path dE is equal to zero. As an important corollary of the fact that it is an exact differential, dE may be written 1 dE - ( ) dx + ( } dy (2.4) \dx/ v \cy' x where x and y are any other variables of state of the system, for instance any two of P 9 T, V. Thus, for example, /XF\ IT (2.5) A further useful property of exact differential expressions is the Euler reciprocity relation. If an exact differential is written dE = M dV + TV dT, then ar/ r \*VJ T (2 ' 6) This can be seen immediately from the typical case of eq. (2.5), whence eq. (2.6) becomes (3 2 /8FOr) -- (d 2 E/dTdV) since the order of differentiation is immaterial. 5. Adiabatic and isothermal processes. Two kinds of processes occur fre- quently both in laboratory experiments and in thermodynamic arguments. An isothermal process is one that occurs at constant temperature, T constant, dT 0. To approach isothermal conditions, reactions are often carried out in thermostats. In an adiabatic process, heat is neither added to nor taken from the system; i.e., q = 0. For a differential adiabatic process, dq 0, and therefore from eq. (2.2) dE dw. For an adiabatic reversible change in volume, dE ==- P dV. Adiabatic conditions can be approached by careful thermal insulation of the system. High vacuum is the best insulator against heat conduction. Highly polished walls minimize radiation. These principles are combined in Dewar vessels of various types. 6. The heat content or enthalpy. No mechanical work is done during a process carried out at constant volume, since V = constant, dV 0, w 0. It follows that the increase in energy % equals the heat absorbed at constant volume. A-? r (2.7) If pressure is held constant, as for example in experiments carried out under atmospheric pressure, A = E 2 E l = q w q P(V 2 K x ) or 1 See, e.g., Granviller, Smith, Longley, Calculus (Boston: Ginn, 1934), p. 412. Sec. 7] THE FIRST LAW OF THERMODYNAMICS 31 ( 2 + PV*) (Ei + PV\) = <!P> where q p is the heat absorbed at constant pressure. We now define a new function, called the enthalpy or heat content 2 by H - E -\ PV (2.8) Then A// = H 2 - H - q p (2.9) The increase in enthalpy equals the heat absorbed at constant pressure. It will be noted that the enthalpy H 9 like the energy *, is a function of the state of the system alone, and is independent of the path by which that state is reached. This fact follows immediately from the definition in eq. (2.8), since ", P, and V are all state functions. 7. Heat capacities. Heat capacities may be measured either at constant volume or at constant pressure. From the definitions in eqs. (1.36), (2.7), and (2.9): heat capacity at constant volume: C v -= ~ - I I (2.10) aT \oTfy heat capacity at constant pressure: C P = -~ \ ) (2.11) aT \dT/p The capital letters C v and C P are used to represent the heat capacities per mole. Unless otherwise specified, all thermodynamic quantities that are extensive in character will be referred to the molar basis. The heat capacity at constant pressure C P is always larger than that at constant volume C F , because at constant pressure part of the heat added to a substance is used in the work of expanding it, whereas at constant volume aH of the added heat produces a rise in temperature. An important equation for the difference C P C v can be obtained as follows: C " - C * = (I'),- (D K = (fH' + P ()- (D F (2 ' 12) Since, dE -- /3\ /3\ /3K\ p\ Substituting this value in eq. (2.12), we find v l/dy\ (2.13) The term P(dV/dT) P may be seen to represent the contribution to the specific heat C P caused by the expansion of the system against the external 2 Note carefully that heat content H and heat capacity dqjdT are two entirely different functions. The similarity in nomenclature is unfortunate, and the term enthalpy is therefore to be preferred to heat content. 32 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 pressure P. The other term (dE/dV) T (dy/3T)j> is the contribution from the work done in expansion against the internal cohesive or repulsive forces of the substance, represented by a change of the energy with volume at constant temperature. The term (dE/3V) T is called the internal pressure? In the case of liquids and solids, which have strong cohesive forces, this term is large. In the case of gases, on the other hand, the term (dE/dV) T is usually small compared with P. In fact, the first attempts to measure (d/dY) T for gases failed to detect it at all. These experiments were carried out by Joule in 1843. 8. The Joule experiment. Joule's drawing of his apparatus is reproduced in Fig. 2.1, and he described the experiment as follows. 4 I provided another copper receiver () which had a capacity of 134 cubic inches. ... I had a piece D attached, in the center of which there was a bore J of an inch diameter, which could be closed per- fectly by means of a proper stopcock. . . . Having filled the receiver R with about 22 atmospheres of dry air and having exhausted the receiver E by means of an air pump, I screwed them together and put them into a tin can con- taining 161 Ib. of water. The water was first thoroughly stirred, and its temperature taken by the same delicate thermometer which was made use of in the former experiments on mechanical equivalent of heat. The stopcock was then opened by means of a proper key, and the air allowed to pass from the full into the empty re- ceiver until equilibrium was established between the two. Lastly, the water was again stirred and Fig. 2.1. The Joule experiment. its temperature carefully noted. Joule then presented a table of experimental data, showing that there was no measurable temperature change, and arrived at the conclusion that "no change of temperature occurs when air is allowed to expand in such a manner as not to develop mechanical power" (i.e., so as to do no external work). The expansion in Joule's experiment, with the air rushing from R into the evacuated vessel , is a typical irreversible process. Inequalities of tem- perature and pressure arise throughout the system, but eventually an equi- librium state is reached. There has been no change in the internal energy of the gas since no work was done by or on it, and it has exchanged no heat with the surrounding water (otherwise the temperature of the water would have changed). Therefore AE 0. Experimentally it is found that Ar 0. It may therefore be concluded that the internal energy must depend only on the temperature and not on the volume. More mathematically expressed: 3 Note that just as d/<V, the derivative of the energy with respect to a displacement, is a force, the derivative with respect to volume, 5E/DK, is a force per unit area or a pressure. 4 Phil. A%., 1843, p.- 263. Sec. 9] THE FIRST LAW OF THERMODYNAMICS 33 while </K>0 Since and it follows that Joule's experiment, however, was not capable of detecting small effects, since the heat capacity of his water calorimeter was extremely large compared with that of the gas used. 9. The Joule-Thomson experiment. William Thomson (Lord Kelvin) suggested a better procedure, and working with Joule, carried out a series of experiments between 1852 and 1862. Their apparatus is shown schematically in Fig. 2.2. The principle involved throttling the gas flow from the high pressure A to the low pressure C side by interposing a porous plug B. In their first trials, this plug consisted of a silk handkerchief; in later work, porous meerschaum was used. In this way, by the time the gas emerges into C it has already reached equilibrium and its temperature can be measured directly. The entire system is thermally insulated, so that the process is an adiabatic one, and q 0. Suppose that the fore pressure in A is /\, the back pressure in C is P 2 > and the volumes per mole of gas at these pressures are V and K 2 , respec- tively. The work per mole done on the gas in forcing it through the plug is then P^, and the work done by the gas in expanding on the other side is P 2 V 2 . The net work done by the gas is therefore w P 2 V 2 P^V^ It follows that a Joule-Thomson expansion occurs at constant enthalpy, since A E 2 E l q w --= w E 2 EI PI V\ ^2 ^2 E 2 + P 2 V 2 ^E,+ P.V, Fig. 2.2. The Joule-Thomson experi- ment. The Joule-Thomson coefficient, /i JmTf9 is defined as the change of temperature with pressure at constant enthalpy: /^r\ (2.14) This quantity is measured directly from the temperature change A7 of the gas as it undergoes a pressure drop A/> through the porous plug. Some 34 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 experimental values of the J.-T. coefficients, which are functions of tem- perature and pressure, are collected in Table 2.1. TABLE 2.1 JOULE-THOMSON COEFFICIENTS FOR CARBON DIOXIDE* /* (C per atm) Tempera- ture (K) Pressure (atm) 1 10 40 60 80 100 i 220 2.2855 2.3035 250 1.6885 1.6954 1.7570 275 1.3455 1.3455 1.3470 300 \ 1.1070 1.1045 1.0840 1.0175 0.9675 325 0.9425 0.9375 0.9075 0.8025 0.7230 0.6165 0.5220 350 0.8195 0.8150 0.7850 0.6780 0.6020 0.5210 0.4340 380 0.7080 0.7045 0.6780 0.5835 0.5165 0.4505 0.3855 400 0.6475 0.6440 0.6210 0.5375 0.4790 0.4225 0.3635 * From John H. Perry, Chemical Engineers' Handbook (New York: McGraw-Hill, 1941). Rearranged from Int. Crit. Tables, vol. 5, where further data may be found. A positive //< corresponds to cooling on expansion, a negative \i to warm- ing. Most gases at room temperatures are cooled by a J.-T. expansion. Hydrogen, however, is warmed if its initial temperature is above -80C, but if it is first cooled below 80C it can then be cooled further by a J.-T. effect. The temperature 80C at which jn ^ is called the Joule-Thomson inversion temperature for hydrogen. Inversion temperatures for other gases, except helium, lie considerably higher. 10. Application of the First Law to ideal gases. An analysis of the theory of the Joule-Thomson experiment must be postponed until the Second Law of Thermodynamics has been studied in the next chapter. It may be said, however, that the porous-plug experiments showed that Joule's original con- clusion that (9/OK) T ^ for all gases was too broad. Real gases may have a considerable internal pressure and work must be done against the cohesive forces when they expand. An ideal gas may now be defined in thermodynamic terms as follows: (1) The internal pressure (9/3K) T = 0. (2) The gas obeys Boyle's Law, PV = constant at constant T. It follows from eq. (2.5) that the energy of an ideal gas is a function of its temperature alone. Thus dE -~- (3E/dT) y dT = C v dT and C v = dE/dT. The heat capacity of an ideal gas also depends only on its temperature. These conclusions greatly simplify the thermodynamics of ideal gases, so that many thermodynamic discussions are carried on in terms of the ideal gas model. Some examples follow: Sec. 10] THE FIRST LAW OF THERMODYNAMICS 35 Difference in heat capacities. When eq. (2.13) is applied to an ideal gas, it becomes Then, since PV = RT /9F\ _R and Cp C v = R (2.15) Heat capacities are usually given in units of calories per degree per mole, and, in these units, R - 8.3144/4.1840 = 1. 9872 caldeg- 1 mole- 1 Temperature changes. Since dE C v dT Likewise for an ideal gas: dH = C P dT and A// = H 2 - H =j*' C v dT (2.17) Isothermal volume or pressure change. For an isothermal change in an ideal gas, the internal energy remains constant. Since dT - and -0, and dq = dw = P dV Since p = f 2 P \ dq = \ Ji Ji dv RT- i V or = w- *nn-^ = RTln (2.18) Since the volume change is carried out reversibly, P always having its equi- librium value RT/V, the work in eq. (2.18) is the maximum work done in an expansion, or the minimum work needed to effect a compression. The equa- tion tells us that the work required to compress a gas from 10 atm to 100 atm is just the same as that required to compress it from 1 atm to 10 atm. 36 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 Reversible adiabatic expansion. In this case, dq = 0, and dE = dw ~ ~PdV. From eq. (2.16) dw = C v dT For a finite change w \C V dT J i We may write eq. (2.19) as C v dT + P dV - _ dT dV Whence (2.19) (2.20) (2.21) Integrating between 7\ and 7^, and ^ and K 2 , the initial and final tempera- tures and volumes, we have C v In J + R In =-- (2.22) ' 1 ^1 This integration assumes that C v is a constant, not a function of T. We may substitute for R from eq. (2.15), and using the conventional symbol y for the heat capacity ratio C^/Cy we find ISOTHERMAL ^v ADIABATIC (y- 7\ n Therefore, Since, for an ideal gas, (2.23) Fig. 2.3. Isothermal and adiabatic expansions. (2.24) It has been shown, therefore, that for a reversible adiabatic expansion of an ideal gas PV Y ^ constant (2.25) We recall that for an isothermal expansion PV constant. These equations are plotted in Fig. 2.3. A given pressure fall produces a lesser volume increase in the adiabatic case, owing to the attendant fall in temperature during the adiabatic expansion. 11. Examples of ideal-gas calculations. Let us take 10 liters of gas at and 10 atm. We therefore have 100/22.414 4.457 moles. We shall calculate the final volume and the work done in three different expansions to a final pressure of 1 atm. The heat capacity is assumed to be C v = $R, independent of temperature. Isothermal reversible expansion. In this case the final volume V 2 = P^Pi - (10)(10)/(1) - 100 liters Sec. 11] THE FIRST LAW OF THERMODYNAMICS 37 The work done by the gas in expanding equals the heat absorbed by the gas from its environment. From eq. (2.18), for n moles, y q ^ w nRTln - V\ - (4.457)(8.314)(273.2)(2.303) log (10) -23,3 10 joules Adiabatlc reversible expansion. The final volume is calculated from eq. (2.24), with C P ($R f R) 5 Thus ^2 H ~ I K! (10) 3/5 - 10 - 39.8 liters \ * 2 The final temperature is obtained from P 2 V 2 nRT 2 : P,V, (1)(39.8) Tz ~ n'R (4.457X0.08205) m * K For an adiabatic process, q =- 0, and A q u ^- - iv. Also, since C r is constant, eq. (2.16) gives A - rtCjAr n%R(T 2 7\) 9125 joules The work done by the gas on expansion is therefore 9125 joules. Irreversible adiabatic expansion. Suppose the pressure is suddenly released to 1 atm and the gas expands adiabatically against this constant pressure. Since this is not a reversible expansion, eq. (2.24) cannot be applied. Since q = 0, A ~w. The value of A depends only on initial and final states: A- - w =-/iC r (7* 2 - 7\) Also, for a constant pressure expansion, we have from eq. (1.39), Equating the two expressions for vv, we obtain The only unknown is T 2 : -^(r-2732)=l^ 2 2 f \ 1 10 > T z - 174.8K Then A = vv =- f Rn(\14.S 273.2) =- 5470 joules 38 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 Note that there is considerably less cooling of the gas and less work done in the irreversible adiabatic expansion than in the reversible expansion. 12. Thermochemistry heats of reaction. Thermochemistry is the study of the heat effects accompanying chemical reactions, the formation of solutions, and changes in state of aggregation such as melting or vaporization. Physico- chemical changes can be classified as endothermic, accompanied by the absorption of heat, or exothermic, accompanied by the evolution of heat. A typical example of an exothermic reaction is the burning of hydrogen: H 2 f- i- O 2 - H 2 O (gas) f 57,780 cal at 18C A typical endothermic reaction would be the reverse of this, the decom- position of water vapor: H 2 O - H 2 -f I O 2 - 57,780 cal at 18C Heats of reaction may be measured at constant pressure or at constant volume. An example of the first type of experiment is the determination of the heat evolved when the reaction takes place at atmospheric pressure in an open vessel. If the reaction is carried out in a closed autoclave or bomb, the constant-volume condition holds. By convention, reaction heats are considered positive when heat is ab- sorbed by the system. Thus an exothermic reaction has a negative "heat of reaction." From eq. (2.7) the heat of reaction at constant volume, Q v - A F (2.26) From eq. (2.9) the heat of reaction at constant pressure, Q P - A//,> - A + P AK (2.27) The heat of reaction at constant volume is greater than that at constant pressure by an amount equal to the external work done by the system in the latter case. In reactions involving only liquids or solids AFis so small that usually P AK is negligible and Q v & Q P . For gas reactions, however, the P A V terms may be appreciable. The heat change in a chemical reaction can best be represented by writing the chemical equation for the reaction, specifying the states of all the reactants and products, and then appending the heat change, noting the temperature at which it is measured. Since most reactions are carried out under essentially constant pressure conditions, A// is usually chosen to represent the heat of reaction. Some examples follow: (1) SO 2 (1 atm) + | O 2 (1 atm) - SO 3 (1 atm) A// 298 - -10,300 cal (2) CO 2 (1 atm) + H 2 (1 atm) - CO (I atm) + H 2 O (1 atm) A// 29 8 - 9860 cal (3) AgBr (cryst>+ \ C1 2 (1 atm) - AgCl (cryst) + \ Br 2 (liq) A//008- -6490 cal Sec. 13] THE FIRST LAW OF THERMODYNAMICS 39 As an immediate consequence of the First Law, A or A// for any chemical reaction is independent of the path; that is, independent of any intermediate reactions that may occur. This principle was first established experimentally by G. H. Hess (1840), and is called The Law of Constant Heat Summation. It is often possible, therefore, to calculate the heat of a reaction from measurements on quite different reactions. For example: : (1) COC1 2 -! H 2 S =-- 2 HCl + COS A// 298 - -42,950 cai (2) COS + H 2 S - H 2 O (g) + CS 2 (1) A// 298 - +3980 cai (3) COC1 2 | 2 H 2 S - 2 HCl + H 2 O (g) + CS 2 (1) A #298 = "38,970 cai 13. Heats of formation. A convenient standard state for a substance may be taken to be the state in which it is stable at 25C and 1 atm pressure; thus, oxygen as O 2 (g), sulfur as S (rhombic crystal), mercury as Hg (1), and so on. By convention, the enthalpies of the chemical elements in this standard state are set equal to zero. The standard enthalpy of any compound is then the heat of the reaction by which it is formed from its elements, reactants and products all being in the standard state of 25C and 1 atm. For example: (1) S + O 2 - SO 2 A// 298 - -70,960 cai (2) 2 Al + | O 2 - A1 2 O 23 A// 298 - -380,000 cai The superscript zero indicates we are writing a standard heat of formation with reactants and products at 1 atm; the absolute temperature is written as a subscript. Thermochemical data are conveniently tabulated as heats of formation. A few examples, selected from a recent compilation of the National Bureau of Standards, 5 are given in Table 2.2. The standard heat of any reaction at 25C is then readily found as 'the difference between the standard heats of formation of the products and of the reactants. TABLE 2.2 STANDARD HEATS OF FORMATION AT 25C Compound State A/^298.1 (kcal/mole) Compound State A#298.16 (kcal/mole) H 2 g -57.7979 H 2 S g -4.815 H 2 1 -68.3174 H 2 SO 4 -193.91 H 2 2 g -31.83 SO, g -70.96 HF g -64.2 S0 3 g -94.45 HCl g -22.063 CO g -26.415? HBr g -8.66 CO 2 g -94.0518 HI g + 6.20 SOC1 2 1 -49.2 HI0 3 c -57.03 S 2 C1 2 g -5.70 5 The Bureau is publishing a comprehensive collection of thermodynamic data, copies of which are to be deposited in every scientific library ("Selected Values of Chemical Thermodynamic Properties'*). 40 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 Many of our thermochemical data have been obtained from measure- ments of heats of combustion. If the heats of formation of all its combustion products are known, the heat of formation of a compound can be calculated from its heat of combustion. For example (1) C 2 H 6 j I O 2 - 2 CO 2 ~| 3 H 2 O (1) A// 298 - -373.8 kcal (2) C (graphite) f O 2 -- CO 2 A// 298 = -94.5 kcai (3) H 2 + I O 2 - H 2 O (1) A// 298 - -68.3 kcai (4) 2 C f 3 H 2 -- C 2 H 6 A// 298 --= -20.1 kcal The data in Table 2.3 were obtained from combustion heats by F. D. Rossini and his co-workers at the National Bureau of Standards. The standard state of carbon has been taken to be graphite. When changes in state of aggregation occur, the appropriate latent heat must be added. For example: S (rh) { O 2 SO 2 A// 298 -70.96 kcal S (rh) S (mono) A// 298 ^ -0.08 kcal S (mono) + O 2 - SO 2 A// 298 . 70.88 kcal 14. Experimental measurements of reaction heats. 6 The measurement of the heat of a reaction consists essentially of a careful determination of the amount of the chemical reaction that produces a definite measured change in the calorimeter, and then the measurement of the amount of electrical energy required to effect exactly the same change. The change in question is usually a temperature change. A notable exception is in the ice calorimeter, in which one measures the volume change produced by the melting of ice, and thereby calculates the heat evolution from the known latent heat of fusion of ice. The A// values in Table 2.3 were obtained by means of a combustion- bomb calorimeter. It is estimated that the limit of accuracy with the present apparatus and technique is 2 parts in 10,000. Measurements with a bomb calorimeter naturally yield A values, which are converted to A//'s via eq. (2.27). A thermochemical problem of great interest in recent years has been the difference in the energies of various organic compounds, especially the hydro- carbons. It is evident that extremely precise work will be necessary to evaluate such differences from combustion data. For example, the heat contents of the five isomers of hexane differ by 1 to 5 kcal per mole, while the heats of combustion of the hexanes are around 1000 kcal per mole; even a 0.1 per cent uncertainty in the combustion heats would lead to about a 50 per cent un- certainty in the energy differences. Important information about such small a Clear detailed descriptions of the experimental equipment and procedures can be found in the publications of F. D. Rossini and his group at the National Bureau of Stan- dards, J. Res. ofN.B.S., rf,. 1 (1930); 13, 469 (1934); 27, 289 (1941). Sec. 15] THE FIRST LAW OF THERMODYNAMICS 41 TABLE 2.3 HEATS OF FORMATION OF GASEOUS HYDROCARBONS Substance Paraffins : Methane Ethane Propane rt-Butane Isobutane w-Pentane 2-Methylbutane Tetramethylmethane Monolefines: Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene 1-Pentene Diolefines: Allene 1,3-Butadiene 1,3-Pentadiene 1,4-Pentadiene Acetylenes : Acetylene Methylacetylene Dimethylacetylene Formula CH 4 C 2 H C 4 H 10 C 4 H, C,H 12 QH!! C 2 H 4 C,H. C 4 H 8 C 4 H fl C,H 8 j (cal/mole) 17,8651 74 -20,191 L 108 24,750 f 124 29,715 153 31,350 I 132 34,7394, 213 36,671 I 153 39,410 L 227 12,556 1 67 4956 t 110 383 JL 180 -1388 J 180 - 2338 180 - 3205 J. 165 -4644 .{ 300 46,046 260 26,865 -> 240 18,885 -f- 300 25,565 j : 300 54,228 J. 235 44,309 }- 240 35,221 -{- 355 energy differences can be obtained for unsaturated hydrocarbons by measure- ment of their heats of hydrogenation. This method has been developed to a high precision by G. B. Kistiakowsky and his co-workers at Harvard. 7 It is evident that in calorimetric experiments for example, in a deter- mination of a heat of combustion the chemical reaction studied may actually occur at a very elevated temperature. One measures, however, the net temperature rise after equilibrium has been reached, and this usually amounts to only a few degrees, owing to the high heat capacity of the calori- meter. Since AE or A// depends only on the initial and final states, one actually measures the A or A//, therefore, at around 25C, even though temperatures of over 2000C may have been attained during the actual combustion process. 15. Heats of solution. In many chemical reactions, one or more of the reactants are in solution, and the investigation of heats of solution is an important branch of thermochemistry. It is necessary to distinguish the integral heat of solution and the differential heat of solution. The distinction 7 Kistiakowsky, et al., /. Am. Cfiem. Soc., 57, 876 (1935). 42 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 between these two terms can best be understood by means of a practical example. If one mole of alcohol (C 2 H 5 OH) is dissolved in nine moles of water, the final solution contains 10 moles per cent of alcohol. The heat absorbed is the integral heat of solution per mole of alcohol to form a solution of this final composition. If the mole of alcohol is dissolved in four moles of water, the integral heat of solution has a different value, corresponding to the formation 3000 m 2000 o 2 I "? 1000 V 10 20 30 40 50 60 70 80 MOLES WATER/MOLES ALCOHOL Fig. 2.4. Heat of solution of ethyl alcohol in water at 0C. of a 20 mole per cent solution. The difference between any two integral heats of solution yields a value for the integral heat of dilution. The example can be written in the form of thermochemical equations as follows: (1) C 2 H 5 OH + 9 H 2 O - C 2 H 5 OH (10 mole % solution) A// 273 = -2300 (2) C 2 H 5 OH -f- 4 H 2 O - C 2 H 5 OH (20% solution) (3) C 2 H 5 OH (20% solution) + 5 H 2 O -= C 2 H 5 OH (10% solution; A// 273 - -1500 A// 273 = -800 The heat of dilution from 20 to 10 per cent amounts to 800 cal per mole. It is evident that the heat evolved ( A//) when a mole of alcohol is dissolved in water depends upon the final concentration of the solution. If one plots the measured integral heat of solution against the ratio moles water per mole alcohol (njn a \ the curve in Fig. 2.4 is obtained. As the solution becomes more and more dilute, njn a approaches infinity. The asymptotic value of the heat of solution is called the heat of solution at infinite dilution, A//^, For alcohol in water at 0C, this amounts to 3350 Sec. 16] THE FIRST LAW OF THERMODYNAMICS 43 calories. The values of A// 80lution generally become quite constant with in- creasing dilution, so that measured values in dilute solutions are usually close to A//^. Often one finds literature values for which the dilution is not specified. These are written, for aqueous solution, simply as in the following example: NaCl + x H 2 O - NaCl (aq) h 1260 cal : In the absence of more detailed information, such values may be taken to give approximately the A// at infinite dilution. The integral solution heats provide an average A// over a range of con- centrations. For example, if alcohol is added to water to make a 50 mole per cent solution, the first alcohol added gives a heat essentially that for the solute dissolving in pure water, whereas the last alcohol is added to a solution of about 50 per cent concentration. For theoretical purposes, it is often necessary to know what the A// would be for the solution of solute in a solution of definite fixed concentration. Let us imagine a tremendous volume of solution of definite composition and add one more mole of solute to it. We can then suppose that this addition causes no detectable change in the concentration. The heat absorbed in this kind of solution process is the differential heat of solution. The same quantity can be defined in terms of a very small addition of dn moles of solute to a solution, the heat absorbed per mole being dq/dn and the composition of the solution remaining un- changed. Methods of evaluating the differential heat will be considered in Chapter 6. 16. Temperature dependence of reaction heats. Reaction heats depend on the temperature and pressure at which they are measured. We may write the energy change in a chemical reaction as ~ ^reactants 3 pr0(1 From eq. (2.10), - = C rprod - C Freact = AC r (2.28) Similarly, f , = C 'i* ' Q'react - AQ, (2.29) Integrating, at a constant pressure of 1 atm, so that A// is the standard A//, we obtain A// T> - A// Ti - ACp dT (2.30) JT, These equations were first ^et forth by G. R. Kirchhoff in 1858. They state that the difference between the heats of reaction at 7\ and at T 2 is equal to the difference in the amounts of heat that must be added to reactants and 44 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 products at constant pressure to raise them from 7\ to T 2 . This conclusion is an immediate consequence of the First Law of Thermodynamics. In order to apply eq. (2.30), expressions are required for the heat capaci- ties of reactants and products over the temperature range of interest. Over a short range, these may often be taken as practically constant, and we obtain: A// r A// TI ---= ACV(7 2 Tj) More generally, the experimental heat-capacity data will be represented by a power series: CV -a + bT-\- cT 2 f . . . (2.31) Examples of such heat-capacity equations are given in Table 2.4. These three-term equations fit the experimental data to within about 0.5 per cent TABLE 2.4 HEAT CAPACITY OF GASES (273-1 500K)* C P = a 4- bT i cT 2 (C/. in calories per deg per mole) Gas b x 10 3 c x 10 7 H, 6.9469 -0.1999 4.808 o; 6.148 3.102 -9.23 CI 2 7.5755 2.4244 -0.650 Br 2 8.4228 0.9739 -3.555 N., 6.524 1.250 -0.01 CO 6.420 1.665 -1.96 HCl 6.7319 0.4352 3.697 HBr 6.5776 0.9549 1.581 H,O 7.256 2.298 2.83 C0 2 6.214 10.396 -35.45 Benzene 0.283 77.936 -262.96 n-Hexane 7.313 104.906 -323.97 CH 4 3.381 18.044 -43.00 * H. M. Spencer, /. Am. Chem. Soc., 67, 1858 (1945). Spencer and Justice, ibid., 56, 2311 (1934). over a temperature range from 0C to 1250C. When the series expression for AC/> is substituted 8 in eq. (2.30), the integration can be carried out analytically. Thus at constant pressure, for the standard enthalpy change, rf(A//) - AQ, dT - (A + BT + CT 2 + . . .)dT A// - A// -f -AT + i BT* + J Cr 3 + . . . (2.32) Here A// is the constant of integration. 9 Any one measurement of A// at 8 For a typical reaction, N 8 -f- ; J H 2 - NH 3 ; AC P = C PNHi - i C^ t - J PH . 9 If the heat-capacity equations are valid to 0K, we may note that at T = 0, A/f = A// , so that the integration constant can be interpreted as the enthalpy change in the reaction at 0K. Sec. 16] THE FIRST LAW OF THERMODYNAMICS 45 a known temperature T makes it possible to evaluate the constant A// in eq. (2.32). Then the A// at any other temperature can be calculated from the equation. If the heat capacities are given in the form of a C P vs. T curve, a graphical integration is often convenient. Recently rather extensive enthalpy tables have become available, which give H as a function of T over a wide range of temperatures. The use of these tables makes direct reference to the heat capacities unnecessary. 16. Chemical affinity. Much of the earlier work on reaction heats was done by Julius Thomsen and Marcellin Berthelot, in the latter part of the nineteenth century. They were inspired to carry out a vast program of thermochemical measurements by the conviction that the heat of reaction was the quantitative measure of the chemical affinity of the reactants. In the words of Berthelot, in his Essai de Mecanique chimique (1878): Every chemical change accomplished without the intervention of an external energy tends toward the production of the body or the system of bodies that sets free the most heat. This principle is incorrect. It would imply that no endothermic reaction could occur spontaneously, and it fails to consider the reversibility of chemi- cal reactions. In order to understand the true nature of chemical affinity and of .he driving force in chemical reactions, it is necessary to go beyond the Firs. Law of Thermodynamics, and to investigate the consequences of the second fundamental law that governs the interrelations of work and heat. PROBLEMS 1. Calculate A and A// when 100 liters of helium at STP are heated to 100C in a closed container. Assume gas is ideal with C v ^R. 2. One mole of ideal gas at 25C is expanded adiabatically and reversibly from 20 atm to 1 atm. What is the final temperature of the gas, assuming Cy = \R1 3. 100 g of nitrogen at 25C arc held by a piston under 30 atm pressure. The pressure is suddenly released to 10 atm and the gas adiabatically expands. If C v for nitrogen - 4.95 cal per deg, calculate the final tempera- ture of the gas. What are A and A// for the change? Assume gas is ideal. 4. At its boiling point (100C) the density of liquid water is 0.9584 g per cc; of water vapor, 0.5977 g per liter. Calculate the maximum work done when a mole of water is vaporized at the boiling point. How does this compare with the latent heat of vaporization of water? 5. If the Joule-Thomson coefficient is /< t/ T ^ 1.084 deg per atm and the heat capacity C P = 8.75 cal per mole deg, calculate the change in enthalpy A// when 50 g of CO 2 at 25C and 1 atm pressure are isothermally com- pressed to 10 atm pressure. What would the value be for an ideal gas? 46 THE FIRST LAW OF THERMODYNAMICS [Chap. 2 6. Using the heat-capacity equation in Table 2.4, calculate the heat required to raise the temperature of one mole of HBr from to 500C. 7. In a laboratory experiment in calorimetry 100 cc of 0.500 TV acetic acid are mixed with 100 cc of 0.500 N sodium hydroxide in a calorimeter. The temperature rises from 25.00 to 27.55C. The effective heat capacity of the calorimeter is 36 cal per deg. The specific heat of 0.250 N sodium acetate solution is 0.963 cal deg- 1 g- 1 and its density, 1.034g per cc. Calculate the heat of neutralization of acetic acid per mole. 8. Assuming ideal gas behavior, calculate the values of &E 29Q for SO 3 (g), H 2 O (g), and HCl (g) from the A// 298 values in Table 2 2. 9. From the heats of formation in Table 2.3, calculate A// 298 for the following cracking reactions: C 2 H 6 + H 2 - 2 CH 4 /i-C 4 H 10 | 3 H 2 -- 4 CH 4 iso-C 4 H 10 + 3 H 2 - 4 CH 4 10. The heat of sublimation of graphite to carbon atoms has been esti- mated as 170 kcal per mole. The dissociation of molecular hydrogen into atoms, H 2 - 2 H, has A// - 103.2 kcal per mole. From these data and the value for the heat of formation of methane, calculate the A// for C (g) + 4 H (g) CH 4 (g). One fourth of this value is a measure of the "energy of the C H bond" in methane. 11. Assuming that the energy of the C H bond in ethane, C 2 H 6 , is the same as in methane (Problem 10) estimate the energy of the C C bond in ethane from the heat of formation in Table 2.3. 12. When w-hexane is passed over a chromia catalyst at 500C, benzene is formed: C 6 H 14 (g) - C 6 H 6 (g) f 4 H 2 , A// 298 ^ 59.78 kcal per mole. Calculate A// for the reaction at 500C (Table 2.4). 13. Derive a general expression for A// of the water gas reaction (H 2 + CO 2 -= H 2 O f CO) as a function of temperature. Use it to calculate A// at 500K and 1000K. 14. From the curve in Fig. 2.4, estimate the heat evolved when 1 kg of a 10 per cent (by weight) solution of ethanol in water is blended with 1 kg of a 75 per cent solution of ethanol in water. 15. If a compound is burned under adiabatic conditions so that all the heat evolved is utilized in heating the product gases, the maximum tempera- ture attained is called the adiabatic flame temperature. Calculate this tem- perature for the burning of ethane with twice the amount of air (80 per cent N 2 , 20 per cent O 2 ) needed for complete combustion to CO 2 and H 2 O. Use heat capacities in Table 2.4, but neglect the terms cT 2 . 16. Show that for a van der Waals gas, (3E/3K) r a/Y 2 . 17. Show that (dEpP) v - 0C r /a. Chap. 2] THE FIRST LAW OF THERMODYNAMICS 47 REFERENCES BOOKS See Chapter 1, p. 25. ARTICLES 1. Parks, G. S., J. Chem. Ed., 26, 262-66 (1949), "Remarks on the History of Thermochemistry." 2. Menger, K., Am. J. Phys. 18, 89 (1950), "The Mathematics of Elementary Thermodynamics." 3. Sturtevant, J. M., Article on "Calorimetry" in Physical Methods of Organic Chemistry, vol. 1, 311-435, edited by A. Weissberger (New York: Interscience, 1945). CHAPTER 3 The Second Law of Thermodynamics 1. The efficiency of heat engines. The experiments of Joule helped to disprove the theory of "caloric" by demonstrating that heat was not a "substance" conserved in physical processes, since it could be generated by mechanical work. The reverse transformation, the conversion of heat into useful work, had been of greater interest to the practical engineer ever since the development of the steam engine by James Watt in 1769. Such an engine operates essentially as follows: A source of heat (e.g., a coal or wood fire) is used to heat a "working substance" (e.g., steam), causing it to expand through an appropriate valve into a cylinder fitted with a piston. The ex- pansion drives the piston forward, and by suitable coupling mechanical work can be obtained from the engine. The working substance is cooled by the expansion, and this cooled working substance is withdrawn from the cylinder through a valve. A flywheel arrangement returns the piston to its original position, in readiness for another expansion stroke. In simplest terms, therefore, any such heat engine withdraws heat from a heat source, or hot reservoir, converts some of this heat into work, and discards the remainder to a heat sink or cold reservoir. In practice there are necessarily frictional losses of work in the various moving components of the engine. The first theoretical discussions of these engines were expressed in terms of the caloric hypothesis. The principal problem was to understand the factors governing the efficiency e of the engine, which was measured by the ratio of useful work output w to the heat input q 2 . w e = - (3.1) ?2 A remarkable advance towards the solution of this problem was made in 1824 by a young French engineer, Sadi Carnot, in a monograph, Reflexions sur la Puissance motrice du Feu. 2. The Carnot cycle. The Carnot cycle represents the operation of an idealized engine in which heat is transferred from a hot reservoir at tem- perature / 2 i s Partly converted into work, and partly discarded to a cold reservoir at temperature t v (Fig. 3. la.) The working substance through which these operations are carried out is returned at the end to the same state that it initially occupied, so that the entire process constitutes a com- plete cycle. We have written the temperatures as t l and t 2 to indicate that they are empirical temperatures, measured on any convenient scale what- soever. The various steps in the cycle are carried out reversibly. Sec. 2] THE SECOND LAW OF THERMODYNAMICS 49 To make the operation more definite, we may consider the working sub- stance to be a gas, and the cyclic process may be represented by the indicator HOT RESERVOIR t2 (SOURCE) <*2 WORK 1 2 V, V 4 (0) (b) Fig. 3.1. The essential features of the heat engine (a) and the Carnot cycle for its operation shown on an indicator diagram (b). diagram of Fig. 3.1b. The steps in the working of the engine for one complete cycle are then : (1) Withdrawal of heat --^- q 2 from a hot reservoir at temperature t 2 by the isothermal reversible expansion of the gas from V v to V 2 . Work done by gas H^. (2) Adiabatic reversible expansion from V 2 to K 3 , during which q = 0, gas does work w 2 and cools from / 2 to t r (3) Isothermal reversible compression at t l from K 3 to F 4 . Work done by the gas w 3 . Heat q l absorbed by the cold reservoir at t v (4) Adiabatic reversible compression from K 4 to V 19 gas warming from t l to t 2 . Work done by gas ^ n> 4 , q = 0. The First Law of Thermodynamics requires that for the cyclic process A = 0. Now A is the sum of all the heat added to the gas, q = q 2 q ly less the sum of all the work done by the gas, w = \v l + w 2 vv 3 ~ vv 4 . A* ^ q w ^ q 2 ~ q l w ^= The net work done by the engine is equal, therefore, to the heat taken from the hot reservoir less the heat that is returned to the cold reservoir: w = #2 ~~ 9i- The efficiency of the engine is: - <tl 92 Since every step in this cycle is carried out reversibly, the maximum 50 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 possible work is obtained for the particular working substance and tem- peratures considered. 1 Consider now another engine operating, for example, with a different working substance. Let us assume that this second engine, working between the same two empirical temperatures / 2 and t l9 is more efficient than engine 1 ; that is, it can deliver a greater amount of work, w' > w, from the same amount of heat q 2 taken from the hot reservoir. (See Fig. 3. la.) It could accomplish this only by discarding less heat, q < q l9 to the cold reservoir. Let us now imagine that, after the completion of a cycle by this sup- posedly more efficient engine, the original engine is run in reverse. It therefore acts as a heat pump. Since the original Carnot cycle is reversible, all the heat and work terms are changed in sign but not in magnitude. The heat pump takes in q l of heat from the cold reservoir, and by the expenditure of work - w delivers q 2 of heat to the hot reservoir. For the first process (engine 2) w' q 2 q^ For the second process (engine 1) - w - q 2 + q l Therefore, the net result is: w' w = q l q^ Since w' > w, and q l > <//, the net result of the combined operation of these two engines is that an amount of heat, q q l <?/, has been abstracted from a heat reservoir at constant temperature t l and an amount of work w" = w' H' has been obtained from it, without any other change what- soever taking place. In this result there is nothing contrary to the First Law of Thermo- dynamics, for energy has been neither created nor destroyed. The work done would be equivalent to the heat extracted from the reservoir. Nevertheless, in all of human history, nobody has ever observed the isothermal conversion of heat into work without any concomitant change in the system. Think what it would imply. It would not be necessary for a ship to carry fuel: this wonderful device would enable it to use a small fraction of the immense thermal energy of the ocean to turn its propellers and run its dynamos. Such a continuous extraction of useful work from the heat of our environment has been called "perpetual motion of the second kind," whereas the production of work from nothing at all was called "perpetual motion of the first kind." The impossibility of the latter is postulated by the First Law of Thermodynamics; the impossibility of the former is postulated by the Second Law. If the supposedly more efficient Carnot engine delivered the same amount of work w as the original engine, it would need to withdraw less heat q 2 < q 2 1 In the isothermal steps, the maximum work is obtained on expansion and the mini- mum work done in compression of the gas (cf. p. 23). In the adiabatic steps A" = w, and the work terms are constant once the initial and final states are fixed. Sec. 3] THE SECOND LAW OF THERMODYNAMICS 51 from the hot reservoir. Then the result of running engine 2 forward and engine 1 in reverse, as a heat pump, would be (2) w^ fc'-fc' (1) _ W = -ft+ft ?2 - ? 2 = ft ~ ft ^ <7 This amounts to the transfer of heat </ from the cold reservoir at t l to the hot reservoir at t 2 without any other change in the system. There is nothing in this conclusion contrary to the First Law, but it is even more obviously contrary to human experience than is perpetual motion of the second kind. We know that heat always flows from the hotter to the colder region. If we place a hot body and a cold body together, the hot one never grows hotter while the cold one becomes colder. We know in fact that considerable work must be expended to refrigerate something, to pump heat out of it. Heat never flows uphill, i.e., against a temperature gradient, of its own accord. 3. Th^Second Law of Thermodynamics- This Second Law may be ex- pressed precisely in various equivalent forms. For example: The principle of Thomson. It is impossible by a cyclic process to take heat from a reservoir and convert it into work without, in the same operation. transferring heat from a hot to a cold reservoir. The principle of Clausius. It is impossible to transfer heat from a cold to a warm reservoir without, in the same process, converting a certain amount of work into heat. Returning to Carnot's cycle, we have seen that the supposition that one reversible cycle may exist that is more efficient than another has led to results contradicting human experience as embodied in the Second Law of Thermo- dynamics. We therefore conclude that all reversible Carnot cycles operating between the same initial and final temperatures must have the same efficiency. Since the cycles are reversible, this efficiency is the maximum possible. It is completely independent of the working substance and is a function only of the working temperatures : *--f(ti,*J 4. The thermodynamic temperature scale. The principle of Clausius may be rephrased as "heat never flows spontaneously, i.e., without the expenditure of work, from a colder to a hotter body." This statement contains essentially a definition of temperature, and we may recall that the temperature concept was first introduced as a result of the observation that all bodies gradually reach a state of thermal equilibrium. Lord Kelvin was the first to use the Second Law to define a thermo- dynamic temperature scale, which is completely independent of any thermo- metric substance. The Carnot theorem on the efficiency of a reversible cycle 52 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 may be written: Efficiency (independent of working substance) = (q 2 qi)/q 2 = /'('i. '2). or 1 - ft/ft =/'(^i, > 2 )- Therefore -=/('i, /a) (3-3) ft We have written /'('i, / 2 ) an d/('i> '2) 1 ~-/'('i '2) to indicate unspecified functions of / t and / 2 - Consider two Carnot cycles such that: qjq 2 =/(^i ^)J ft/?3 "/(^ '3)- They must be equivalent to a third cycle, operating between / t and / 3 , with ^ A'i '3)- Therefore ^^^Wa) ^ * 2 But, if this condition is satisfied, we can write: J(t l9 t 3 ) = F(t^)IF(t^\f(t^ t 3 ) = F(t 2 )/F(t 3 ). That is, the efficiency function, f(t l9 / 2 ), is the quotient of a function of t l alone and a function of t 2 alone. It follows that * = (3.4) Lord Kelvin decided to use eq. (3.4) as the basis of a thermodynamic temperature scale. He took the functions F(t^) and F(/ 2 ) to have the simplest possible form, namely, 7\ and To. Thus a temperature ratio on the Kelvin scale was defined as equal to the ratio of the heat absorbed to the heat rejected in the working of a reversible Carnot cycle. ** = P (3-5) <7i 7*i The efficiency of the cycle, eq. (3.2), then becomes The zero point of the thermodynamic scale is physically fixed as the temperature of the cold reservoir at which the efficiency becomes equal to unity, i.e., the heat engine is perfectly efficient. From eq. (3.6), in the limit as 7\->0, <?-> 1. The efficiency calculated from eq. (3.6) is the maximum thermal efficiency that can be approached by a heat engine. Since it is calculated for a reversible Carnot cycle, it represents an ideal that actual irreversible cycles can never achieve. Thus with a heat source at 120C and a sink at 20C, the maximum thermal efficiency is 100/393 =- 25.4 per cent. If the heat source is at 220 and the sink still at 20, the efficiency is raised to 200/493 = 40.6 per cent. It is easy to see why the trend in power plant design has been to higher tem- peratures for the heat source. In practice, the efficiency of steam engines seldom exceeds 80 per cent of the theoretical value. Steam turbines generally Sec. 5] THE SECOND LAW OF THERMODYNAMICS 53 can operate somewhat closer to their maximum thermal efficiencies, since they have fewer moving parts and consequently lower frictional losses. 5. Application to ideal gases. Temperature on the Kelvin, or thermo- dynamic, scale has been denoted by the symbol T, which is the same symbol used previously for the absolute ideal gas scale. It can be shown that these scales are indeed numerically the same by running a Carnot cycle with an ideal gas as the working substance. Applying eqs. (2.18) and (2.20) to the four steps: (1) Isothermal expansion: \\\ ~ q 2 RT 2 In K 2 /K t C T (2) Adiabatic expansion: w 2 ~- * C v dT; q J TI (3) Isothermal compression: u 3 q i RT In VJV% CT (4) Adiabatic compression: \v 4 = * C v dT; q -- j TI By summation of these terms, the net work obtained is w --= l + w 2 RT 2 ln V 2 /y i + RT\\n Since, from eq. (2.22), K,/^ - K 3 /K 4 , w - R(T 2 - T,) In ^ - T, 92 7 2 Comparison with eq. (3.6) completes the proof of the identity of the ideal gas and thermodynamic temperature scales. 6. Entropy. Equation (3.6) for a reversible Carnot cycle operating be- tween T 2 and 7\ irrespective of the working substance may be rewritten ?2 7\ Now it can be shown that any cyclic process can be broken down into a number of Carnot cycles. Consider the perfectly general ABA of Fig. 3.2. The area of the figure has been divided into a number of Carnot cycles by the crosshatched system of isothermals and adiabatics. The outside bound- aries of these little cycles form the heavy zigzag curve which follows quite closely the path of the general cycle ABA. The inside portions of the little Carnot cycles cancel out, since each section is traversed once in the for- ward direction and once in the reverse direction. For example, consider the Fig. 3.2. General cycle broken down isothermal xy which belongs to an into Carnot cycles. ISOTHERMALS ADIABATICS 54 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 expansion in the small cycle /?, and to a compression in the small cycle a, all the work and heat terms arising from it thereby being canceled. If eq. (3.7) is now applied to all these little Carnot cycles, we have for the zigzag segments V q\T = 0. As the Carnot cycles are made smaller and smaller, the boundary curve approaches more and more closely to that for the general cyclic process ABA. In the limit, for differential Carnot cycles, the area enclosed by the crooked boundary becomes identical with the area of the cycle ABA. We can then replace the summation of finite terms by the integration of differentials and obtain 2 (3.8) r- This equation holds true for any reversible cyclic process whatsoever. Fig. 3-3. Carnot cycle on a TS diagram. It may be recalled (p. 30) that the vanishing of the cyclic integral means that the integrand is a perfect differential of some function of the state of the system. This new function is defined by *3. T (for a reversible process) (3.9) Thus, . , c c_i_c c o - *-J/? A ~> 'i I'i The function 5 was first introduced by Clausius in 1850, and is called the entropy. Equation (3.9) indicates that when the inexact differential expression dq is multiplied by 1/r, it becomes an exact differential; the factor \JT is CB called an integrating multiplier. The integral dq KV is dependent on the f ft * A. path, whereas I dq rev /T is independent of the path. This, in itself, is an alternative statement of the Second Law of Thermodynamics. It is interesting to consider the TS diagram in Fig. 3.3, which is analogous to the PV diagram of Fig. 1.8. In the PV case, the area under the curve is a 2 See P. S. Epstein, Textbook of Thermodynamics (New York: Wiley, 1938), p. 57. Sec. 7] THE SECOND LAW OF THERMODYNAMICS 55 measure of the work done in traversing the indicated path. In the TS diagram, the area under the curve is a measure of the heat added to the system. Tem- perature and pressure are intensity factors ; entropy and volume are capacity factors. The products P dV and T dS both have the dimensions of energy. 7. The inequality of Clausius. Equation (3.8) was obtained for a reversible cycle. Clausius showed that for a cycle into which irreversibility enters at any stage, the integral of dq\T is always less than zero. ?<0 (3.10) The proof is evident from the fact that the efficiency of an irreversible Carnot cycle is always less than that of a reversible cycle operating between the same two temperatures. For the irreversible case, we therefore conclude from eq. (3.6) that </2 " T <2 Then, instead of eq. (3.7), we find that *_*<(> T, 7', This relation is extended to the general cycle, by following the argument based on Fig. (3.2). Instead of eq. (3.8), which applies to the reversible case, we obtain the inequality of Clausius, given by eq. (3.10). 8. Entropy changes in an ideal gas. The calculation of entropy changes in an ideal gas is particularly simple because in this case (3/<)K) T 0, and heat or work terms due to cohesive forces need not be considered at any point. For a reversible process in an ideal gas, the First Law requires that RT dV dq - dE + PdV~ Cy dT -\ y- Therefore, ^ = ^ + ^ (3.,,) On integration, AS 1 = S 2 - S l =- J 2 C v d\n T f J 2 RdlnV If C r is independent of temperature, AS- C F lnp+ Rln^ (3.12) 7\ V l For the special case of a temperature change at constant volume, the increase in entropy with increase in temperature is therefore AS- C F ln^ (3.13) If the temperature of one mole of ideal gas with C y ^ 3 is^ doubled, the 56 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 entropy is increased by 3 In 2 --- 2.08 calories per degree, or 2.08 entropy units (eu). For the case of an isothermal expansion, the entropy increase becomes AS- /?ln~- R\n Pl (3.14) YI PZ If one mole of ideal gas is expanded to twice its original volume, its entropy is increased by R In 2 1.38 eu. 9. Entropy changes in isolated systems. The change in entropy in going from a state A to a state B is always the same, irrespective of the path between A and B, since the entropy is a function of the state of the system alone. It makes no difference whether the path is reversible or irreversible. Only in case the path is reversible, however, is the entropy change given by AS S tt -S A --j ^ (3.15) In order to evaluate the entropy change for an irreversible process, it is necessary to devise a reversible method for going from the same initial to the same final state, and then to apply eq. (3.15). In any completely isolated system we are restricted to adiabatic processes, since no heat can either enter or leave such a system. 3 For a reversible process in an isolated system, therefore, dq and dS dq/T 0, or S --- constant. If one part of the system increases in entropy, the remaining part must decrease by an exactly equal amount. A fundamental example of an irreversible process is the transfer of heat from a hot to a colder body. We can make use of an ideal gas to carry out the transfer reversibly, and thereby calculate the entropy change. The gas is placed in thermal contact with the hot body at T 2 and expanded reversibly and isothermally until it takes up heat equal to q. To simplify the argument, it is assumed that the bodies have heat capacities so large that changes in their temperatures on adding or withdrawing heat q are negligible. The gas is then removed from contact with the hot reservoir and allowed to expand reversibly and adiabatically until its temperature falls to T v Next it is placed in contact with the colder body at 7\ and compressed isothermally until it gives up heat equal to q. The hot reservoir has now lost entropy = q/T 2 , whereas the cold reservoir has gained entropy ^ q/T r The net entropy change of the reservoirs has therefore been AS - <//7\ - q/T 2 . Since T 2 > 7\, AS > 0, and the entropy has increased. The entropy of the ideal gas, however, has decreased by an exactly equal amount, so that for the entire isolated system of ideal gas plus heat reservoirs, AS for the reversible process. If the heat transfer had 3 The completely isolated system is, of course, a figment of imagination. Perhaps our whole universe might be considered as an isolated system, but no small section of it can be rigorously isolated. As usual, -the precision and sensitivity of our experiments must be allowed to determine how the system is to be defined. Sec. 9] THE SECOND LAW OF THERMODYNAMICS 57 been carried out irreversibly, for example by placing the two bodies in direct thermal contact and allowing heat cj to flow along the finite temperature gradient thus established, there would have been no compensating entropy decrease. The entropy of the isolated system would have increased during the irreversible process, by the amount AS <//7\ qlT 2 . We shall now prove that the entropv of an isolated system always increases during an irreversible process. The proof of this theorem is based on the inequality of Clausius. Consider in Fig. (3.4) a perfectly general irreversible process in an isolated system, leading from state A to state B. It is represented by the dashed line. Next consider that the system is returned to its initial state A by a reversible path represented by the solid line from B to A. During this reversible Fig- 3.4. A cyclic process. process, the system need not be isolated, and can exchange heat and work with its environment. Since the entire cycle is in part irreversible, cq. (3.10) applies, and Writing the cycle in terms of its two sections, we obtain ^<0 (3.16) The first integral is equal to zero, since during the process A >- B the system is by hypothesis isolated and therefore no transfer of heat is possible. The second integral, from eq. (3.15), is equal to S t S H . Therefore eq. (3.16) becomes S A -- S H < 0, or SH S A > We have therefore proved that the entropy of the final state B is always greater than that of the initial state A, if A passes to B by an irreversible process in an isolated system. Since all naturally occurring processes are irreversible, any change that actually occurs spontaneously in nature is accompanied by a net increase in entropy. This conclusion led Clausius to his famous concise statement of the laws of thermodynamics. "The energy of the universe is a constant; the entropy of the universe tends always towards a maximum." This increasing tendency of the entropy has also been expressed as a principle of the degradation of energy, by which it becomes less available for useful work. Thus temperature differences tend to become leveled out, mountains tend to become plains, fuel supplies become exhausted, and work is frittered away into heat by frictional losses. Interesting philosophical 58 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 discussions have arisen from the entropy concept, notably the suggestion of Sir Arthur Eddington that, because of its continuously increasing character, "entropy is time's arrow"; that is, the constantly increasing entropy of the universe is the physical basis of our concept of time. The "meaning" of entropy will be displayed in another aspect when we discuss its statistical interpretation. 10. Change of entropy in changes of state of aggregation. As an example of a change in state of aggregation we may take the melting of a solid. At a fixed pressure, the melting point is a definite temperature T m at which solid and liquid are in equilibrium. In order to change some of the solid to liquid, heat must be added to the system. As long as both solid and liquid are present, this added heat does not change the temperature of the system, but is absorbed by the system as the latent heat of fusion X f of the solid. Since the change occurs at constant pressure, the latent heat, by eq. (2.9), equals the difference in enthalpy between liquid and solid. Per mole of substance, A, -- A// 7 -- //i, (|U j d //solid At the melting point, liquid and solid exist together in equilibrium. The addition of a little heat would melt some of the solid, the removal of a little heat would solidify some of the liquid, but the equilibrium between solid and liquid would be maintained. The Litent heat is necessarily a reversible heat, because the process of melting follows a path consisting of successive equilibrium states. We can therefore evaluate the entropy of fusion AS/ by a direct application of the relation A5 - </ r ev/^ which applies to any rever- sible isothermal process. A//, T f ^liquid Ssolid ~ AS, - ^ (3.17) For example, 4 A//, for ice is 1430 cal per mole, so that AS, = 1430/273.2 = 5.25 cal deg" 1 mole" 1 . By an exactly similar argument the entropy of vaporization AS y , the latent heat of vaporization A// v , and the boiling point T b are related by -A _ A// * ^vapor ~~ ^liquid ~" AS V ,= (3.18) A similar equation holds for a change from one form of a polymorphic solid to another, if the change occurs at a T and P at which the two forms are in equilibrium, and if there is a latent heat A associated with the trans- formation. For example, grey tin and white tin are in equilibrium at 13C and 1 atm, and A = 500 cal. Then AS, = 500/286 = 1.75 cal deg- 1 mole- 1 . 11. Entropy and equilibrium. Now that the entropy function has been defined and a method outlined for the evaluation of entropy changes, we have gained a powerful tool for our attack on the fundamental problem of 4 Further typical data are 'shown in Table 14.1 in sec. 14.8. Sec. 12] THE SECOND LAW OF THERMODYNAMICS 59 physicochemical equilibrium. In our introductory chapter, the position of equilibrium in purely mechanical systems was shown to be the position of minimum potential energy. What is the criterion for equilibrium in a thermo- dynamic system? Any spontaneously occurring change in an isolated system is accom- panied by an increase in entropy. From the First Law of Thermodynamics we know that energy can be neither created nor destroyed, so that the internal energy of an isolated system must be constant. The only way such a system could gain or lose energy would be by some interaction with its surroundings, but the absence of any such interaction is just what we mean when we say that the system is "isolated" no work is done on it; no heat flows across its boundaries. If we restrict work to PV work (expansion or compression), and exclude linear or surface effects, it follows also that the volume of an isolated system must remain constant. An isolated system may be defined, therefore, as a system of constant energy and constant volume. The first sentence of this paragraph can thus be rephrased: In a system at constant E and K, any spontaneous change is accompanied by an increase in entropy. Now a system is said to be at equilibrium when it has no further tendency to change its properties. The entropy of an isolated system will increase until no further spontaneous changes can occur. When the entropy reaches its maximum, the system no longer changes: the equilibrium has been attained. A criterion for thermodynamic equilibrium is therefore the following: In a system at constant energy and volume, the entropy is a maximum. At constant E and K, the S is a maximum. If instead of a system at constant E and K, a system at constant 5 and V is considered, the equilibrium criterion takes the following form: At constant S and V, the E is a minimum. This is just the condition applicable in ordinary mechanics, in which thermal effects are excluded. The drive, or perhaps better the drift, of physicochemical systems toward equilibrium is therefore compounded of two factors. One is the tendency toward minimum energy, the bottom of the potential energy curve. The other is the tendency toward maximum entropy. Only if E is held con- stant can S achieve its maximum; only if S is held constant can E achieve its minimum. What happens when E and 5 are forced to strike a compromise? 12. The free energy and work functions. Chemical reactions are rarely studied under constant entropy or constant energy conditions. Usually the physical chemist places his systems in thermostats and investigates them under conditions of approximately constant temperature and pressure. Sometimes changes at constant volume and temperature are followed, for example, in bomb calorimeters. It is most desirable, therefore, to obtain criteria for thermodynamic equilibrium that will be applicable under these practical conditions. 60 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 To this end, two new functions have been invented, defined by the following equations: A - E - TS (3.19) F-- H TS (3.20) A is called the work function', F is called the free energy.^ Both A and F, by their definitions in terms of state functions, are themselves functions of the state of the system alone. For a change at constant temperature, A/I - A 7AS (3.21) If this change is carried out reversibly, T AS q, and A/J A - q or - A/* - u Wx (3.22) The work is the maximum obtainable since the process is reversible. When the system isothermally performs maximum work u' mttx , its work function decreases by A/f. In any naturally occurring process, which is more or less irreversible, the work obtained is always less than the decrease in A. From cqs. (3.19) and (3.20), since H E \ PV, F---- A \ PV (3.23) For a change at constant pressure, AF - &A \ P AF (3.24) From eqs. (3.22) and (3.24), at constant temperature and pressure, -AF ,,' max P&Y (3.25) The decrease in free energy equals the maximum work less the work done by the expansion of the system at constant pressure. This work of expansion is always equal to P(V^ VJ P AK no matter how the change occurs, reversibly or irreversibly, provided the external pressure is kept constant. The net work over and above this is given by - AF/or a reversible process. For an irreversible process the net work is always less than A/ 7 . It may be zero as, for example, in a chemical reaction carried out in such a way that it yields no net work. Thus the combustion of gasoline in an automobile engine yields net work, but burning the same gasoline in a calorimeter yields none. The value of AFfor the change is the same in either case, provided the initial and final states are the same. A helpful interpretation of the entropy can be obtained in terms of the new functions A and F. From eqs. (3.19) and (3.20), we can write for a change at constant temperature, A/4- A-TAS (3.21) AF ----- A// - T AS (3.26) 5 Sometimes A is called the Helmholtz free energy, and F the Gibbs free energy or thermodynamic potential. Sec. 13] THE SECOND LAW OF THERMODYNAMICS 61 The change in the work function in an isothermal process equals the change in the energy minus a quantity TAS that may be called the unavailable energy. Similarly, the change in free energy equals the total change in en- thalpy minus the unavailable energy. 13. Free energy and equilibrium. The free energy function F may be used to define a condition for equilibrium in a form that is more directly applicable to experimental situations than the criteria in terms of the entropy. We have seen that for a reversible process occurring at constant temperature and pressure the net work done by the system is equal to the decrease in free energy. For a differential change, therefore, under these reversible (i.e., equilibrium) conditions at constant temperature and pressure, dF^ -</w net (3.27) Now most chemical laboratory experiments are carried out under such conditions that no work is obtained from the system or added to the system except the ordinary PV work, 6 so that dw net ^ 0. In these cases the equili- brium criterion becomes simply: At constant T and P, dF (3.28) This may be stated as follows: Any change in a system at equilibrium at constant temperature and pressure is such that the free energy remains constant. Thus we have obtained an answer to the question of how the drive to- ward maximum entropy and the drive toward minimum energy reach a compromise as a system tends toward equilibrium. From eq. (3.26) it is evident that an increase in S and a decrease in H both tend to lower the free energy. Therefore the third criterion for equilibrium can be written: at constant T and P, the F is a minimum. A similar discussion of eq. (3.19) provides the equilibrium condition at constant temperature and volume: at constant T and K, the A is a minimum. These are the equilibrium conditions that are of greatest use in most chemical applications. 14. Pressure dependence of the free energy. From eq. (3.20), F = H TS = E + PV TS. Differentiating, we obtain dF = dE+PdY + VdP - TdS - S dT Since dE = TdS - P dV cJF = VdP - SdT (3.29) Therefore, I I - V (3.30) For an isothermal change from state (1) to state (2): F 2 - F, = bF^dF^l VdP (3.31) 6 Notable exceptions are experiments with electrochemical cells, in which electric work may be exchanged with the system. A detailed discussion is given in Chapter 15. 62 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 In order to integrate this equation, the variation of V with P must be known for the substance being studied. Then if the free energy is known at one pressure, it can be calculated for any other pressure. If a suitable equa- tion of state is available, it can be solved for V as a function of P, and eq. (3.31) can be integrated after substituting this f(P) for V. In the simple case of the ideal gas, V - RT/P, and F 2 F t - AF RT\n^ (3.32) This gives the increase in free energy on compression, or decrease on ex- pansion. For example, if one mole of an ideal gas is compressed isother- mally at 300K to twice its original pressure, its free energy is increased by 1.98 x 300 In 2 - 413 calories. 15. Temperature dependence of free energy. From eq. (3.29), at constant pressure, 377 " S (3 ' 33) To integrate this equation, we must know S as a function of temperature. This question is considered in the next section. An alternative expression can be obtained by combining eq. (3.33) with eq. (3.20): /DF\ F H W7 7 > ~T~ (3.34) For isothermal changes in a system, the variation of AF with temperature 7 is then /DAF\ , AF A// (IT \r ~ AS =- - T - < 3 - 35 > This is called the Gibbs-Helmholtz equation. It permits us to calculate the change in enthalpy A// from a knowledge of AF and the temperature co- efficient of AF. Since </(AF) AF d /AF\ d( dT \T ) " T " dT T 2 the Gibbs-Helmholtz equation can be written in the alternative forms: A// T ' ' Or, = A// L 3(1 IT) \,, 7 For example the free energy change AF of a chemical reaction might be studied at a series of different constant temperatures, always under the same constant pressure. The equation predicts how the observed AF depends on the temperature at which the reaction is studied. Sec. 16] THE. SECOND LAW OF THERMODYNAMICS 63 Thus the slope of the plot of &F/Tvs. 1/7 is A//, the change in enthalpy. Important applications of these equations to chemical reactions will be con- sidered in the next chapter. They are especially important because many chemical processes are carried out in thermostats under practically constant atmospheric pressure. 16. Variation of entropy with temperature and pressure. Besides its useful- ness in the formulation of equilibrium conditions, the free-energy function can be used to derive important relations between the other thermodynamic variables. Consider, for example, the mathematical identity \3Tjp D: By virtue of eqs. (3.30) and (3.33), this identity yields an expression for the Cp pressure coefficient of the entropy: 8 T" ( <>> Thus at constant temperature, dS IP, so that f J vaK\ f 1 '- AS- L_ <//>-= - a* J/>, \dTj r Jl\ dP (3.38) Fig. 3.5, Graphical evaluation of the entropy change with temperature. To evaluate this integral, the equation of state or other PVT data must be available. For an ideal gas, (3Fpr) 7 > R/P. In this case eq. (3.37) becomes dS RdlnP, or AS =- RlnPJP^^ Rln VJV 19 as already shown in Section 3.8. The temperature variation of the entropy can be calculated as follows: At constant pressure, _ dq __ dH C P dT ~ T 7 " T ~ f~ At constant volume, C v dT //c _ ^ - _ . ^ ~ T ~ T ~ Thus at constant pressure, S=\C P dlnT+ const ~dT+ const; AS 8 Alternatively, apply Euler's rule to eq. (3.29). (3.39) (3.40) = r " Jr, f dT (3.41) 64 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 When C/* is known as a function of 7", the entropy change is evaluated by the integration in eq. (3.41). This integration is often conveniently carried out graphically, as in Fig. 3.5: if C V \T is plotted against 7, the area under the curve is a measure of the entropy change. The entropy change is also the area under the curve of C P vs. In T. 17. The entropy of mixing. Consider two gases at a pressure P. If these gases are brought together at constant temperature and pressure, they will become mixed spontaneously by interdiffusion. The spontaneous process will be associated with an increase in entropy. This entropy of mixing is of interest in a number of applications, and it can be calculated as follows. In the final mixture of gases the partial pressure of gas (1) is P 1 = A^P, of gas (2), P 2 X 2 P, where X l and X 2 are the mole fractions. 9 The AS of mixing is equal to the AS required to expand each gas from its initial pressure P to its partial pressure in the gas mixture. On the basis of one mole of ideal gas mixture, AS XT.R In - P - f X 2 R In ~ - X^ In 1 X 2 R In AS - -R(X l In X l + X 2 In X 2 ) This result can be extended to any number of gases in a mixture, yielding AS- -R2X t lnX t (3.42) The equation is only approximately valid for liquid and solid solutions. Let us calculate the entropy of mixing of the elements in air, taking the composition to be 79 per cent N 2 , 20 per cent O 2 , and 1 per cent argon. AS - -/?(0.79 In 0.79 + 0.20 In 0.20 + 0.01 In 0.01) 1.10 cal per deg per mole of mixture 18. The calculation of thermodynamic relations. One great utility of thermodynamics is that it enables us by means of a few simple paper-and- pencil operations to avoid many tedious and difficult laboratory experiments. The general aim is to reduce the body of thermodynamic data to relations in terms of readily measurable functions. Thus the coefficients (3K/3r) P , (3P/37V, and (3K/3P) r can usually be measured by straightforward experi- ments. The results are often expressed implicitly in the equation of state for the substance, of the general formf(P, V, T) = 0. The heat capacity at constant pressure C P is usually measured directly and C v can then be calculated from it and the equation of state. Thermo- dynamics itself does not provide any theoretical interpretation of heat 9 See Chapter 6, Section 1.. Sec. 18] THE SECOND LAW OF THERMODYNAMICS 65 capacities, the magnitudes of which depend on the structures and con- stitutions of the substances considered. The basic thermodynamic relations may be reduced to a few fundamental equations: (1) H -^ E + PV (2) A -- E TS (3) F=E+-Py-TS (4) dE^TdS PdV (5) dH = TdS+ VdP (6) dA - -SdT- PdV (7) dF- SdT f- VdP Since dA and dF are perfect differentials, they obey the Euler condition eq. (2.6), and therefore from (6) and (7) (8) (*S (9) QS/aP) T - - By the definition of the heat capacities, (10) C t , - (11) ? ~ These eleven equations are the starting point for the evaluation of all others. 10 The relation dE TdS P dV may be considered as a convenient ex- pression of the combined First and Second Laws of Thermodynamics. By differentiating it with respect to volume at constant temperature, ($EfiV) T - T(3S/dV) T - P. Then, since (3S/dV) T -- (dPpT) r , This equation has often been called a thermodynamic equation of state, since it provides a relationship among P, T, K, and the energy E that is valid for all substances. To be sure, all thermodynamic equations are in a sense equations of state, since they are relations between state variables, but equations like eq. (3.43) are particularly useful because they are closely related to the ordinary PVT data. It is now possible by means of eq. (3.43) to prove the statement in the previous chapter that a gas that obeys the equation PV ~ RT has a zero internal pressure, (dE/3V) T . For such a gas T(3P/dT) y =-- RTJV --=-- P, so that An equation similar to eq. (3.43) can be obtained in terms of the enthalpy instead of the energy: (3.44) 10 A. Tobolsky, /. Chem. Phys., 10, 644 (1942), gives a useful general method. 66 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 An important application of this equation is the theoretical discussion of the Joule-Thomson experiment. Since it follows from eq. (3.44) that TQVfiT),. - V (3.45) It is apparent that the Joule-Thomson effect can be either a warming or a cooling of the substance, depending on the relative magnitudes of the two terms in the numerator of eq. (3.45). In general, a gas will have one or more inversion points at which the sign of the coefficient changes as it passes through zero. The condition for an inversion point is that =- V p A coefficient of thermal expansion is defined by 1 so that the Joule-Thomson coefficient vanishes when K -- <xK o r. For an ideal gas this is always true (Law of Gay-Lussac) so that //./.T. is always zero in this case. For other equations of state, it is possible to derive /i JmTm from eq. (3.45) without direct measurement, if C P data are available. These considerations are very important in the design of equipment for the liquefaction of gases. Usually, the gas is cooled by doing external work in an adiabatic expansion until it is below its inversion point, after which further cooling is accomplished by a Joule-Thomson expansion. A further discussion of the methods used for attaining very low temperatures will be postponed till the next chapter. We shall then see that these low-temperature studies have an important bearing on the problem of chemical equilibrium. PROBLEMS 1. A steam engine operates between 120 and 30C. What is the minimum amount of heat that must be withdrawn from the hot reservoir to obtain 1000 joules of work? 2. Compare the maximum thermal efficiencies of heat engines operating with (a) steam between 130C and 40C, (b) mercury vapor between 380C and 50C. 3. A cooling system is designed to maintain a refrigerator at 20C in a room at ambient temperature of 25C. The heat transfer into the refrigera- tor is estimated as 10 4 joules per min. If the refrigerating unit is assumed to Chap. 3] THE SECOND LAW OF THERMODYNAMICS 67 operate at 50 per cent of its maximum thermal efficiency, estimate the power (in watts) required to operate the unit. 4. Prove that it is impossible for two reversible adiabatics on a P-V diagram to intersect. 5. One mole of an ideal gas is heated at constant pressure from 25 to 300C. Calculate the entropy change AS if C v -= $R. 6. Find the increase in , //, 5, A, and Fin expanding 1.0 liter of an ideal gas at 25C to 100 liter at the same temperature. 7. Ten grams of carbon monoxide at 0C are adiabatically and reversibly compressed from 1 atm to 20 atm. Calculate A, A//, AS for the change in the gas. Assume C v = 4.95 cal per deg mole and ideal gas behavior. Would it be possible to calculate AF from the data provided? 8. At 5C the vapor pressure of ice is 3.012 mm and that of supercooled liquid water is 3.163 mm. Calculate the AFper mole for the transition water -> ice at -5C. 9. One mole of an ideal gas, initially at 100C and 10 atm, is adiabatically expanded against a constant pressure of 5 atm until equilibrium is reattained. If c r -= 4.50 -f 0.0057 calculate A, A//, AS for the change in the gas. 10. Calculate AS when 10 g of ice at 0C are added to 50 g of water at 40C in an isolated system. The latent heat of fusion of ice is 79.7 cal per g; the specific heat of water, 1 .00 cal per g deg. 11. The following data are available for water: latent heat of vaporization 9630 cal per mole; latent heat of fusion 1435 cal per mole. Molar heat capacities: solid, C P = -0.50 + 0.030 T\ liquid, C P = 18.0; vapor, C P = 7.256 + 2.30 x 10~ 3 r+ 2.83 x 10~ 7 r 2 . Calculate AS when one mole of water at 100K is heated at constant pressure of 1 atm to 500K. 12. Derive an expression for the Joule-Thomson coefficient of a van der Waals gas. 13. Calculate the AS per liter of solution when pure N 2 , H 2 , and NH 3 gases are mixed to form a solution having the final composition 20 per cent N 2 , 50 per cent H 2 , and 30 per cent NH 3 (at S.T.P.). 14. Prove that a gas that obeys Boyle's Law and has zero internal pressure follows the equation of state, PV = RT. 15. For each of the following processes, state which of the quantities A, A//, AS, AF, A/* are equal to zero. (a) An ideal gas is taken around a Carnot cycle. (b) H 2 and O 2 react to form H 2 O in a thermally isolated bomb. (c) A nonideal gas is expanded through a throttling valve. (d) Liquid water is vaporized at 100C and 1 atm pressure. 16. Derive the expression (3///3P) r = T(dSfiP) T + V. 11. Derive: (2C P /dP) T = -T(yv/dT*) M >. 68 THE SECOND LAW OF THERMODYNAMICS [Chap. 3 18. Evaluate the following coefficients for (a) an ideal gas; (b) a van der Waals gas: (yppT*) y ; (3/aP) T ; (<>PfiV) 8 \ (9 2 K/arV 19. Derive expressions for: (a) (dA/dP) T in terms of P and V\ (b) (dF/dT)^ in terms of A and T. 20. Bridgman obtained the following volumes for methanol under high pressure, relative to a volume 1.0000 at 0C and I kg per cm 2 : P, kg/cm 2 1 500 1000 2000 3000 4000 5000 Vol. at 20 1.0238 0.9823 0.9530 0.9087 0.8792 0.8551 0.8354 Vol. at 50 1.0610 1.0096 0.9763 0.9271 0.8947 0.8687 0.8476 Use these data to estimate the AS when 1 mole of methanol at 35C and 1 kg per cm 2 pressure is compressed isothermally to 5000 kg per cm 2 . REFERENCES BOOKS See Chapter 1, p. 25. ARTICLES 1. Buchdahl, H. A., Am. J. Phys., 17, 41-46 (1949), "Principle of Cara- theodory." 2. Crawford, F. H., Am. J. Phys., 17, 1-5 (1949), "Jacobian Methods in Thermodynamics." 3. Darrow, K. K., Am. J. Phys., 12, 183-96 (1944), "Concept Of Entropy." 4. Dyson, F. J., Scientific American, 191, 58-63 (1954), "What is Heat?" 5. LaMer, V. K., O. Foss, and H. Reiss, Ann. N. Y. Acad. Sci., 51, 605-26 (1949), "Thermodynamic Theory of J. N. Br0nsted." CHAPTER 4 Thermodynamics and Chemical Equilibrium 1. Chemical affinity. The problem of chemical affinity may be sum- marized in the question, "What are the factors that determine the position of equilibrium in chemical reactions?" The earliest reflections on this subject were those of the ancient al- chemists, who endowed their chemicals with almost human natures, and answered simply that reactions occurred when the reactants loved each other. Robert Boyle, in The Sceptical Chymyst (1661), commented upon these theories without enthusiasm: "I look upon amity and enmity as affections of intelligent beings, and I have not yet found it explained by any, how those appetites can be placed in bodies inanimate and devoid of knowledge or of so much as sense." Isaac Newton's interest in gravitational attractions led him to consider also the problem of chemical interaction, which he thought might spring from the same causes. Thus in 1701, he surveyed some of the existing experimental knowledge, as follows: When oil of vitriol is mix'd with a little water . . . in the form of spirit of vitriol, and this spirit being poured upon iron, copper, or salt of tartar, unites with the body and lets go the water, doth not this show that the acid spirit is attracted by the water, and more attracted by the fix'd body than by the water, and therefore lets go the water to close with the fix'd body? And is it not also from a natural attrac- tion that the spirits of soot and sea-salt unite and compose the particles of sal- ammoniac . . . and that the particles of mercury uniting with the acid particles of spirit of salt compose mercury sublimate, and with particles of sulphur, compose cinnaber . . . and that in subliming cinnaber from salt of tartar, or from quick lime, the sulphur by a stronger attraction of the salt or lime lets go the mercury, and stays with the fix'd body ? Such considerations achieved a more systematic form in the early "Tables of Affinity," such as that of Etienne Geoffroy in 1718, which re- corded the order in which acids would expel weaker acids from combination with bases. Claude Louis de Berthollet, in 1801, pointed out in his famous book, Essai de statique chimique, that these tables were wrong in principle, since the quantity of reagent present plays a most important role, and a reaction can be reversed by adding a sufficient excess of one of the products. While serving as scientific adviser to Napoleon with the expedition to Egypt in 1799, he noted the deposition of sodium carbonate along the shores of the salt lakes there. The reaction Na 2 CO 3 + CaCl 2 = CaCO 3 + 2 NaCl as carried out in the laboratory was known to proceed to completion as the CaCO 3 was 69 70 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 precipitated. Berthollet recognized that, under the peculiar conditions of large excess of sodium chloride that occurred in the evaporating brines, the reaction could be reversed, converting the limestone into sodium carbonate. Berthollet, unfortunately, pushed his theorizing too far, and finally main- tained that the actual composition of chemical compounds could be changed by varying the proportions of the reaction mixture. In the ensuing contro- versy with Louis Proust the Law of Definite Proportions was well established, but Berthollet's ideas on chemical equilibrium, the good with the bad, were discredited, and consequently neglected for some fifty years. 1 It is curious that the correct form of what we now know as the Law of Chemical Equilibrium was arrived at as the result of a series of studies of chemical reaction rates, and not of equilibria at all. In 1850, Ludwig Wilhelmy investigated the hydrolysis of sugar with acids and found that the rate was proportional to the concentration of sugar remaining undecomposed. In 1862, Marcellin Berthelot and Pean de St. Gilles reported similar results in their famous paper 2 on the hydrolysis of esters, data from which are shown in Table 4.1. The effect on the products of varying the concentrations of the reactants is readily apparent. TABLE 4.1 DATA OF BERTHELOT AND ST. GILLES ON THE REACTION C 2 H 5 OH -}- CH 3 COOH ^ CH 3 COOC 2 H 5 ! H 2 (One mole of acetic acid is mixed with varying amounts of alcohol, and the amount of ester present at equilibrium is found) Moles of Alcohol Moles of Ester Produced Equilibrium Constant [EtAc][H 2 0] [EtOH][HAc] 0.05 0.049 2.62 0.18 0.171 3.92 0.50 0.414 3.40 1.00 0.667 4.00 2.00 0.858 4.52 8.00 0.966 3.75 In 1863, the Norwegian chemists C. M. Guldberg and P. Waage expressed these relations in a very general form and applied the results to the problem of chemical equilibrium. They recognized that chemical equilibrium is a dynamic and not a static condition. It is characterized not by the cessation of all reaction but by the fact that the rates of the forward and reverse reactions have become the same. Consider the general reaction, A + B ^ C + D. According to the "law of mass action," the rate of the forward reaction is proportional to the 1 We now recognize many examples of definite departures from stoichiometric com- position in various inorganic compounds such as metallic oxides and sulfides, which are appropriately called "berthollide compounds." 2 Ann. chim. phys., [3] 65, 385 (1862). Sec. 2] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 71 concentrations of A and of B. If these are written as (A) and (/?), K forward = k\ (A)(B). Similarly, K backward = k*> (Q(D). At equilibrium, therefore, ^forward ~ ^backward so tnat Thus (C}(D} ThUS > More generally, if the reaction is aA + bB cC + dD, at equilibrium < Equation (4.1) is a statement of Guldberg and Waage's Law of Chemical Equilibrium. The constant K is called the equilibrium constant of the reaction. It provides a quantitative expression for the dependence of chemical affinity on the concentrations of reactants and products. By convention, the con- centration terms for the reaction products are always placed in the numerator of the expression for the equilibrium constant. Actually, this work of Guldberg and Waage does not constitute a general proof of the equilibrium law, since it is based on a very special type of rate equation, which is certainly not always obeyed, as we shall see when we take up the study of chemical kinetics. Their recognition that chemical affinity is influenced by two factors, the "concentration effect" and what might be called the "specific affinity," depending on the chemical nature of the reacting species, their temperature, and pressure, was nevertheless very important. The equilibrium law will subsequently be derived from thermodynamic principles. 2. Free energy and chemical affinity. The free-energy function described in Chapter 3 provides the true measure of chemical affinity under conditions of constant temperature and pressure. The free-energy change in a chemical reaction can be defined as AF ^ F produt . t8 ^reactants- When the free-energy change is zero, there is no net work obtainable by any change or reaction at constant temperature and pressure. The system is in a state of equilibrium. When the free-energy change is positive for a proposed reaction, net work must be put into the system to effect the reaction, otherwise it cannot take place. When the free-energy change is negative, the reaction can proceed spontaneously with the accomplishment of useful net work. The larger the amount of this work that can be accomplished, the farther removed is the reaction from equilibrium. For this reason, AF has often been called the driving force of the reaction. From the statement of the equilibrium law, it is evident that this driving force depends on the concentrations of the re- actants and products. It also depends on their specific chemical constitution, and on the temperature and pressure, which determine the molar free-energy values of reactants and products. If we consider a reaction at constant temperature, e.g., one conducted in 72 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 a thermostat, AF = A// -f T AS. The driving force is seen to be made up of two parts, a A// term and a 7 AS term. The A// term is the reaction heat at constant pressure, and the T AS term is the heat change when the process is carried out reversibly. The difference is the amount of reaction heat that can be converted into useful net work, i.e., total heat minus unavailable heat. If a reaction at constant volume and temperature is considered, the decrease in the work function, A/l = AF + T AS, should be used as the proper measure of the affinity of the reactants, or the driving force of the reaction. The constant volume condition is much less usual in laboratory practice. It is now apparent why the principle of Berthelot and Thomsen (p. 45) was wrong. They considered only one of the two factors that make up the driving force of a chemical reaction, namely, the heat of reaction. They neglected the T AS term. The reason for the apparent validity of their prin- ciple was that for many reactions the A// term far outweighs the T AS term. This is especially so at low temperatures; at higher temperatures the TAS term naturally increases. The fact that the driving force for a reaction is large (AF is a large nega- tive quantity) does not mean that the reaction will necessarily occur under any given conditions. An example is a bulb of hydrogen and oxygen on the laboratory shelf. For the reaction, H 2 + \ O 2 == H 2 O (g), AF 298 = 54,638 cal. Despite the large negative AF, the reaction mixture can be kept for years without any detectable formation of water vapor. If, after ten years on the shelf, a pinch of platinum-sponge catalyst is added, the reaction takes place with explosive violence. The necessary affinity was certainly there, but the rate of attainment of equilibrium depended on entirely different factors. Another example is the resistance to oxidation of such extremely active metals as aluminum and magnesium. 2 Mg + O 2 (l atm) = 2 MgO (c); AF 298 136,370 cal. In this case, after the metal is exposed to air it becomes covered with a very thin layer of oxide and further reaction occurs at an immeasurably slow rate since the reactants must diffuse through the oxide film. Thus the equilibrium condition is never attained. The incendiary bomb and the thermit reaction, on the other hand, remind us that the large AF for this reaction is a valid measure of the great affinity of the reactants. 3. Free-energy and cell reactions. Reactions occurring in electrochemical cells with the production of electric energy are of especial interest in the discussion of free-energy changes, since they can be carried out under con- ditions that are almost ideally reversible. This practical reversibility is achieved by balancing the electromotive force of the cell by an opposing emf which is imperceptibly less than that of the cell. Such a procedure can be accomplished with the laboratory potentiometer, in which an external source of emf, such as a battery, is balanced against the standard cell. The Sec. 3] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 73 arrangement for this "compensation method" is shown in Fig. 4.1. When the opposing emf 's are balanced by adjustment of the slide wire 5, there is no detectable deflection of the galvanometer G. An electrochemical cell converts chemical free energy into electric free energy. The electric energy is given by the product of the emf of the cell times the amount of electricity flowing through it. Michael Faraday showed, in 1834, that a given amount of electricity was always produced by or would produce the same amount of chemical reaction. For one chemical equivalent of reaction the associated amount of electricity is called the Faraday, ^", CELL AR and is equal to 96,519 coulombs. Thus the electric energy available per mole of reaction equals zS^ ", where z is the number of equivalents per mole and S is the emf of the cell. A GALVANOMETER( convenient energy unit is therefore the volt-coulomb or joule. When the reaction is carried out UNKNOWN emt reversibly, this energy is the maxi- _. . . _ . ,, . r / ., , , p- 7 , . Fig. 4.1. Compensation method for mum available, or the net woi k w . mea suring the emf of a cell without drawing If the reaction is carried out at a current from it. When there is no deflection finite rate, some of the energy is of galvanometer ff x - (SXISS')tf t . expended in overcoming the electric resistance of the cell, appearing as heat. This Joule heat, 7 2 /?, is the electrical analogue of the frictional heat produced in irreversible mechanical pro- cesses. We may now write, if ^ is the reversible emf, (4.2) This equation provides a direct method for evaluating the free-energy change in the cell reaction. If we know the temperature coefficient of the emf of the cell, we can also calculate A// and AS for the reaction by means of eq. (3.35), which on combination with eq. (4.2) yields the relations A T-f >r<3r \ JP T \ AC T^ (A. "\\ l\rt = 2^ \6 1 -TIL], IAO Z^ (4..J) \ #77 dT In a later chapter, devoted to electrochemistry, we shall see that it is possible to carry out many changes by means of reversible cells, and thereby to evaluate AF and A// for the changes from measurements of the emf and its temperature coefficient. A cell that is occasionally used as a laboratory standard of emf is the Clark cell shown in Fig. 4.2. The reaction in this cell is Zn -f Hg 2 SO 4 ZnSO 4 -f 2 Hg, or more simply, Zn -f 2 Hg+ = Zn++ + 2 Hg. The emf of the cell is 1 .4324 volts at 1 5C and the temperature coefficient dSjdT = 0.00119 volt per degree. It can therefore be calculated that for the cell 74 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 ZINC SULFATE SOLUTION ZINC AMALGAM SOLID Hg 2 S0 4 MERCURY Fig. 4.2. A typical electrochemical cell: the Clark cell. reaction AF = (-1.4324 x 2 x 96,519 - -276,510 joule. From eq. (4.3), AS- (-0.00119 x 2 x 96,519) - -229.7 joule deg- 1 mole" 1 . Whence, A// - AF f T AS = -276,510 - 66,200 - -342,710 joule. The value of A// obtained from thermochemical data is 339,500, in good agreement with the electrochemical value. Since the temperature coefficient is negative, heat is given up to the surroundings during the working of this cell, and the net work obtainable, A/% is less than the heat of the reaction. There are other cells for which the temperature coefficient is positive. These cells absorb heat from the environ- ment, and their work output, under re- versible conditions, is greater than the heat of the reaction. These relationships, discovered theo- retically by Willard Gibbs in 1876, were first applied to experimental cases by Helmholtz in 1882. Before that time it was thought, reasoning from the First Law, that the maximum work output that could be achieved was the conversion of all of the heat of reaction into work. The Gibbs- Helmholtz treatment shows clearly that the work output is governed by the value of AFfor the cell reaction, not by that of A// The working cell can either reversibly absorb heat from or furnish heat to its environment. This reversible heat change then appears as the T AS term in the free-energy expression. 4. Standard free energies. In Chapter 2 (p. 39) the definition of standard states was introduced in order to simplify calculations with energies and enthalpies. Similar conventions are very helpful for use with free-energy data. Various choices of the standard state have been made, one that is frequently used being the state of the substance under one atmosphere pressure. This is a useful definition for gas reactions ; for reactions in solution, other choices of standard state may be more convenient and will be introduced as needed. A superscript zero will be used to indicate a standard state of 1 atm pressure. The absolute temperature will be written as a subscript. The most stable form of an element in the standard state (1 atm pressure) and at a temperature of 25C will by convention be assigned a free energy of zero. The standard free energy of formation of a compound is the free energy of the reaction by which it is formed from its elements, when all the reactants and products are in the standard state. For example: H 2 (I atm) + i O 2 (1 atm) - H 2 O (g; 1 atm) AF 298 = -54,638 S (rhombic crystal) + 3 F 2 (1 atm) - SF 6 (g; 1 atm) AF 298 = -235,000 Sec. 5] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 75 In this way it is possible to make tabulations of standard free energies such as that given by Latimer, 3 examples from which are collected in Table 4.2. Some of these free-energy values are determined directly from reversible cell emf's but most are obtained by other methods to be described later. TABLE 4.2 STANDARD FREE ENERGIES OF FORMATION OF CHEMICAL COMPOUNDS AT 25C Compound State AF 298 (kcaljmole) Compound State AF2,8 (kcallmole) AgCl c -26.22 H 2 g -54.638 Agl c -15.81 H 2 2 g -24.73 CaCl 2 aq -195.36 H 2 2 1 -28.23 CaC0 3 c -207.22 H 2 S g -7.87 CH 4 g - 12.09 NaCl c -91.70 C 2 H 2 g 50.0 NH 3 g -3.94 C 2 H 4 g 16.28 N 2 O g 24.93 C 2 H 6 g -7.79 NO g 20.66 CO g -32.79 N0 2 g 12.27 C0 2 g -94.24 N 2 4 g 23.44 CuO c -30.4 3 g 39.4 Cu 2 c -35.15 SO 2 g -71.74 H 2 1 -56.693 HCOOH 1 -86.4 Free-energy equations can be added and subtracted just as thermo- chemical equations are, so that the free energy of any reaction can be cal- culated from the sum of the free energies of the products minus the sum of the free energies of the reactants. \F C V F V F ^-* r ~~ Z, r products Z, r reactants If we adopt the convention that moles of products are positive and moles of reactants negative in the summation, this equation can be written concisely as AF - 2 ", F> (4.4) For example: Cu 2 (c) T- NO (g) ^ 2 CuO (c) + i N 2 (g) From Table 4.2, AF = 2 (-30.4) + i (0) - 20.66 - (-35.15) - -46.31 kcal 5. Free energy and equilibrium constant of ideal gas reactions. Many im- portant applications of equilibrium theory are in the field of homogeneous gas reactions, that is, reactions taking place entirely between gaseous pro- ducts and reactants. To a good approximation in many such cases, the gases may be considered to obey the ideal gas laws. The variation at constant temperature of the free energy of an ideal gas 8 W. M. Latimer, The Oxidation States of the Elements, 2nd ed. (New York: Prentice- Hall, 1952). 76 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 is given from eq. (3.29) as dF = V dP = RTdln P. Integrating from F and P, the free energy and pressure in the chosen standard state, to F and P, the values in any other state, F - F = RTln (P/P). Since P = 1 atm, this becomes F-F = RT\nP (4.5) Equation (4.5) gives the free energy of one mole of an ideal gas at pressure P and temperature 7, minus its free energy in a standard state at P = I atm and temperature T. If an ideal mixture of ideal gases is considered, Dalton's Law of Partial Pressures must be obeyed, and the total pressure is the sum of the pressures that the gases would exert if each one occupied the entire volume by itself. These pressures are called the partial pressures of the gases in the mixture, Pj, 7*2 ... P n . Thus if /? t is the number of moles of gas / in the mixture, RT P-lP, = -fI", (4.6) For each individual gas / in the mixture eq. (4.5) can be written F l -F? = RTlnP, (4.7) For n t moles, n t (F l F) = RTn t In P t . For a chemical reaction, therefore, from eq. (4.4), AF - AF = RT 2 n t In P t (4.8) If we now consider the pressures P t to be the equilibrium pressures in the gas mixture, AF must equal zero for the reaction at equilibrium. Thu we obtain the important relation - AF = RT 2 n t In P** (4.9) AF or 2 , ^ P** - - Since AF is a function of the temperature alone, the left side of this ex- pression is equal to a constant at constant temperature. For a typical reaction, aA + bB --= cC 4- dD, the summation can be written out as This expression is simply the logarithm of the equilibrium constant in terms of partial pressures, K p . Equation (4.9) therefore becomes -AF- RTlnK, (4.10) The analysis in this section has now established two important results. The constancy of the expression Sec. 6] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 77 at equilibrium has been proved by thermodynamic arguments. This con- stitutes a thermodynamic proof of the Law of Chemical Equilibrium. Second, an explicit expression has been derived, eq. (4.10), which relates the equili- brium constant to the standard free-energy change in the chemical reaction. We are now able, from thermodynamic data, to calculate the equilibrium constant, and thus the concentration of products from any given concentra- tion of reactants. This was one of the fundamental problems that chemical thermodynamics aimed to answer. Sometimes the equilibrium constant is expressed explicitly in terms of concentrations c t . For an ideal gas PI n^RT/V) c { RT. Substituting in eq. (4.11), we find CA^B K p = K c (RT)* n (4.12) Here K c is the equilibrium constant in terms of concentrations (e.g., moles per liter) and A is the number of moles of products less that of reactants in the stoichio metric equation for the reaction. Another way of expressing the concentrations of the reacting species is in terms of mole fractions. The mole fraction of component / in a mixture is defined by *, - Y n ( 4 - 13 > It is the number of moles of a component / in the mixture divided by the total number of moles of all the components. It follows that P t = Therefore the equilibrium constant in terms of the mole fractions is Since K 9 for ideal gases is independent of pressure, it is evident that K x is a function of pressure except when A/2 = 0. It is thus a "constant" only with respect to variations of the A"s at constant T and P. 6, The measurement of homogeneous gas equilibria. The experimental methods for measuring gaseous equilibria can be classified as either static or dynamic. In the static method, known amounts of the reactants are introduced into suitable reaction vessels, which are closed and kept in a thermostat until equilibrium has been attained. The contents of the vessels are then analyzed in order to determine the equilibrium concentrations. If the reaction proceeds very slowly at temperatures below those chosen for the experiment, it is sometimes possible to "freeze the equilibrium" by chilling the reaction vessel rapidly. The vessel may then be opened and the contents analyzed chemically. 78 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 This was the procedure used by Max Bodenstein 4 in his classic investiga- tion of the hydrogen-iodine equilibrium: H 2 + I 2 = 2 HI. The reaction pro- ducts were treated with an excess of standard alkali; iodide and iodine were determined by titration, and the hydrogen gas was collected and its volume measured. For the formation of hydrogen iodide, A/? = 0; there is no change in the number of moles during the reaction. Therefore K v = K c K x . If the initial numbers of moles of H 2 and I 2 are a and b, respectively, they will be reduced to a x and b x with the formation of 2x moles of HI. The total number of moles at equilibrium is therefore a -f b + c, where c is the initial number of moles of HI. Accordingly the equilibrium constant can be written - 2 *) The (a + b + c) terms required to convert "number of moles" into "mole fraction" have been canceled out between numerator and denominator. In a run at 448C, Bodenstein mixed 22.13 cc at STP of H 2 with 16.18 of I 2 , and found 25.72 cc of HI at equilibrium. Hence K~ - _ 25/72i - - -215 (22.13 - 12.86)(16.18 - 12.86) In the dynamic method for studying equilibria, the reactant gases are passed through a thermostated hot tube at a rate slow enough to allow complete attainment of equilibrium. This condition can be tested by making runs at successively lower flow rates, until there is no longer any change in the observed extent of reaction. The effluent gases are rapidly chilled and then analyzed. Sometimes a catalyst is included in the hot zone to speed the attainment of equilibrium. This is a safer method if a suitable catalyst is available, since it minimizes the possibility of any back reaction occurring after the gases leave the reaction chamber. The catalyst changes the rate of a reaction, not the position of final equilibrium. These flow methods were extensively used by W. Nernst and F. Haber (around 1900) in their pioneer work on technically important gas reactions. An example is the "water-gas equilibrium," which has been studied both with and without an iron catalyst. 5 The reaction is H 2 + C0 2 - H 2 + CO, and K 9 = ^ HiQ f CQ If we consider an original mixture containing a moles of H 2 , b moles of CO 2 , c moles of H 2 O, and d moles of CO, the analysis of the data is as follows. 4 Z.physik. Chem., 22, \ (1897); 29, 295 (1899). 5 Z. anorg. Chem., 38. 5 (1904). Sec. 7] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 79 Constituent Original Moles Moles H 2 a a X C0 2 b b x H 2 c c + x CO d d + x At Equilibrium Mole Fraction Partial Pressure a x/(a -f b + c -f d) ((a - x)ln]P b - x/(a + b + c + d) Kb - x)/n]P (c + x)l(a b + c- f d) [(c + x)ln\P (d + x)l(a 4- b + c + d) [(d + x)/n]P Total Moles at Equilibrium a + b j rc + d = n Substituting the partial pressure expressions, we obtain ^ The values for the equilibrium composition, obtained by analysis of the product gases, have been used to calculate the constants in Table 4.3. TABLE 4.3 THE WATER GAS EQUILIBRIUM H 2 -f CO 2 = H 2 O -f CO; temperature 986C Initial Composition (moles per cent) Equilibrium Composition (moles per cent) K. C0 2 H, C0 2 H 2 CO = H 2 O 10.1 89.9 0.69 80.52 9.40 .59 30.1 69.9 7.15 46.93 22.96 .57 49.1 51.9 21.44 22.85 27.86 .58 60.9 39.1 34.43 12.68 26.43 .61 70.3 29.7 47.51 6.86 22.82 .60 It is often possible to calculate the equilibrium constant for a reaction from the known values of the constants of other reactions. This is a principle of great practical utility. For example, from the dissociation of water vapor and the water-gas equilibrium one can calculate the equilibrium constant for the dissociation of carbon dioxide. H a O CO 2 CO 2 = H H 2 - H 2 O 2 CO CO + O 2 It is apparent that AT/ -= K V 'K V . 7. The principle of Le Chatelier. The effects of such variables as pressure, temperature, and concentration on the position of chemical equilibrium have been succinctly summarized by Henry Le Chatelier (1888). "Any change in 80 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 one of the variables that determine the state of a system in equilibrium causes a shift in the position of equilibrium in a direction that tends to counteract the change in the variable under consideration." This is a prin- ciple of broad and general utility, and it can be applied not only to chemical equilibria but to equilibrium states in any physical system. It is indeed possible that it can be applied also with good success in the psychological, economic, and sociological fields. The principle indicates, for example, that if heat is evolved in a chemical reaction, increasing the temperature tends to reverse the reaction; if the volume decreases in a reaction, increasing the pressure shifts the equilibrium position farther toward the product side. Quantitative expressions for the effect of variables such as temperature and pressure on the position of equilibrium will now be obtained by thermodynamic methods. 8. Pressure dependence of equilibrium constant. The equilibrium constants K p and K c are independent of the pressure for ideal gases; the constant K x is pressure-dependent. Since K x = K P P An , In K x -= In K v - A In P. dP " P RT ' When a reaction occurs without any change in the total number of moles of gas in the system, A/? = 0. An example is the previously considered water gas reaction. In these instances the constant K p is the same as K x or K c , and for ideal gases the position of equilibrium does not depend on the total pressure. When AH is not equal to zero, the pressure dependence of K x is given by eq. (4.15). When there is a decrease in the mole number (A/z < 0) and thus a decrease in the volume, K x increases with increasing pressure. If there is an increase in n and V (A/7 > 0), K x decreases with increasing pressure. An important class of reactions for which A -- is that of dissociation- association equilibria. An extensively studied example is the dissociation of nitrogen tetroxide into the dioxide, N 2 O 4 2 NO 2 . In this case, K p = PxoJP$ t o t ' Jf one m l e f N 2 O 4 is dissociated at equilibrium to a fractional extent a, 2a moles of NO 2 are produced. The total number of moles at equilibrium is then (1 a) ~\- 2a = 1 + a. It follows that (\-a)/(\+a) 1-0 2 Since for this reaction AA? -^ ~f 1, p ~ * When a is small compared to unity, this expression predicts that the degree of dissociation a shall vary inversely as the square root of the pressure. Experimentally it is found that N 2 O 4 is appreciably dissociated even at Sec. 9] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 81 room temperatures. As a result, the observed pressure is greater than that predicted by the ideal gas law for a mole of N 2 ^4 s * nce ea h m l e yields 1 -\- a moles of gas after dissociation. Thus P (ideal) RTfV, whereas P (observed) - (1 + a)RT/V. Hence a - (K/*r)(/> ob8 - /> ldcal ). This behavior provides a very simple means for measuring a. For example, in an experiment at 318K and 1 atm pressure, a is found to be 0.38. There- fore K x = 4(0.38) 2 /(1 - 0.38 2 ) - 0.67. At 10 atm pressure, K x - 0.067 and a is 0.128. Among the most interesting dissociation reactions are those of the elementary gases. The equilibrium constants for a few of these are collected in Table 4.4. TABLE 4.4 EQUILIBRIUM CONSTANTS OF DISSOCIATION REACTIONS Temp. K. rig 2 H M 2 ^ M C1 2 - 2 Cl Br 2 - 2 Br 600 1.4 x 10~ 37 3.6 x 10~ 33 1.3 x 10- 66 4.8 x 10~ 16 6.18 x 10~ 12 800 9.2 X 10~ 27 1.2 x 10~ 23 5.1 x 10~ 41 1.04 x 10~ 10 1.02 x 10- 7 1000 3.3 x 10- 20 7.0 x 10~ 18 1.3 x 10~ 31 2.45 x 10~ 7 3.58 x 10~ 5 1200 8.0 x 10~ 16 5.05 x 10' 14 2.4 x 10~ 26 2.48 x 10~ 5 1.81 x 10- 8 1400 1.1 x 10~ 12 2.96 x 10- 11 7.5 x 10~ 21 8.80 x 10~ 4 3.03 x I0~ a 1600 2.5 x 10~ 10 3.59 x 10~ 9 1.8 x 10- 17 1.29 x 10~ 2 2.55 x 10- 1 1800 1.7 x 10- 8 1.52 x 10~ 7 7.6 x 10- 16 0.106 2000 5.2 x 10- 7 3.10 x 10- 9.8 x 10~ 13 0.570 9. Effect of an inert gas on equilibrium. In reactions in which there is no change in the total number of moles, AH = 0, and the addition of an inert gas cannot affect the composition of the equilibrium mixture. If, however, A ^ 0, the inert gas must be included in calculating the mole fractions and the total pressure P. Let us consider as an example the technically important gas reaction, SO 2 + \ O 2 = SO 3 . In this case A = |, and K p = K X P~ 1/2 . Let the initial reactant mixture contain a moles of SO 2 , b moles of O 2 , and c moles of inert gas, for example N 2 . If y moles of SO 3 are formed at equi- librium, the equilibrium mole fractions are b - (y/2) y Here n is the total number of moles at equilibrium: n a + b + c (y/2). The equilibrium constant, K = K P 1/2 = * y/n yn 1/2 [(a [a - 82 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 It follows that = / a y /?S 3 __ "so, where w s()i , w 80a , w 0i , /? are the equilibrium mole numbers. Let us now consider three cases. (1) If the pressure is increased by com- pressing the system without addition of gas from outside, n is constant, and as P increases, n$ Jn$ 0t a l so increases. (2) If an inert gas is added at constant volume, both n and P increase in the same ratio, so that the equilibrium conversion of SO 2 to SO 3 , w SO| //7 SOi remains unchanged. (3) If an inert gas is added at constant pressure, n is increased while P remains constant, and this dilution of the mixture with the inert gas decreases the extent of con- version /f so > s <v This reaction is exothermic, and therefore increasing the temperature decreases the formation of products. The practical problem is to run the reaction at a temperature high enough to secure a sufficiently rapid velocity, without reaching so high a temperature that the equilibrium lies too far to the left. In practice, a temperature around 500C is chosen, with a platinum or vanadium-pentoxide catalyst to accelerate the reaction. The equilibrium constant from 700 to 1200K is represented quite well by the equation In K p = (22,6QO/ RT) - (21.36/7?). At 800K, therefore, K v - 33.4. Let us now consider two different gas mixtures, the first containing 20 per cent SO 2 and 80 per cent O 2 at 1 atm pressure, and a second containing in addition a considerable admixture of nitrogen, e.g., 2 per cent SO 2 , 8 per cent O 2 , 90 per cent N 2 , at 1 atm pressure. Letting y moles SO 3 at equi- librium, we obtain: I II K s - ffpP 1 / 2 = 33.4 K x - K p P l l 2 = 33.4 y y 1 ~ (y/2) 1 - (y/2) 0.2 -y roSj- O/2)] 1 / 1 0.02 - y ["0.08 -(y/2)] 1 / 2 1 - (y/2) I 1 - '(y/2) J 1 - (y/2) I 1 - (^T J / - 2.000/ -I- 0.681^ - 0.0641 =0 / - 0.1985/ + 6.81 X 10~ 3 j y = 0.190 64.06 X 10~ 6 = y - 0.0180 95 % conversion of SO 2 to SO 3 90 % conversion of SO 2 to SO 3 The cubic equations that arise in problems like these are probably best solved by successive approximations. Beginning with a reasonable value guessed for the percentage conversion, a sufficiently accurate solution can usually be obtained after three or four trials. Sec. 10] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 83 10. Temperature dependence of the equilibrium constant. An expression for the variation of K P with temperature is derived by combining eqs. (4.10) and (3.36). Since v (4.10) and therefore A// c r 2 dT (3.36) (4.16) It is apparent that if the reaction is endothermic (A// positive) the equilibrium constant increases with temperature; if the reaction is exothermic (A// negative) the equilibrium con- stant decreases as the temperature is raised. Equation (4.16) can also be written: iH (4.17) d(\IT) R Thus if In K p is plotted against \/T 9 the slope of the curve at any point is equal to A// //?. As an example of this treatment, the data for the varia- tion with temperature of the 2 HI --= H 2 + I 2 equilibrium are plotted in Fig. 4.3. The curve is almost a straight line, indicating that A// is approxi- mately constant for the reaction over the experimental temperature range. The value calculated from the slope at 400C is A// = 7080 cal. It is also possible to measure the equilibrium constant at one tempera- ture and with a value of A// obtained from thermochemical data to calculate the constant at other temperatures. Equation (4.16) can be integrated, giving In 3.3U \ C f\f\ \ \ \ \ \ A on \ t SLOPE w* - V * ^n ^ ..... h- h .v \ V L25 150 1.75 Fig. 4.3. The variation with temper- ature of K f = PH 2 Pi.JPm". (Data of Bodenstein.) Since, over a short temperature range, A/f may often be taken as approxi- mately constant, one obtains In KJTj -A// R (4.18) 84 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 If the variations of the heat capacities of the reactants and products are known as functions of temperature, an explicit expression for the tempera- ture dependence of A// can be derived from Kirchhoff 's equation (2.29). This expression for A// as a function of temperature can then be substituted into eq. (4.16), whereupon integration yields an explicit equation for K 9 as a function of temperature. This has the form In K, - - A// //?r + A In T + BT + CT* . . . + I (4.19) In this case, as usual, the value of the integration constant / can be deter- mined if the value of K p is known at any one temperature, either experiment- ally or by calculation from AF. It will be recalled that one value of A// is needed to determine A// , the integration constant of the Kirchhoff equation. To summarize, from a knowledge of the heat capacities of the reactants and products, and of one value each for A// and A^,it is possible to calculate the equilibrium constant at any temperature. As an example, consider the calculation of the constant for the water- gas reaction as a function of the temperature. CO + H 2 (g) - H 2 + C0 2 ; K v - ^ co> -CO r H,0 From Table 4.2, the standard free-energy change at 25C is: A/r 298 = -94,240 - (-54,640 - 32,790) - -6810 6810 298^ From the enthalpies of formation on page 39, A# 298 - -94,050 - (-57,800 - 26,420) = -9830 The heat capacity table on page 44 yields for this reaction Thus In K V298 = - = 1 1.48, or K v298 = 9.55 X 10* = -0.515 + 6.23 x 10~ 3 r- 29.9 x 10~ 7 r 2 From eq. (2.32), A// = A// - 0.5157+ 3.12 x 10~ 3 r 2 - 10.0 x 1Q-T 3 Substituting A// = -9830, T = 298K, and solving for A// , we get A// = 9921. Then the temperature dependence of the equilibrium constant, eq. (4.19), becomes By inserting the value ofln K v at 298K, the integration constant can be Sec. 11] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 85 evaluated as / = 3.97. The final expression for K v as a function of tem- perature is, therefore, In K - -3.97 + 9 - - 0.259 In T + 1.56 x I0~ 3 r - 2.53 x 10~ 7 r 2 For example, at 800K, In K 9 = 1.63, K v - 5.10. 11. Equilibrium constants from thermal data. We have now seen how a knowledge of the heat of reaction and of the temperature variation of the heat capacities of reactants and products allows us to calculate the equi- librium constant at any temperature, provided there is a single experimental measurement of either K 9 or AF at some one temperature. If an independent method is available for finding the integration constant /in eq. (4.19), it will be possible to calculate K 9 without any recourse to experimental measure- ments of the equilibrium or of the free-energy change. This calculation would be equivalent to the evaluation of the entropy change, AS , from thermal data alone, i.e., heats of reaction and heat capacities. If we know AS and A//, K p can be found from AF - A// - TA5. From eq. (3.41), the entropy per mole of a substance at temperature T is given by 5 = f r C P rflnr+S where 5 is the entropy at 0K. 6 If any changes of state occur between the temperature limits, the associated entropy changes should be added. For a gas at temperature Tthe general expression for the entropy therefore becomes o 'Q, cryst din T + ^^ +J^C P ^dln T A// r T + -=r SE + Cp^dlnT+S, (4.20) * b J T /' Jo All these terms can be measured except the constant S . The evaluation of this constant becomes possible by virtue of the third fundamental law of thermodynamics. 12. The approach to absolute zero. The laws of thermodynamics are in- ductive in character. They are broad generalizations having an experimental basis in certain human frustrations. Our failure to invent a perpetual-motion machine has led us to postulate the First Law of Thermodynamics. Our failure ever to observe a spontaneous flow of heat from a cold to a hotter body or to obtain perpetual motion of the second kind has led to the state- ment of the Second Law. The Third Law of Thermodynamics can be based on our failure to attain the absolute zero of temperature. A detailed study of refrigeration principles indicates that the absolute zero can never be reached. 8 Be careful not to confuse 5, the entropy in the standard state of 1 atm pressure, and S" , the entropy at 0K. 86 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 Most cryogenic systems have depended on the cooling of a gas by an adiabatic expansion. This effect was first described by Clement and Desormes in 1819. If a container of compressed air is vented to the atmosphere, the outrushing gas must do work to push back the gas ahead of it. If the process is carried out rapidly enough, it is essentially adiabatic, and the gas is cooled by the expansion. To obtain continuous refrigeration, some kind of cyclic process must be devised; simply opening a valve on a tank of compressed gas is obviously unsatisfactory. 7 Two methods of controlled expansion can be utilized: (1) a Joule-Thomson expansion through a throttling valve; (2) an expansion against a constraining piston. In the latter case, the gas does work against the external force and also against its internal cohesive forces. In the Joule- Thomson case, only the internal forces are operative, and these change in sign as the gas passes through an inversion point. It was shown on page 66 that in order to obtain cooling // t/ T = (\/C P )[T(dV/dT) P V] must be positive. In 1860, Sir William Siemens devised a countercurrent heat exchanger, which greatly enhanced the utility of the Joule-Thomson method. This was applied in the Linde process for the production of liquid air. Chilled com- pressed gas is cooled further by passage through a throttling valve. The expanded gas passes back over the inlet tube, cooling the unexpanded gas. When the cooling is sufficient to cause condensation, the liquid air can be drawn off at the bottom of the apparatus. Liquid nitrogen boils at 77K, liquid oxygen at 90K, and they are easily separated by fractional distillation. In order to liquefy hydrogen, it is necessary to chill it below its Joule- Thomson inversion temperature at 193K; the Linde process can then be used to bring it below its critical temperature at 33K. The production of liquid hydrogen was first achieved in this way by James- Dewar in 1898. The boiling point of hydrogen is 22K. In 1908, Kammerlingh-Onnes, founder of Leiden's famous cryogenic laboratory, used liquid hydrogen to cool helium below its inversion point at 100K, and then liquefied it by an adaptation of the Joule-Thomson principle. Temperatures as low as 0.84K have been obtained with liquid helium boiling under reduced pressures. This temperature is about the limit of this method, since enormous pumps become necessary to carry off the gaseous helium. Let us consider more carefully this cooling produced by evaporating liquid from a thermally isolated system. The change in state, liquid -> vapor, is a change from the liquid, a state of low entropy and low energy, to the vapor, a state of higher entropy and higher energy. The increase in entropy on evaporation can be equated to A// vap /r. Since the system is thermally 7 This method is used, however; in a laboratory device for making small quantities of "dry ice,** solid carbon dioxide. Sec. 13] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 87 isolated, the necessary heat of vaporization can come only from the liquid itself. Thus the temperature of the liquid must fall as the adiabatic evaporation proceeds. In 1926, a new refrigeration principle was proposed independently by W. F. Giauque 8 and P. Debye. This is the adiabatic demagnetization method. Certain rare earth salts have a high paramagnetic susceptibility? i.e., in a magnetic field they tend to become highly magnetized, but when the field is removed, they lose their magnetism immediately. In 1933, Giauque per- formed the following experiment. A sample of gadolinium sulfate was cooled to 1.5K in a magnetic field of 8000 oersteds, and then thermally isolated. The field was suddenly shut off. The salt lost its magnetism spontaneously. Since this was a spontaneous process, it was accompanied by an increase in the entropy of the salt. The magnetized state is a state of lower energy and lower entropy than the demagnetized state. The change, magnetized -> de- magnetized, is therefore analogous to the change, liquid -> vapor, discussed in the preceding paragraph. If the demagnetization occurs in a thermally isolated system, the temperature of the salt must fall. When the field was turned off in Giauque's experiment, the temperature fell to 0.25K. In 1950, workers at Leiden 10 reached a temperature of 0.0014K by this method. Even the measurement of these low temperatures is a problem of some magnitude. The helium vapor-pressure thermometer is satisfactory down to about 1K. Below this, the Curie- Weiss expression for the paramagnetic susceptibility, # ^= const/r, can be used to define a temperature scale. The fact that we have approached to within a few thousandths of a degree of absolute zero does not mean that the remaining step will soon be taken. On the contrary, it is the detailed analysis of these low-temperature experiments that indicates most definitely that zero degrees Kelvin is absolutely unattainable. The Third Law of Thermodynamics will, therefore, be postulated as follows: "It is impossible by any procedure, no matter how idealized, to reduce the temperature of any system to the absolute zero in a finite number of operations." 11 13. The Third Law of Thermodynamics. How does the Third Law answer the question of the value of the entropy of a substance at T = 0K, the integration constant S Q in eq. (4.20)? Since absolute zero is unattainable, it would be more precise to ask what is the limit of S as T approaches 0. Consider a completely general process, written as a -> b. This may be a chemical reaction, a change in temperature, a change in the magnetization, 8 /. Am. Chem. Soc., 49, 1870 (1927). 9 Cf. Sec. 1 1-20. 10 D. de Klerk, M. J. Steenland, and C. J. Gorter, Physica, 16, 571 (1950). 11 R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics (London: Cam- bridge, 1940), p. 224. 88 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 or the like. The entropies of the system in the two different states a and b can be written as : (4.21) S aQ and S bQ are the limiting entropy values as T approaches zero. Let us start with the system a at a temperature T f and allow the process a -> b to take place adiabatically and reversibly, the final temperature being T" . The entropy must remain constant, so that S a = S b , or +J7 Q d In T = 5 &0 +/J' C b dlnT In order for the temperature T" in the final state to equal zero, it would be necessary to have S bQ -S a0 ^*' C a d\nT (4.22) As T -> 0, C n -> 0. Now if 5 50 > 5 a0 it is possible to choose an initial T' that satisfies this equation, since the integral is a positive quantity. In this way the process a -> b could be used to reach the absolute zero starting from this T 1 . This conclusion, however, would be a direct contradiction of the Third Law, the principle of the unattainability of absolute zero. The only escape is to declare that S b0 cannot be greater than 5 a0 . Then there can be no 7" that satisfies the condition (4.22). The same reasoning, based upon the reverse process b -> a, can be used to show that 5* a0 cannot be greater than S bQ . Since S aQ can be neither greater than nor less than S 60 , it must be equal to S bQ . In order to conform with the principle of the unattainability of absolute zero, therefore, it is necessary to have 5 rt0 -5 60 or AS = (4.23) This equation indicates that for any change in a thermodynamic system the limiting value of AS as one approaches absolute zero is equal to zero. The change in question may be a chemical reaction, a change in physical state such as magnetized ^ demagnetized, or in general any change that can in principle be carried out reversibly. This requirement of a possible reversible process is necessary, since otherwise there would be no way of evaluating the AS for the change being considered. 12 The statement in eq. (4.23) is the 12 This restriction may be a little too severe. In one-component systems, changes of one polymorphic crystal to another may also have A5 = 0. Examples are white tin -> grey tin, diamond -> graphite, monochnic sulfur -> rhombic sulfur, zinc blende -> wurtzite. The heat capacity of the metastable form can be measured at low temperatures, and by extrapolation to 0K and assuming 5 0, it is possible to obtain a "Third-Law entropy,*' as defined in the next section. Sec. 14] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 89 famous heat theorem first proposed by Walther Nernst in 1906. It has served as a useful statement of the Third Law of Thermodynamics. Certain types of systems therefore do not fall within the scope of eq. (4.23). For example, any reaction that changed the identity of the chemical elements, i.e., nuclear transmutation, would not be included, since there is no thermo- dynamic method of calculating AS for such a change. This restriction, of course, does not affect chemical thermodynamics in any way, since the nuclei of the elements retain their identities in any chemical change. Another class of changes that must be excluded from eq. (4.23) comprises those in which the system passes from a metastable to a more stable state. Such changes are essentially irreversible and can proceed in one direction only, namely, toward the more stable states. Certain systems can become "frozen" in nonequilibrium states at low temperatures. Examples are glasses, which can be regarded as supercooled liquids, and solid solutions and alloys, in which there is a residual entropy of mixing. At sufficiently low tempera- tures, the glass is metastable with respect to the crystalline silicates of which it is composed, and the solid solutions are less stable than a mixture of pure crystalline metals. Yet the rate of attainment of equilibrium becomes so slow in the very cold solids that transformations to the more stable states do not occur. Such systems have an extra entropy, which can be considered as an entropy of mixing, and this may persist at the lowest tem- peratures attainable experimentally. This fact does not contradict eq. (4.23) because a change such as "metastable glass ~> crystalline silicates" cannot be carried out by a reversible isothermal path. Hence these metastable states are said to be "nonaccessible," and the changes do not lie within the scope of eq. (4.23). These cases will be discussed later from a statistical point of view in Chapter 12. 14. Third-law entropies. Only changes or differences in entropy have any physical meaning in thermodynamics. When we speak of the entropy of a substance at a certain temperature, we really mean the difference between its entropy at that temperature and its entropy at some other temperature, usually 0K. Since the chemical elements are unchanged in any physico- chemical process, we can assign any arbitrary values to their entropies at 0K without affecting in any way the values of AS for any chemical change. It is most convenient, therefore, to take the value of 5 for all the chemical elements as equal to zero. This is a convention first proposed by Max Planck in 1912. It then follows, from eq. (4.23), that the entropies of all pure chemical compounds in their stable states at 0K are also zero, because for their formation from the elements, AS = 0. This formulation is equivalent to setting the constant 5 in eq. (4.20) equal to zero. It is now possible to use heat-capacity data extrapolated to 0K to deter- mine so-called third-law entropies, which can be used in equilibrium calcula- tions. As an example, the determination of the standard entropy, S Q 29B , for 90 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 TABLE 4.5 EVALUATION OF ENTROPY OF HYDROGEN CHLORIDE FROM HEAT-CAPACITY MEASUREMENTS Contribution 1. Extrapolation from 0-1 6K (Debye Theory, Sec. 13-23) 2. lC,,d\r\ Tfor Solid I from 16 J ~98.36 3. Transition, Solid I Solid II, 2843/98.36 4. JO</ln 7 for Solid II from 98.36 r -l 58.91 5. Fusion, 476.0/158.91 6. JCV/ln Tfor Liquid from 158.91-! 88.07 7. Vaporization, 3860/188.07 8. JCW In Tfor Gas from 188.07-298.15K caljdeg mole 0.30 7.06 2.89 5.05 3.00 2.36 20.52 3.22 S ~~~ 5 -44.40 0. 10 hydrogen chloride gas is shown in Table 4.5. The value S 2 98 44.4 eu is that for HC1 at 25C and 1 atm pressure. A small correction due to non- ideality of the gas raises the figure to 44.7. A number of third-law-entropies are collected in Table 4.6. TABLE 4.6 THIRD-LAW ENTROPIES (Substances in the Standard State at 25C) Substance H 2 D 2 He N 2 2 C1 2 HC1 CO Mercury Bromine Water Methanol Ethanol C (diamond) C (graphite) S (rhombic) S (monoclinic) Ag Cu Fe Na (calldeg mole) 31.2 34.4 29.8 45.8 49.0 53.2 44.7 47.3 17.8 18.4 16.8 30.3 38.4 0.6 1.4 7.6 7.8 10.2 8.0 6.7 12.3 Gases Liquids Solids Substance C0 2 H 2 NH 3 S0 2 CH 4 C 2 H 2 C 2 H 4 C 2 H, Benzene Toluene Diethylether rt-Hexane Cyclphexane K I 2 NaCl KCl KBr KI AgCl Hg 2 Cl 2 298 (calldeg mole) 51.1 45.2 46.4 59.2 44.5 48.0 52.5 55.0 41.9 52.4 60.5 70.6 49.2 16.5 14.0 17.2 19.9 22.5 23.4 23.4 46.4 Sec. i5] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 91 The standard entropy change AS in a chemical reaction can be calculated immediately, if the standard entropies of products and reactants are known. AS - 2 ", S? One of the most satisfactory experimental checks of the Third Law is pro- vided by the comparison of AS values obtained in this way from low- temperature heat capacity measurements, with AS values derived either from measured equilibrium constants and reaction heats or from the tem- perature coefficients of celt emf's eq. (4.3). Examples of such comparisons are shown in Table 4.7. The Third Law is now considered to be on a firm experimental basis. Its full meaning will become clearer when its statistical interpretation is considered in a later chapter. The great utility of Third Law measurements in the calculation of chemical equilibria has led to an intensive development of low-temperature heat-capacity techniques, using liquid hydrogen as a refrigerant. The ex- perimental procedure consists essentially in a careful measurement of the temperature rise that is caused in an insulated sample by a carefully measured energy input. We have now seen how thermodynamics has been able to answer the old question of chemical affinity by providing a quantitative method for calculating (from thermal data alone) the position of equilibrium in chemical reactions. TABLE 4.7 CHECKS OF THE THIRD LAW OF THERMODYNAMICS Reaction Ag (c) f i Br 2 (1) - AgBr (c) Ag (c) + * C1 2 (g) - AgCl (c) Zn (c) + J 2 (g) = ZnO (c) C M 2 (g) = CO (g) CaC0 3 (c) - CaO (c) + CO 2 (g) Temp. (K) Third Law AS (cal/deg mole) Experimental AS Method 265.9 -3.01 0.40 -3.02 0.10 emf 298.16 -13.85 0.25 -13.73 0.10 emf 298.16 -24.07 0.25 -24.24 0.05 K and Atf 298.16 -20.01 0.40 -2 1.38 0.05 #andA# 298.16 38.40 0.20 38.03 0.20 K and Atf 15. General theory of chemical equilibrium: the chemical potential. We have so far confined our attention to equilibria involving ideal gases. The relations discovered are of great utility, and are accurate enough for the discussion of most homogeneous gas equilibria. Some gas reactions, how- ever, are carried out under such conditions that the ideal gas laws are no longer a good approximation. Examples include the high-pressure syntheses of ammonia and methanol. In addition, there are the great number of chemical reactions that occur in condensed phases such as liquid or even solid solutions. In order to treat these reactions especially, a more general equilibrium theory will be needed. 92 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 The composition of a system in which a chemical reaction is taking place is continually changing, and the state of the system is not defined by specifying merely the pressure, volume, and temperature. In order to discuss the changes of composition it is necessary to introduce, in addition to P 9 V, and T, new variables that are a measure of the amount of each chemical constituent of the system. As usual, the mole will be chosen as the chemical measure, with the symbols n l9 w 2 n 3 . . . n l representing the number of moles of constituent 1, 2, 3, or i. It then follows that each thermodynamic function depends on these /i/s as well as on P, V, and T. Thus, E - E(P, V, T, n t ); F = F(P, V, T, n,\ etc. Consequently, a perfect differential, for example of the free energy, becomes By eq. (3.29) dF ~ ~S dT + V dP for any system of constant composition, i.e., when all dn % 0. Therefore = -SdT + VdP + - dn, (4.25) T,P,n, The coefficient (dF/dn t ) T P n> , first introduced by Gibbs, has been given a special name because of its great importance in chemical thermodynamics. It is called the chemical potential, and is written as (4 ' 26) It is the change of the free energy with change in number of moles t of component /, the temperature, the pressure, and the number of moles of all other components in the system being kept constant. Using the new symbol, eq. (4.25) becomes dF - -5 dT + VdP + 2 Hi dn, (4.27) i At constant temperature and pressure, </F=2^<**. (4-28) The condition for equilibrium, dF 0, then becomes I to **< = <> (4.29) i For an ideal gas, the chemical potential is simply the free energy per mole at pressure P t . Therefore from eq. (4.7), (4.30) The value of //, for the ideal gas is the same whether the ideal gas is pure gas at a pressure P t or is in an ideal gas mixture 13 at partial pressure P t . If, 13 This statement is a definition of an ideal gas mixture. To be precise, one must distin- guish an ideal gas mixture from a mixture of ideal gases. There might be specific interactions between two ideal gases that would cause their mixture to deviate from ideality. Sec. 16] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 93 however, the gas mixture is not ideal, this identity no longer holds true. Various interaction forces come into operation, and the evaluation of ^ { becomes a separate experimental problem in each case. 16. The fugacity. Because relations such as eq. (4.30) lead to equations of such simple form in the development of the theory of chemical equilibrium, it is convenient to introduce a new function, called the fugacity of the sub- stance, that preserves the form of eq. (4.30) even for nonideal systems. Therefore we write dfi = VdP = RTdlnf, and /i, - - RTln^ J i where/- is the fugacity of the substance, and/? is its fugacity in the standard state. It now becomes desirable to change the definition of the standard state so that instead of the state of unit pressure, it becomes the state of unit fugacity,/? = 1. Then Vi-tf-RTlnfi (4.31) Now the treatment of equilibrium in Section 4-5 can be carried through in terms of the fugacity and chemical potential. This leads to an expression for the equilibrium constant which is true in general, not only for real (nonideal) gases but also for substances in any state of aggregation what- soever: f cf d f ~~ f af b JA JB -A// - RT \nK f (4.32) The fugacity of a pure gas or of a gas in a mixture can be evaluated if sufficiently detailed PVT data, are available. This discussion will be limited to an illustration of the method for determining the fugacity of a pure gas. In this case, dF=dp=VdP (4.33) If the gas is ideal, V = RT/P. For a nonideal gas, this is no longer true. We may write a - F ideal - K real - (RT/P) - K, whence V = (RT/P) ~ a. Sub- stituting this expression into eq. (4.33), we find RTdlnf - dF=dfA = RTctlnP - adP The equation is integrated from P = to P. RT\ f dlnf=RTf P d\nP-( P adP J/,p=o J JP-O Jo As its pressure approaches zero, a gas approaches ideality, and for an ideal gas the fugacity equals the pressure, /= P [cf. eqs. (4.30 and (4.31)]. The lower limits of the first two integrals must therefore be equal, so that we obtain RT\nf= RT In P -J P <2 dP (4.34) 94 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 This equation enables us to evaluate the fugacity at any pressure and temperature, provided PVT data for the gas are available. If the deviation from ideality of the gas volume is plotted against P, the integral in eq. (4.34) can be evaluated graphically. Alternatively, an equation of state can be used to calculate an expression for a as a function of P, making it possible to evaluate the integral by analytical methods. The fugacity may be thought of as a sort of idealized pressure, which measures the true escaping tendency of a gas. In Chapter 1, it was pointed V 2 6 8 10 12 14 16 18 20 22 24 REDUCED PRESSURF-Ffe Fig. 4.4. Variation of activity coefficient with reduced pressure at various reduced temperatures. out that the deviations of gases from ideality are approximately determined by their closeness to the critical point. This behavior is confirmed by the fact that at the same reduced pressures all gases have approximately the same ratio of fugacity to pressure. The ratio of fugacity to pressure is called the activity coefficient, y =f/P. Figure 4.4 shows a family of curves 14 relating the activity coefficient of a gas to its reduced pressure P K at various values of the reduced temperature T lf . To the approximation that the law of corre- sponding states is valid, all gases have the same value of y when they are in corresponding states, i.e., at equal P R and T I{ . This is a very useful principle, 14 Newton, Ind. Eng. Chem., 27, 302 (1935). Graphs for other ranges of P R and T R are included in this paper. Sec. 17] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 95 for it allows us to estimate the fugacity of a gas solely from a knowledge of its critical constants. 17. Use of fugacity in equilibrium calculations. Among the industrially important gas reactions that are carried out under high pressures is the synthesis of ammonia: | N 2 + $ H 2 = NH 3 . This reaction has been carefully investigated up to 1000 atm by Larson and Dodge. 15 The per cent of NH 3 in equilibrium with a three-to-one H 2 -N 2 mixture at 450C and various total pressures is shown in Table 4.8. In the third column of the table are the values of K p = PyuJPy^Pa?' 2 calculated from these data. Since K p for ideal gases should be independent of the pressure, these results indicate considerable deviations from ideality at the higher pressures. Let us therefore calculate the equilibrium constant K f using Newton's graphs to obtain the fugacities. We are therefore adopting the approximation that the fugacity of a gas in a mixture is determined only by the temperature and by the total pressure of the gas mixture. Consider the calculation of the activity coefficients at 450C (723K) and 600 atm. PC T C P R T R r N 2 '33.5 126 17.9 5.74 1.35 H 2 . 12.8 33.3 46.8 21.7 1.19 NH 3 . . . 111.5 405.6 5.38 1.78 0.85 The activity coefficients y are read from the graphs, at the proper values of reduced pressure P H and reduced temperature T R . (Only the NH 3 values are found in Fig. 4.4; the complete graphs must be consulted for the other gases.) TABLE 4.8 EQUILIBRIUM IN THE AMMONIA SYNTHESIS AT 450C WITH 3 : 1 RATIO OF H 2 TO N 2 Total Per cent Pressure NH 3 at K v K Y (atm) Equilibrium 10 2.04 0.00659 0.995 0.( 30 5.80 0.00676 0.975 O.C 50 9.17 0.00690 0.945 O.C 100 16.36 0.00725 0.880 O.C 300 35.5 0.00884 0.688 O.C 600 ,53.6 0.01294 0.497 O.C 1000 69.4 0.02328 0.434 O.C Since the fugacity/^ yP, we can write in general K f = K Y K^ where in this case K v = ^H^N^H, 372 - The values of K v and K f are shown in Table 4.8. There is a marked improvement in the constancy of K f as com- pared with K v . Only at 1000 atm does the approximate treatment of the fugacities appear to fail. To carry out an exact thermodynamic treatment, it 15 /. Am. Chem. Soc., 45, 2918 (1923); 46, 367 (1924). 96 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 would be necessary to calculate the fugacity of each gas in the particular mixture under study. This would require very extensive PVT data on the mixture. Often, knowing AF for the reaction, we wish to calculate the equilibrium concentrations in a reaction mixture. The procedure is to obtain K f from AF = RTln K f , to estimate K y from the graphs, and then to calculate the partial pressures from K p K f /K y . PROBLEMS 1. The emf of the cadmium-calomel cell in which the reaction is Cd + Hg 2 ++ = Cd++ + 2 Hg, can be represented by: ? = 0.6708 - 1.02 x 10~ 4 (/ 25) 2.4 x 10~ 6 (r - 25) 2 , where t is the centigrade temperature. Calculate AF, AS, and A// for the cell reaction at 45C. 2. From the standard free energies in Table 4.2 calculate A/ 70 and K v at 25C for the following reactions : (a) N 2 O + 4 H 2 - 2 NH 3 + H 2 O (g) (b) H 2 2 (g)-H 2 0(g) + |0 2 (c) CO + H 2 O (1) = HCOOH (1) 3. At 900K the reaction C 2 H 6 = C 2 H 4 + H 2 has A// - 34.42, AF = 5.35 kcal. Calculate the per cent H 2 present at equilibrium if pure C 2 H 6 is passed over a dehydrogenation catalyst at this temperature and 1 atm pressure. Estimate the per cent H 2 at equilibrium at 1000K. 4. If an initial mixture of 10 per cent C 2 H 4 , 10 per cent C 2 H 6 , and 80 per cent N 2 is passed over the catalyst at 900K and 1 atm, what is the per cent composition of effluent gas at equilibrium? What if the same mixture is used at 100 atm? (Cf. data in Problem 3.) 5. The equilibrium LaCl 3 (s) + H 2 O (g) - LaOCl (s) + 2 HCl (g). [/. Am. Chem. Soc. 9 74, 2349 (1952)] was found to have K p - 0.63 at 804K, and 0.125 at 733K. Estimate A// for the reaction. If the equilibrium HCI vapor pressure at 900K is 2.0mm estimate the equilibrium H 2 O vapor pressure. 6. From the data in Table 4.4, calculate the heat of dissociation of O 2 into 2 O at 1000K. Similarly, calculate A// 1000 for H 2 = 2 H. Assuming atomic H and O are ideal gases with C P = f /*, and using the Q>'s for H 2 and O 2 in Table 2.4, calculate A// 298 for 2 H + O = H 2 O (g). The heat of formation of H 2 O(g) is 57.80 kcal. One-half the heat calculated in this problem is a measure of the "strength of the O H bond" in water. 7. For the reaction N 2 O 4 2 NO 2 , calculate K P , K x , K c at 25C. and 1 atm from the free energies of formation of the compounds (Table 4.2). 8. PC1 5 vapor decomposes on heating according to PC1 5 = PC1 3 + C1 2 . The density of a sample t>f partially dissociated PC1 5 at 1 atm and 230C was 15 0.89 20 2.07 30 4.59 40 6.90 50 8.53 60 9.76 70 10.70 80 11.47 90 12.10 100 12.62 120 13.56 140 14.45 160 15.31 180 16.19 200 17.08 220 17.98 240 18.88 260 25.01 280 25.17 300 25.35 Chap. 4] THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 97 found to be 4.80 g per liter. Calculate the degree of dissociation a and AF for the dissociation at 230C. 9. The following results were obtained for the degree of dissociation of CO 2 (CO 2 = CO + i O 2 ) at 1 atm: K . . . 1000 1400 2000 a . . . 2.0 x 10~ 7 1.27 x 10~ 4 1.55 x 10~ 2 What is AS for the reaction at 1400K? 10. The free energy of formation of H 2 S is given by AF = 19,200 + 0.94rin T - 0.001 65T 2 - 0.00000037r 3 + 1-657. H 2 + J S 2 (g) - H 2 S (g). If H 2 S at 1 atm is passed through a tube heated to 1200K, what is per cent H 2 in the gas at equilibrium? 11. Jones and Giauque obtained the following values for C P of nitro- methane. 16 K C P K C P The melting point is 244.7K, heat of fusion 2319cal per mole. The vapor pressure of the liquid at 298. 1K is 3.666 cm. The heat of vaporization at 298. 1K is 9147 cat per mole. Calculate the Third-Law entropy of CH 3 NO 2 gas at 298. 1K and 1 atm pressure (assuming ideal gas behavior). 12. Using the Third-Law entropies in Table 4.6 and the standard heats of formation calculate the equilibrium constants at 25C of the following reactions : H 2 + C1 2 - 2 HC1 CH 4 + 2 2 = C0 2 + 2 H 2 (g) 2Ag(s) + Cl 2 -2AgCi(s) 13. For the reaction CO + 2 H 2 - CH 3 OH (g), AF - -3220 cal at 700K. Calculate the per cent CH 3 OH at equilibrium with a 2 : 1 mixture of H 2 + CO at a pressure of 600 atm using (a) ideal gas law, (b) Newton's fugacity charts. 14. At high temperature and pressure, a quite good equation of state for gases is P(V b) = RT. Calculate the fugacity of N 2 at 1000 atm and 1000C according to this equation, if b = 39.1 cc per mole. 15. Show that T,F,n, \s,P f n, W t ' S,V,n, 16. Amagat measured the molar volume of CO 2 at 60C. Pressure, atm . . 13.01 35.42 53.65 74.68 85.35 Volume, cc . . 2000.0 666.7 400.0 250.0 200.0 16 /. Am. Chem. Soc., 69, 983 (1947). 98 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [Chap. 4 Calculate the activity coefficient y = f/P for CO 2 at 60C and pressures of 10, 20, 40, and 80 atm. 17. When rt-pentane is passed over an isomerization catalyst at 600 K, the following reactions occur : (A) CH 3 CH 2 CH 2 CH 2 CH 3 - CH 3 CH(CH 3 )CH 2 CH 3 (B) - C(CH 3 ) 4 (C) The free energies of formation at 600K are: (A) 33.79, (B) 32.66, (C) 35.08 kcal per mole. Calculate the composition of the mixture when complete equilibrium is attained. 18. For the reaction 3 CuCl (g) - Cu 3 C) 3 (g), Brewer and Lofgren [J. Am. Chem. Soc., 72, 3038 (1950)] found AF - -126,400 - 12.5iriog T + 104.7 T. What are the A// and AS of reaction at 2000K? What is the equilibrium mole fraction of trimer in the gas at 1 atm and 2000K? REFERENCES BOOKS 1. Kubaschewski, O., and E. L. Evans, Metallurgical Thermochemistry (London: Butterworth, 1951). 2. Putnam, P. C., Energy in the Future (New York: Van Nostrand, 1953). 3. Squire, C. F., Low Temperature Physics (New York: McGraw-Hill, 1953). 4. Wenner, R. R., Thermochemical Calculations (New York: McGraw-Hill, 1941). Also see Chapter 1, p. 25. ARTICLES 1. Chem. Revs., 39, 357-481 (1946), "Symposium on Low Temperature Research." 2. Daniels, F., Scientific American, 191, 58-63 (1954), "High Temperature Chemistry," 3. Huffman, H. M.,' Chem. Revs., 40, 1-14 (1947), "Low Temperature Calorimetry." 4. Lemay, P., and R. Oesper, /. Chem. Ed., 23, 158-65, 230-36 (1946), "Claude Berthollet " 5. Oesper, R., /. Chem. Ed., 21, 263-64 (1944), "H. Kammerlingh-Onnes." 6. Urey, H. C., /. Chem. Soc., 562-81 (1947), "Thermodynamic Properties of Isotopic Substances." 7. Walden, P., /. Chem. Ed., 31, 27-33 (1954), "Beginnings of the Doctrine of Chemical Affinity." 8. Watson, R. G., Research, 7, 34-40 (1954), "Electrochemical Generation of Electricity." CHAPTER 5 Changes of State 1. Phase equilibria. Among the applications of thermodynamics is the study of the equilibrium conditions for changes such as the melting of ice, the solution of sugar, the vaporization of benzene, or the transformation of monoclinic to rhombic sulfur. Certain fundamental principles are applicable to all such phenomena, which are examples of "changes in state of aggrega- tion" or "phase changes." The word phase is derived from the Greek (pa.ai<t, meaning "appearance." If a system is "uniform throughout, not only in chemical composition, but also in physical state," 1 it is said to be homogeneous, or to consist of only one phase. Examples are a volume of air, a noggin of rum, or a cake of ice. Mere difference in shape or in degree of subdivision is not enough to deter- mine a new phase. Thus a mass of cracked ice is still only one phase. 2 A system consisting of more than one phase is called heterogeneous. Each physically or chemically different, homogeneous, and mechanically separable part of a system constitutes a distinct phase. Thus a glassful of water with cracked ice in it is a two-phase system. The contents of a flask of liquid benzene in contact with benzene vapor and air is a two-phase system; if we add a spoonful of sugar (practically insoluble in benzene) we obtain a three-phase system: a solid, a liquid, and a vapor phase. In systems consisting entirely of gases, only one phase can exist at equi- librium, since all gases are miscible in all proportions (unless, of course, a chemical reaction intervenes, e.g., NH 3 + HC1). With liquids, depending on their mutual miscibility, one, two, or more phases can arise. Many different solid phases can coexist. 2. Components. The composition of a system may be completely de- scribed in terms of the "components" that are present in it. The ordinary meaning of the word "component" is somewhat restricted in this technical usage. We wish to impose a requirement of economy on our description of the system. This is done by using the minimum number of chemically distinct constituents necessary to describe the composition of each phase in the system. The constituents so chosen are the components. If the concentrations of the components are stated for each phase, then the concentrations in each phase of any and all substances present in the system are uniquely fixed. This definition may be expressed more elegantly by saying that the com- 1 J. Willard Gibbs. 2 This is because we are assuming, at this stage in our analysis, that a variable surface area has no appreciable effect on the properties of a substance. QO 100 CHANGES OF STATE [Chap. 5 ponents are those constituents whose concentrations may be independently varied in the various phases. Consider, for example, a system consisting of liquid water in contact with its vapor. We know that water is composed of hydrogen and oxygen, but these elements are always present in fixed and definite proportions. The system therefore contains one component only. Another example is the system consisting of calcium carbonate, calcium oxide, and carbon dioxide. A chemical reaction between these compounds is possible, CaCO 3 CaO + CO 2 . In this case, three phases are present, gaseous CO 2 , solid CaCO 3 and CaO. Two components are required in order" to describe the composition of all of these phases, the most suitable choice being CaO and CO 2 . A less obvious example is the system formed by water and two salts without a common ion, e.g., H 2 O, NaCl, KBr. As a result of interaction between ions in solution four different salts, or their hydrates, may occur in solid phases, namely NaCl, KBr, NaBr, KC1. In order to specify the com- position of all possible phases, four components are necessary, consisting of water and three of the possible salts. This fixes the concentrations of three of the four ions in any phase, and the fourth is fixed by the requirement of over-all electrical neutrality. Careful examination of each individual system is necessary in order to decide the best choice of components. It is generally wise to choose as com- ponents those constituents that cannot be converted into one another by reactions occurring within the system. Thus CaCO 3 and CaO would be a possible choice for the CaCO 3 CaO + CO 2 system, but a poor choice because the concentrations of CO 2 would have to be expressed by negative quantities. While the identity of the components is subject to some degree of choice, the number of components is always definitely fixed for any given case. Even the last statement should perhaps be modified, because the actual choice of the number of components depends on how precisely one wishes to describe a system. In the water system, there is always some dissociation of water vapor into hydrogen and oxygen. At moderate temperatures, this dissociation is of no consequence in any experimental measurements, and to consider it in deciding the number of components would be unduly scrupu- lous. 3 The precision with which experimental data on the system can be obtained should be allowed to decide borderline cases. 3. Degrees of freedom. For the complete description of a system, the numerical values of certain variables' must be reported. These variables are 3 It is worth noting that the mere dissociation of water into hydrogen and oxygen does not create new components, because the proportion of H 2 to O 2 is always fixed at 2:1, since we exclude the possibility that additional H 2 or O 2 can be added to the system. The reason why an extra component, either H 2 or O 2 , might conceivably be required is that H 2 and O 2 dissolve to different extents in the water, so that their ratio is no longer fixed at 2:1 in each phase. Sec. 4] CHANGES OF STATE 101 chosen from among the "state functions" of the system, such as pressure, temperature, volume, energy, entropy, and the concentrations of the various components in the different phases. Values for all of the possible variables need not be explicitly stated, for a knowledge of some of them definitely determines the values of the others. For any complete description, however, at least one capacity factor is required, since otherwise the mass of the system is undetermined, and one is not able, for example, to distinguish between a system containing a ton of water and one containing a few drops. An important feature of equilibria between phases is that they are in- dependent of the actual amounts of the phases that may be present. 4 Thus the vapor pressure of water above liquid water in no way depends on the volume of the vessel or on whether a few milliliters or many gallons of water are in equilibrium with the vapor phase. Similarly, the concentration of a saturated solution of salt in water is a fixed and definite quantity, regardless of whether a large or a small excess of undissolved salt is present. In discussing phase equilibria, we therefore need not consider the capacity factors, which express the absolute bulk of any phase. We consider only the intensity factors, such as temperature, pressure, and concentrations. Of these variables a certain number may be independently varied, but the rest are fixed by the values chosen for the independent variables and by the thermo- dynamic requirements for equilibrium. The number of the intensive state variables that can be independently varied without changing the number of phases is called the number of degrees of freedom of the system, or sometimes the variance. For example, the state of a certain amount of a pure gas may be specified completely by any two of the variables, pressure, temperature, and density. If any two of these are known, the third can be calculated. This is therefore a system with iwo degrees of freedom, or a bi variant system. In the system "water water vapor," only one variable need be specified to determine the state. At any given temperature, the pressure of vapor in equilibrium with liquid water is fixed in value. This system has one degree of freedom, or is said to be univariant. 4. Conditions for equilibrium between phases. In a system containing several phases, certain thermodynamic requirements for the existence of equilibrium may be derived. For thermal equilibrium it is necessary that the temperatures of all the phases be the same. Otherwise, heat would flow from one phase to another. This intuitively recognized condition may be proved by considering two phases a and /? at temperatures r a , T ft . The condition for equilibrium at constant volume and composition is given on p. 59 as dS 0. Let 5 a and S ft be the entropies of the two phases, and suppose there were a transfer of heat dq from a to /? at equilibrium. 4 This statement is proved in the next Section. It is true as long as surface area variations are left out of consideration. (See Chapter 16.) 102 CHANGES OF STATE [Chap. 5 Then dS = dS* + dS ft = or - -| + -| - whence 7 a - 7* (5.1) For mechanical equilibrium it is necessary that the pressures of all the phases be the same. Otherwise, one phase would increase in volume at the expense of another. This condition may be derived from the equilibrium condition at constant over-all volume and temperature, dA 0. Suppose one phase expanded into another by 6V. Then or P - Pft (5.2) In addition to the conditions given by eqs. (5.1) and (5.2), a condition is needed that expresses the requirements of chemical equilibrium. Let us con- sider the system with phases a and ft maintained at constant temperature and pressure, and denote by n*, /?/*, the numbers of moles of some particular component / in the two phases. From eq. (3.28) the equilibrium condition becomes dF 0, or dF | - tf*-=0 (5.3) Suppose that a process occurred by which dn t moles of component / were taken from phase a and added to phase ft. (This process might be a chemical reaction or a change in aggregation-state.) Then, by virtue of eq. (4.28), eq. (5.3) becomes -^ /i f to, + /*//!, = or ^ - p* (5.4) This is the general condition for equilibrium with respect to transport of matter between phases, including chemical equilibrium between phases. For any component / in the system, the value of the chemical potential /^ must be the same in every phase. An important symmetry between the various equilibrium conditions is apparent in the following summary: Capacity Intensity Equilibrium factor factor condition S T T* = TP V P P = Pft 5. The phase rule. Between 1875 and 1878, Josiah Willard Gibbs, Pro- fessor of Mathematical Physics at Yale University, published in the Trans- actions of the Connecticut Academy of Sciences a series of papers entitled "On the Equilibrium of Heterogeneous Substances." In these papers Gibbs disclosed the entire science, of heterogeneous equilibrium with a beauty and preciseness never before and seldom since seen in thermodynamic studies. Sec. 5] CHANGES OF STATE 103 Subsequent investigators have had little to do save to provide experimental illustrations for Gibbs's equations. The Gibbs phase rule provides a general relationship among the degrees of freedom of a system/, the number of phases /?, and the number of com- ponents c. This relationship always is f^c-p + 2 (5.5) The derivation proceeds as follows: The number of degrees of freedom is equal to the number of intensive variables required to describe a system, minus the number that cannot be independently varied. The state of a system containing p phases and c com- ponents is specified at equilibrium if we specify the temperature, the pressure, and the amounts of each component in each phase. The total variables required in order to do this are therefore pc -\- 2. Let n* denote the number of moles of a component / in a phase a. Since the size of the system, or the actual amount of material in any phase, does not affect the equilibrium, we are really incerested in the relative amounts of the components in the different phases and not in their absolute amounts. Therefore, instead of the mole numbers n* 9 the mole fractions X? should be used. These are given by For each phase, the sum of the mole fractions equals unity. Xf + X,' + AV [-... -+ X? = 1 or 2 X>" =- 1 (5-6) I If all but one mole fraction are specified, that one can be calculated from eq. (5.6). If there are/? phases, there are/? equations similar to eq. (5.6), and therefore p mole fractions that need not be specified since they can be cal- culated. The total number of independent variables to be specified is thus pc + 2 p or p(c 1) + 2. At equilibrium, the eqs. (5.4) impose a set of further restraints on the system by requiring that the chemical potentials of each component be the same in every phase. These conditions are expressed by a set of equations such as : V* = fit = /*!" = ... ^ - ^ = ^ - - ' - (5.7) Each equality sign in this set of equations signifies a condition imposed on the system, decreasing its variance by one. Inspection shows that there are therefore c(p 1) of these conditions. 104 CHANGES OF STATE [Chap. 5 The degrees of freedom equal the total required variables minus the restraining conditions. Therefore f-=p(c- l) + 2-c(/>- 1) f=c-p + 2 (5.8) 6. Systems of one component water. In the remainder of this chapter, systems of one component will be considered. These systems comprise the study of the conditions of equilibrium in changes in the state of aggregation of pure substances. From the phase rule, when c* !,/= 3 /?, and three different cases are possible: p ^ l,/ r - 2 bi variant system p --= 2,f^= 1 univariant system p _z_- 3,/ invariant system These situations may be illustrated by the water system, with its three familiar phases, ice, water, and steam. Since the maximum number of degrees of freedom is two, any one-component system can be represented by a two- dimensional diagram. The most convenient variables are the pressure and the temperature. The water system is shown in Fig. 5.1. .0075 100 TEMPERATURE - *C 374 Fig. 5.1. The water system schematic. (Not drawn to scale.) The diagram is divided into three areas, the fields of existence of ice, water, and steam. Within these single-phase areas, the system is bivariant, and pressure and temperature may be independently varied. Sec. 7] CHANGES OF STATE 105 Separating the areas are lines connecting the points at which two phases may coexist at equilibrium. Thus the curve AC dividing the liquid from the vapor region is the familiar vapor-pressure curve of liquid water. At any given temperature there is one and only one pressure at which water vapor is in equilibrium with liquid water. The system is univariant, having one degree of freedom. The curve AC has a natural upper limit at the point C, which is the critical point, beyond which the liquid phase is no longer distinguishable from the vapor phase. Similarly, the curve AB is the sublimation-pressure curve of ice, giving the pressure of water vapor in equilibrium with solid ice, and dividing the ice region from the vapor region. The curve AD divides the solid-ice region from the liquid-water region. It shows how the melting temperature of ice or the freezing temperature of water varies with the pressure. It is still an open question whether such curves, at sufficiently high pressures, ever have a natural upper limit beyond which solid and liquid are indistinguishable. These three curves intersect at a point A, at which solid, liquid, and vapor are simultaneously at equilibrium. This point, which occurs at 0.0075C and 4.579 mm pressure, is called a triple point. Since three phases coexist, the system is invariant. There are no degrees of freedom and neither pressure nor temperature can be altered even slightly without causing the disappear- ance of one of the phases. It should be noted that this triple point is not the same as the ordinary melting point of ice, which by definition is the temperature at which ice and water are in equilibrium under an applied pressure of 1 atm or 760 mm. This temperature is, by definition, 0C. Liquid water may be cooled below its freezing point without solidifying. In AE we have drawn the vapor-pressure curve of this supercooled water, which is a continuous extension of curve AC. It is shown as a dotted line on the diagram since it represents a metastable system. Note that the meta- stable vapor pressure of supercooled water is higher than the vapor pressure of ice. The slope of the curve A D, the melting-point curve, is worth remarking. It shows that the melting point of ice is decreased by increasing pressure. This is a rather unusual behavior; only bismuth and antimony among common substances behave similarly. These substances expand on freezing. Therefore the Le Chatelier principle demands that increasing the pressure should lower the melting point. The popularity of ice skating and the flow of glaciers are among the consequences of the peculiar slope of the melting point curve for ice. For most substances, the density of the solid is greater than that of the liquid, and by Le Chatelier's principle, increase in pressure raises the melting point. 7. The Clapeyron-CIausius equation. There are two fundamental theoreti- cal equations that govern much of the field of phase equilibrium. The first 106 CHANGES OF STATE [Chap. 5 is the Gibbs phase rule, which determines the general pattern of the phase diagram. The second is the Clapeyron-Clausius equation, which determines the slopes of the lines in the diagram. It is a quantitative expression for the Le Chatelier principle as it applies to heterogeneous systems. First proposed by the French engineer Clapeyron in 1834, it was placed on a firm thermo- dynamic foundation by Clausius, some thirty years later. From eq. (5.4) the condition for equilibrium of a component / between two phases, a and /7, is ju^ -= ///. For a system of one component, the chemical potentials // are identical with the free energies per mole F, so that F* F ft at equilibrium. Consider two different equilibrium states, at slightly separated temperatures and pressures : (1) T,P, F* - F ft . (2) T + dT, P + dP, F* + dF* - F ft \- dF?. It follows that dF* - dF fl . The change in F with T and P is given by eq. (3.29), dF = V dP - S dT. Therefore, V dP - 5 a dT - V & dP - S? dT 9 or dP_S'-S^AS dT VP - K a AK ^ ' ' If the heat of the phase transformation is /I, AS is simply A/T where T is the temperature at which the phase change is occurring. The Clapeyron- Clausius equation is now obtained as ^ - -A (5 10) dT (5 ' 10) This equation is applicable to any change of state: fusion, vaporization, sublimation, and changes between crystalline forms, provided the appro- priate latent heat is employed. In order to integrate the equation exactly, it would be necessary to know both X and AK as functions of temperature and pressure. 5 The latter corre- sponds to a knowledge of the densities of the two phases over the desired temperature range. In most calculations over short temperature ranges, however, both X and AKmay be taken as constants. In the case of the change "liquid ^ vapor," several approximations are possible, leading to a simpler equation than eq. (5.10), dT Neglecting the volume of the liquid compared with that of the vapor, and assuming ideal gas behavior for the latter, one obtains d In P _ A vap ~W = Kf* (5 ' 12) 6 A good discussion of the temperature variation of A is given by Guggenheim, Modern Thermodynamics y p. 57. The variation with pressure of A and A Kis much less than that with temperature. Sec. 8] CHANGES OF STATE 107 A similar equation would be a good approximation for the sublimation curve. Just as was shown for eq. (3.36), this may also be written wTn = ~R (5 - 13) If the logarithm of the vapor pressure is plotted against 1/r, the slope of the curve at any point multiplied by R yields a value for the heat of vapori- zation. In many cases, since X is effectively constant over short temperature ranges, a straight-line plot is obtained. This fact is useful to remember in extrapolating vapor pressure data. When A is taken as constant, the integrated form of eq. (5.12) is ln -h -*(?-) (5 - 14) An approximate value for A vap can often be obtained from Troutorfs Rule (1884): ^ & 22 cal deg" 1 mole" 1 The rule is followed fairly well by many nonpolar liquids (Sec. 14-8). It is equivalent to the statement that the entropy of vaporization is approximately the same for all such liquids. 8. Vapor pressure and external pressure. It is of interest to consider the effect of an increased hydrostatic pressure on the vapor pressure of a liquid. Let us suppose that an external hydrostatic pressure P p is imposed on a liquid of molar volume V v Let the vapor pressure be P, and the molar volume of the vapor V g . Then at equilibrium at constant temperature: </F vap = rfF llq or V g dP - V, dP e or ~ Vl (5.15) dl f V fl This is sometimes called the Gibbs equation. If the vapor is an ideal gas, this equation becomes "(-.SrH Since the molar volume of the liquid does not vary greatly with pressure, this equation may be integrated approximately, assuming constant V t : n \r f n n '\ In theory, one can measure the vapor pressure of a liquid under an applied hydrostatic pressure in only two ways: (1) with an atmosphere of "inert" gas; (2) with an ideal membrane semipermeable to the vapor. In 108 CHANGES OF STATE [Chap. 5 practice, the inert gas will dissolve in the liquid, so that the application of the Gibbs equation to the problem is dubious. The second case is treated in the theory of osmotic pressure. As an example of the use of eq. (5.16), let us calculate the vapor pressure of mercury under an external pressure of 1000 atm at 100C. The density is 13.352 gem- 3 ; hence V, = M/p = 200.61/13.352 - 15.025 cm 3 , and P l 15.025(1000-1) In >l = lilQrx 373.2 Therefore, Pi/P 2 = 1.633. The vapor pressure at 1 atm is 0.273 mm, so that the calculated vapor pressure at 1000 atm is 0.455 mm. 9. Experimental measurement of vapor pressure. Many different experi- mental arrangements have been employed in vapor-pressure measurements. One of the most convenient static methods is the Smith- Menzies isoteniscope shown in Fig. 5.2. The bulb and short attached U-tube are filled with the TO AIR THERMOMETER ISOTENISCOPE TO VACUUM BALLAST VOLUME THERMOSTAT MANOMETER Fig. 5.2. Vapor pressure measurement with isoteniscope. liquid to be studied, which is allowed to boil vigorously until all air is re- moved from the sample side of the U-tube. At each temperature the external pressure is adjusted until the arms of the differential U-tube manometer are level, and the pressure and temperature are then recorded. The gas-saturation method was used extensively by Ramsay and Young. An inert gas is passed through the liquid maintained in a thermostat. The volume of gas used is measured, and its final vapor content or the loss in weight of the substance being studied is determined. If care is taken to ensure saturation of the flowing gas, the vapor pressure of the liquid may readily be calculated. Some experimentally measured vapor pressures are collected 6 in Table 5.1. 6 A very complete compilation is given by D. R. Stull, Ind. Eng. Chem., 39, 517-550 (1947). Sec. 10] CHANGES OF STATE TABLE 5.1 TYPICAL VAPOR PRESSURE DATA 109 Vapor Pressure in Millimeters of Mercury Temp. CO CC1 4 CH 3 COOH C 2 H 5 OH (C 2 H 5 ) 2 C 7 H 16 QH 5 -CH 8 :H 2 o 12.2 185.3 11.45 4.579 10 -- 23.6 291.7 20.5 9.209 20 91 11.7 43.9 442.2 35.5 17.535 30 143.0 20.6 78.8 647.3 58.35 36.7 31.824 40 215.8 34.8 135.3 921.3 92.05 59.1 55.324 50 317.1 56.6 222.2 1277 140.9 92.6 92.51 60 450.8 88.9 352.7 208.9 139.5 149.38 70 622.3 136.0 542.5 302.3 202.4 233.7 80 843 202.3 812.6 426.6 289.7 355.1 90 1122 293.7 1187 588.8 404.6 525.76 100 1463 417.1 795.2 557.2 760.00 ! 10. Solid-solid transformations the sulfur system. Sulfur provides the classical example of a one-component system displaying a solid-solid trans- formation. The phenomenon of polymorphism, discovered by Mitscherlich in 1821, is the occurrence of the same chemical substance in two or more different crystalline forms. In the case of elements, it is called a I lot ropy. Sulfur occurs in a low-temperature rhombic form and a high-temperature monoclinic form. The phase diagram for the system is shown in Fig. 5.3. The pressure scale in this diagram has been made logarithmic in order to bring the interesting low-pressure regions into prominence. The curve AB is the vapor-pressure curve of solid rhombic sulfur. At point B it intersects the vapor-pressure curve of monoclinic sulfur BE, and also the transformation curve for rhombic-monoclinic sulfur, BD. This inter- section determines the triple point B, at which rhombic and monoclinic sulfur and sulfur vapor coexist. Since there are three phases and one component, f= c p-^-2 3 3 0, and point B is an invariant point. It occurs at 0.01 mm pressure and 95.5C. The density of monoclinic sulfur is less than that of rhombic sulfur, and therefore the transition temperature (S r ^ S m ) increases with increasing pressure. Monoclinic sulfur melts under its own vapor pressure of 0.025 mm at 120C, the point E on the diagram. From E to the critical point F there extends the vapor-pressure curve of liquid sulfur EF. Also from , there extends the curve ED, the melting-point curve of monoclinic sulfur. The density of liquid sulfur is less than that of the monoclinic solid, the usual situation in a solid-liquid transformation, and hence ED slopes to the right as shown. The point E is a triple point, S m -S liq -S vap . The slope of BD is greater than that of ED, so that these curves intersect 110 CHANGES OF STATE [Chap. 5 at Z), forming a third triple point on the diagram, S f -S m -S llq . This occurs at 155 and 1290atm. At pressures higher than this, rhombic sulfur is again the stable solid form, and DG is the melting-point curve of rhombic sulfur in this high-pressure region. The range of stable existence of monoclinic sulfur is confined to the totally enclosed area BED. Besides the stable equilibria represented by the solid lines, a number of metastable equilibria are easily observed. If rhombic sulfur is heated quite rapidly, it will pass by the transition point B without change and finally melt 10' IO 10 10* RHOMBIC 80 90 100 110 120 130 140 150 160 TEMPERATURE Fig. 5.3. The sulfur system. to liquid sulfur at 1 14C (point //). The curve BH is the metastable vapor- pressure curve of rhombic sulfur, and the curve EH is the metastable vapor pressure curve of supercooled liquid sulfur. Extending from H to D is the metastable rhombic melting-point curve. Point H is a metastable triple point, S r -S liq -S vap . All these metastable equilibria are quite easily studied because of the extreme sluggishness that characterizes the rate of attainment of equilibrium between solid phases. In this discussion of the sulfur system, the well-known equilibrium between SA and S /4 in liquid sulfur has not been taken into consideration. If this occurrence of two distinct forms of liquid sulfur is considered, the sulfur Sec. 11] CHANGES OF STATE 111 system can no longer be treated as a simple one-component system, but becomes a "pseudobinary" system. 7 11. Enantiotropism and monotropism. The transformation of monoclinic to rhombic sulfur under equilibrium conditions of temperature and pressure is perfectly reversible. This fact is, of course, familiar, since the transforma- tion curve represents a set of stable equilibrium conditions. Such a change between two solid forms, occurring in a region of the phase diagram where both are stable, is called an enantiotropic change. On the other hand, there are cases in which the transformation of one solid form to another is irreversible. The classical example occurs in the VAPOR ENANTIOTROPISM VAPOR MONOTROPISM T Fig. 5.4. Enantiotropic and monotropic changes. phosphorus system, in the relations between white (cubic) phosphorus and violet (hexagonal) phosphorus. When white phosphorus is heated, trans- formation into violet phosphorus occurs at an appreciable rate at tem- peratures above 260; but solid violet phosphorus is never observed to change into solid white phosphorus under any conditions. In order to obtain white phosphorus, it is necessary to vaporize the violet variety, whereupon the vapor condenses to white phosphorus. Such an irreversible solid-state transformation is called a monotropic change. It may be characterized by saying that one form is metastable with respect to the other at all temperatures up to its melting point. The situa- tion is shown schematically in Fig. 5.4. The transition point (metastable) between the two solid forms in this case lies above the melting point of either form. 7 If SA and S^ came to equilibrium quickly when the T or P of the liquid was changed, the sulfur system would still have only one component (unary system) as explained in foot- note 3. If SA and S^ were present in fixed proportions, which did not change with rand P, because the time of transformation was very long compared with the time of the experiment, the sulfur system would have two components (binary system). In fact it appears that the time of transformation is roughly comparable with the time of most experiments, so that the observed behavior is partly unary and partly binary, being called "pseudobinary." 112 CHANGES OF STATE [Chap. 5 Actually, the phosphorus case is complicated by the occurrence of several molecular species, P 2 , P 4 , P % , and so on, so that considerations based on a one-component system must be applied with caution. 12. Second-order transitions. The usual change of state (solid to liquid, liquid to vapor, etc.) is called & first-order transition. At the transition tem- perature T t at constant pressure, the free energies of the two forms are equal, but there is a discontinuous change in the slope of the F vs. T curve for the substance at T t . Since (3F/3T) S, there is therefore a break in the S vs. T curve, the value of AS at T t being related to the observed latent heat for the transition by AS = XjT t . There is also a discontinuous change in volume AF, since the densities of the two forms are not the same. A number of transitions have been studied in which no latent heat or density change can be detected. Examples are the transformation of certain metals from ferromagnetic to paramagnetic solids at their Curie points, the transition of some metals at low temperatures to a condition of electric superconductivity, and the transition observed in helium from one liquid form to another. 8 In these cases, there is a change in slope, but no dis- continuity, in the S vs. T curve at T t . As a result, there is a break AC P in the heat capacity curve, since C p = T(dS/dT) r . Such a change is called a second-order transition. 13. High-pressure studies. It is only a truism that our attitude toward the physical world is conditioned by the scale of magnitudes provided in our terrestrial environment. We tend, for example, to classify pressures or temperatures as high or low by comparing them with the fifteen pounds per square inch and 70F of a spring day in the laboratory, despite the fact that almost all the matter in the universe exists under conditions very different from these. Thus, even at the center of the earth, by no means a large astro- nomical body, the pressure is around 1,200,000 atm, and substances at this pressure would have properties quite unlike those to which we are accus- tomed. At the center of a comparatively small star, like our sun, the pressure would be around ten billion atmospheres. The pioneer work of Gustav Tammann on high-pressure measurements has been greatly extended over the past twenty years by P. W. Bridgman and his associates at Harvard. Pressures up to 400,000 atm have been achieved and methods have been developed for measuring the properties of substances at 100,000 atm. 9 The attainment of such pressures has been made possible by the con- struction of pressure vessels of alloys such as Carboloy, and by the use of a multiple-chamber technique. The container for the substance to be studied is enclosed in another vessel, and pressure is applied both inside and outside the inner container, usually by means of hydraulic presses. Thus although 8 W. H. Keesom, Helium (Amsterdam: Elsevier, 1942). * For details see P. W. Bridgman, The Physics of High Pressures (London: Bell & Co., 1949), and his review article, Rev. Mod. Phys., 18, 1 (1946). Sec. 13] CHANGES OF STATE 113 the absolute pressure in the inner vessel may be 100,000 atm, the pressure differential that its walls must sustain is only 50,000 atm. High-pressure measurements on water yielded some of the most inter- esting results, which are shown in the phase diagram of Fig. 5.5. The melting point of ordinary ice (ice I) falls on compression, until a value of 22.0C is reached at 2040 atm. Further increase in pressure results in the transforma- tion of ice I into a new modification, ice III, whose melting point increases 9000 8000- 7000 - 6000 - 5000 - 1 4000 - 3000- 2000 - 1000 - -20 20 TEMPERATURE -*C Fig. 5.5. Water system at high pressures. with pressure. Altogether six different polymorphic forms of ice have been found. There are six triple points shown on the water diagram. Ice VII is an extreme high-pressure form not shown on the diagram; at a pressure of around 20,000 atm, liquid water freezes to ice VII at about 100C. Ice IV is not shown. Its existence was indicated by the work of Tammann, but it was not confirmed by Bridgman. PROBLEMS 1. From the following data, roughly sketch the phase diagram for carbon dioxide: critical point at 31C and 73 atm; triple point (solid-liquid-vapor) at 57 and 5.3 atm; solid is denser than liquid at the triple point. Label all regions on the diagram. 2. Roughly sketch the phase diagram of acetic acid, from the data: (a) The low-pressure a form melts at 16.6C under its own vapor pressure of 9. 1 mm. 114 CHANGES OF STATE [Chap. 5 (b) There is a high-pressure /? form that is denser than the a, but both a and /? are denser than the liquid. (c) The normal boiling point of liquid is 1 18C. (d) Phases a, /?, liquid are in equilibrium at 55C and 2000 atm. 3. Sketch the liquid-solid regions of the phase diagram of urethane. There are three solid forms, a, /?, y. The triple points and the volume changes AF in cc per kg at the triple points are as follows: (a) liq, a, /? P - 2270 atm / - 66C AK (I - a) = 25.3 (I - ft - 35.5 ( a _ ft ^ 10.2 (b) liq, ft y P - 4090 atm / - 77C A V: (I - ft) = 18.4 (1 - y) - 64.0 0? - y) - 45.6 (c) a, /?, y - P = 3290 atm / - 25.5C A K: (a - ft - 9.2 (/? - y) - 48.2 (a - y) - 57.4 4. The density /> of ice at 1 atm and 0C is 0.917 g per cc. Water under the same conditions has p -=-- l.OOOg per cc. Estimate the melting point of ice under a pressure of 400 atm assuming that p for both ice and water is practically constant over the temperature and pressure range. 5. Bridgman found the following melting points / (C) and volume changes on melting AK(cc per g) for Na: P, kg/cm 2 . 1 2000 4000 6000 / . 97.6 114.2 129.8 142.5 AK . . 0.0279 0.0236 0.0207 0.0187 Estimate the heat of fusion of sodium at 3000 atm. 6. Estimate the vapor pressure of mercury at 25C assuming that the liquid obeys Trouton's rule. The normal boiling point is 356.9C. 7. The vapor pressure of solid iodine is 0.25 mm and its density 4.93 at 20C. Assuming the Gibbs equation to hold, calculate the vapor pressure of iodine under a 1000-atm argon pressure. 8. In a determination of the vapor pressure of ethyl acetate by the gas saturation method 100 liters of nitrogen (STP) were passed through a saturator containing ethyl acetate at 0C, which lost a weight of 12.8g. Calculate vapor pressure at 0C. 9. The vapor pressures of liquid gallium are as follows: /, C . 1029 1154 1350 P, mm . . 0.01 0.1 1.0 Calculate A//, AF, and AS for the vaporization of gallium at 1154C. 10. At 25C, the heat of combustion of diamond is 94.484 kcal per mole and that of graphite is 94.030. The molar entropies are 0.5829 and 1.3609 cal per deg mole, respectively. Find the AFfor the transition graphite -> diamond at 25C and 1 atm. The densities are 3.513 g per cc for diamond and 2.260 Chap. 5] CHANGES OF STATE 115 for graphite. Estimate the pressure at which the two forms would be in equilibrium at 25C. You may assume the densities to be independent of pressure. 11. Sketch graphs of F, S, V, Q> against T at constant P, and P at constant T, for typical first- and second-order phase transitions. 12. From the data in Table 5.1, plot log P vs. T~ l for water and calculate the latent heats of vaporization of water at 20 and at 80C. REFERENCES BOOKS 1. Bridgman, P. W., The Physics of High Pressures (London: Bell, 1949). 2. Findlay, A., The Phase Rule (New York: Dover, 1945). 3. Marsh, J. S., Principles of Phase Diagrams (New York: McGraw-Hill, 1935). 4. Ricci, J. E., The Phase Rule and Heterogeneous Equilibrium (New York: Van Nostrand, 1951). 5. Tammann, G., The States of Aggregation (New York: Van Nostrand, 1925). 6. Wheeler, L. P.,Josiah WillardGibbs(New Haven: Yale Univ. Press, 1953). ARTICLES 1. Bridgman, P. W., Science in Progress, vol. Ill, 108-46 (New Haven: Yale Univ. Press, 1942), "Recent Work in the Field of High Pressures." 2. Garner, W. E.,/. Chem. Soc., 1961-1973 (1952), "The Tammann Memor- ial Lecture." 3. Staveley, L. A. K., Quart. Rev., 3, 65-81 (1949), "Transitions in Solids and Liquids." 4. Swietoslawski, W., J. Chem. Ed., 23, 183-85 (1946), "Phase Rule and the Action of Gravity." 5. Ubbelohde, A. R., Quart. Rev., 4, 356-81 (1950), "Melting and Crystal Structure." CHAPTER 6 Solutions and Phase Equilibria 1. The description of solutions. As soon as systems of two or more com- ponents are studied, the properties of solutions must be considered, for a solution is by definition any phase containing more than one component. This phase may be gaseous, liquid, or solid. Gases are in general miscible in all proportions, so that all mixtures of gases, at equilibrium, are solutions. Liquids often dissolve a wide variety of gases, solids, or other liquids, and the composition of these liquid solutions can be varied over a wide or narrow range depending on the particular solubility relationships in the individual system. Solid solutions are formed when a gas, a liquid, or another solid dissolves in a solid. They are often characterized by very limited concentra- tion ranges, although pairs of solids are known, for example copper and nickel, that are mutually soluble in all proportions. It is often convenient in discussing solutions to call some components the solvents and others the solutes. It should be recognized, however, that the only distinction between solute and solvent is a verbal one, although the solvent is usually taken to be the constituent present in excess. The concentration relations in solutions are expressed in a variety of units. The more important of these are summarized in Table 6.1. TABLE 6.1 CONCENTRATION OF SOLUTIONS Name Symbol Definition Molar Molai Volume molal Weight per cent Mole fraction c m m f % X Moles of solute in 1 liter solution Moles of solute in 1000 g solvent Moles of solute in 1 liter solvent Grams of solute in 100 g solution Moles of solute divided by total number of moles of all components 2. Partial molar quantities: partial molar volume. The equilibrium prop- erties of solutions are described in terms of the thermodynamic state func- tions, such as P, T, K, ", 5, F, //. One of the most important problems in the theory of solutions is how these properties depend on the concentrations of the various components. In discussing this question, it will be assumed that the solution is kept at constant over-all pressure and temperature. Consider a solution containing n A moles of A and n B moles of B. Let the volume of the solution be K, and assume that this volume is so large that Sec. 2] SOLUTIONS AND PHASE EQUILIBRIA 117 the addition of one extra mole of A or of B does not change the concentration of the solution to an appreciable extent. The change in volume caused by adding one mole of A to this large amount of solution is then called the partial molar volume of A in the solution at the specified pressure, tempera- ture, and concentration, and is denoted by the symbol V A . It is the change of volume K, with moles of A, n A , at constant temperature, pressure; and moles of B, and is therefore written as One reason for introducing such a function is that the volume of a solution is not, in general, simply the sum of the volumes of the individual components. For example, if 100ml of alcohol are mixed at 25C with 100 ml of water, the volume of the solution is not 200 ml, but about 190 ml. The volume change on mixing depends on the relative amount of each component in the solution. \fdn A moles of A and dn B moles of B are added to a solution, the increase in volume at constant temperature and pressure is given by the complete differential, -(")*, + () *. A'* M WB'* A or dV --= V A dn A + P yy dn tt (6.2) This expression can be integrated, which corresponds physically to increasing the volume of the solution without changing its composition, V A and V n hence being held constant. 1 The result is V = V A n A + V B n B (6.3) This equation tells us that the volume of the solution equals the number of moles of A times the partial molar volume of A, plus the number of moles of B times the partial molar volume of B. On differentiation, eq. (6.3) yields <*V= VA dn A + n A d? A + V B dn B + n B dV B By comparison with eq. (6.2), it follows that or dV A - - dV B (6.4) n A Equation (6.4) is one example of the Gibbs-Duhem equation. This par- ticular application is in terms of the partial molar volumes, but any other 1 Mathematically, the integration is equivalent to the application of Euler's theorem to the homogeneous differential expression. See D. V. Widder, Advanced Calculus (New York: Prentice-Hall, 1947), p. 15. 118 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 partial molar quantity may be substituted for the volume. These partial molar quantities can be defined for any extensive state function. For example: Sl . a . WA'T,I',H. ' ' n The partial molar quantities are themselves intensity factors, since they are capacity factors per mole. The partial molar free energy is the chemical potential /i. All the thermodynamic relations derived in earlier chapters can be applied to the partial molar quantities. For example: /v ' (6 ' 5) The general thermodynamic theory of solutions is expressed in terms of these partial molar functions and their derivatives just as the theory for pure substances is based on the ordinary thermodynamic functions. 3. The determination of partial molar quantities. The evaluation of the partial quantities will now be described, using the partial molar volume as an example. The methods for ff A , S A , F A9 and so on, are exactly similar. The partial molar volume V A , defined by eq. (6.1), is equal to the slope of the curve obtained when the volume of the solution is plotted against the molal concentration of A. This follows since the molal concentration m A is the number of moles of A in a constant quantity, namely 1000 grams, of solvent B. The determination of partial molar volumes by this slope method is rather inaccurate; the method of intercepts is therefore usually preferred. To employ this method, a quantity is defined, called the molar volume of the solution v, which is the volume of the solution divided by the total number of moles of the various constituents. For a two-component solution: "A + "B Then, Y = v (n A + n B ) and P A = Now the derivative with respect to mole number of A, n A , is transformed into a derivative with respect to mole fraction of B, X B . dv sinc e XK = ^> 1^1 =- JL ("A + ' Sec. 3] SOLUTIONS AND PHASE EQUILIBRIA 119 Thus eq. (6.6) becomes : V A = v V n B dv n A + n s dX B x dv A B Ti7~ (6.7) The application of this equation is illustrated in Fig. 6.1, where v for a solution is plotted against the mole fraction. The slope S^ is drawn tangent to the curve at point P, corresponding to a definite mole fraction X B ' . The line ^iA 2 is drawn through P parallel to O^O 2 . Therefore the distance Si 2 XB i MOLE FRACTION OF B-X B Fig. 6.1. Determination of partial molar volumes intercept method. O l A l v, the molar volume corresponding to X B . The distance S l A l is equal to the slope at X B multiplied by X B , i.e., to the term in eq. (6.7), X n (dv/dX B ). It follows that O^ = O l A l S l A l equals V A , the partial molar volume of A in the solution. It can readily be shown that the intercept on the other axis, O 2 5 2 , is the partial molar volume of B, P B . This con- venient intercept method is the one usually used to determine partial molar quantities. It is not restricted to volumes, but can be applied to any extensive state function, 5, H, E, F 9 and so on, given the necessary data. It can also be applied to heats of solution, and the partial molar heats of solution so obtained are the same as the differential heats described in Chapter 2. If the variation with concentration of a partial molar quantity is known for one component in a binary solution, the Gibbs-Duhem equation (6.4) permits the calculation of the variation for the other component. This cal- culation can be accomplished by graphical integration of eq. (6.4). For example: 120 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 where X is the mole fraction. If X B \X A is plotted against V n , the area under the curve gives the change in V A between the upper and lower limits of integration. The V A of pure A is simply the molar volume of pure A, and this can be used as the starting point for the evaluation of V A at any other concentration. 4. The ideal solution Raoult's Law. The concept of the "ideal gas" has played a most important role in discussions of the thermodynamics of gases and vapors. Many cases of practical interest are treated adequately by means of the ideal gas approximations, and even systems deviating largely from ideality are conveniently referred to the norm of behavior set by the ideal case. It would be most helpful to find some similar concept to act as a guide in the theory of solutions, and fortunately this is indeed possible. Because they are very much more condensed than gases, liquid or solid solutions cannot be expected to behave ideally in the sense of obeying an equation of state such as the ideal gas law. Ideality in a gas implies a complete absence of cohesive forces; the internal pressure, (3E/c>V) T 0. Ideality in a solution is defined by complete uniformity of cohesive forces. If there are two com- ponents A and B, the forces between A and A, B and /?, and A and B are all the same. A property of great importance in the discussion of solutions is the vapor pressure of a component above the solution. This partial vapor pressure may be taken as a good measure of the escaping tendency of the given species from the solution. The exact measure of this escaping tendency is the fugacity, which becomes equal to the partial pressure when the vapor behaves as an ideal gas. The tendency of a component to escape from solution into the vapor phase is a very direct reflection of the physical state of affairs within the solution, 2 so that by studying the escaping tendencies, or partial vapor pressures, as functions of temperature, pressure, and concentration, we obtain a description of the properties of the solution. This method is a direct consequence of the relation between chemical potential and fugacity. If we have a solution, say of A and J9, the chemical potential of A in the solution must be equal to the chemical potential of A in the vapor phase. This is related to the fugacity by eq. (4.31), since If we know the pressure, the temperature, and the chemical potentials of the various components, we then have a complete thermodynamic descrip- tion of a system, except for the absolute amounts of the various phases. The partial vapor pressures are important because they are an approximate indication of the chemical potentials. A solution is said to be ideal if the escaping tendency of each component 2 One may think of an analogy in which a nation represents a solution and its citizens the molecules. If life in the nation is a good one, the tendency to emigrate will be low. This presupposes, of course, the absence of artificial barriers. Sec. 4] SOLUTIONS AND PHASE EQUILIBRIA 121 is proportional to the mole fraction of that component in the solution. It is helpful to look at this concept from a molecular point of view. Consider an ideal solution of A and B. The definition of ideality then implies that a molecule of A in the solution will have the same tendency to escape into the vapor whether it is surrounded entirely by other A molecules, entirely by B molecules, or partly by A and partly by B molecules. This means that the intermolecular forces between A and A, A and B, and B and B, are all essentially the same. It is immaterial to the behavior of a molecule what sort of neighbors it has. The escaping tendency of component A from such 200 180 160 0140 E 6120 a: 60 <40 20 TOTAL VAPOR *-- ^PRESSURE . A C 2 H 4 Br 2 .2 3 .4 .5 .6 .7 .8 .9 1.0 MOLE FRACTION OF C 3 H 6 Br 2 B C 3 H 6 Br 2 Fig. 6.2. Pressures of vapors above solutions of ethylene bromide and propylene bromide at 85C. The solutions follow Raoult's Law. an ideal solution, as measured by its partial vapor pressure, is accordingly the same as that from pure liquid A, except that it is proportionately reduced on account of the lowered fraction of A molecules in the solution. This law of behavior for the ideal solution was first given by Francois Marie Raoult in 1886, being based on experimental vapor-pressure data. It can be expressed as PA = *A PA (6-9) Here P A is the partial vapor pressure of A above a solution in which its mole fraction is X A , and P A is the vapor pressure of pure liquid A at the same temperature. If the component B added to pure A lowers the vapor pressure, eq. (6.9) can be written in terms of a relative vapor pressure lowering, (6.10) 1 22 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 This form of the equation is especially useful for solutions of a relatively involatile solute in a volatile solvent. The vapor pressures of the system ethylene bromide propylene bromide are plotted in Fig. 6.2. The experimental results almost coincide with the theoretical curves predicted by eq. (6.9). The agreement with Raoult's Law in this instance is very close. Only in exceptional cases are solutions found that follow Raoult's Law closely over an extended range of concentrations. This is because ideality in solutions implies a complete similarity of interaction between the com- ponents, which can rarely be achieved. This equality of interaction leads to two thermodynamic conditions: (1) there can be no heat of solution; (2) there can be no volume change on mixing. Hence, AF 80lution - and A// 80hltion - 0. 5. Equilibria in ideal solutions. If we wish to avoid the assumption that the saturated vapor above a solution behaves as an ideal gas, Raoult's Law may be written fA~X A ft (6.11) where / 4 and/jj' are the fugacities of A in the solution, and in pure A. It is evident from eq. (6.8) that dp -- RTd\nf A RTd\nX A (6.12) Then, following the sort of development given in Section 4-5, one obtains for the equilibrium constant in an ideal solution -AF --= RT\r\K x with K <'-'r* (6 ' 13) A A A B for the typical case. 6. Henry's Law. Consider a solution of component B, which may be called the solute, in A 9 the solvent. If the solution is sufficiently diluted, a condition ultimately is attained in which each molecule of B is effectively completely surrounded by component A. The solute B is then in a uniform environment irrespective of the fact that A and B may form solutions that are far from ideal at higher concentrations. In such a very dilute solution, the escaping tendency of B from its uniform environment is proportional to its mole fraction, but the proportionality constant k no longer is P#. We may write PB - kX B (6.14) This equation was established and extensively tested by William Henry in 1803 in a series of measurements of the pressure dependence of the solu- bility of gases in liquids. Some results of this type are collected in Table 6.2. The fc's are almost constajit, so that Henry's Law is nearly but not exactly obeyed. Sec. 7] SOLUTIONS AND PHASE EQUILIBRIA 123 TABLE 6.2 THE SOLUBILITY OF GASES IN WATER (ILLUSTRATING HENRY'S LAW P B = kX B , THE GAS PRESSURE BEING IN ATM AND X B BEING THE MOLE FRACTION) Partial Pressure Henry's Law Constant (k x 10~ 4 ) * (atm) N 2 at 19.4 O 2 at 23 H 2 at 23 1 1.18 8.24 4.58 2.63 8.32 4.59 7.76 3.95 8.41 4.60 7.77 5.26 8.49 4.68 7.81 6.58 8.59 4.73 7.89 7.90 8.74 4.80 8.00 9.20 8.86 4.88 8.16 As an example, let us calculate the volume of oxygen (at STP) dissolved in 1 liter of water in equilibrium with air at 23. From eq. (6.14) the mole fraction of O 2 is X n = P#/k. Since P B = 0.20, and from the table k = 4.58 x 10 4 , X n - 4.36 x 10~ 6 . In 1 liter of H 2 O there are 1000/18 = 55.6 moles. Thus X R n n l(n u + 55.6), or n B = 2 A3 x 10~ 4 . This number of moles of oxygen equals 5.45 cc at STP. Henry's Law is not restricted to gas-liquid systems, but is followed by a wide variety of fairly dilute solutions and by all solutions in the limit of extreme dilution. 7. Two-component systems. For systems of two components the phase rule, /= c p + 2, becomes /= 4 p. The following cases are possible: p ^= [ , / ^ 3 trivariant system p 2, / 2 bivariant system p 3, f= 1 univariant system p = 4, f ~ invariant system The maximum number of degrees of freedom is three. A complete graphical representation of a two-component system therefore requires a three-dimensional diagram, with coordinates corresponding to pressure, temperature, and composition. Since a three-dimensional representation is usually inconvenient, one variable is held constant while the behavior of the other two is plotted. In this way, plane graphs are obtained showing pressure vs. composition at constant temperature, temperature vs. com- position at constant pressure, or pressure vs. temperature at constant composition. 8. Pressure-composition diagrams. The example of a (P-X) diagram in Fig. 6.3a shows the system ethylene bromide-propylene bromide, which obeys Raoiilt's Law quite closely over the entire range of compositions. The straight upper line represents the dependence of the total vapor pressure 124 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 above the solution on the mole fraction in the liquid. The curved lower line represents the dependence of the pressure on the composition of the vapor. Consider a liquid of composition X 2 at a pressure P 2 (point C on the diagram). This point lies in a one-phase region, so there are three degrees of freedom. One of these degrees is used by the requirement of constant tem- perature for the diagram. Thus for any arbitrary composition X 2 , the liquid solution at constant T can exist over a range of different pressures. As the pressure is decreased along the dotted line of constant com- position, nothing happens until the liquidus curve is reached at B. At this point liquid begins to vaporize. The vapor that is formed is richer than the 140 138 P. 136 134 132 VAPOR .1 .2 3 .4 .5 .6 .7 .8 .9 1.0 A *B~" B C 2 H 4 Br 2 Fig. 6.3a. Pressure-composition (mole 01 2345. 6 789 10 Fig. 6.3b. Temperature-composition fraction) diagram for system obeying diagram for system obeying Raoult's Raoult's Law. Law. liquid in the more volatile component, ethylene bromide. The composition of the first vapor to appear is given by the point A on the vapor curve. As the pressure is further reduced below B, a two-phase region on the diagram is entered. This represents the region of stable coexistence of liquid and vapor. The dotted line passing horizontally through a typical point D in the two-phase region is called a tie line', it connects the liquid and vapor compositions that are in equilibrium. The over-all composition of the system is X 2 . This is made up of liquid having a composition X { and vapor having a composition X 3 . The relative amounts of the liquid and vapor phases required to yield the over-all composition are given by the lever rule: if (/) is the fraction of liquid and (v) the fraction 3 of vapor, (/)/(r) - (Jjf 3 - X Z )/(X 2 - A\). This rule 3 Since a mole fraction diagram is being used, (v) is the fraction of the total number of moles that is vapor. On a weight fraction diagram, (v} would be the weight fraction that is vapor. Sec. 9] SOLUTIONS AND PHASE EQUILIBRIA 125 is readily proved: It is evident that X 2 -= (l)X l I [1 ~ (I)]X& or (7)^ (X 2 - *s)/(*i- *s)- Similarly (v) - 1 - (/) (X,- X 2 )/(X l - JIT,). Hence (/)/() =(*3 - *,)/(* a - A\), the lever rule. As the pressure is still further decreased along BF more and more liquid is vaporized till finally, at F, no liquid remains. Further decrease in pressure then proceeds in the one-phase, all-vapor region. In the two-phase region, the system is bi variant. One of the degrees of freedom is used by the requirement of constant temperature; only one remains. When the pressure is fixed in this region, therefore, the compositions of both the liquid and the vapor phases are also definitely fixed. They are given, as has been seen, by the end points of the tie line. 9. Temperature-composition diagrams. The temperature-composition dia- gram of the liquid-vapor equilibrium is the boiling-point diagram of the solutions at the constant pressure chosen. If the pressure is one atmosphere, the boiling points are the normal ones. The boiling-point diagram for a solution in which the solvent obeys Raoult's Law can be calculated if the vapor pressures of the pure com- ponents are known as functions of temperature (Fig. 6.3b). The two end points of the boiling-point diagram shown in Fig. 6.3b are the temperatures at which the pure components have a vapor pressure of 760 mm, viz., 131. 5C and 141. 6C. The composition of the solution that boils anywhere between these two temperatures, say at 135C, is found as follows: According to Raoult's Law, letting X A be the mole fraction of C 2 H 4 Br 2 , 760 -= P A X A + P K (\ - X A ). At 135, the vapor pressure of C 2 H 4 Br 2 is 835mm, of C 3 H 6 Br 2 , 652mm. Thus, 760 - 835 X A -f 652(1 - X A \ or X A 0.590, X n ^ 0.410. This gives one intermediate point on the liquidus curve; the others are calculated in the same way. The composition of the vapor is given by Dalton's Law: The vapor-composition curve is therefore readily constructed from the liquidus curve. 10. Fractional distillation. The application of the boiling-point diagram to a simplified representation of distillation is shown in Fig. 6.3b* The solution of composition X begins to boil at temperature T v The first vapor that is formed has a composition Y, richer in the more volatile component. If this is condensed and reboiled, vapor of composition Z is obtained. This process is repeated until the distillate is composed of pure component A. In practical cases, the successive fractions will each cover a range of com- positions, but the vertical lines in Fig. 6.3b, may be considered to represent average compositions within these ranges. 126 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 _ CONDENSER LIQUID FEED A fractionating column is a device that carries out automatically the successive condensations and vaporizations required for fractional distilla- tion. An especially clear example of this is the "bubble-cap" type of column in Fig. 6.4. As the vapor ascends from the boiler, it bubbles through a film of liquid on the first plate. This liquid is somewhat cooler than that in the boiler, so that a partial condensation takes place. The vapor that leaves the first plate is therefore richer than the vapor from the boiler in the more volatile component. A similar enrich- ment takes place on each succeeding plate. Each attainment of equilibrium between liquid and vapor corresponds to one of the steps in Fig. 6.3b. The efficiency of a distilling column is measured by the number of such equilibrium stages that it achieves. Each such stage is called a theoretical plate. In a well designed bubble-cap column, each unit acts very nearly as one theoretical plate. The performance of various types of packed columns is also described in terms of theoretical plates. The separation of liquids whose boiling points lie close together requires a column with a con- siderable number of theoretical plates. The number actually required de- pends on the cut that is taken from the head of the column, the ratio of distillate taken off to that returned to the column. 4 11. Boiling-point elevation. If a small amount of a nonvolatile solute is dissolved in a volatile solvent, the solution being sufficiently dilute to behave ideally, the lowering of the vapor pressure can be calculated from eq. (6.10). As a consequence of the lowered vapor pressure, the boiling point of the solution is higher than that of the pure solvent. This fact is evident on inspection of the vapor pressure curves in Fig. 6.5. The condition for equilibrium of a component A, the volatile solvent, between the liquid and vapor phases is simply ju, A v = [i A l . From eq. (6.12), t*A l PA + RTln X A , where fi A is the chemical potential of pure liquid A, i.e., fJL A when X A 1. At the boiling point the pressure is 1 atm, so that /// = iff* th e chemical potential of pure A vapor at 1 atm. Therefore (l*A 9 = PA I ) becomes JU A = p% + RT\n X A . For the pure component A 9 the chemical potentials // are identical with the molar free energies F. Hence, BOILER Fig. 6.4. Schematic draw- ing of bubble-cap column. 4 For details of methods for determining the number of theoretical plates in a column, see C. S. Robinson and E. R. Gilliland, Fractional Distillation (New York: McGraw-Hill, Sec. 11] SOLUTIONS AND PHASE EQUILIBRIA 127 From eq. (3.36), 5(F/T)/3T = ~H/T\ so that differentiating the above yields H? - H? = - Since H^ H% is the molar heat of vaporization, RT 2 I ATM SOLUTION - PURE SOLID PURE LIQUID - PURE SOLID rwrt <JWL.II/ i rwr\u s>v*>i_iu EQUILIBRIUM^/ f EQUILIBRIUM / L f_ + _____ / ___ / ^ ,^/, ^<?/ '</:*^- ^:<y |m wo FR DER B.P EL. Fig. 6.5. Diagram showing the elevation of the boiling point caused by addition of a nonvolatile solute to a pure liquid. Taking A as constant over the temperature range, this equation is integrated between the limits set by the pure solvent (X A = 1, T = 7^) and the solution R r TT Q When the boiling-point elevation is not large, TT Q can be replaced by T 2 . If X B is the mole fraction of solute, the term on the left can be written In (I X B \ and then expanded in a power series. Writing A7^ for the boiling-point elevation, T jT , we obtain RTJ i X 128 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 When the solution is dilute, X B is a small fraction whose higher powers may be neglected. Then, RT 2 AT^y-*-** (6.15) ^vap In the dilute solutions for which eq. (6.15) is valid, it is also a good approximation to replace X B by (W B M A )/(W A M S ); W B , M B> and W Ay M A being the masses and molecular weights of solute and solvent. Then, - _ B *vap W A M B ' / vap W A M B where / vap is the latent heat of vaporization per gram. Finally W B \W A M B is set equal to w/1000, m being the weight molal concentration, moles of solute per 1000 grams of solvent. Thus, and K B is called the molal boiling-point elevation constant. For example, for water T Q = 373.2, / vap 538 cal per g. Hence (1.986)(373.2)* KB = 1538X1000T = ' 5 4 For benzene, K B - 2.67; for acetone, 1.67, etc. The expression (6.16) is used frequently for molecular-weight determination from the boiling-point elevation. From K B and the measured T B , we calculate m, and then the molecular weight from M B 1000 W B jmW A . For many combinations of solute and solvent, perfectly normal molecular weights are obtained. In certain instances, however, there is apparently an association or dissociation of the solute molecules in the solution. For example, the molec- ular weight of benzoic acid in acetone solution is found to be equal to the formula weight of 122.1. In 1 per cent solution in benzene, benzoic acid has an apparent molecular weight of 242. This indicates that the acid is to a considerable extent dimerized into double molecules. The extent of associa- tion is greater in more concentrated solutions, as is required by the Le Chatelier principle. From molecular-weight determinations at different concentra- tions, it is possible to calculate the equilibrium constant of the reaction (C 6 H 5 COOH) 2 = 2 C 6 H 5 COOH. 12. Solid and liquid phases in equilibrium. The properties of solutions related to the vapor pressure are called colligatwe from the Latin, colligatus, collected together. They are properties which depend on the collection of particles present, that is, on the number of particles, rather than on the kind. A colligative property amenable to the same sort of treatment as the boiling- point elevation is the depression of the freezing point. That this also has its origin in the lowering of the vapor pressure in solutions can be seen by Sec. 12] SOLUTIONS AND PHASE EQUILIBRIA 129 inspection of Fig. 6.5. The freezing point of pure solvent, T mo is lowered to T m in the solution. It should be understood that "freezing-point depression curve" and "solubility curve" are merely two different names for the same thing that is, a temperature vs. composition curve for a solid-liquid equilibrium at some constant pressure, usually chosen as one atmosphere. Such a diagram is shown in Fig. 6.13 (p. 147) for the system benzene-chloroform. The curve CE may be considered to illustrate either (1) the depression of the freezing point of benzene by the addition of chloroform, or (2) the solubility of solid benzene in the solution. Both interpretations are fundamentally equivalent: in one case, we consider Tas a function of c; in the other, c as a function of T. The lowest point E on the solid-liquid diagram is called the eutectic point (evT'^KTO?, "easily melted"). In this diagram, the solid phases that separate out are shown as pure benzene (A) on one side and pure chloroform (B) on the other. It becomes evident in the next section that this is not exactly correct, since there is usually at least a slight solid solution of the second component B in the solid component A. Nevertheless the absence of any solid solution is in many cases a good enough approximation. The equation for the freezing-point depression, or the solubility equation for ideal solutions, is derived by essentially the same method used for the boiling-point elevation. In order for a pure solid A to be in equilibrium with a solution containing A, it is necessary that the chemical potentials of A be the same in the two phases, JU A * = fi A l . From eq. (6.12) the chemical potential of component A in an ideal solution is JU A I = fi A -f RT\t\ X A , where p, A is the chemical potential of pure liquid A. Thus the equilibrium condition can be written H A * = fi A f RTln X A . Now jti A 8 and fj, A are simply the molar free energies of pure solid and pure liquid, hence Z^/--=l*XA (6.17) Since we have d(F/T)/3T = H/T* from eq. (3.36), differentiation of eq. (6.17) with respect to T yields 1 2 ^/l V (6.18) RT* RT* dT Integrating this expression from T Q9 the freezing point of pure A, mole fraction unity, to T, the temperature at which pure solid A is in equilibrium with solution of mole fraction X A , we obtain 5 ^fus 5 It is a good approximation to take A fu8 independent of T over moderate ranges of temperature. 130 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 This is the equation for the temperature variation of the solubility X A of a pure solid in an ideal solution. As an example, let us calculate the solubility of naphthalene in an ideal solution at 25C. Naphthalene melts at 80C, and its heat of fusion at the melting point is 4610 cal per mole. Thus, from eq. (6.19), 4610 (353.2- 1 - 298.2- 1 ) - 2.303 log X A 1.986 X A - 0.298 This is the mole fraction of naphthalene in any ideal solution, whatever the solvent may be. Actually, the solution will approach ideality only if the solvent is rather similar in chemical and physical properties to the solute. Typical experimental values for the solubility X A of naphthalene in various solvents at 25C are as follows: chlorobenzene, 0.317; benzene, 0.296; toluene, 0.286; acetone, 0.224; hexane, 0.125. The simplification of eq. (6.19) for dilute solutions follows from the same approximations used in the boiling-point elevation case. The final expression for the depression of the freezing point &T F -^ T T is RT K - For example: water, K F = 1.855; benzene, 5.12; camphor, 40.0, and so on. Because of its exceptionally large K F , camphor is used in a micro method for molecular-weight determination by freezing-point depression. 13. The Distribution Law. The equilibrium condition for a component A distributed between two phases a and ft is p A * /^/, From eq. (6.8), f A = //. If the solutions are ideal, Raoult's Law is followed, and f A = XA/A* where f A is the fugacity of pure A (equal, if the vapor is an ideal gas/ to the vapor pressure P A . Thus X A *f A = JT//J, or X A * - Xj, and as long as the solutions are ideal, the solute A must be distributed equally between them. If the solutions do not follow Raoult's Law, but are sufficiently dilute to follow Henry's Law,/^ k A X A , and it follows that y a k a JT, = F? - * < 6 - 21 ) A A K A The ratio of the Henry's Law constants, K D , is called the distribution constant (or distribution coefficient). Thus K D is a function of temperature and pressure. Equation (6.21) is one form of the Nernst Distribution Law* In a dilute solution, X A = n A j(n A + n B ) ^ n A jn u & c A M B /\QQQp Ii , where C A is the ordinary molar concentration and M B and p B are the mole- cular weight and density of the solvent. With this approximation, the ratio W. Nernst, Z. physik. Chem., 8, 110 (1891). Sec. 14] SOLUTIONS AND PHASE EQUILIBRIA 131 of mole fractions is proportional to the ratio of molar concentrations, and eq. (6.21) becomes ^4 = K D (6-22) CA P A test of the Law in this form, for the distribution of iodine between water and carbon bisulfide may be seen in Table 6.3. TABLE 6.3 DISTRIBUTION OF I 2 BETWEEN CS 2 AND H 2 O AT 1 8 c a g I 2 per liter CS 2 . 174 129 66 41 7.6 cP g I 2 per liter H 2 O 0.41 0.32 0.16 0.10 0.017 K D ' - c*/cP . 420 400 410 410 440 If association, dissociation, or chemical reaction of the distributed com- ponent takes place in either phase, modification of the Distribution Law is required. For example, if a solute S is partly dimerized to S 2 molecules in both phases, there will be two distribution equations, one for monomer and one for dimer, but the two distribution constants will not be independent, being related through the dissociation constants of the dimers. Solvent extraction is an important method for the isolation of pure organic compounds. Apparatus has been developed by L. C. Craig 7 at the Rockefeller Institute to carry out continuously hundreds of successive stages of extraction by the so-called "countercurrent distribution method." 14. Osmotic pressure. The classical trio of colligative properties, of which boiling-point elevation and freezing-point depression are the first two members, is completed by the phenomenon of osmotic pressure. In 1748, the Abbe Nollet described an experiment in which a solution of "spirits of wine" was placed in a cylinder, the mouth of which was closed with an animal bladder and immersed in pure water. The bladder was observed to swell greatly and sometimes even to burst. The animal membrane is semipermeable; water can pass through it, but alcohol cannot. The in- creased pressure in the tube, caused by diffusion of water into the solution, was called the osmotic pressure (from the Greek, coer/jos "impulse"). The first detailed quantitative study of osmotic pressure is found in a series of researches by W. Pfeffer, published in 1877. Ten years earlier, Moritz Traube had observed that colloidal films of cupric ferrocyanide acted as semipermeable membranes. PfefTer deposited this colloidal precipitate within the pores of earthenware pots, by soaking them first in copper sulfate and then in potassium ferrocyanide solution. Some typical results of measure- ments using such artificial membranes are summarized in Table 6.4. 7 L. C. Craig and D. Craig, "Extraction and Distribution," in Techniques of Organic Chemistry, ed. by A. Weissberger (New York: Interscience, 1950). 132 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 In 1885 J. H. van't Hoff pointed out that in dilute solutions the osmotic pressure FI obeyed the relationship II V = nRT, or II = cRT (6.23) where c = w/Kis the concentration of solute in moles per liter. The validity of the equation can be judged by comparison of the calculated and experi- mental values of II in Table 6.4. TABLE 6.4 OSMOTIC PRESSURES OF SOLUTIONS OF SUCROSE IN WATER AT 20 Molal Molar Observed Calculated Osmotic Pressure Concen- Concen- Osmotic tration tration Pressure (m) (c) (atm) Eq. (6.23) Eq. (6.27) Eq. (6.25) 0.1 0.098 2.59 2.36 2.40 2.44 0.2 0.192 5.06 4.63 4.81 5.46 0.3 0.282 7.61 6.80 7.21 7.82 0.4 0.370 10.14 8.90 9.62 10.22 0.5 0.453 12.75 10.9 12.0 12.62 0.6 0.533 15.39 12.8 14.4 15.00 0.7 0.610 18.13 14.7 16.8 17.40 0.8 0.685 20.91 16.5 19.2 19.77 0.9 0.757 23.72 18.2 21.6 22.15 1.0 0.825 26.64 19.8 24.0 24.48 The essential requirements for the existence of an osmotic pressure are two. There must be two solutions of different concentrations (or a pure solvent and a solution) and there must be a semipermeable membrane separating these solutions. A simple illustration can be found in the case of a gaseous solution of hydrogen and nitrogen. Thin palladium foil is appre- ciably permeable to hydrogen, but practically impermeable to nitrogen. If pure nitrogen is put on one side of a palladium barrier and a solution of nitrogen and hydrogen on the other side, the requirements for osmosis are satisfied. Hydrogen flows through the palladium from the hydrogen-rich to the hydrogen-poor side of the membrane. This flow continues until the chemical potential of the H 2 , /%, * s ^ e same on both sides of the barrier. In this example, the nature of the semipermeable membrane is rather clear. Hydrogen molecules are catalytically dissociated into hydrogen atoms at the palladium surface, and these atoms, perhaps in the form of protons and electrons, diffuse through the barrier. A solution mechanism of some kind probably is responsible for many cases of semipermeability. For example, it seems reasonable that protein membranes, like those employed by Nollet, can dissolve water but not alcohol. In other cases, the membrane may act as a sieve, or as a bundle of capil- laries. The cross sections of these capillaries may be very small, so that they Sec. 15] SOLUTIONS AND PHASE EQUILIBRIA 133 can be permeated by small molecules like water, but not by large molecules like carbohydrates or proteins. Irrespective of the mechanism by which the semipermeable membrane operates, the final result is the same. Osmotic flow continues until the chemical potential of the diffusing component is the same on both sides, of the barrier. If the flow takes place into a closed volume, the pressure therein necessarily increases. The final equilibrium osmotic pressure can be cal- culated by thermodynamic methods. It is the pressure that must be applied to the solution in order to prevent flow of solvent across the semipermeable membrane from the pure solvent into the solution. The same effect can be produced by applying a negative pressure or tension to the pure solvent. 15. Measurement of osmotic pressure. We are principally indebted to two groups of workers for precise measurements of osmotic pressure: H. N. Morse, J. C. W. Frazer, and their colleagues at Johns Hopkins, and the Earl of Berkeley and E. G. J. Hartley at Oxford. 8 CAPILLARY MANOMETER FOR PRESSURE MEASUREMENT SOLUTION =-_.- - POROUS CELL- IMPREGNATED WITH Cu 2 Fe(CN) 6 (o) APPLIED PRESSURE CAPILLARY FOR MEASURING FLOW THROUGH CELL* PRESSURE GAUGE SOLUTION' L WATER ^IMPREGNATED CELL (b) Fig. 6.6. Osmotic pressure measurements: (a) method of Frazer; (b) method of Berkeley and Hartley. The method used by the Hopkins group is shown in (a), Fig. 6.6. The porous cell impregnated with copper ferrocyanide is filled with water and immersed in a vessel containing the aqueous solution. The pressure is measured by means of an attached manometer. The system is allowed to stand until there is no further increase in pressure. Then the osmotic pres- sure is just balanced by the hydrostatic pressure in the column of solution. The pressures studied extended up to several hundred atmospheres, and a 8 An excellent detailed discussion of this work is to be found in J. C. W. Frazer's article, 'The Laws of Dilute Solutions' 1 in A Treatise on Physical Chemistry, 2nd ed., edited by H. S. Taylor (New York: Van Nostrand, 1931), pp. 353-414. 134 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 number of ingenious methods of measurement were developed. These included the calculation of the pressure from the change in the refractive index of water on compression, and the application of piezoelectric gauges. The English workers used the apparatus shown schematically in (b), Fig. 6.6. Instead of waiting for equilibrium to be established and then reading the pressure, they applied an external pressure to the solution just sufficient to balance the osmotic pressure. This balance could be made very precisely by observing the level of liquid in the capillary tube, which would fall rapidly if there was any flow of solvent into the solution. 16. Osmotic pressure and vapor pressure. Consider a pure solvent A that is separated from a solution of B in A by a membrane permeable to A alone. At equilibrium an osmotic pressure FI has developed. The condition for equilibrium is that the chemical potential of A is the same on both sides of the membrane, /if // /. Thus the }t A in the solution must equal that of the pure A. There are two factors tending to make the value of p A in the solution different from that in pure A. These factors must therefore have exactly equal and opposite effects on fi A . The first is the change in p A pro- duced by dilution of A in the solution. This change causes a lowering of p A equal to A/* =- RT\nP 4 /P A [eq. (6.8) with /=/>]. Exactly counteracting this is the increase in p A in the solution due to the imposed pressure II. From eq. (6.5) dp PdP, so that A/ J n V A dP. At equilibrium, therefore, in order that p A in solution should equal p A in the pure liquid, J* 1 V A dP -= -RT\n(P A /P A ). If it is assumed that the partial molar volume V A is independent of pressure, i.e., the solution is practically incompressible, PJT = firing (6.24) The significance of this equation can be stated as follows: the osmotic pressure is the external pressure that must be applied to the solution to raise the vapor pressure of solvent A to that of pure A. In most cases, also, the partial molar volume of solvent in solution V A can be well approximated by the molar volume of the pure liquid V A . In the special case of an ideal solution, eq. (6.24) becomes HV A - -RT\nX A (6.25) By replacing X A by (1 X B ) and expanding as in Section 6-11, the dilute solution formula is obtained: HV A = RTX B (6.26) Since the solution is dilute, RT n II w -S *< RTm' (6.27) This is the equation used by Frazer and Morse as a better approximation Sec. 17] SOLUTIONS AND PHASE EQUILIBRIA 135 than the van't HofT equation (6.23). As the solution becomes very dilute, m' the volume molal concentration approaches c the molar concentration, and we find as the end product of the series of approximations Ft = RTc (6.23) The adequacy with which eqs. (6.23), (6.25), and (6.27) represent the experimental data can be judged from the comparisons in Table 6.4. 9 17. Deviations from Raoult's Law. Only a very few of the many liquid solutions that have been investigated follow Raoult's Law over the complete 600 500 E E u -400 ( ID CO C/> (T a 200 100 400 320 240 160 80 f).0 .2 .4 .6 .8 CH 2 (OCH 3 ) 2 MO LE FRACTION CSa (a) 1.0 ^0 .2 A .6 .8 CS 2 (CH 3 ) 2 CO MOLE FRACTION CHC1 3 CH Cl * (b) Fig. 6.7. (a) Positive deviation from Raoult's Law the PX diagram of carbon bisulfide-methylal system, (b) Negative deviation from Raould's Law the PX diagram of chloroform-acetone system. range of concentrations. It is for this reason that the greatest practical application of the ideal equations is made in the treatment of dilute solutions, in which the solvent obeys Raoult's Law and the solute obeys Henry's Law. Nevertheless, one of the most instructive ways of qualitatively discussing the properties of nonideal solutions is in terms of their deviations from ideality. The first extensive series of vapor-pressure measurements, per- mitting such comparisons, were those made by Jan von Zawidski, around 1900. Two general types of deviation were distinguished. An example exhibiting a positive deviation from Raoult's Law is the system carbon bisulfide- methylal, whose vapor-pressure-composition diagram is shown in (a), Fig. 6.7. An ideal solution would follow the dashed lines. The positive 9 The osmotic pressures of solutions of high polymers and proteins provide some of the best data on their thermodynamic properties. A typical investigation is that of Shick, Doty, and Zimm, /. Am. Chetn. Soc., 72, 530 (1950). 136 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 deviation is characterized by vapor pressures higher than those calculated for ideal solutions. The escaping tendencies of the components in the solution are accordingly higher than the escaping tendencies in the individual pure liquids. This effect has been ascribed to cohesive forces between unlike components smaller than those within the pure liquids, resulting in a trend away from complete miscibility. To put it naively, the components are happier by themselves than when they are mixed together; they are unsociable. These are metaphorical expressions; a scientific translation is obtained by equating a happy com- ponent to one in a state of low free energy. One would expect that this incipient immiscibility would be reflected in an increase in volume on mixing and also in an absorption of heat on mixing. The other general type of departure from Raoult's Law is the negative deviation. This type is illustrated by the system chloroform-acetone in (b), Fig. 6.7. In this case, the escaping tendency of a component from solution is less than it would be from the pure liquid. This fact may be interpreted as being the result of greater attractive forces between the unlike molecules in solution than between the like molecules in the pure liquids. In some cases, actual association or compound formation may occur in the solution. As a result, in cases of negative deviation, a contraction in volume and an evolution of heat are to be expected on mixing. In some cases of deviation from ideality, the simple picture of varying cohesive forces may not be adequate. For example, positive deviations are often observed in aqueous solutions. Pure water is itself strongly associated and addition of a second component may cause partial depolymerization of the water. This would lead to an increased partial vapor pressure. A sufficiently great positive deviation from ideality may lead to a maxi- mum in the PX diagram, and a sufficiently great negative deviation, to a minimum. An illustration of this behavior is shown in (a), Fig. 6.8. It is now no longer meaningful to say that the vapor is richer than the liquid in the "more volatile component." The following more general statement (Kono- valov's Rule) is employed: the vapor is richer than the liquid with which it is in equilibrium in that component by addition of which to the system the vapor pressure is raised. At a maximum or minimum in the vapor-pressure curve, the vapor and the liquid must have the same composition. 18. Boiling-point diagrams. The PX diagram in (a), Fig. 6.8, has its counterpart in the boiling-point diagram in (b), Fig. 6.8. A minimum in the PX curve necessarily leads to a maximum in the TX curve. A well known example is the system HC1-H 2 O, which has a maximum boiling point (at 760 mm) of 108.58 at a concentration of 20.222 per cent HC1. A solution with the composition corresponding to a maximum or minimum point on the boiling-point diagram is called an azeotropic solution (c>, "to boil"; arppTros, "unchanging"), since there is no change in composition on boiling. Such solutions cannot be separated by isobaric Sec. 19] SOLUTIONS AND PHASE EQUILIBRIA 137 distillation. It was, in fact, thought at one time that they were real chemical compounds, but changing the pressure changes the composition of the azeotropic solution. The distillation of a system with a maximum boiling point can be dis- cussed by reference to (b), Fig. 6.8. If the temperature of a solution having the composition / is raised, it begins to boil at the temperature t v The first vapor that distills has the composition y, richer in component A than is the original liquid. The residual solution therefore becomes richer in B; and if the vapor is continuously removed the boiling point of the residue rises, as LIQUID VAPOR VAPOR LIQUID X (a) A v I X (b) I' B Fig. 6.8. Large negative deviation from Raoult's Law. The PX curve has a minimum; the TX curve has a maximum. its composition moves along the liquidus curve from / toward m. If a frac- tional distillation is carried out, a final separation into pure A and the azeotropic solution is achieved. Similarly a solution of original composition /' can be separated into pure B and azeotrope. 19. Partial miscibility. If the positive deviations from Raoult's Law become sufficiently large, the components may no longer form a continuous series of solutions. As successive portions of one component are added to the other, a limiting solubility is finally reached, beyond which two distinct liquid phases are formed. Usually, but not always, increasing temperature tends to promote solubility, as the thermal kinetic energy conquers the reluctance of the components to mix freely. In otjier words, the T AS term in AF = A// T AS* becomes more important. A solution that displays a large positive deviation from ideality at elevated temperatures therefore frequently splits into two phases when it is cooled. A PC diagram for a partially miscible liquid system, such as aniline and water, is shown in (a), Fig. 6.9. The point x lies in the two-phase region and corresponds to a system of two liquid solutions, one a dilute solution of aniline in water having the composition y, and the other a dilute solution of 138 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 water in aniline having the composition z. These are called conjugate solutions. The relative amounts of the two phases are given by the ratios of the distances along the tie line, xy/xz. Applying the phase rule to this two-phase region: since p 2 and c -- 2, the system is bivariant. Because of the requirement of constant temperature imposed on the PC diagram, only one degree of freedom remains. Once the pressure is fixed, the compositions of both phases are fixed, which is indeed what the diagram indicates. The over-all com- position x is of course not fixed, since this depends on the relative amounts of the two conjugate solutions, with which the phase rule is not concerned. P 760mm VAPOR VAPOR A B A WATER COMPOSITION ANILINE WATER B COMPOSITION ANILINE (a) (b) Fig. 6.9. Schematic diagrams for aniline-water system, showing limited solubility of liquids, (a) PC diagram, (b) TC diagram. Let us follow the sequence of events as the pressure is gradually reduced along the line of constant composition, or isopleth, xx', At the point P, vapor having a composition corresponding to point Q begins to appear. There are now three phases coexisting in equilibrium, so that the system is invariant. If the volume available to the vapor is increased, the amount of the vapor phase will increase, at constant pressure, until all the aniline-rich solution, of composition /?, has vaporized. When this process is complete, there will remain a vapor of composition Q and a solution of composition N 9 so that the system becomes univariant again as the pressure falls below that at P. Since the vapor that is formed is richer in aniline, the composition of the residual solution becomes rjcher in water. The liquid composition moves along the line NL, and the vapor composition moves along QL until all the Sec. 20] SOLUTIONS AND PHASE EQUILIBRIA 139 liquid has been transformed into vapor, at the point M. After this, further decrease in pressure proceeds at constant vapor composition along MX' . It may be noted that the two conjugate solutions N and R have the same total vapor pressure and the same vapor composition. It follows that the partial" vapor pressure of component A above a dilute solution of A in B is the same as the vapor pressure of A above the dilute solution of B in A. For example, if benzene and water are mixed at 25C, two immiscible layers are formed, one containing 0.09 per cent C 6 H 6 and 99.91 per cent H 2 O, the other 99.81 per cent C 6 H 6 and 0.19 per cent H 2 O. The partial pressure of benzene above either of these solutions is the same, namely 85 mm. In (a), Fig. 6.9, the lines NN' and RR' are almost vertical, since the solubility limits are only slightly de- pendent on pressure. Change in tem- perature, on the other hand, may greatly affect the mutual solubility of two liquids. In (b), Fig. 6.9, the TC diagram for the water-aniline system is drawn for the constant pressure of one atmosphere (normal-boiling-point diagram). Increasing the temperature tends to close the solubilitv gap, the difference between the concentrations of the two conjugate solutions. The interpretation of the solubility gap can be given in terms of the free energy of the system. At some constant temperature, let us plot the molar free energy of the system, defined as F =-- F/(n A -f- n B ), against the mole fraction of B, X B , for both the a and ft phases. In Fig. 6.9b, for example, these phases would be the two immiscible liquid solutions. The diagram obtained, Fig. 6.10, is an exact analog of Fig. 6.1, which was used for the determination of partial molar volumes. In this case, the intercept of the common tangent to the two F vs. X curves gives the value of the partial molar free energies, or chemical potentials, of the two components. At this composition, therefore, /// = /^/, and // yy a /y f /, i.e., the condition for equilibrium of components A and B between the two phases is fulfilled. The corresponding mole fractions represent the phase-boundary compositions; at any composition between X' B and X" B , the system will split into two distinct phases, since in this way it can reach its minimum free energy. For X B < X' B , however, pure phase a gives the lowest free energy, and for X B > X" B , pure phase ft. 20. Condensed-liquid systems. In (b), Fig. 6.9, the variation of solubility with temperature is shown for only one pressure. At high enough tempera- tures boiling occurs, and it is therefore not possible to trace the ultimate Fig. 6.10. Partial miscibility deter- mined by free energy. 140 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 course of the solubility curves. One might expect that the solubility gap would close completely if a high enough temperature could be reached before the onset of boiling. This expectation is represented by the dashed line in (b), Fig. 6.9. A number of condensed systems have been studied, which illustrate com- plete liquid-liquid solubility curves. A classical example is the phenol-water system of Fig. 6.11 (a). At the temperature and composition indicated by the point x, two phases coexist, the conjugate solutions represented by y and 70 60 50 .40 30 20 10 c X' *- d L 180 160 140 M20 IOO 4 V ^ Y / 60 \ -- - 50 30 20 10 r\ J TWO RE hH PH/ IGlOf N -PH .GIO VSE TWO -PHASE ~ REGION ' X X *EGIC ^SE )N \ ii ^H ONE RE (\SE SI 180 60 40 ^ ONE -PHASE REGION ) 20 40 60 80 100 ^0 20 40 60 80 100 tv t) 20 40 60 8O 10 PER CENT WATER PER CENT PER CENT NICOTINE TRIETHYLAMINE (a) (b) (c) Fig. 6.11. Partial miscibility of two liquids, (a) phenol-water system, (b) tnethylamine-water system, (c) nicotine-water system. z. The relative amounts of the two phases are proportional, as usual, to the segments of the tie line. As the temperature is increased along the isopleth XX \ the amount of the phenol-rich phase decreases and the amount of water-rich phase increases. Finally at Y the compositions of the two phases become identical, the phenol-rich phase disappears completely, and at temperatures above Y there is only one solution. This gradual disappearance of one solution is characteristic of systems having all compositions except one. The exception is the composition corre- sponding to the maximum in the TC curve. This composition is called the critical composition and the temperature at the maximum is the critical solution temperature or upper consolute temperature. If a two-phase system having the critical composition is gradually heated [line CC in (a), Fig. 6.11] there is no gradual disappearance of one phase. Even in the immediate neighborhood of the maximum d, the ratio of the segments of the tie line remains practically constant. The compositions of the two conjugate solu- tions gradually approach each other, until, at the point d, the boundary line between the two phases suddenly disappears and a single-phase system remains. Sec. 21] SOLUTIONS AND PHASE EQUILIBRIA 141 As the critical temperature is slowly approached from above, a most curious phenomenon is observed. Just before the single homogeneous phase passes over into two distinct phases, the solution is diffused by a pearly opalescence. This critical opalescence is believed to be caused by the scatter- ing of light from small regions of slightly differing density, which are formed in the liquid in the incipient separation of the two phases. Strangely enough, some systems exhibit a lower consolute temperature. At high temperatures, two partially miscible solutions are present, which become completely intersoluble when sufficiently cooled. An example is the triethylamine-water system in (b), Fig. 6.11, with a lower consolute tem- perature of 18.5 at 1 atm pressure. It is almost impossible to locate the critical composition exactly, since lowering the temperature a fraction of a degree greatly increases the solubility. This somewhat weird behavior suggests that large negative deviations from Raoult's Law (e.g., compound formation) become sufficient at the lower temperatures to counteract the positive deviations responsible for the immiscibility. Finally, systems have been found with both upper and lower consolute temperatures. These are most common at elevated pressures, and indeed one would expect all systems with a lower consolute temperature to display an upper one at sufficiently high temperature and pressure. An atmospheric- pressure example is the nicotine-water system of Fig. 6.11 (c). Having come to solutions of this type, we have run the gamut of deviations from ideality. 21. Thermodynamics of nonideal solutions: the activity. A complete thermodynamic description of a solution, except for its amount, can be expressed in terms of the temperature, the pressure, and the chemical poten- tials of the various components. All the other thermodynamic functions can be derived from these. For a single pure ideal gas, the change in chemical potential is given from eq. (4.33) as dp RTdln P. By integration we obtain ^ = // + RT In P, where ju is the chemical potential of the gas at one atmosphere pressure. For a pure gas, this equation is identical with F F + RTlnP, where F is the free energy per mole. If the gas is not ideal, the fugacity is defined by the equation ft = JLL + RTlnf. Such an equation holds also for any component in a mixture of gases (gaseous solution). The constant p is a function of temperature alone. It is the chemical potential of the gas in its standard state of unit fugacity, or the standard free energy of the gas. The same equation is valid for a component in a liquid or solid solution, since at equilibrium the chemical potential must be the same in the con- densed phase as in the vapor. For a component A, (6.8) If the vapor above the solution can be considered to behave as an ideal 142 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 gas, f A P A , and /LI A /t A + RTinP A . For an ideal solution, P A = XA/A = X A P A> and therefore (6.28) nX A The two constant terms can be combined, giving P.i = 1% + *T\n X A (6.29) This is the expression for the chemical potential in an ideal solution, p A being the chemical potential of A when X A - \ ; i.e., of pure liquid A. It should be clearly understood that p, A is a function of both temperature and pressure, in contrast with JU A in eq. (6.8). This is because the vapor pressure of the pure liquid, P A in eq. (6.28), is a function of both temperature and over-all pressure (p. 107). In the discussion of nonideal solutions we can always use the chemical potential, obtained from eq. (6.8) in terms of the partial vapor pressure or fugacity. Sometimes, however, it is convenient to introduce a new function, the activity a, which was invented by G. N. Lewis. It is defined as follows so as to preserve the form of eq. (6.29), p A - fi A 4 RTlna A (6.30) or P^ I- where y a/X is called the activity coefficient. One advantage of the activity coefficient is that it indicates at a glance the magnitude of the deviation from ideality in the solution. In terms of the activity, Raoult's Law becomes simply a = X, or y = 1 . Comparing eq. (6.30) with eqs. (6.28) and (6.8), we find that *A -4 (6.31) J A The activity is accordingly the ratio of the fugacity to the fugacity in the standard state. We have implicitly taken this standard state to be pure A, but other definitions might have been used. For a gas/J = 1 and therefore a A ~f A , the activity equals the fugacity. Equation (6.31) provides the most direct method of determining the activity of a component in a solution. It is usually sufficiently accurate to ignore gas imperfections and set the fugacity ratio equal to the vapor pressure ratio, so that a A = P A /P A . Some activities calculated in this way from vapor-pressure data are collected in Table 6.5. Once the activity of one component has been obtained as a function of concentration, the activity of the other component in a binary solution can be calpulated from the Gibbs-Duhem equation. Sec. 22] SOLUTIONS AND PHASE EQUILIBRIA 143 TABLE 6.5 ACTIVITIES OF WATER AND SUCROSE IN THEIR SOLUTIONS AT 50C OBTAINED FROM VAPOR PRESSURE LOWERING AND THE GIBBS-DUHEM EQUATION Mole Fraction of Water Activity of Water Mole Fraction of Sucrose Activity of Sucrose X A <*A X* a B 0.9940 0.9939 0.0060 0.0060T 0.9864 0.9934 0.0136 0.0136 0.9826 0.9799 0.0174 0.0197 0.9762 0.9697 0.0238 0.0302 0.9665 0.9617 0.0335 0.0481 0.9559 0.9477 0.0441 0.0716 0.9439 0.9299 0.0561 0.1037 0.9323 0.9043 0.0677 0.1390 0.9098 0.8758 0.0902 0.2190 0.8911 0.8140 0.1089 0.3045 Corresponding with eq. (6.4) for the partial molar volumes, we have for the partial molar free energies or chemical potentials, d / l A ^ - n A From eq. (6.30), d\n a A --= d In a,, If a B is known as a function of X J}9 a A can be obtained by a graphical integration. Activities can also be calculated from any of the colligative properties related to the vapor pressure. The details of these calculations are to be found in various treatises on thermodynamics. 10 22. Chemical equilibria in nonideal solutions. The activity function defined in eq. (6.30) is useful in discussing the equilibrium constants of reactions in solution. It is readily proved (cf. p. 76) that for the schematic reaction aA + bB^cC + dD aS a r (6.32) and A/* - RTln K a In terms of activity coefficients and mole fractions, "a A. a A. & y a y b -~r"x 7 A 7s A A A B In an ideal solution, all the activity coefficients become equal to unity, 10 G. N. Lewis and M. Randall, Thermodynamics and Free Energy of Chemical Sub- stances (New York: McGraw-Hill, 1923), p. 278. 144 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 and the equilibrium constant is simply K x . Extensive data on the activity coefficients of components in solutions of nonelectrolytes are not available, and the most important applications of eq. (6.32) have been made in electro- lytic solutions, which will be discussed in Chapter 15. 23. Gas-solid equilibria. The varieties of heterogeneous equilibrium that have been considered so far have almost all been chosen from systems involving liquid and vapor phases only. Some systems of the solid-vapor and solid-liquid types will now be described. Most of the examples will be chosen 200 160 160 140 : 120 =JOO 60 60 40 20 SATURATED SOLUTION * CuS0 4 -5H20 t VAPOR CuS0 4 '5l CuS0 4 -3H 2 ' + 2H 2 100 90 80 70 60 > 40 30 20 10 20 30 40 50 60 70 80 90 KX) 110 120 t,c (0) t -50*C CuS0 4 3H 2 GuS0 4 + H 2 - H 2 12345 MOLES H 2 0/MOLE CuS0 4 IN SOLID (b) Fig. 6.12. The system CuSO 4 ~H 2 O. from two-component systems, with only a brief introduction to three- component phase diagrams. A two-component gas-solid system in which there is no appreciable solid-solution formation is exemplified by: CaCO 3 ^ CaO f CO 2 . Since c 2, the degrees of freedom are/^ 4 p. If the two solid phases are present, together with the gaseous phase CO 2 , the system is univariant, /=4 3=1. At a given temperature, the pressure of CO 2 has a fixed value. For example, if CO 2 is admitted to a sample of CaO at 700C, there is no absorption of gas until a pressure of 25 mm is reached; then the CaO takes up CO 2 at constant pressure until it is completely converted into CaCO 3 , whereupon further addition of CO 2 again results in an increase in pressure. The pressure-temperature diagram for such a system is therefore similar to the vapor-pressure curve of a pure liquid or solid. The CO 2 pressure has been loosely called the "dissociation pressure of CaCO 3 ." Since the pressure has a definite value only when the vapor phase is in equilibrium with both solid phases, it is really necessary to speak of the "dissociation pressure in the system CaCO 3 -CaO-CO 2 ." Sec. 24] SOLUTIONS AND PHASE EQUILIBRIA 145 The necessity of specifying both the solid phases is to be emphasized in systems formed by various salts, their hydrates, and water vapor. The case of copper sulfate-water is shown in (a), Fig. 6.12, on a PT diagram, and in (b), Fig. 6.12, on a PC diagram. As long as only the two phases are present, a salt hydrate can exist in equilibrium with water vapor at any temperature if the pressure of water vapor is (1) above the dissociation pressure to lower hydrate or anhydrous salt and (2) below the dissociation pressure of the next higher hydrate or the vapor pressure of the saturated solution. State- ments in the older literature that a given hydrate "loses water at 1 10C" are devoid of precise meaning. When the pressure of water vapor falls below the dissociation pressure for the system, efflorescence occurs, as the hydrate loses water and its surface becomes covered with a layer of lower hydrate or anhydrous salt. When the vapor pressure exceeds that of the saturated aqueous solution, deliquescence occurs, and the surface of hydrate becomes covered with a layer of saturated solution. 24. Equilibrium constant in solid-gas reactions. The equilibrium constant for a reaction involving solid phases can be discussed conveniently by con- sidering a typical reaction of this kind, the reduction of zinc oxide by carbon monoxide, ZnO (s) f CO -> Zn (g) + CO 2 . The equilibrium constant in terms of activities can be written as follows: K a = gzn * C( \ AF - -RTln K a (6.33) The activity is the ratio of the fugacity under the experimental conditions to the fugacity in a standard state, f A lf^ The standard state of a pure solid component is taken to be its state as a pure solid at one atmosphere pressure. The fugacity of the solid varies so slightly with pressure that over a con- siderable range of pressure, f A lf% for a solid is effectively a constant equal to unity. Making this very good approximation, the expression in eq. (6.33) becomes a co /co If the gases are considered to be ideal the activity ratio equals the partial pressure ratio, and K 9 = /WW^co- This discussion leads to the following general rule: no terms involving pure solid or liquid components need be included in equilibrium constants for solid-gas or liquid-gas reactions, unless very high precision is required, in which case there may be a small pressure correction to K v or K f . Equilibrium data for the reduction of zinc oxide are given in Table 6.6. 25. Solid-liquid equilibria: simple eutectic diagrams. For two-component solid-liquid equilibria in which the liquids are completely intersoluble in all proportions and there is no appreciable solids-solid solubility, the simple 146 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 TABLE 6.6 THE EQUILIBRIUM ZnO -f CO ^ Zn h CO 2 (TOTAL PRESSURE = 760 MM) Equilibrium Concentrations in Vapor Temp. (C) K = PznPcoJPco (atm) Per cent CO Per cent CO 2 -= Per cent Zn 427 99.98 0.01 1.00 x 10~ 8 627 99.12 0.44 1.95 x 10~ 5 827 91.76 4.12 1.84 x 10~ 3 1027 66.92 16.54 4.08 x 10~ 2 1227 30.84 34.58 3.87 x 10- 1 1427 9.8 45.1 2.07 diagram of Fig. 6.13 is obtained. Examples of systems of this type are collected in Table 6.7. TABLE 6.7 SYSTEMS WITH SIMPLE EUTECTIC DIAGRAMS SUCH AS FIG. 6.13 Eutectic Component A M. pt. A Component B M. pt. B . v *"-'/ \ ^-v Mol | C per cent B CHBr 3 7.5 C 6 H 8 5.5 -26 50 CHC1 3 63 C 6 H 5 NH 2 -6 -71 24 Picric acid 122 TNT 80 60 64 Sb 630 Pb 326 246 81 Cd 321 Bi 271 144 55 KC1 790 AgCI 451 306 69 Si 1412 Al 657 578 89 Be 1282 Si 1412 1090 32 Consider the behavior of a solution of composition X on cooling along the isopleth XX' . When point P is reached, pure solid A begins to separate from the solution. As a result, the residual solution becomes richer in the other component B, its composition falling along the line PE. At any point Q in the two-phase region, the relative amounts of pure A and residual solution are given as usual by the ratio of the tie-line segments. When point R is reached, the residual solution has the eutectic composition E. Further cooling now results in the simultaneous precipitation of a mixture of A and B in relative amounts corresponding to E. The eutectic point is an invariant point on a constant pressure diagram; since three phases are in equilibrium,/^ c p + 2 = 2 --/? + 2 --= 4 3=1, and the single degree of freedom is used by the choice of the constant- pressure condition. Sec. 26] SOLUTIONS AND PHASE EQUILIBRIA 147 Microscopic examination of alloys often reveals a structure indicating that they have been formed from a melt by a cooling process similar to that considered along the isopleth XX' of Fig. 6.13. Crystallites of pure metal are found dispersed in a matrix of finely divided eutectic mixture. An example taken from the antimony- lead system is shown in the photo- micrograph of Fig. 6.14. 26. Cooling curves. The method of cooling curves is one of the most useful for the experimental study of solid-liquid systems. A two-compo- nent system is heated until a homo- geneous melt is obtained. A thermo- couple, or other convenient device for temperature measurement, is - PER CENT B - Fig. 6.13. Simple eutectic diagram for two components, A and B, completely inter- soluble as liquids but with negligible solid- solid solubility. immersed in the liquid, which is kept in a fairly well insulated container. As the system slowly cools, the temperature is recorded at regular time intervals. Examples of such curves for the system shown in Fig. 6.13 are drawn in Fig. 6.15. The curve a for pure A exhibits a gradual decline until the melting point of A is reached. It then remains perfectly flat as long as solid and liquid A Fig. 6.14. Photomicrograph at 50X of 80 per cent Pb-20 per cent Sb, showing crystals of Sb in a eutectic matrix. (Courtesy Professor Arthur Phillips, Yale University.) are both present, and resumes its decline only after all the liquid has solidified. The curve for cooling along the isopleth XX' is shown in b. The decline as the homogeneous melt is cooled becomes suddenly less steep when the tem- perature is reached corresponding to point P, where the first solid begins to 148 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 separate from the solution. This change of slope is a consequence of the liberation of latent heat of fusion during the solidification of A. The more gradual decline continues until the eutectic temperature is reached. Then the cooling curve becomes absolutely flat. This is because the eutectic point in a two-component system, just as the melting point of one component, is an invariant point at constant over-all pressure. If the composition of the COMPOSITION ALONG \EUTECTIC \COMPOSITION (0) TIME Fig. 6.15. Cooling curves for various compositions on the simple eutectic diagram of Fig. 6.13. system chosen initially happened to be the same as that of the eutectic, the cooling curve would be that drawn in c. The duration of the constant-temperature period at the eutectic tempera- ture is called the eutectic halt. This halt is a maximum for a melt having the eutectic composition. Each cooling curve determination yields one point on the TC diagram (point of initial break in slope) in addition to a value for the eutectic tempera- ture. By these methods, the entire diagram can be constructed. 27. Compound formation. If aniline and phenol are melted together in equimolar proportions, a definite compound crystallizes on cooling, C 6 H 5 OH-C 6 H 5 NH 2 . Pure phenol melts at 40C, pure aniline at -6.1C, and the compound melts at 3lC. The complete TC diagram for this system, in Fig. 6.16, is typical of many instances in which stable compounds occur as solid phases. The most convenient way of looking at such a diagram is to imagine it to be made up of two diagrams of the simple eutectic type placed side by side. In this case, one such diagram would be the phenol-compound diagram, and the other the aniline-compound diagram. The phases corresponding with the various regions of the diagram are labeled. A maximum such as the point C is said to indicate the formation of a Sec. 28] SOLUTIONS AND PHASE EQUILIBRIA 149 compound with a congruent melting point, since if a solid having the com- position C 6 H 5 OH-C 6 H 5 NH 2 is heated to 31C, it melts to a liquid of identical composition. Compounds with congruent melting points are readily detected- o UJ< S 1 liJ 40 30 20 O lr'0 -10 -20, (b) rSOLID PHENOL [ + SOLUTION SOLID COMPOUND SOLUTIONJ SOLID PHENOL + SOLID COMPOUND SOLUTION SOLID ANILINE + SOLID COMPOUND .1 .2 .3 4 .5 .6 .7 .8 .9 1.0 CgHsOH MOLE FRACTION ANILINE C 6 H 5 NH 2 (0) Fig. 6.16. The system phenol-aniline. by the cooling-curve method. A liquid having the composition of the com- pound exhibits no eutectic halt, behaving in every respect like a single pure component. 28. Solid compounds with incongruent melting points. In some systems, solid compounds are formed that do not melt to a liquid having the same composition, but instead decompose before such a melting point is reached. An example is the silica-alumina system (Fig. 6.17), which includes a com- pound, 3Al 2 O 3 -SiO 2 , called mul/ite. If a melt containing 40 per cent A1 2 O 3 is prepared and cooled slowly, solid mullite begins to separate at about 1780C. If some of this solid com- pound is removed and reheated along the line XX', it decomposes at 1800C into solid corundum and a liquid solution (melt) having the composition P. Thus: 3Al 2 O 3 -SiO 2 -* A1 2 O 3 + solution. Such a change is called incongruent melting, since the composition of the liquid differs from that of the solid. The point P is called the incongruent melting point or the peritectic point (rrjKTo*, "melting"; TTC/M, "around"). The suitability of this name becomes evident if one follows the course of events as a solution with composition 3Al 2 O 3 -SiO 2 is gradually cooled along XX' . When the point M is reached, 150 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 solid corundum (A1 2 O 3 ) begins to separate from the melt, whose com- position therefore becomes richer in SiO 2 , falling along the line MP. When the temperature falls below that of the peritectic at />, the following change occurs: liquid + corundum --> mullite. The solid A1 2 O 3 that has separated 2100 60 3A1203 80 100 Si02 A\203 w PER CENT A1 2 03 Fig. 6.17. System displaying peritectic. reacts with the surrounding melt to form the compound mullite. If a specimen taken at a point such as Q is examined, the solid material is found to consist of two phases, a core of corundum surrounded by a coating of mullite. It was from this characteristic appearance that the term "peritectic" originated. 29. Solid solutions. Solid solutions are in theory no different from other kinds of solution: they are simply solid phases containing more than one 1500 1452 I000 10 20 30 4p 50 60 70 80 90 100 Cu wt PER CENT NICKEL Ni Fig. 6.18. The copper-nickel system a continuous series of solid solutions. component. The phase rule makes no distinction between the kind of phase (gas, liquid, or solid) that occurs, being concerned only with how many Sec. 33] SOLUTIONS AND PHASE EQUILIBRIA 155 pure tin melts at 232C and pure bismuth at 268C, their eutectic being at 133C and 42 per cent Sn. The Sn-Bi eutectic temperature is lowered by the addition of lead to a minimum at 96C and a composition of 32 per cent Pb, 16 per cent Sn, 52 per cent Bi. This is the ternary eutectic point. Pb Pb Sn Bi Sn SOLID Pb + SOLUTION SOLID Pb ^ L + SOLIDSn Pb /-SOLID Pb + SOLUTION SOLUTION 325 a 315 (0) (b) SOLID Pb + SOLID S SOLUTION 182 133 (0 (d) SOLID Pb 4- SOLUTION SOLID Pb + SOLID Bi SOLUTION Bi Fig. 6.22. The system Pb-Sn-Bi : three-dimensional diagram and iso- thermal sections. Without using a solid model, the behavior of this system is best illustrated by a series of isothermal sections, shown in Fig. 6.22. Above 325C (a), the melting point of pure lead, there is a single liquid solution. At around 315C (b) the system consists of solid Pb and solution. The section at 182C (c) indicates the binary eutectic of Sn and Pb. Below this temperature, solid Pb and solid Sn both separate from the solution. At 133C the binary eutectic between Sn and Bi is reached (d). Finally, in (e) at 100C there is shown a section slightly above the ternary eutectic. The subject of ternary diagrams is an extended and very important one, and only a few of the introductory aspects have been mentioned. For further details some of the special treatises that are available should be consulted. 11 11 J. S. Marsh, Principles of Phase Diagrams (New York: McGraw-Hill, 1935); G. Mas- sing, Introduction to the Theory of Three Component-Systems (New York: Reinhold, 1944). 156 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 PROBLEMS 1. Solutions are prepared at 25C containing 1000 g of water and n moles of NaCl. The volume is found to vary with n as V =- 1001.38 + 16.6253 + 1.7738 3/2 + 0.1194fl 2 . Draw a graph showing the partial molar volumes of H 2 O and NaCl in the solution as a function of the molality from to 2 molal. 2. In the International Critical Tables (vol. Ill, p. 58) there is an extensive table of densities of HNO 3 H 2 O solutions. Use these data to calculate, by the graphical method of Fig. 6.1, the partial molar volumes of H 2 O and HNO 3 in 10, 20, 30, and 40 per cent solutions at 25. 3. When 2 g of nonvolatile hydrocarbon containing 94.4 per cent C is dissolved in 100 g benzene, the vapor pressure of benzene at 20C is lowered from 74.66 mm to 74.01 mm. Calculate the empirical formula of the hydro- carbon. 4. Pure water is saturated with a 2 : 1 mixture of hydrogen and oxygen at a total pressure of 5 atm. The water is then boiled to remove all the gases. Calculate the per cent composition of the gases driven off (after drying). Use data from Table 6.2. 5. Water and nitrobenzene can be considered to be immiscible liquids. Their vapor pressures are: H 2 O, 92.5mm at 50C; 760mm at 100C; C 6 H 5 NO 2 , 22.4 mm at 10QC; 148 mm at 150C. Estimate the boiling point of a mixture of water and nitrobenzene at 1 atm pressure. In a steam dis- tillation at 1 atm how many grams of steam would be condensed to obtain one gram of nitrobenzene in the distillate? 6. The following data were obtained for the boiling points at 1 atm of solutions of CC1 4 in C 2 C1 4 : Mole fraction CCI 4 inliq. . 0.000 0.100 0.200 0.400 0.600 0.800 1.000 Mole fraction CCI 4 invap. . 0.000 0.469 0.670 0.861 0.918 0.958 1.000 Boiling point C . . 120.8 108.5 100.8 89.3 83.5 79.9 76.9 If half of a solution 30 mole per cent in CC1 4 is distilled, what is the com- position of the distillate? If a solution 50 mole per cent in CC1 4 is distilled until the residue is 20 mole per cent CC1 4 , what is approximate composition of the distillate? 7. A compound insoluble in water is steam distilled at 97.0C, the dis- tillate containing 68 wt. per cent H 2 O. The vapor pressure of water is 682 mm at 97. What is the molecular weight of the compound? 8. When hexaphenylethane is dissolved in benzene, the f.pt. depression of a 2 per cent solution is 0.219C; the b.pt. elevation is 0.135. Calculate the heat of dissociation of hexaphenylethane into triphenylmethyl radicals. Chap. 6] SOLUTIONS AND PHASE EQUILIBRIA 1 57 9. Calculate the weight of (a) methanol, (b) ethylene glycol which, when dissolved in 4.0 liters of water, would just prevent the formation of ice at -10C. 10. The solubility of picric acid in benzene is: /, C . . 5 10 15 20 25 3t g/100gC 6 H 6 . 3.70 5.37 7.29 9.56 12.66 21.38 The melting points of benzene and picric acid are 5.5 and 121.8C. Calculate the heat of fusion of picric acid. 11. The osmotic pressure at 25C of a solution of /Mactoglobulin con- taining 1.346 g protein per 100 cc solution was found to be 9.91 cm of water. Estimate the molecular weight of the protein. 12. For the ideal solutions of ethylene bromide and propylenc bromide (p. 124), draw a curve showing how the mole fraction of C 2 H 4 Br 2 in the vapor varies with that in the liquid. Use this curve to estimate the number of theoretical plates required in a column in order to yield a distillate with mole fraction of C 2 H 4 Br 2 0.9 from a solution of mole fraction 0. 1 . Assume total reflux. 13. Calculate the distribution coefficient K for piperidine between water and benzene at 20C, given : g solute/ 1 00 cc water layer . . 0.635 1.023 1.635 2.694 g solute/100 cc benzene layer . . 0.550 0.898 1.450 2.325 14. A solution of 3.795 g sulfur in 100 g carbon bisulfide (b.pt. CS 2 46.30C; A// vap = 6400 cal per mole) boils at 46.66C. What is the formula of the sulfur molecule in the solution? 15. The melting points and heats of fusion of 0, /?, m dinitrobenzenes are: 116.9, 173.5, 89.8, and 3905, 3345, 4280 cal per mole [Johnston, J. Phys. Chem., 29, 882, 1041 (1925)]. Assuming the ideal solubility law, calculate the ternary eutectic temperature and composition for mixtures of o, m, p com- pounds. 16. The following boiling points are obtained for solutions of oxygen and nitrogen at 1 atm: b.pt., K . . 77.3 78.0 79.0 80.0 82.0 84.0 86.0 88.0 90.1 Mole % O in liq. . 8.1 21.6 33.4 52.2 66.2 77.8 88.5 100 Mole%Oinvap. 2.2 6.8 12.0 23.6 36.9 52.2 69.6 100 Draw the TX diagram. If 90 per cent of a mixture containing 20 per cent O 2 and 80 per cent N 2 is distilled, what will be the composition of the residual liquid and its b.pt. ? Plot an activity a vs. mole fraction X diagram from the data. 17. For a two-component system (A, B) show that: 158 SOLUTIONS AND PHASE EQUILIBRIA [Chap. 6 18. Redder and Barratt [/. Chem. Soc., 537 (1933)] measured the vapor pressures of potassium amalgams at 387.5C, at which temperature the vapor pressure of K is 3.25 mm, of Hg 1280 mm. Mole % K in liq. . 41.1 46.8 50.0 56.1 63.0 72.0 PofHg, mm . 31.87 17.30 13.00 9.11 6.53 3.70 PofK, mm . 0.348 0.68 1.07 1.69 2.26 2.95 Calculate the activity coefficients of K and Hg in the amalgams and plot them vs. the composition in the range studied. 19. The equilibrium pressures for the system CaSO 4 -2 H 2 O = CaSO 4 + 2 H 2 O, and the vapor pressures of pure water, at various temperatures are: /, C 50 55 60 65 CaSO 4 system, mm . .80 109 149 204 H 2 O, mm .... 92 118 149 188 The solubility of CaSO 4 in water is so low that the vapor pressure of the saturated solution can be taken to equal that of pure water. (a) State what happens on heating the dihydrate in a previously evacu- ated sealed tube from 50 to 65C. (b) What solid phase separates when a solution of CaSO 4 is evaporated at 65, at 55C? (c) What solid phase separates on evaporating at 55 if, when the solution becomes saturated, enough CaCl 2 is added to reduce its v.p. by 10 per cent? 20. Data for the Au-Te system: wt. % Te 10 20 30 40 42 50 56.4 60 70 82.5 90 100 f.pt., C 1063 940 855 710 480 447 458 464 460 448 416 425 453 Sketch the phase diagram. Label all regions carefully. Describe what happens when a melt containing 50 per cent Te is cooled slowly. 21. The dissociation pressure of galactose monohydrate is given by Iog 10 />(mrn) - 7.04 - 1780/7. Calculate AF, A//, AS , at 25C for the dissociation. 22. The solubility of glycine in liquid ammonia was found to be: Moles per liter . . . 0.20 0.65 2.52 /, C -77 -63 -45 Estimate the heat of solution per mole. 23. For the free energies of formation of Cu 2 O and CuO the following equations are cited : Cu 2 0: AF = -40,720 + 1.1771n T - 1.545 x lQr*T* -f 85.77 1 / 2 -f 6.977 CuO: AF = -37,680 + 1.757 In T - 2.73 X 10- 8 7 2 + 85.77' / 2 + 9.497 What product is formed when O 2 at 10 mm pressure is passed over copper at900C? REFERENCES BOOKS 1. Brick, R. M., and A. Phillips, Structure and Properties of Alloys (New York: McGraw-Hill, 1949). Chap. 6] SOLUTIONS AND PHASE EQUILIBRIA 159 2. Carney, T. P., Laboratory Fractional Distillation (New York: Macmillan, 1949). 3. Guggenheim, E. A., Mixtures (New York: Oxford, 1952). 4. Hildebrand, J. H., and R. L. Scott, Solubility of Nonelectrolytes (New York: Reinhold, 1950). 5. Hume-Rothery, W., J. W. Christian, and W. B. Pearson, Metallurgical Equilibrium Diagrams (London: Inst. of Physics, 1952). 6. Shand, S. J., Rocks for Chemists (London: Murby, 1952). 7. Wagner, C, Thermodynamics of Alloys (Cambridge, Mass: Addison- Wesley, 1952). 8. Weissberger, A. (editor), Physical Methods of Organic Chemistry , vol. I (New York: Interscience, 1950). Articles on determination of melting point, boiling point, solubility, osmotic pressure. ARTICLES 1. Chem. Rev., 44, 1-233 (1949), "Symposium on Thermodynamics of Solutions." 2. Fleer, K. B., /. Chem. Ed., 22, 588-92 (1945), "Azeotropism." 3. Hildebrand, J. H., J. Chem. Ed., 25, 74-77 (1948), "Ammonia as a Solvent." 4. Teller, A. J., Chem. Eng., 61, 168-88 (1954), "Binary Distillation." CHAPTER 7 The Kinetic Theory 1. The beginning of the atom. Thermodynamics is a science that takes things more or less as it finds them. It deals with pressures, volumes, tem- peratures, and energies, and the relations between them, without seeking to elucidate further the nature of these entities. For thermodynamics, matter is a Continuous substance, and energy behaves in many ways like an incom- pressible, weightless fluid. The analysis of nature provided by thermo- dynamics is very effective in a rather limited field. Almost from the beginning of human thought, however, man has tried to achieve an insight into the structure of things, and to find an indestructible reality beneath the ever- changing appearances of natural phenomena. The best example of this endeavor has been the development of the atomic theory. The word atom is derived from the Greek aro/io?, meaning "indivisible" ; the atoms were believed to be the ultimate and eternal particles of which all material things were made. Our knowledge of Greek atomism comes mainly from the long poem of the Roman, Lucretius, De Rerwn Natura "Concerning the Nature of Things," written in the first century before Christ. Lucretius expounded the theories of Epicurus and of Democritus : The same letters, variously selected and combined Signify heaven, earth, sea, rivers, sun, Most having some letters in common. But the different subjects are distinguished By the arrangement of letters to form the words. So likewise in the things themselves, When the intervals, passages, connections, weights, Impulses, collisions, movements, order, And position of the atoms interchange, So also must the things formed from them change. The properties of substances were determined by the form of their atoms. Atoms of iron were hard and strong with spines that locked them together into a solid; atoms of water were smooth and slippery like poppy seeds; atoms of salt were sharp and pointed and pricked the tongue; whirling atoms of air pervaded all matter. Later philosophers were inclined to discredit the atomic theory. They found it hard to explain the many qualities of materials, color, form, taste, and odor, in terms of naked, colorless, tasteless, odorless atoms. Many followed the lead of Heraclitus and Aristotle, considering matter to be formed from the four "elements," earth, air, fire, and water, in varying 160 Sec. 2] THE KINETIC THEORY 161 proportions. Among the alchemists there came into favor the tria prima of Paracelsus (1493-1541), who wrote: Know, then, that all the seven metals are born from a threefold matter, namely, Mercury, Sulphur, and Salt, but with distinct and peculiar colorings. Atoms were almost forgotten till the seventeenth century, as the al- chemists sought the philosopher's stone by which the "principles" could be blended to make gold. 2. The renascence of the atom. The writings of Descartes (1596-1650) helped to restore the idea of a corpuscular structure of matter. Gassendi (1592-1655) introduced many of the concepts of the present atomic theory; his atoms were rigid, moved at random in a void, and collided with one another. These ideas were extended by Hooke, who first proposed (1678) that the "elasticity" of a gas was the result of collisions of its atoms with the retaining walls. The necessary philosophic background for the rapid development of atomism was now provided by John Locke. In his Essay on Human Under- standing (1690), he took up the old problem of how the atoms could account for all the qualities perceived by the senses in material things. The qualities of things were divided into two classes. The primary qualities were those of shape, size, motion, and situation. These were the properties inherent in the corpuscles or atoms that make up matter. Secondary qualities, such as color, odor, and taste, existed only in the mind of the observer. They arose when certain arrangements of the atoms of matter interacted with other arrange- ments of atoms in the sense organs of the observer. Thus a "hot object" might produce a change in the size, motion, or situation of the corpuscles of the skin, which then produces in the mind the sensations of warmth or of pain. The consequences of Locke's empiricism have been admirably summarized by J. C. Gregory. 1 The doctrine of qualities was a curiously dichotomized version of perception. A snowflake, as perceived, was half in the mind and half out of it, for its shape was seen but its whiteness was only in the mind. . . . This had quick consequences for philosophy. . . . The division between science and philosophy began about the time of Locke, as the one turned, with its experimental appliances, to the study of the corpuscular mechanism, and the other explored the mind and its ideas. The severance had begun between science and philosophy and, although it only gradually progressed into the nineteenth century cleft between them, when the seventeenth century closed, physical science was taking the physical world for her domain, and philosophy was taking the mental world for hers. In the early part of the eighteenth century, the idea of the atom became widely accepted. Newton wrote in 1718: It seems probable to me that God in the beginning formed matter in solid, massy, hard, impenetrable, movable particles, of such sizes and figures, and with such other properties, and in such proportion, as most conduced to the end for which He formed them. 1 A Short History of Atomism (London: A. & C. Black, Ltd., 1931). 162 THE KINETIC THEORY [Chap. 7 Newton suggested, incorrectly, that the pressure of a gas was due to repulsive forces between its constituent atoms. In 1738, Daniel Bernoulli correctly derived Boyle's Law by considering the collisions of atoms with the container wall. 3. Atoms and molecules. Boyle had discarded the alchemical notion of elements and defined them as substances that had not been decomposed in the laboratory. Until the work of Antoine Lavoisier from 1772 to 1783, however, chemical thought was completely dominated by the phlogiston theory of Georg Stahl, which was actually a survival of alchemical concep- tions. With Lavoisier's work the elements took on their modern meaning, and chemistry became a quantitative science. The Law of Definite Propor- tions and The Law of Multiple Proportions had become fairly well established by 1808, when John Dalton published his New System of Chemical Philosophy . Dalton proposed that the atoms of each element had a characteristic atomic weight, and that these atoms were the combining units in chemical reactions. This hypothesis provided a clear explanation for the Laws of Definite and Multiple Proportions. Dalton had no unequivocal way of assigning atomic weights, and he made the unfounded assumption that in the most common compound between two elements, one atom of each was combined. According to this system, water would be HO, and ammonia NH. If the atomic weight of hydrogen was set equal to unity, the analytical data would then give O = 8, N - 4.5, in Dalton's system. At about this time, Gay-Lussac was studying the chemical combinations of gases, and he found that the ratios of the volumes of the reacting gases were small whole numbers. This discovery provided a more logical method for assigning atomic weights. Gay-Lussac, Berzelius, and others felt that the volume occupied by the atoms of a gas must be very small compared with the total gas volume, so that equal volumes of gas should contain equal numbers of atoms. The weights of such equal volumes would therefore be proportional to the atomic weights. This idea was received coldly by Dalton and many of his contemporaries, who pointed to reactions such as that which they wrote as N + O NO. Experimentally the nitric oxide was found to occupy the same volume as the nitrogen and oxygen from which it was formed, although it evidently contained only half as many "atoms." 2 Not till 1860 was the solution to this problem understood by most chemists, although half a century earlier it had been given by Amadeo Avogadro. In 1811, he published in the Journal de physique an article that clearly drew the distinction between the molecule and the atom. The "atoms" of hydrogen, oxygen, and nitrogen are in reality "molecules" containing two atoms each. Equal volumes of gases should contain the same number of molecules (Avogadro's Principle). Since a molecular weight in grams-(mole) of any substance contains the same number of molecules, 'according to Avogadro's Principle the molar 2 The elementary corpuscles of compounds were then called "atoms" of the compound. Sec. 4] THE KINETIC THEORY 163 volumes of all gases should be the same. The extent to which real gases conform to this rule may be seen from the molar volumes in Table 7.1 cal- culated from the measured gas densities. For an ideal gas at 0C and 1 atm, the molar volume would be 22,414 cc. The number of molecules in one mole is now called Avogadro's Number N. TABLE 7.1 MOLAR VOLUMES OF GASES IN cc AT 0C AND 1 ATM PRESSURE Hydrogen . . 22,432 Helium . . . 22,396 Methane . . . 22,377 Nitrogen . . . 22,403 Oxygen . . . 22,392 Ammonia . . 22,094 Argon . . . 22,390 Chlorine . . . 22,063 Carbon dioxide . . 22,263 Ethane . . . 22,172 Ethylene . . . 22,246 Acetylene . . . 22,085 The work of Avogadro was almost completely neglected until it was forcefully presented by Cannizzaro at the Karlsruhe Conference in 1860. The reason for this neglect was probably the deeply rooted feeling that chemical combination occurred by virtue of an affinity between unlike ele- ments. After the electrical discoveries of Galvani and Volta, this affinity was generally ascribed to the attraction between unlike charges. The idea that two identical atoms of hydrogen might combine into the compound molecule H 2 was abhorrent to the chemical philosophy of the early nineteenth century. 4. The kinetic theory of heat. Even among the most primitive peoples the connection between heat and motion was known through frictional phenomena. As the kinetic theory became accepted during the seventeenth century, the identification of heat with the mechanical motion of the atoms or corpuscles became quite common. Francis Bacon (1561-1626) wrote: When I say of motion that it is the genus of which heat is a species I would be understood to mean, not that heat generates motion or that motion generates heat (though both are true in certain cases) but that heat itself, its essence and quiddity, is motion and nothing else. . . . Heat is a motion of expansion, not uniformly of the whole body together, but in the smaller parts of it ... the body acquires a motion alternative, perpetually quivering, striving, and struggling, and initiated by repercussion, whence springs the fury of fire and heat. Although such ideas were widely discussed during the intervening years, the caloric theory, considering heat as a weightless fluid, was the working hypothesis of most natural philosophers until the quantitative work of Rum- ford and Joule brought about the general adoption of the mechanical theory. This theory was rapidly developed by Boltzmann, Maxwell, Clausius, and others, from 1860 to 1890. According to the tenets of the kinetic theory, both temperature and pressure are thus manifestations of molecular motion. Temperature is a measure of the average translational kinetic energy of the molecules, and 164 THE KINETIC THEORY [Chap. 7 pressure arises from the average force resulting from repeated impacts of molecules with the containing walls. 5. The pressure of a gas. The simplest kinetic-theory model of a gas assumes that the volume occupied by the molecules may be neglected com- pletely compared to the total volume. It is further assumed that the molecules behave like rigid spheres, with no forces of attraction or repulsion between them except during actual collisions. In order to calculate the pressure in terms of molecular quantities, let us consider a volume of gas contained in a cubical box of side /. The velocity c of any molecule may be resolved into components u, v, and w, parallel to the three mutually perpendicular axes X, Y, and Z, so that its magnitude is given by C 2^_ U 2 +V 2 + W 2 (7.1) Collisions between a molecule and the walls are assumed to be perfectly elastic; the angle of incidence equals the angle of reflexion, and the velocity changes in direction but not in magnitude. At each collision with a wall that is perpendicular to X, the velocity component u changes sign from } u to - w, or vice versa; the momentum component of the molecule accord- ingly changes from imw to ^mu, where m is the mass of the molecule. The magnitude of the change in momentum is therefore 2 mu. The number of collisions in unit time with the two walls perpendicular to X is equal to w//, and thus the change in the X component of momentum in unit time is 2mu - (u/l) -- 2mu 2 /l. If there are N molecules in the box, the change in momentum in unit time becomes 2(/Ww( 2 )//), where (w 2 ) is the average value of the square of velocity component 3 u. This rate of change of momentum is simply the force exerted by the molecules colliding against the two container walls normal to X, whose area is 2/ 2 . Since pressure is defined as the force normal to unit area, _ 2Nm(u 2 ) Nm(rf) P ^ 21* -I " ~ V~ Now there is nothing to distinguish the magnitude of one particular component from another in eq. (7.1) so that on the average (u 2 ) = (v 2 ) = (w 2 ). Thus 3(w 2 ) (c 2 ) and the expression for the pressure becomes (7.2) V ' 3K The quantity (c 2 ) is called the mean square speed of the molecules, and may be given the special symbol C 2 . Then C = (c 2 ) 172 is called the root mean 3 Not to be confused with the square of the average value of the velocity component, which would be written (w) 2 . In this derivation we are averaging w 2 , not //. Sec. 6] - THE KINETIC THEORY 165 square speed. The total translational kinetic energy E K of the molecules is iNmC*. Therefore from eq. (7.2): PV = \NrnC* ^%E K (7.3) Since the total kinetic energy is a constant, unchanged by the elastic collisions, eq. (7.3) is equivalent to Boyle's Law. If several different molecular species are present in a gas mixture, their kinetic energies are additive. From eq. (7.3), therefore, the total pressure is the sum of the pressures each gas would exert if it occupied the entire volume alone. This is Dalton's Law of Partial Pressures. 4 6. Kinetic energy and temperature. The concept of temperature was first introduced in connection with the study of thermal equilibrium. When two bodies are-placed in contact, energy flows from one to the other until a state of equilibrium is reached. The two bodies are then at the same temperature. We have found that the temperature can be measured conveniently by means of an ideal-gas thermometer, this empirical scale being identical with the thermodynamic scale derived from the Second Law. A distinction was drawn in thermodynamics between mechanical work and heat. According to the kinetic theory, the transformation of mechanical work into heat is simply a degradation of large-scale motion into motion on the molecular scale. An increase in the temperature of a body is equivalent to an increase in the average translational kinetic energy of its constituent molecules. We may express this mathematically by saying that the tempera- ture is a function of E K alone, T -^ f(E K ). We know that this function must have the special form T %E K /R, or E K - $RT (7.4) so that eq. (7.3) may be consistent with the ideal-gas relation, PV RT. Temperature is thus not only a function of, but in fact proportional to, the average translational kinetic energy of the molecules. The kinetic-theory interpretation of absolute zero is thus the complete cessation of all molecular motion the zero point of kinetic energy. 5 The average translational kinetic energy may be resolved into components in the three degrees of freedom corresponding to velocities parallel to the three rectangular coordinates. Thus, for one mole of gas, where TV is Avogadro's Number, E K = For each translational degree of freedom, therefore, from eq. (7.4), E' K - \Nm^f) - \RT (7.5) * PV = l(E Kl \ E K2 + ...); P,V - \E Kl \ P 2 y =- E Therefore, P = /\ + P 2 + . . ., Dalton's Law. 5 It will be seen later that this picture has been somewhat changed by quantum theory, which requires a small residual energy even at the absolute zero. 166 THE KINETIC THEORY [Chap. 7 This is a special case of a more general theorem known as the Principle of the Equipartition of Energy. 7. Molecular speeds. Equation (7.3) may be written C 2 =--= 3/> (7.6) P where /> -= NmjV is the density of the gas. From eqs. (7.3) and (7.4) we obtain for the root mean square speed C, if M is the molecular weight, 2 3RT 3RT ^ ~ ~ ~ The average speed c, as we shall see later, differs only slightly from the root mean square speed : From eq. (7.6), (7.7), or (7.8), we can readily calculate average or root mean square speeds of the molecules of any gas at any temperature. Some results are shown in Table 7.2. The average molecular speed of hydrogen at 25C is 1768 m per sec or 3955 mi per hr, about the speed of a rifle bullet. The average speed of a mercury vapor atom would be only about 400 mi per *hr. TABLE 7.2 AVERAGE SPEEDS OF GAS MOLECULES AT 0C Meters/sec . 1692.0 . 1196.0 . 170.0 . 600.6 . 454.2 . 425.1 . 566.5 We note that, in accordance with the principle of equipartition of energy, at any constant temperature the lighter molecules have the higher average speeds. This principle extends even to the phenomena of Brownian motion, where the dancing particles are some thousand times heavier than the molecules colliding with them, but nevertheless have the same average kinetic energy. 8. Molecular effusion. A direct experimental illustration of the different average speeds of molecules of different gases can be obtained from the phenomenon called molecular effusion. Consider the arrangement shown in (a), Fig. 7.1. Molecules from a vessel of gas under pressure are permitted to escape through a tiny orifice, so small that the distribution of the velocities Gas Meters/sec Gas Ammonia Argon Benzene . Carbon dioxide . 582.7 . 380.8 . 272.2 . 362.5 Hydrogen Deuterium Mercury . Methane . Carbon monoxide Chlorine . Helium . 454.5 . 285.6 . 1204.0 Nitrogen . Oxygen . Water Sec. 8] THE KINETIC THEORY 167 of the gas molecules remaining in the vessel is not affected in any way; that is, no appreciable mass flow in the direction of the orifice is set up. The number of molecules escaping in unit time is then equal to the number that, in their random motion, happen to hit the orifice, and this number is pro- portional to the average molecular speed. In (b), Fig. 7.1 is shown an enlarged view of the orifice, having an area ds. If all the molecules were moving directly perpendicular to the opening with their mean speed r, in one second all those molecules would hit the opening that were contained in an element of volume of base ds and height c, or volume c ds, for a molecule at a distance c will just reach the orifice at (b) Fig. 7.1. Effusion of gases. the end of one second. If there are n molecules per cc, the number striking would be nc ds. To a first approximation only one-sixth of all the molecules are moving toward the opening, since there are six different possible direc- tions of translation corresponding to the three rectangular axes. The number of molecules streaming through the orifice would therefore be \nc ds, or per unit area \nc. Actually, the problem is considerably more complicated, since half the molecules have a component of motion toward the area, and one must average over all the different possible directions of motion. This gives the result: number of molecules striking unit area per second = number of molecules effusing through unit area per second = \nc. It is instructive to consider how this result is obtained, since the averaging method is typical of many kinetic-theory calculations. This derivation will be the only one in the chapter that makes any pretense of exactitude, and may therefore serve also to inculcate a proper suspicion of the cursory methods used to obtain subsequent equations. If the direction of the molecules is no longer normal to the wall, instead of the situation of Fig. 7. 1 , we have that of Fig. 7.2(a). For^any given direction the number of molecules hitting ds in unit time will be those contained in a cylinder of base ds and slant height c. The volume of this cylinder is c cos ds, and the number of molecules in it is nc cos 6 ds. The next step is to discover how many molecules out of the total have 168 THE KINETIC THEORY [Chap. 7 NORMAL velocities in the specified direction. The velocities of the molecules will be referred to a system of polar coordinates [Fig. 7.2.(b)] with its origin at the wall of the vessel. We call such a representation a plot of the molecular velocities in "velocity space." The distance from the origin c defines the magnitude of the velocity, and the angles and <f> represent its direction. Any particular direction from the origin is specified by the differential solid angle doj. The fraction of the total number of molecules having their velocities within this particular spread of directions is */a>/477 since 4?r is the total solid angle subtended by the surface of a sphere. In polar co- (c) (a) (b) Fig. 7.2. Calculation of gaseous effusion. Element of solid angle is shown in (c). ordinates this solid angle is given 6 by sin 6 dO d<f>. The number of molecules hit- ting the surface ds in unit time from the given direction (6, (/>) be- comes ( 1 /47r)nc cos sin 6 dO dc/> ds. Or, for unit surface, it is (l/47r)rtccos sin 6 dO d<j>. In order to find the total number striking from all directions, dn'/dt, this expression must be integrated : dn f w/2 f 2 " 1 -y = --nccosO sin d<f> dO at Jo Jo 4-7T The limits of integration of <f> are from to 2?r, corresponding to ail the directions around the circle at any given 0. Then is integrated from to 7T/2. The final result for the number of molecules striking unit area in unit time is then ^ - i nc (7.9) The steps of the derivation may be reviewed by referring to Fig. 7.2. If p is the gas density, the weight of gas that effuses in unit time is From eq. (7.8) dW^ _ ~dt " dW _ / ~dt ~~ n vl/2 P \bM) (7.10) (7.11) 6 G. P. Harriwell, Principles of Electricity and Electromagnetism (New York: McGraw- Hill, 1949), p. 649. Sec. 9] THE KINETIC THEORY 169 For the volume rate of flow, e.g., cc per sec per cm 2 , 1/2 dt - (--V \l-nMJ (7.12) It follows that at constant temperature the rate of effusion varies in- versely as the square root of the molecular weight. Thomas Graham (1848) was the first to obtain experimental evidence for this law, which is now named in his honor. Some of his data are shown in Table 7.3. TABLE 7.3 THE EFFUSION OF GASES* Gas Air . Nitrogen , Oxygen Hydrogen Carbon dioxide . Relative Velocity of Effusion Observed Calculated from (7.12) 0) 1.0160 0.9503 3.6070 0.8354 (1) 1.0146 0.9510 3.7994 0.8087 * Source: Graham, "On the Motion of Gases," Phil. Trans. Roy. Soc. (London), 136, 573 (1846). It appears from Graham's work, and also from that of later experi- menters, that eq. (7.12) is not perfectly obeyed. It fails rapidly when one goes to higher pressures and larger orifices. Under these conditions the molecules can collide many times with one another in passing through the orifice, and a hydrodynamic flow towards the orifice is set up throughout the container, leading to the formation of a jet of escaping gas. 7 It is evident from eq. (7.12) that the effusive-flow process provides a good method for separating gases of different molecular weights. By using permeable barriers with very fine pores, important applications have been made in the separation of isotopes. Because the lengths of the pores are considerably greater than their diameters, the flow of gases through such barriers does not follow the simple orifice-effusion equation. The dependence on molecular weight is the same, since each molecule passes through the barrier independently of any others. 9. Imperfect gases van der Waals' equation, The calculated properties of the perfect gas of the kinetic theory are the same as the experimental properties of the ideal gas of thermodynamics. It might be expected then that extension and modification of the simplified model of the perfect gas should provide an explanation for observed deviations from ideal-gas behavior. 7 For a discussion of jet flow, see H. W. Liepmann and A. E. Puckett, Introduction to Aerodynamics of a Compressible Fluid (New York: Wiley, 1947), pp. 32 et seq. 170 THE KINETIC THEORY [Chap. 7 The first improvement of the model is to abandon the assumption that the volume of the molecules themselves can be completely neglected in com- parison with the total gas volume. The effect of the finite volume of the molecules is to decrease the available void space in which the molecules are free to move. Instead of the V in the perfect gas equation, we must write V b where b is called the excluded volume. This is not just equal to the volume occupied by the molecules, but actually to four times that volume. This may be seen in a qualitative way by considering the two molecules of Fig. 7.3 (a), regarded as impenetrable spheres each with a diameter d. The QD o o. o o (o) o o o (b) Fig. 7.3. Corrections to perfect gas law. (a) Excluded volume, (b) Intermolecular forces. centers of these two molecules cannot approach each other more closely than the distance d\ the excluded volume for the pair is therefore a sphere of radius d 2r (where r is the radius of a molecule). This volume is JTrrf 3 -- 8 . |77r 3 per pair, or 4 . ^vrr 3 which equals 4V m per molecule (where V m is the volume of the molecule). The consideration of the finite molecular volumes leads therefore to a gas equation of the form: P(V b) ---- RT. A second correction to the perfect gas formula comes from consideration of the forces of cohesion between the molecules. We recall that the thermodynamic definition of the ideal gas includes the requirement that (dE/dV) T 0. If this condition is not fulfilled, when the gas is expanded work must be done against the cohesive forces between the molecules. The way in which these attractive forces enter into the gas equation may be seen by considering Fig. 7.3. (b). The molecules completely surrounded by other gas molecules are in a uniform field of force, whereas the molecules near to or colliding with the container walls experience a net inward pull towards the body of the gas. This tends to decrease the pressure compared to that which would be exerted by molecules in the absence of such attractive forces. The total inward pull is proportional to the number of surface-layer molecules being pulled, and to the number of molecules in the inner layer of the gas that are doing the pulling. Both factors are proportional to the Sec. 10] THE KINETIC THEORY 171 density of the gas, giving a pull proportional to p 2 , or equal to c/> 2 , where c is a constant. Since the density is inversely proportional to the volume at any given pressure and temperature, the pull may also be written a/V 2 . This amount must therefore be added to the pressure to make up for the effect of the attractive forces. Then, (Y- b)^RT (7.13) This is the famous equation of state first given by van der Waals in 1873. It provides a good representation of the behavior of gases at moderate densities, but deviations become very appreciable at higher densities. The values of the constants a and b are obtained from the experimental PVT data at moderate densities, or more usually from the critical constants of the gas. Some of these values were collected in Table 1.1 on p. 14. Equation (7.13) may also be written in the form PV-^RT+bP -^+^ 2 (7.14) The way in which this equation serves to interpret PV vs. P data may be seen from an examination of the compressibility factor curves at different temperatures, shown in Fig. 1.5 (p. 15). At sufficiently high temperatures the intermolecular potential energy, which is not temperature dependent, becomes negligible compared to the kinetic energy of the molecules, which increases with temperature. Then the equation reduces to PV -= RT + bP. At lower temperatures, the effect of intermolecular forces becomes more appreciable. Then, at moderate pressures the ~a\ V term becomes important, and there are corresponding declines in the PV vs. P curves. At still higher pressures, however, the term +ab/V 2 predominates, and the curves eventually rise again. 10. Collisions between molecules. Now that the oversimplification that the molecules of a gas occupy no volume themselves has been abandoned, it is possible to consider further the phenomena that depend on collisions between the molecules. Let us suppose that all the molecules have a diameter d, and consider as in Fig. 7.4 the approach of a molecule A toward another molecule B. A "collision" occurs whenever the distance between their centers becomes as small as d. Let us imagine the center of A to be surrounded by a sphere of radius d. A collision occurs whenever the center 'of another molecule comes within this sphere. If A is traveling with the average speed c, its "sphere of influence" sweeps out in unit time a volume nd^c. Since this volume contains n molecules per cc, there are -nnd^c collisions experienced by A per second. A more exact calculation takes into consideration that only the speed of 172 THE KINETIC THEORY [Chap. 7 a molecule relative to other moving molecules determines the number of collisions Z x that it experiences. This fact leads to the expression Zj -^ V27wd 2 c (7.15) The origin of the A/2 factor may be seen by considering, in Fig. 7.5, the relative velocities of two molecules just before or just after a collision. The 2d Fig. 7.4. Molecular collisions. limiting cases are the head-on collision and the grazing collision. The average case appears to be the 90 collision, after which the magnitude of the relative velocity is V2c. If we now examine the similar motions of all the molecules, the total (b) (C) Fig. 7.5. Relative speeds, (a) Head-on collision. (b) Grazing collision, (c) Right-angle collision. number of collisions per second of all the n molecules contained in one cc of gas is found, from eq. (7.15), to be Z n - (7.16) The factor J is introduced so that each collision is not counted twice (once as A hits B, and once as B hits A). 11. Mean free paths. An important quantity in kinetic theory is the average distance a molecule travels between collisions. This is called the mean free path. The average number of collisions experienced by one mole- cule in one second is, from eq. (7.15)^ -=- \^2irnd 2 c. In this time the Sec. 12] THE KINETIC THEORY 173 molecule has traveled a distance c. The mean free path A is therefore c/Z lt or (7.17) In order to calculate the mean free path, we must know the molecular diameter d. This might be obtained, for example, from the van der Waals b ( 4V m ) if the value of Avogadro's Number N were known. So far, our development of kinetic theory has provided no method for obtaining this number. The theory of gas viscosity as developed by James Clerk Maxwell presents a key to this problem, besides affording one of the most striking demonstrations of the powers of the kinetic theory of gases. 12. The viscosity of a gas. The concept of viscosity is first met in problems of fluid flow, treated by hydrodynamics and aerodynamics, as a measure of -AREA Fig. 7.6. Viscosity of fluids. the frictional resistance that a fluid in motion offers to an applied shearing force. The nature of this resistance may be seen from Fig. 7.6 (a). If a fluid is flowing past a stationary plane surface, the layer of fluid adjacent to the plane boundary is stagnant; successive layers have increasingly higher velocities. The frictional force /, resisting the relative motion of any two adjacent layers, is proportional to S, the area of the interface between them, and to dvjdr, the velocity gradient between them. This is Newton's Law of Viscous Flow, dv f^viS-j- (7.18) dr The proportionality constant r\ is called the coefficient of viscosity. It is evident that the dimensions of rj are ml~ l t~~ l . In the COS system, the unit is g per cm sec, called the poise. 174 THE KINETIC THEORY [Chap. 7 The kind of flow governed by this relationship is called laminar or streamline flow. It is evidently quite different in character from the effusive (or diffusive) flow previously discussed, since it is a massive flow of fluid, in which there is superimposed on all the random molecular velocities a com- ponent of velocity in the direction of flow. An especially important case of viscous flow is the flow through pipes or tubes when the diameter of the tube is large compared with the mean free path in the fluid. The study of flow through tubes has been the basis for many of the experimental determinations cf the viscosity coefficient. The theory of the process was first worked out by J. L. Poiseuille, in 1844. Consider an incompressible fluid flowing through a tube of circular cross section with radius R and length L. The fluid at the walls of the tube is assumed to be stagnant, and the rate of flow increases to a maximum at the center of the tube [see Fig. 7.6 (b)]. Let v be the linear velocity at any distance r from the axis of the tube. A cylinder of fluid of rad'"<; r experiences a viscous drag given by eq. (7.18) as V dv i r Jr --1J/ *L For steady flow, this force must be exactly balanced by the force driving the fluid in this cylinder through the tube. Since pressure is the force per unit area, the driving force is _ /, = wVi - J*a) where P l is the fore pressure and P% the back pressure. Thus, for steady flow, f r f r fp _ p \ On integration, v =--- ----- \ r -- r 2 + const & According to our hypothesis, v = when /-=/?; this boundary condition enables us to determine the integration constant, so that we obtain finally The total volume of fluid flowing through the tube per second is calculated by integrating over each element of cross-sectional area, given by 2-nr dr [see Fig. 7.6 (c)]. Thus dv f\ J ^Pi ~^ 2 )* 4 ~dt^k 2 dr ^ "" " *>^ ( 7 - 19 > ZLrj Sec. 13] THE KINETIC THEORY 175 This is Poiseuille's equation. It was derived for an incompressible fluid and therefore may be satisfactorily applied to liquids but not to gases. For gases, the volume is a strong function of the pressure. The average pressure along the tube is (P l + ^ 2 )/2- If ^o i s tne pressure at which the volume is measured, the equation becomes dV dt (7.20) By measuring the volume rate of flow through a tube of known dimen- sions, the viscosity i] of the gas can be determined. Some results of such measurements are collected in Table 7.4. TABLE 7.4 TRANSPORT PHENOMENA IN GASES (At 0C and 1 aim) Thermal Gas Mean Free Path A, A Viscosity //, Poise x 10~ 6 Conduc- tivity *, cal/g secC Specific Heat, c v cal/g r]C v /K x 10 6 Ammonia 441 97.6 51.3 0.399 0.76 Argon .... 635 213 38.8 0.0750 0.41 Carbon dioxide 397 138 34.3 0.153 0.62 Carbon monoxide 584 168 56.3 0.177 0.53 Chlorine 287 123 18.3 0.0818 0.55 Ethylene 345 93.3 40.7 0.286 0.66 Helium 1798 190 336 0.743 0.42 Hydrogen Nitrogen 1123 600 84.2 167 406 58.0 2.40 0.176 0.50 0.51 Oxygen 647 192 58.9 0.155 0.51 13. Kinetic theory of gas viscosity. The kinetic picture of gas viscosity has been represented by the following analogy: Two railroad trains are moving in the same direction, but at different speeds, on parallel tracks. The passengers on these trains amuse themselves by jumping back and forth from one to the other. When a passenger jumps from the more rapidly moving train to the slower one he transports momentum of amount mv, where m is his mass and v the velocity of his train. He tends to speed up the more slowly moving train when he lands upon it. A passenger who jumps from the slower to the faster train, on the other hand, tends to slow it down. The net result of the jumping game is thus a tendency to equalize the velocities of the two trains. An observer from afar who could not see the jumpers might simply note this result as a frictional drag between the trains. The mechanism by which one layer of flowing gas exerts a viscous drag 176 THE KINETIC THEORY [Chap. 7 on an adjacent layer is exactly similar, the gas molecules taking the role of the playful passengers. Consider in Fig. 7.7 a gas in a state of laminar flow parallel to the Y axis. Its velocity increases from zero at the plane x to larger and larger values of v with increasing x. If a molecule at P crosses to g, in one of its free paths between collisions, it will bring to Q, on the average, an amount of momentum which is less than that common to the position Q by virtue of its distance along the A' axis. Conversely, if a molecule travels from Q to P it will transport to the lower, more slowly moving layer, an amount of momentum in excess of that belonging to that layer. The net result of the random thermal motions of the molecules is to decrease the average velocities of the molecules in the layer at Q and to increase those in the layer at P. This transport of momentum tends to counteract the velocity gradient set up by the shear forces acting on the gas. The length of the mean free path X may be taken as the average dis- tance over which momentum is trans- ferred. 8 If the velocity gradient is du/dx, the difference in velocity be- tween the two ends of the free path is X du/dx. A molecule of mass m, passing from the upper to the lower layer, thus transports momentum equal to wA du/dx. On the average, one-third of the molecules are moving up and down; if n is the number of molecules per cc and c their average speed, the number traveling up and down per second per square cm is -J- nc. The momentum transport per second is then \nc mX(du/dx)* This momentum change with time is equivalent to the frictional force of eq. (7.18) which was/^~ r)(du/dx) per unit area. Hence Fig. 7.7. Kinetic theory of gas viscosity. du ~T dx du - - dx (7.21) The measurement of the viscosity thus allows us to calculate the value of the mean free path A. Some values obtained in this way are included in Table 7.4, in Angstrom units (1 A = 10~ 8 cm). 8 This is not strictly true, and proper averaging indicates f A should be used. 9 The factor J obtained here results from the cancellation of two errors in the derivation. From eq. (7.9) one should take nc as the molecules moving across unit area but proper averaging gives the distance between planes as JA instead of A. Sec. 14] THE KINETIC THEORY 177 By eliminating A between eqs. (7.17) and (7.21), one obtains n - -^- (7.22) 3\/2W 2 This equation indicates that the viscosity of a gas is independent of its density. This seemingly improbable result was predicted by Maxwell on purely theoretical grounds, and its subsequent experimental verification was one of the great triumphs of the kinetic theory. The physical reason for the result is clear from the preceding derivation: At lower densities, fewer molecules jump from layer to layer in the flowing gas, but, because of the longer free paths, each jump carries proportionately greater momentum. For imperfect gases, the equation fails and the viscosity increases with density. The second important conclusion from eq. (7.22) is that the viscosity of a gas increases with increasing temperatuie, linearly with the \/T. This con- clusion has been well confirmed by the experimental results, although the viscosity increases somewhat more rapidly than predicted by the \/T law. 14. Thermal conductivity and diffusion. Gas viscosity depends on the transport of momentum across a momentum (velocity) gradient. It is a typical transport phenomenon. An exactly similar theoretical treatment is applicable to thermal conductivity and to diffusion. The thermal conductivity of gases is a consequence of the transport of kinetic energy across a tem- perature (i.e., kinetic energy) gradient. Diffusion of gases is the transport of mass across a concentration gradient. The thermal conductivity coefficient K is defined as the heat flow per unit time q, per unit temperature gradient across unit cross-sectional area, i.e., by c dT q =. K- S- ax By comparison with eq. (7.21), dT 1 , de K nc/. ax 3 ax where de/dx is the gradient of e, the average kinetic energy per molecule. Now de dT de where m is the molecular mass and c v is the specific heat (heat capacity per gram). It follows that K -^ lnmc v ch \pc v cX ^ r\c v (7.23) Some thermal conductivity coefficients are included in Table 7.4. It should be emphasized that, even for an ideal gas, the simple theory is approxi- mate, since it assumes that all the molecules are moving with the same speed, c, and that energy is exchanged completely at each collision. 178 THE KINETIC THEORY [Chap. 7 The treatment of diffusion is again similar. Generally one deals with the diffusion in a mixture of two different gases. The diffusion coefficient D is the number of molecules per second crossing unit area under unit con- centration gradient. It is found to be 10 D - -J Vi*2 +- iV a *i where X l and X 2 are the mole fractions of the two gases in the mixture. If the two kinds of molecules are essentially the same, for example radioactive chlorine in normal chlorine, the self-diffusion coefficient is obtained as D &c (7.24) The results of the simple mean-free-path treatments of the transport processes may be summarized as follows: Process Transport of Simple Theoretical Expression CGS Units of Coefficient Viscosity Thermal conductivity . Diffusion . Momentum mv Kinetic energy J/w> 2 Mass /// ry - \ P ck K -- \pckc v D ~ \Xc g/cm sec ergs/cm sec degree cm 2 /sec Now van der Waals' b is given by b - - 47V 15. Avogadro's number and molecular dimensions. Equation (7.22) may be written, from eq. (7.8), Me 2V1RTM 71 ^ (7.25) (7.26) Let us substitute the appropriate values for the hydrogen molecule, H 2 , all in CGS units. M= 2.016 6-26.6 ?7-0.93 x 10~ 4 r= 298K R = 8.314 x 10 7 Solving for d, we find d -=-- 2.2 x 10~ 8 cm. 6 3 Multiplying these two equations, and solving for d, 10 For example, see E. H. Kennard, Kinetic Theory of Gases (New York: McGraw-HilJ, 1938), p. 188. Sec. 15] THE KINETIC THEORY 179 This value may be substituted back into eq. (7.25) to obtain a value for Avogadro's Number TV equal to about 10 24 . Because of the known approximations involved in the van der Waals formula, this value of TV is only approximate. It is nevertheless of the correct order of magnitude, and it is interesting that the value can be obtained purely from kinetic-theory calculations. Later methods, which will be dis- cussed in a subsequent chapter, give the value TV 6.02 x 10 23 . We may use this figure to obtain more accurate values for molecular diameters from viscosity or thermal conductivity measurements. Some of these values are shown in Table 7.5, together with values obtained from van der Waals' b, and by the following somewhat different method. TABLE 7.5 MOLECULAR DIAMETERS (Angstrom Units) From From From From Molecule Gas van der Molecular Closest Viscosity Waals' b Refraction* Packing A 2.86 2.86 2.96 3.83 CO 3.80 3.16 4.30 CO 2 4.60 3.24 2.86 C1 2 3.70 3.30 3.30 4.65 He 2.00 2.48 1.48 H 2 2.18 2.76 1.86 Kr 3.18 3.14 3.34 4.02 Hg 3.60 2.38 _ Ne 2.34 2.66 3.20 N 2 3.16 3.14 2.40 4.00 2 2.96 2.90 2.34 3.75 H 2 2.72 2.88 2.26 * The theory of this method is discussed in Section 1 1-18. In the solid state the molecules are closely packed together. If we assume that these molecules are spherical in shape, the closest possible packing of spheres leaves a void space of 26 per cent of the total volume. The volume occupied by a mole of molecules is M/p, where M is the molecular weight and p the density of the solid. For spherical molecules we may therefore write (7r/6)Nd* = Q.14(M/p). Values of d obtained from this equation may be expected to be good approximations for the nionatomic gases (He, Ne, A, Kr) and for spherical molecules like CH 4 , CC1 4 . The equation is only roughly applicable to diatomic molecules like N 2 or O 2 . The rather diverse values often obtained for molecular diameters calcu- lated by different methods are indications of the inadequacy of a rigid-sphere model, even for very simple molecules. The extreme minuteness of the molecules and the tremendous size of the 180 THE KINETIC THEORY [Chap. 7 Avogadro Number N are strikingly shown by two popular illustrations given by Sir James Jeans. If the molecules in a glass of water were turned into grains of sand, there would be enough sand produced to cover the whole United States to a depth of about 100 feet. A man breathes out about 400 cc at each breath, or about 10 22 molecules. The earth's atmosphere contains about 10 44 molecules. Thus, one molecule is the same fraction of a breath of air as the breath is of the entire atmosphere. If the last breath of Julius Caesar has become scattered throughout the entire atmosphere, the chances are that we inhale one molecule from it in each breath we take. 16. The softening of the atom. We noted before that the viscosity of a gas increases more rapidly with temperature than is predicted by the \/T law. This is because the molecules are not actually hard spheres, but must be regarded as being somewhat soft, or surrounded by fields of force. This is true even for the atom-molecules of the inert gases. The higher the tem- perature, the faster the molecules are moving, and hence the further one molecule can penetrate into the field of force of another, before it is repelled or bounced away. The molecular diameter thus appears to be smaller at higher temperatures. This correction has been embodied in a formula due to Sutherland (1893) d*--<Il\\\-) (7.27) Here d^ and C are constants, d^ being interpreted as the value of d as T approaches infinity. More recent work has sought to express the temperature coefficient of the viscosity in terms of the laws of force between the molecules. Thus here, just as in the discussion of the equation of state, the qualitative picture of rigid molecules must be modified to consider the fields of force between molecules. We recall from Chapter 1 that forces may be represented as derivatives of a potential-energy function,/^ (<3(7/cV), and a representation of this function serves to illustrate the nature of the forces. In Fig. 7.8 we have drawn the mutual potential energy of pairs of molecules of several different gases. We may imagine the motion of one molecule as it approaches rapidly toward another to be represented by that of a billiard ball rolled with con- siderable force along a track having the shape of the potential curve. As the molecule approaches another it is accelerated at first, but then slowed down as it reaches the steep ascending portion of the curve. Finally it is brought to a halt when its kinetic energy is completely used up, and it rolls back down and out the curve again. Since the kinetic energy is almost always greater than the depth of the potential-energy trough, there is little chance of a molecule's becoming trapped therein. (If it did, another collision would soon knock it out again.) This softening of the original kinetic-theory picture of the atom as a Sec. 17] THE KINETIC THEORY 181 hard rigid sphere was of the greatest significance. It immediately suggested that the atoms could not be the ultimate building units in the construction of matter, and that man must seek still further for an indestructible reality to explain the behavior of material things. So far in this chapter we have dealt with average properties of large collections of molecules : average velocities, mean free paths, viscosity, and DISTANCE BETWEEN MOLECULES -15 Fig. 7.8. Mutual potential energy of pairs of molecules. so on. In what follows, the contributions of the individual molecules to these averages will be considered in some detail. 17. The distribution of molecular velocities. The molecules of a gas in their constant motion collide many times with one another, and -these collisions provide the mechanism through which the velocities of individual molecules are continually changing. As a result, there exists a distribution of velocities among the molecules; most have velocities with magnitudes close to the average, and relatively few have velocities much above or much below the average. 182 THE KINETIC THEORY [Chap. 7 I sq cm .__f d * A molecule may acquire an exceptionally high speed as the result of a series of especially favorable collisions. The theory of probability shows that the chance of a molecule's experiencing a series of n lucky hits is pro- portional to a factor of the form e~ an , where a is a constant. 11 Thus the probability of the molecule's having the energy E above the average energy is likewise proportional to e~ bE . The exact derivation of this factor may be carried out in several ways, and the problems involved in the distribution of velocities, and hence of kinetic energies, among the molecules, form one of the most important parts of the kinetic theory. 18. The barometric formula. It is common knowledge that the density of the earth's atmosphere decreases with increas- ing altitude. If one makes the simplifying assumption that a column of gas extending upward into the atmosphere is at constant temperature, a formula may be derived for this variation of gas pressure in the gravita- tional field. The situation is pictured in Fig. 7.9. The weight of a thin layer of gas of thick- ness dx and one cm 2 cross section is its mass Fig. 7.9. Barometric formula, times the acceleration due to gravity, or pg dx, where /> is the gas density. The difference in pressure between the upper and lower boundaries of the layer is ( -dP/dx)dx, equal to the weight of the layer of unit cross section. Thus dP = pgdx A ^ PM ror an ideal gas, p -= D _ Kl Therefore RT dx Integrating between the limits P P Q at x 0, and P P at x --= x, P Mgx In RT n r> _ -MgxIRT ("l *)Q\ r r e \i .LO) Now, Mgx is simply the gravitational potential energy at the point x, which may be written as E^ per mole. Then P^P Q e~ E IHT (7.29) If, instead of the molar energy, we consider that of the individual mole- cule, e p , eq. (7.29) becomes . P = P e" e * lltr (7.30) 11 If the chance of one lucky hit is 1/c, the chance P for n in a row is P - (I/c) w . Then Sec. 19] THE KINETIC THEORY 183 The constant k is called the Bohzmann constant. Ft is the gas constant per molecule. Equation (7.30) is but one special case of a very general expression derived by L. Boltzmann in 1886. This states that if A? O is the number of molecules in any given state, the number n in a state whose potential energy is e above that of the given state is n = n^e- hlkT (7.31) 19. The distribution of kinetic energies. 12 To analyze more closely the kinetic picture underlying the barometric formula, let us consider the in- dividual gas molecules moving with their diverse velocities in the earth's gravitational field. The velocity components parallel to the earth's surface (in the y and z directions along which no field exists) are not now of interest and only the vertical or x component u need be considered. The motion of a molecule with an upward velocity u is just like that of a ball thrown vertically into the air. If its initial velocity is w , it will rise with continuously decreasing speed, as its kinetic energy is transformed into potential energy according to the equation mgx i/w/ 2 iww 2 At the height, x =- u Q 2 /2g, it will stop, and then fall back to earth. The gravitational field acts as a device that breaks up the mixture of various molecular velocities into a "spectrum" of velocities. The slowest molecules can rise only a short distance; the faster ones can rise propor- tionately higher. By determining the number of molecules that can reach any given height, we can likewise determine how many had a given initial velocity component. As is to be expected from the physical picture of the process, the dis- tribution of kinetic energies k among the molecules must follow an ex- ponential law just as the potential energy distribution does. Representing the fraction of molecules having a velocity between u and u f du by dnfn^ this law may be written from eq. (7.31) as - Ae~^ lkT du (7.32) "o Here A is a constant whose value is yet to be determined. 20. Consequences of the distribution law. This distribution law is com- pletely unaffected by collisions between molecules, since a collision results only in an interchange of velocity components between two molecules. Expressions exactly similar to eq. (7.32) must also hold for the velocity 12 The method suggested here is given in detail by K. F. Herzfeld in H. S. Taylor's Treatise on Physical Chemistry, 2nd ed. (New York: Van Nostrand, 1931), p. 93. 184 THE KINETIC THEORY [Chap. 7 distributions in the y and z directions, since it is necessary only to imagine some sort of potential field in these directions in order to analyze the velocities into their spectrum. I50 r OJ ca I ~ (T UJ 50 1C 200 400 600 800 1000 2000 u, METERS /SECOND Fig. 7.10. One-dimensional velocity distribution (nitrogen at 0C). The constant A 9 in eq. (7.32), may be evaluated from the fact that the sum of all the fractions of molecules in all the velocity ranges must be unity. Thus, integrating over all possible velocities from oo to + oo, we have r+oo _ WM /2JIT A e du -~ 1 J - oo mu 2 2 2kT ^ X +oo A ' ^ ' ^ Letting 2^7^\i/2 /*+ V/ J-c Since Therefore, eq. (7.32) becomes 1/2 (7.33) This function is shown plotted in Fig. 7.10. It will be noted that the fraction of the molecules with a velocity component in a given range declines Sec. 20] THE KINETIC THEORY 185 at first slowly and then rapidly as the velocity is increased. From the curve and from a consideration of eq. (7.33), it is evident that as long as Jmw 2 < kT the fraction of molecules having a velocity u falls off slowly with increasing u. When %mu 2 = \QkT, the fraction has decreased to e~ 10 , or 5 x 10~ 5 times its value at \rn\f 1 = kT. Thus only a very small proportion of any lot of mole- cules can have kinetic energies much greater than kT per degree of freedom. If, instead of a one-dimensional gas (one degree of freedom of trans- lation), a two-dimensional gas is considered, it can be proved 13 that the probability of a molecule having a given x velocity component u in no way depends on the value of its y component v. The fraction of the molecules having simultaneously velocity components between u and u + du, and v and v -f- dv, is then simply the product of the two individual probabilities. dn I m \ ^ = \27Tkf) (7.34) This sort of distribution may be graphically represented as in Fig. 7.11, where a coordinate system with u and v axes has been drawn. Any point in the (w, v) plane represents a simul- taneous value of u and v\ the plane is a two-dimensional velocity space similar to that used on p. 168. The dots have been drawn so as to represent schematically the density of points in this space, i.e., the relative frequency of occurrence of sets of simultaneous values of u and v. The graph bears a striking re- semblance to a target that has been peppered with shots by a marks- man aiming at the bull's-eye. In the molecular case, each individual Fig . 1Mm Distribution of points in two- molecular-velocity component, u or dimensional velocity space: v x = u; v v - v. v, aims at .the value zero. The resulting distribution represents the statistical summary of the results. The more skilful the marksman, the more closely will his results cluster around the center of the target. For the molecules, the skill of the marksman has its analogue in the temperature of the gas. The lower the temperature, the better the chance a molecular-velocity component has of coming close to zero. If, instead of the individual components u and v, the resultant speed c is considered, where c 2 = u 2 + v 2 9 it is evident that its most probable value is not zero. This is because the number of ways in which c can be made up 18 For a discussion of this theorem see, for example, J. Jeans, Introduction to the Kinetic Theory of Gases (London: Cambridge, 1940), p. 1Q5. 186 THE KINETIC THEORY [Chap. 7 from u and v increases in direct proportion with c, whereas at first the prob- ability of any value of u or v declines rather slowly with increasing velocity. From Fig. 7.11, it appears that the distribution of c, regardless of direc- tion, is obtainable by integrating over the annular area between c and c | dc, which is 2nc dc. The required fraction is then dn m .,.., . __ e (7.35) 21. Distribution law in three dimensions. The three-dimensional distribu- tion law may now be obtained by a simple extension of this treatment. The 200 1000 2000 C, METERS /SECOND Fig. 7.12. Distribution of molecular speeds (nitrogen). 3000 fraction of molecules having simultaneously a velocity component between u and u + du, v and v + dv, and w and w + dw, is dn_ 1 "" m \2irkTj (7.36) Sec. 22] THE KINETIC THEORY 187 We wish an expression for the number with a speed between c and c + dc, regardless of direction, where c 2 = u 2 + v 2 + w 2 . These are the molecules whose velocity points lie within a spherical shell of thickness dc at a distance c from the origin. The volume of this shell is 477-cVc, and therefore the desired distribution function is * This is the usual expression of the distribution equation derived by James Clerk Maxwell in 1860. The equation is plotted in Fig. 7.12 at several different temperatures, showing how the curve becomes broader and less peaked at the higher tem- peratures, as relatively more molecules acquire kinetic energies greater than the average of f kT. 22. The average speed. The average value f of any property r of the molecules is obtained by multiplying each value of r, r,, by the number of molecules n l having this value, adding these products, and then dividing by the total number of molecules. Thus where 2 n i =~ n o * s the total number of molecules. In case n is known as a continuously varying function of r, n(r), instead of the summations of eq. (7.38) we have the integrations Jo rdn \ r ) 1 Too -r - = --L rdn w ( 7 - 39 > ' dn(r) "o j This formula may be illustrated by the calculation of the average mole- cular speed c. Using eq. (7.37), we have c - - f c dn - 47r (^r\ m I" e -""V^V j c n Q J o xlirkT] J o The evaluation of this integral can be obtained 14 from 1 >~~ ( 2a* Making the appropriate substitutions, we find c '=O 1/2 <> 14 Letting x 2 = z, Too if 00 l/C~ a2: \ 00 1 Jo e ' aXxdx ^2J^ e aZdz = 2 I'^/o = 2a Too I pux J Q e d 188 THE KINETIC THEORY [Chap. 7 Similarly, the average kinetic energy can readily be evaluated as \ & dn 2/7 Jo This yields - \kT (7.41) 23. The equipartition of energy. Equation (7.41) gives the average trans- lational kinetic energy of a molecule in a gas. It will be noted that the average energy is independent of the mass of the molecule. Per mole of gas, /C(t,a,,) - INkT- $RT (7.42) For a monatomic gas, like helium, argon, or mercury vapor, this translational kinetic energy is the total kinetic energy of the gas. For diatomic gases, like N 2 or C1 2 , and polyatomic gases, like CH 4 or N 2 O, there may also be energy associated with rotational and vibrational motions. A useful model for a molecule is obtained by supposing that the masses of the constituent atoms are concentrated at points. As will be seen in Chapter 9, almost all the atomic mass is in fact concentrated in a tiny nucleus, the radius of which is about 10 13 cm. Since the over-all dimensions of molecules are of the order of 10~ 8 cm, a model based on point masses is physically most reasonable. Consider a molecule composed of n atoms. In order to represent the instantaneous locations in space of A? mass points, we should require 3/7 coordinates. The number of coordinates required to locate all the mass points (atoms) in a molecule is called the number of its degrees of freedom. Thus a molecule of n atoms has 3/7 degrees of freedom. The atoms within each molecule move through space as a connected entity, and we can represent the translational motion of the molecule as a whole by the motion of the center of mass of its constituent atoms. Three coordinates (degrees of freedom) are required to represent the instantaneous position of the center of mass. The remaining (3/7 3) coordinates represent the so-called internal degrees of freedom. The internal degrees of freedom may be further subdivided into rotations and vibrations. Since the molecule has moments of inertia / about suitably chosen axes, it can be set into rotation about these axes. If its angular velocity about an axis is (, the rotational kinetic energy is i/o> 2 . The vibratory motion, in which one atom in a molecule oscillates about an equilibrium separation from another, is associated with both kinetic and potential energies, being in this respect exactly like the vibration of an ordinary spring. The vibrational kinetic energy is also represented by a quadratic expression, *,mv 2 . The vibrational potential energy can in some cases be represented also by a quadratic expression, but in the coordinates q rather than in the velocities, for example, i/o/ 2 . Each vibrational degree of freedom would then contribute two quadratic r terms to the total energy of the molecule. By an extension of the derivation leading to eq. (7.41), it can be shown Sec. 24] THE KINETIC THEORY 189 that each of these quadratic terms that comprise the total energy of the molecule has an average value of \kT. This conclusion, a direct consequence of the Maxwell-Boltzmann distribution law, is the most general expression of the Principle of Equipartition of Energy. 24. Rotation and vibration of diatomic molecules. The rotation of a di- atomic molecule may be visualized by reference to the so-called dumbbell model in Fig. 7.13, which might represent a molecule such as H 2 , N 2 , HCI, (b) Fig. 7.13. Dumbbell rotator. or CO. The masses of the atoms, m r and w 2 , are concentrated at points, distant r x and r 2 , respectively, from the center of mass. The molecule there- fore has moments of inertia about the X and Z axes, but not about the Y axis on which the mass points lie. The energy of rotation of a rigid body is given by rot - Uco* (7.43) where o> is the angular velocity of rotation, and / is the moment of inertia. For the dumbbell model, / = w^ 2 + w 2 r 2 2 . The distances r x and r 2 from the center of mass are m 2 mi Thus (7.44) The quantity (7.45) is called the reduced mass of the molecule. The rotational motion is equivalent to that of a mass p at a distance r from the intersection of the axes. Only two coordinates are required to describe such a rotation com- pletely; for example, two angles 6 and <f> suffice to fix the orientation of the rotator in space. There are thus two degrees of freedom for the rotation of a dumbbell-like structure. According to the principle of the equipartition of 190 THE KINETIC THEORY [Chap. 7 energy, the average rotational energy should therefore be rot = - RT. The simplest model for a vibrating diatomic molecule (Fig. 7.14) is the harmonic oscillator. From mechanics we know that simple harmonic motion occurs when a particle is acted on by a restor- ing force directly proportional to its distance , t*99$w$9999f{pt , ft from the equilibrium position. Thus Fig. 7.14. Harmonic oscil- lator. The constant K is called the force constant. The motion of a particle under the influ- ence of such a restoring force may be represented by a potential energy function U(r). f P u \ f= -\Sr-)^- lcr U(r) - Jicr 2 (7.47) This is the equation of a parabola and the potential-energy curve is drawn in Fig. 7.15. The motion of the partide, as has been pointed out in previous cases, is analogous to that of a ball moving on such a surface. Starting from rest at any position r, it has only potential energy, U = i/cr 2 . As it rolls down the surface, it gains kinetic energy up to a maximum at position r 0, the equi- librium interatomic distance. The kinetic energy is then reconverted to potential energy as the ball rolls up the other side of the incline. The total energy at any time is always a constant, U(r) E vih -r +r ^ Fig. 7.15. Potential curve of harmonic oscillator. It is apparent, therefore, that vibrating molecules when heated can take up energy as both potential and kinetic energy of vibration. The equipartition principle states that the average energy for each vibrational degree of freedom is therefore kT, \kT for the kinetic energy plus \kT for the potential energy. For a diatomic molecule the total average energy per mole therefore becomes ~ ^tnms + rot + ^vib =- $RT + RT + RT - 25. Motions of polyatomic molecules. The motions of polyatomic mole- cules can also be represented by the simple mechanical models of the rigid rotator and the harmonic oscillator. If the molecule contains n atoms, there Sec. 25] THE KINETIC THEORY 191 are (3n 3) internal degrees of freedom. In the case of the diatomic molecule, 3n 3 -- 3. Two of the three internal coordinates are required to represent the rotation, leaving one vibrational coordinate. In the case of a triatomic molecule, 3/7 3 6. In order to divide these six internal degrees of freedom into rotations and vibrations, we must first consider whether the molecule is linear or bent. If it is linear, all the atomic mass points lie on one axis, and there is therefore no moment of inertia about this axis. A linear molecule behaves like a diatomic molecule in regard to rotation, and there are only two rotational degrees of freedom. For a linear triatomic molecule, there are thus 3n 3 2 4 vibrational degrees of freedom. The average energy of the molecules according to the Equipartition Principle would therefore be E ~~~ ^trans I ^rot ^ ^vih =- 3(\RT) + 2(\RT) ^ 4(RT] 6 1 2 RT per mole A nonlinear (bent) triatomic molecule has three principal moments of inertia, and therefore three rotational degrees of freedom. Any nonlinear polatomic molecule has 3/76 vibrational degrees of freedom. For the triatomic case, there are therefore three vibrational degrees of freedom. The average energy according to the Equipartition Principle would be E 3(1 RT) ~\ 3(1 RT) f 3(RT) 6RT per mole Examples of linear triatomic molecules are HCN, CO 2 , and CS 2 . Bent triatomic molecules include H 2 O and SO 2 . The vibratory motion of a collection of mass points bound together by linear restoring forces [i.e., a polyatomic molecule in which the individual atomic displacements obey eq. (7.46)] may be quite complicated. It is always possible, however, to represent the complex vibratory motion by means of a number of simple motions, the so-called normal modes of vibration. In a normal mode of vibration, each atom in the molecule is oscillating with the same frequency. Examples of the normal modes for linear and bent triatomic molecules are shown in Fig. 7.16. The bent molecule has three distinct normal modes, each with a characteristic frequency. The frequencies of course have different numerical values in different compounds. In the case of the linear molecule, there are four normal modes; two correspond to stretching of the molecule (v l9 v 3 ) and two correspond to bending (v 2a , v 2b ). The two bending vibrations differ only in that one is in the plane of the paper and one normal to the plane (denoted by + and ). These vibrations have the same frequency, and are called degenerate vibrations. When we described the translational motions of molecules and their consequences for the kinetic theory of gases, it was desirable at first to employ a very simplified model. The same procedure has been followed in this dis- cussion of the internal molecular motions. Thus diatomic molecules do not 192 THE KINETIC THEORY [Chap. 7 really behave as rigid rotators, since, at rapid rotation speeds, centrifugal force tends to separate the atoms by stretching the bond between them. 9 6 '2a I/I V 2 1/3 Fig. 7.16. Normal modes of vibration of triatomic molecules. Likewise, a more detailed theory shows that the vibrations of the atoms are not strictly harmonic. 26. The equipartition principle and the heat capacity of gases. According to the equipartition principle, a gas on warming should take up energy in all its degrees of freedom, \RT per mole for each translational or rotational coordinate, and RT per mole for each vibration. The heat capacity at con- stant volume, C v --= (3E/DTV, could then be readily calculated from the average energy. From eq. (7.42) the translational contribution to C v is (f)/?. Since R *= 1 .986 cal per degree C, the molar heat capacity is 2.98 cal per degree C. When this figure is compared with the experimental values in Table 7.6, it is found to be confirmed for the monatomic gases, He, Ne, A, Hg, which TABLE 7.6 MOLAR HEAT CAPACITY C v OF GASES Gas He, Ne, A, Hg H 2 . N 2 . . O a . . CI 2 . . H 2 O C0 a . . Temperature (C) -100 2.98 4.18 4.95 4.98 100 400 600 2.98 2.98 2.98 2.98 4.92 4.97 4.99 5.00 4.95 4.96 5.30 5.42 5.00 5.15 5.85 6.19 5.85 5.88 6.24 6.40 6.37 6.82 7.60 6.75 7.68 9.86 10.90 Sec. 27] THE KINETIC THEORY 193 have no internal degrees of freedom. The observed heat capacities of the diatomic and polyatomic gases are always higher, and increase with tem- perature, so that it may be surmised that rotational and vibrational contri- butions are occurring. For a diatomic gas, the equipartition principle predicts an average energy of (%)RT, or C v ()R -= 6.93. This value seems to be approached at high 'temperatures for H 2 , N 2 , O 2 , and C1 2 , but at lower temperatures the experi- mental C v values fall much below the theoretical ones. For polyatomic gases, the discrepancy with the simple theory is even more marked. The equi- partition principle cannot explain why the observed C r is less than predicted, why C v increases with temperature, nor why the C r values differ for the different diatomic gases. The theory is t)ius satisfactory for translational motion, but most unsatisfactory when applied to rotation and vibration. Since the equipartition principle is a direct consequence of the kinetic theory, and in particular of the Maxwell-Boltzmann distribution law, it is evident that an entirely new basic theory will be required to cope with the heat capacity problem. Such a development is found in the quantum theory introduced in Chapter 10. 27. Brownian motion. In 1827, shortly after the invention of the achro- matic lens, the botanist Robert Brown 15 studied pollen grains under his microscope and watched a curious behavior. While examining the form of these particles immersed in water, I observed many of them very evidently in motion; their motion consisting not only of a change of place in the fluid, manifested by alterations of their relative positions, but also not infrequently of a change in form of the particle itself; a contraction or curvature taking place repeatedly about the middle of one side, accompanied by a correspond- ing swelling or convexity on the opposite side of the particle. In a few instances the particle was seen to turn on its longer axis. These motions were such as to satisfy me, after frequently repeated observations, that they arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself. In 1888, G. Gouy proposed that the particles were propelled by collisions with the rapidly moving molecules of the suspension liquid. Jean Perrin recognized that the microscopic particles provide a visible illustration of many aspects of the kinetic theory. The dancing granules should be governed by the same laws as the molecules in a gas. One striking confirmation of this hypothesis was discovered in Perrin's work on the distribution of colloidal particles in a gravitational field, the sedimentation equilibrium. By careful fractional centrifuging, he was able to prepare suspensions of gamboge 16 particles that were spherical in shape and very uniform in size. It was possible to measure the radius of the particles either microscopically or by weighing a counted number. If these granules 15 Brown, Phil. Mag., 4, 161 (1828); 6, 161 (1829); 8, 41 (1830). 18 Gamboge is a gummy material from the desiccation of the latex secreted by garcinia more/la (Indo-China). It is used as a bright yellow water color. 194 THE KINETIC THEORY [Chap. 7 behave in a gravitational field like gas molecules, their equilibrium distribu- tion throughout a suspension should obey the Boltzmann equation (7.48) Instead of m we may write $7rr 3 (p p t ) where r is the radius of the particle, and p and p t are the densities of the gamboge and of the suspending liquid. Then eq. (7.48) becomes - Pi) RT N (7.49) Fig. 7.17. Sedimentation equilibrium. By determining the difference in the numbers of particles at heights separated by h, it is possible to calculate a value for Avogadro's Number N. A drawing of the results of Perrin's micro- scopic examination of the equilibrium distri- bution with granules of gamboge 0.6/* in diameter 17 is shown in Fig. 7.17. The relative change in density observed in 10/j of this suspension is equivalent to that occurring in 6 km of air, a magnification of six hundred million. The calculation from eq. (7.49) resulted in a value of N 6.5 x 10 23 . This value is in good agreement with other determinations, and is evidence that the visible microscopic particles are behaving as giant molecules in accordance with the kinetic theory. These studies were welcomed at the time as a proof of molecular reality. 28. Thermodynamics and Brownian motion. A striking feature of the Brownian motion of microscopic particles is that it never stops, but goes on continuously without any diminution of its activity. This perpetual motion is not in contradiction with the First Law, for the source of the energy that moves the particles is the kinetic energy of the molecules of the suspending liquid. We may assume that in any region where the colloid particles gain kinetic energy, there is a corresponding loss in kinetic energy by the molecules of the fluid, which undergoes a localized cooling. This amounts to perpetual motion of the second kind, for the transformation of heat into mechanical energy is prohibited by the Second Law, unless there is an accompanying transfer of heat from a hot to a cold reservoir. The study of Brownian motion thus reveals an important limitation of the scope of the Second Law, which also allows us to appreciate its true nature. The increase in potential energy in small regions of a colloidal sus- pension is equivalent to a spontaneous decrease in the entropy of the region. On the average, of course, over long periods of time the entropy of the entire 17 1 micron (/<) - 10~ 3 mm = I0~ 6 m. Sec. 29] THE KINETIC THEORY 195 system does not change. In any microscopic region, however, the entropy fluctuates, sometimes increasing and sometimes decreasing. On the macroscopic scale such fluctuations are never observed, and the Second Law is completely valid. No one observing a book lying on a desk would expect to see it spontaneously fly up to the ceiling as it experienced a sudden chill. Yet it is not impossible to imagine a situation in which all the molecules in the book moved spontaneously in a given direction. Such a situation is only extremely improbable, since there are so many molecules in any macroscopic portion of matter. 29. Entropy and probability. The law of the increase of entropy is thus a probability law. When the number of molecules in a system becomes sufficiently small, the probability of observing a spontaneous decrease in entropy becomes appreciable. The relation between entropy and probability may be clarified by con- sidering (Fig. 7.18) two different gases, A and /?, in separate containers. oo oo 00 00 O 00 ~ _ *T. B A+B Fig. 7.18. Increase in randomness and entropy on mixing. Mien the partition is removed the gases diffuse into each other, the process Continuing until they are perfectly mixed. If they were originally mixed, we should never expect them to become spontaneously unmixed by diffusion, since this condition would require the simultaneous adjustment of some 10 24 different velocity components per mole of gas. The mixed condition is the condition of greater randomness, of greater disorder; it is the condition of greater entropy since it arises spontaneously from unmixed conditions. [The entropy of mixing was given in eq. (3.42).] Hence entropy is sometimes considered a measure of the degree of disorder or of randomness in a system. The system of greatest randomness is also the system of highest statistical probability, for there are many arrange- ments of molecules that can comprise a disordered system, and much fewer for an ordered system. When one considers how seldom thirteen spades are dealt in a bridge hand, 18 one can realize how much more probable is the mixed condition in a system containing 10 24 molecules. Mathematically, the probabilities of independent individual events are multiplied together to obtain the probability of the combined event. The 18 Once in 653,013,559,600 deals, if the decks are well-shuffled and the dealers virtuous. 196 THE KINETIC THEORY [Chap. 7 probability of drawing a spade from a pack of cards is 1/4; the probability of drawing two spades in a row is (1/4)(12/51); the probability of drawing the ace of spades is (1/4)0/13) - 1/52. Thus W 12 = WJV<^ Entropy, on the other hand, is an additive function, S 12 = S l + S 2 . This difference enables us to state that the relation between entropy S and probability W must be a logarithmic one. Thus, S - a In W f- b (7.50) The value of the constant a may be derived by analyzing from the view- point of probability a simple change for which the AS is known from thermodynamics. Consider the expansion of one mole of an ideal gas, originally at pressure P l in a container of volume V l9 into an evacuated container of volume K 2 . The final pressure is P 2 and the final volume, K, I V 2 . For this change, -* (7.5,, When the containers are connected, the probability w l of finding one given molecule in the first container is simply the ratio of the volume V l to the total volume V l -(- K 2 or vv t --= V^\(V^ ! K 2 ). Since probabilities are multiplicative, the chance of finding all TV molecules in the first container, I.e., the probability W l of the original state of the system, is Since in the final state all the molecules must be in one or the other of the containers, the probability W 2 ~ \ N 1. Therefore from eq. (7.50), Comparison with eq. (7.51) shows that a is equal to k, the Boltzmann constant. Thus S - k In W + b W AS - S 2 - S l -.-- k In y/ 2 (7.52) W v This relation was first given by Boltzmann in 1896. For physicochemical applications, we are concerned always with entropy changes, and may conveniently set the constant b equal to zero. 19 The application of eq. (7.52) cannot successfully be made until we have more detailed information about the energy states of atoms and molecules. 19 A further discussion of this point is to be found in Chapter 12. Sec. 29] THE KINETIC THEORY 197 This information will allow us to calculate W and hence the entropy and other thermodynamic functions. The relative probability of observing a decrease in entropy of AS below the equilibrium value may be obtained by inverting eq. (7.52): ~ -=- *-**'* (7.53) rreq For one mole of helium, S/k at 273 -^ 4 x 10 25 . The chance of observing an entropy decrease one-millionth of this amount is about e~ w ". It is evi- dent, therefore, that anyone observing a book flying spontaneously into the air is dealing with a poltergeist and not an entropy fluctuation (probably!). Only when the system is very small is there an appreciable chance of ob- serving a large relative decrease in entropy. A further analysis may be made of the driving force of a chemical re- action or other change, AF =^ - A// -f T&S. It is made up of two terms, the heat of the reaction and the increase in randomness times the tempera- ture. The higher the temperature, the greater is the driving force due to the increase in disorder. This may be physically clearer in the converse state- ment: The lower the temperature, the more likely it is that ordered states can persist. The drive toward equilibrium is a drive toward minimum potential energy and toward maximum randomness. In general, both can- not be achieved in the same system under any given set of conditions. The free-energy minimum represents (at constant T and P) the most satisfactory compromise that can be attained. PROBLEMS 1. At what speeds would molecules of hydrogen and oxygen have to leave the surface of (a) the earth, (b) the moon, in order to escape into space? At what temperatures would the average speeds of these molecules equal these "speeds of escape"? The mass of the moon can be taken as -fa that of the earth. 2. Calculate the number of (a) ergs per molecule, (b) kcal per mole corresponding to one electron volt per molecule. The electron volt is the energy acquired by an electron in falling through a potential difference of one volt. What is the mean kinetic energy of a molecule at 25C in ev? What is A: in ev per C? 3. The density of nitrogen at 0C and 3000 atm is 0.835 g per cc. Cal- culate the average distance apart of the centers of the molecules. How does this compare with the molecular diameter calculated from van der Waals' b = 39.1 cc per mole? 4. In the method of Knudsen [Ann. Physik, 29, 179 (1909)], the vapor 198 THE KINETIC THEORY [Chap. 7 pressure is determined by the rate at which the substance, under its equi- librium pressure, diffuses through an orifice. In one experiment, beryllium powder was placed inside a molybdenum bucket having an effusion hole 0.318 cm in diameter. At 1537K, it was found that 0.00888 g of Be effused in 15.2 min. Calculate the vapor pressure of Be at 1537K. 5. Two concentric cylinders are 10cm long, and 2.00 and 2.20cm in diameter. The space between them is filled with nitrogen at 10~ 2 mm pressure. Estimate the heat flow by conduction between the two cylinders when they differ in temperature by 10C. 6. At 25 C what fraction of the molecules in hydrogen gas have a kinetic energy within kT 10 per cent? What fraction at 500C? What fraction of molecules in mercury vapor? 7. Derive an expression for the fraction of molecules in a gas that have an energy greater than a given value E in two degrees of freedom. 8. Show that the most probable speed of a molecule in a gas equals V2kT/m. 9. Derive the expression (\mc 2 ) %kT from the Maxwell distribution law. 10. In a cc of oxygen at 1 atm and 300K, how many molecules have translational kinetic energies greater than 2 electron volts? At 1000K? 11. What is the mean free path of argon at 25C and a pressure of I atm? Of 10- 5 atm? 12. A pinhole 0.2 micron in diameter is punctured in a liter vessel con- taining chlorine gas at 300 K and 1 mm pressure. If the gas effuses into a vacuum, how long will it take for the pressure to fall to 0.5 mm? 13. Perrin studied the distribution of uniform spherical (0.212^ radius) grains of gamboge (p = 1.206) suspended in water at 15C by taking counts on four equidistant horizontal planes across a cell 100/< deep. The relative concentrations of grains at the four levels were level: 5/< 35 // 65 /< 95 ju concentration: 100 47 22.6 12 Estimate Avogadro's Number from these data. 14. Show that the number of collisions per second between unlike mole- cules, A and 8, in one cc of gas is where the reduced mass, JLI (tn A m Ii )/(m A + m B ). In an equimolar mixture of H 2 and I 2 at 500K and 1 atm calculate the number of collisions per sec per cc between H 2 and H 2 , H 2 and I 2 , I 2 and I 2 . For H 2 take d =r 2. 18 A, for U, d -- 3.76 A. Chap. 7] THE KINETIC THEORY 199 15. The f )rce constant of O 2 is 1 1.8 x 10 5 dynes per cm and r (> 1.21 A. Estimate the potential energy per mole at r = 0.8r . 16. Calculate the moments of inertia of the following molecules: (a) NaCl, r -= 2.51 A; (b) H 2 O, 'OH = - 9 57 A, L HOH - 105 3'. 17. In Fig. 7.18, assume that there are 10 white balls and 10 black balls distributed at random between the two containers of equal volume. What is the AS between the random configuration and one in which there are 8 white balls and 2 black balls in the left-hand container, and 2 whites and 8 blacks in the right. Calculate the answer by eq. (7.52) and also by eq. (3.42). What is the explanation of the different answers? 18. In a carefully designed high vacuum system it is possible to reach a pressure as low as 10~ 10 mm. Calculate the mean free path of helium at this pressure and 25C. 19. The permeability constant at 20C of pyrex glass to helium is given as 6.4 x 10~ 12 cc sec" 1 per cm 2 area per mm thickness per cm of Hg pressure difference. The helium content of the atmosphere at sea level is about 5 x 10~ 4 mole per cent. Suppose a 100 cc round pyrex flask with walls 2 mm thick was evacuated to 10~ 10 mm and sealed. What would be the pressure at the end of one year due to inward diffusion of helium? REFERENCES BOOKS 1. Herzfeld, K. F. and H. M. Smallwood, "Kinetic Theory of Ideal Gases," in Treatise on Physical Chemistry, vol. II, edited by H. S. Taylor and S. Glasstone (New York: Van Nostrand, 1951). 2. Jeans, J. H., Introduction to the Kinetic Theory of Gases (London: Cam- bridge, 1940). 3. Kennard, E. H., Kinetic Theory of Gases (New York: McGraw-Hill, 1938). 4. Knudsen, M., The Kinetic Theory of Gases (London: Methuen, 1950). 5. Loeb, L. B., Kinetic Theory of Gases (New York: McGraw-Hill, 1927). ARTICLES 1. Furry, W. H., Am. J. Phys., 16, 63-78 (1948), "Diffusion Phenomena in Gases." 2. Pease, R. N., J. Chem. Ed., 16, 242-47, 366-73 (1939), "The Kinetic Theory of Gases." 3. Rabi, I. L, Science in Progress, vol. IV (New Haven: Yale Univ. Press, 1945), 195-204, "Streams of Atoms." 4. Rodebush, W. H., /. Chem. Ed., 27, 39-43 (1950), "The Dynamics of Gas Flow." 5. Wheeler, T. S., Endeavour, 11, 47-52 (1952), "William Higgins, Chemist." CHAPTER 8 The Structure of the Atom 1. Electricity. The word "electric" was coined in 1600 by Queen Eliza- beth's physician, William Gilbert, from the Greek, r/Aocrpov, "amber." It was applied to bodies that when rubbed with fur acquired the property of attracting to themselves small bits of paper or pith. Gilbert was un- willing to admit the possibiliy of "action at a distance," and in his treatise De Magnete he advanced an ingenious theory for the electrical attraction. An effluvium is exhaled by the amber and is sent forth by friction. Pearls carnelian, agate, jasper, chalcedony, coral, metals, and the like, when rubbed are inactive; but is there nought emitted from them also by heat and friction? There is indeed, but what is emitted from the dense bodies is thick and vaporous [and thus not mobile enough to cause attractions]. A breath, then . . . reaches the body that is to be attracted and as soon as it is reached it is united to the attracting electric. For as no action can be performed by matter save by contact, these electric bodies do not appear to touch, but of necessity something is given out from the one to the other to come into close contact therewith, and to be a cause of incitation to it. Further investigation revealed that materials such as glass, after rubbing with silk, exerted forces opposed to those observed with amber. Two varieties of electricity were thus distinguished, the vitreous and the resinous. Two varieties of effluvia, emanating from the pores of the electrics, were invoked in explanation. Electricity was supposed to be an imponderable fluid similar in many ways to "caloric." Frictional machines for generating high electro- static potentials were devised, and used to charge condensers in the form of Leiden jars. Benjamin Franklin (1747) considerably simplified matters by proposing a one-fluid theory. According to this theory, when bodies are rubbed together they acquire a surplus or deficit of the electric fluid, depending on their relative attraction for it. The resultant difference in charge is responsible for the observed forces. Franklin established the convention that the vitreous type of electricity is positive (fluid in excess), and the resinous type is negative (fluid in defect). In 1791, Luigi Galvani accidentally brought the bare nerve of a partially dissected frog's leg into contact with a discharging electrical machine. The sharp convulsion of the leg muscles led to the discovery of galvanic elec- tricity, for it was soon found that the electric machine was unnecessary and that the twitching could be produced simply by bringing the nerve ending and the end of the leg into contact through a metal strip. The action was Sec. 2] THE STRUCTURE OF THE ATOM 201 enhanced when two dissimilar metals completed the circuit. Galvani, a physician, named the new phenomenon "animal electricity" and believed that it was characteristic only of living tissues. Alessandro Volta, a physicist, Professor of Natural Philosophy at Pavia, soon discovered that the electricity was of inanimate origin; and using dissimilar metals in contact with moist paper, he was able to charge an electroscope. In 1800 he constructed his famous "pile," consisting of many consecutive plates of silver, zinc, and cloth soaked in salt solution. From the terminals of the pile the thitherto static-electrical manifestations of shock and sparks were obtained. The news of Volta's pile was received with an enthusiasm and amazement akin to that occasioned by the uranium pile in 1945. In May of 1800, Nicholson and Carlyle decomposed water into hydrogen and oxygen by means of the electric current, the oxygen appearing at one pole of the pile and the hydrogen at the other. Solutions of various salts were soon decom- posed, and in 1806-1807, Humphry Davy used a pile to isolate sodium and potassium from their hydroxides. The theory that the atoms in a compound were held together by the attraction between unlike charges immediately gained a wide acceptance. 2. Faraday's Laws and electrochemical equivalents. In 1813 Michael Faraday, then 22 years old and a bookbinder's apprentice, went to the Royal Institution as Davy's laboratory assistant. In the following years, he carried out the series of researches that were the foundations of electro- chemistry and electromagnetism. Faraday studied intensively the decomposition of solutions of salts, acids, and bases by the electric current. With the assistance of the Rev. William Whewell, he devised the nomenclature universally used in these studies: electrode, electrolysis, electrolyte, ion, anion, cation. The positive electrode is called the anode (oo>, "path"); the negative ion (IOP, "going"), which moves toward the anode, is called the anion. The positive ion, or cation, moves toward the negative electrode, or cathode. Faraday proceeded to study quantitatively the relation between the amount of electrolysis, or chemical action produced by the current, and the quantity of electricity. The unit of electric quantity is now the coulomb or ampere second. The results were summarized as follows: 1 The chemical power of a current of electricity is in direct proportion to the absolute quantity of electricity which passes. . . . The substances into which these [electrolytes] divide, under the influence of the electric current, form an exceedingly important general class. They are combining bodies, are directly associated with the fundamental parts of the doctrine of chemical affinity; and have each a definite proportion, in which they are always evolved during electrolytic action. I have proposed to call . . . the numbers representing the proportions in which they are evolved electrochemical equivalents. Thus hydrogen, oxygen, chlorine, iodine, lead, 1 Phil. Trans. Roy. Soc., 124, 77 (1834). 202 THE STRUCTURE OF THE ATOM [Chap. 8 tin, are ions\ the three former are anions, the two metals are cations, and 1, 8, 36, 125, 104, 58 are their electrochemical equivalents nearly. Electrochemical equivalents coincide, and are the same, with ordinary chemical equivalents. I think I cannot deceive myself in considering the doctrine of definite electrochemical action as of the utmost importance. It touches by its facts more directly and closely than any former fact, or set of facts, have done, upon the beautiful idea that ordinary chemical affinity is a mere consequence of the electrical attractions of different kinds of matter. . . . A very valuable use of electrochemical equivalents will be to decide, in cases of doubt, what is the true chemical equivalent, or definite proportional, or atomic number [weight] of a body. ... I can have no doubt that, assuming hydrogen as 1, and dismissing small fractions for the simplicity of expression, the equivalent number or atomic weight of oxygen is 8, of chlorine 36, of bromine 78.4, of lead 103.5, of tin 59, etc , notwithstanding that a very high authority doubles several of these numbers. The "high authority" cited was undoubtedly Jons Jakob Berzelius, who was then using atomic weights based on combining volumes and gas-density measurements. Faraday believed that when a substance was decomposed, it always yielded one positive and one negative ion. Since the current liberates from water eight grams of oxygen for each gram of hydrogen, he concluded that the formula was HO and that the atomic weight of oxygen was equal to 8. It will be recalled that the work of Avogadro, which held the key to this problem, was lying forgotten during these years. 3. The development of valence theory. Much new knowledge about the combinations of atoms was being gained by the organic chemists. Especially noteworthy was the work of Alexander Williamson. In 1850 he treated potassium alcoholate with ethyl iodide and obtained ordinary ethyl ether. At that time, most chemists, using O 8, C 6, were writing alcohol as C 4 H 5 OOH, and ether C 4 H 5 O. If O 1 6, C 1 2 were used, the formulas would be Williamson realized that his reaction could be readily explained on this basis as r\ i p if i _ i/i i -25 r\ K f W ~ K1 * C 2 HJ The older system could still be maintained, however, if a two-step reaction was postulated : C 4 H 5 O-OK KO I C 4 H 5 O C 4 H 5 l 4 KO - KI + C 4 H 5 O Williamson settled the question by treating potassium ethylate with methyl iodide. If the reaction proceeded in two steps, he should obtain equal amounts of diethyl and dimethyl ethers: C 4 H 5 O pK ------ KO + C 4 H 5 O ' C 2 H 3 l + 'KO - KI + C 2 H 3 O Sec. 3] THE STRUCTURE OF THE ATOM 203 On the other hand, if the oxygen atom held two radicals, a new compound, methyl ethyl ether, should be the product : 4- 3 - KI + 25 K t- j j- iu+ CH The new compound was indeed obtained. This was the first unequivocal chemical demonstration that the formulas based on C 12, O -- 16, must be correct. The concept of valence was gradually developed as a result of such organic-chemical researches. It should be mentioned that as early as 1819 two other important criteria for establishing atomic weights were proposed. Pierre Dulong and Alexis Petit pointed out that, for most solid elements, especially the metals, the product of the specific heat and the atomic weight appeared to be a constant, with a value of around 6 calories per C. If this relation is accepted as a general principle, it provides a guide by which the proper atomic weight can be selected from a number of multiples. In the same year, Eilhard Mitscherlich published his work on isomor- phism of crystals, based on an examination of such series as the alums and the vitriols. He found that one element could often be substituted for an analogous one in such a series without changing the crystalline form, and concluded that the substitute elements must enter into the compound in the same atomic proportions. Thus if alum is written KA1(SO 4 ) 2 -12 H 2 O, ferric alum must be KFe(SO 4 ) 2 -12 H 2 O, and chrome alum must be KCr(SO 4 ) 2 -12 H 2 O. The analyst is thus enabled to deduce a consistent set of atomic weights for the analogous elements in the crystals. Avogadro's Hypothesis, when resurrected at the 1860 conference, re- solved all remaining doubts, and the old problem of how to determine the atomic weights was finally solved. We now recognize that ions in solution may bear more than one elemen- tary charge, and that the electrochemical equivalent weight is the atomic weight M divided by the number of charges on the ion z. The amount of electricity required to set free one equivalent is called the faraday, and is equal to 96,519 coulombs. The fact that a definite quantity of electric charge, or a small integral multiple thereof, was always associated with each charged atom in solu- tion strongly suggested that electricity was itself atomic in nature. Hence, in 1874, G. Johnstone Stoney addressed the British Association as follows: Nature presents us with a single definite quantity of electricity which is inde- pendent of the particular bodies acted on. To make this clear, I shall express Faraday's Law in the following terms. . . . For each chemical bond which is ruptured within an electrolyte a certain quantity of electricity traverses the electrolyte which is the same in all cases. In 1891, Stoney proposed that this natural unit of electricity should be 204 THE STRUCTURE OF THE ATOM [Chap. 8 given a special name, the electron. Its magnitude could be calculated by dividing the faraday by Avogadro's Number. 4. The Periodic Law. The idea that matter was constituted of some ninety different kinds of fundamental building blocks was not one that could appeal for long to the mind of man. We have seen how during the nineteenth century evidence was being accumulated from various sources, especially the kinetic theory of gases, that the atom was not merely a minute billiard ball, a more detailed structure being required to explain the interactions between atoms. In 1815, William Prout proposed that all atoms were composed of atoms of hydrogen. In evidence for this hypothesis, he noted that all the atomic weights then known were nearly whole numbers. Prout's hypothesis won many converts, but their enthusiasm was lessened by the careful atomic weight determinations of Jean Stas, who found, for example, that chlorine had a weight of 35.46. Attempts to correlate the chemical properties of the elements with their atomic weights continued, but without striking success till after 1860, when unequivocal weights became available. In 1865, John Newlands tabulated the elements in the order of their atomic weights, and noted that every eighth element formed part of a set with very similar chemical properties. This regularity he unfortunately called "The Law of Octaves." The suggested similarity to a musical scale aroused a good deal of scientific sarcasm, and the importance of Newland's observations was drowned in the general merriment. From 1868 to 1870, a series of important papers by Julius Lothar Meyer and Dmitri Mendeleev clearly established the fundamental principles of the Periodic Law. Meyer emphasized the periodic nature of the physical pro- perties of the elements. This periodicity is illustrated by the well-known graph of atomic volume vs. atomic weight. Mendeleev arranged the elements in his famous Periodic Table. This Table immediately systematized inorganic chemistry, made it possible to predict the properties of undiscovered elements, and pointed strongly to the existence of an underlying regularity in atomic architecture. Closer examination revealed certain defects in the arrangement of ele- ments according to their atomic weights. Thus the most careful determina- tions showed that tellurium had a higher atomic weight than iodine, despite the positions in the Table obviously required by their properties. After Sir William Ramsay's discovery of the rare gases (1894-1897), it was found that argon had an atomic weight of 39. 88, which was greater than that of potassium, 39.10. Such exceptipns to the arrangement by weights suggested that the whole truth behind the Periodic Law was not yet realized. Sec. 5] THE STRUCTURE OF THE ATOM 205 5. The discharge of electricity through gases. The answer to this and many other questions about atomic structure was to be found in a quite unexpected quarter the study of the discharge of electricity through gases. William Watson, 2 who proposed a one-fluid theory of electricity at the same time as Franklin, was the first to describe the continuous discharge of an electric machine through a rarefied gas (1748). It was a most delightful spectacle, when the room was darkened, to see the electricity in its passage: to be able to observe not, as in the open air, its brushes or pencils of rays an inch or two in length, but here the corruscations were of the whole length of the tube between the plates, that is to say, thirty-two inches. Progress in the study of the discharge was retarded by the lack of suitable air pumps. In 1855, Geissler invented a mercury pump that permitted the attainment of higher degrees of vacuum. In 1858, Julius Pliicker observed the deflection of the negative glow in a magnetic field and in 1869 his student, Hittorf, found that a shadow was cast by an opaque body placed between the cathode and the fluorescent walls of the tube, suggesting that rays from the cathode were causing the fluorescence. In 1876, Eugen Goldstein called these rays cathode rays and confirmed the observation that they traveled in straight lines and cast shadows. Sir William Crookes (1879) regarded the rays as a torrent of negatively ionized gas molecules repelled from the cathode. The charged particle theory was contested by many who believed the rays were electromagnetic in origin, and thus similar to light waves. This group was led by Heinrich Hertz, who showed that the cathode radiation could pass through thin metal foils, which would be impossible if it were composed of massive particles. Hermann von Helmholtz, however, strongly championed the particle theory; in a lecture before the Chemical Society of London in 1881 he declared : If we accept the hypothesis that the elementary substances are composed of atoms, we cannot avoid concluding that electricity also, positive as well as negative, is divided into definite elementary portions which behave like atoms of electricity. 6. The electron. In 1895, Wilhelm Roentgen discovered that a very pene- trating radiation was emitted from solid bodies placed in the path of cathode rays. An experimental arrangement for the production of these "X rays" is shown in Fig. 8.1. J. J. Thomson in his Recollections and Reflections* has described his first work in this field: It was a most fortunate coincidence that the advent of research students at the Cavendish Laboratory came at the same time as the announcement by Roentgen of his discovery of the X rays. I had a copy of his apparatus made and set up at the Laboratory, and the first thing I did with it was t6 see what effect the passage of 2 Phil. Trans. Roy. Soc., 40, 93 (1748); 44, 362 (1752). 3 G. Bell and Sons, London, 1933. 206 THE STRUCTURE OF THE ATOM [Chap. 8 these rays through a gas would produce on its electrical properties. To my great delight I found that this made it a conductor of electricity, even though the electric force applied to the gas was exceedingly small, whereas the gas when it was not exposed to the rays did not conduct electricity unless the electric force were in- creased many thousandfold. . . The X rays seemed to turn the gas into a gaseous electrolyte. I started at once, in the late autumn of 1895, on working at the electric properties of gases exposed to Roentgen rays, and soon found some interesting and suggestive HIGH VOLTAGE SOURCE \ k ANODE (TARGET) CATHODE ////>., ^X-RAYS Fig. 8.1. Production of X-rays. results. . . . There is an interval when the gas conducts though the rays have ceased to go through it. We studied the properties of the gas in this state, and found that the conductivity was destroyed when the gas passed through a filter of glass wool. A still more interesting discovery was that the conductivity could be filtered out without using any mechanical filter by exposing the conducting gas to electric forces. The first experiments show that the conductivity is due to particles present in the gas, and the second shows that these particles are charged with electricity. The conductivity due to the Roentgen rays is caused by these rays producing in the gas a number of charged particles. 7. The ratio of charge to mass of the cathode particles. J. J. Thomson next turned his attention to the behavior of cathode rays in electric and magnetic fields, 4 using the apparatus shown in Fig. 8.2. Fig. 8.2. Thomson's apparatus for determining e\m of cathode particles. The rays from the cathode C pass through a slit in the anode A, which is a metal plug fitting tightly into the tube and connected with the earth; after passing through a second slit in another earth-connected metal plug B, they travel between two parallel aluminium plates about 5 cm apart; they then fall on the end of the tube 4 Phil. Mag., 44, 293 (1897). Sec. 7] THE STRUCTURE OF THE ATOM 207 and produce a narrow well-defined phosphorescent patch. A scale pasted on the outside of the tube serves to measure the deflection of this patch. At high exhaustions the rays were deflected when the two aluminium plates were connected with the terminal of a battery of small storage cells; the rays were depressed when the upper plate was connected with the negative pole of the battery, the lower with the positive, and raised when the upper plate was connected with the positive, the lower with the negative pole. In an electric field of strength , a particle with charge e will be subject to a force of magnitude Ee. The trajectory of an electron in an electric field of strength E perpendicular to its direction of motion may be illustrated by Fig. 8.3. Deflection of electron in an electric field. the diagram in Fig. 8.3. If m is the mass of the electron, the equations of motion may be written : r in ~~ --- Ee dt* (8.2) With / - as the instant the particle enters the electric field, its velocity in the y direction is zero at / -= 0. This velocity increases while the electron is in the field, while its initial velocity in the x direction, r () , remains constant. Integrating eqs. (8.2) we obtain eE , x =- V = 2m (8.3) Equations (8.3) define a parabolic path, as is evident when t is eliminated from the equations, giving * 8 (8.4) After the electron leaves the field, it travels along a straight line tangent to this parabolic path. In many experimental arrangements, its total path is 208 THE STRUCTURE OF THE ATOM [Chap. 8 considerably longer than the length of the electric field, so that the deflection in the^ direction experienced while in the field is comparatively small com- pared to the total observed deflection. To a good approximation, therefore, the parabolic path can be considered as a circular arc of radius R E , with the force exerted by the field equal to the centrifugal force on the electron in this circular path, eE- ^ (8.5) K K The time required to traverse the field of length / is simply l/v so that the deflection in eq. (8.3) becomes eE / 2 2m r 2 Thus 1 ?* (8.6) m rE The ratio of charge to mass may be calculated from the deflection in the electric field, provided the velocity of the particles is known. This may be obtained by balancing the deflection in the electric field by an opposite deflection in a magnetic field. This magnetic field is applied by the pole pieces of a magnet M mounted outside the apparatus in Fig. 8.2, so that the field is at right angles to both the electric field and to the direction of motion of the cathode rays. A moving charged particle is equivalent to a current of electricity, the strength of the current being the product of the charge on the particle and its velocity. From Ampere's Law, therefore, the magnitude of the force on the moving charge is given by /-= evBunO (8.7) where is the angle between the velocity vector v and the magnetic induction vector B. When the magnetic field is perpendicular to the direction of motion, this equation becomes /- evB (8.8) Figure 8.4 illustrates the directional factors involved. The force on the electron due to the magnetic field is always perpendicular to its direction of motion, and thus a magnetic field can never change the speed of a moving charge, but simply changes its direction. As in eq. (8.5), the force may be equated to the centrifugal force on the electron, which in this case moves in a truly circular path. Thus mv 2 Bev - (8.9) K H If now the force due to the, magnetic field exactly balances that due to the electric field, the phosphorescent patch in Thomson's apparatus will be Sec. 8] THE STRUCTURE OF THE ATOM 209 brought back to its initial position. When this occurs, evB - Ee and v - EjB. When this value is substituted in eq, (8.6) one obtains m 2yE (8.10) The units in this equation may be taken to be those of the absolute practical (MKS) system. The charge e is in coulombs; the electric field E in volts per meter; the magnetic induction B in webers per square meter (1 weber per meter 2 ~ 10 4 gausses); and lengths and masses are in meters and kilograms, respectively. MAGNETIC FIELD B Fig. 8.4. Deflection of moving electron in magnetic field. Thomson found the experimental ratio of charge to mass to be of the order of 10 11 coulombs per kilogram. The most recent value of e/m for the electron is e/m 1.7589 x 10 11 coulomb per kilogram 5.273 x 10 17 esu per gram. The value found for the hydrogen ion, H+, in electrolysis was 1836 times less than this. The most reasonable explanation seemed to be that the mass of the cathode particle was only j- 8 ~ c that of the hydrogen ion ; this presumption was soon confirmed by measurements of e, the charge borne by the particle. 8. The charge of the electron. In 1898, Thomson succeeded in measuring the charge of the cathode particles. Two years before, C. T. R. Wilson had shown that gases rendered ionizing by X rays caused the condensation of clouds of water droplets from an atmosphere supersaturated with water vapor. The ions formed acted as nuclei for the condensation of the water droplets. This principle was later used in the Wilson Cloud Chamber to render visible the trajectories of individual charged corpuscles, and thus made possible much of the experimental development of modern nuclear physics. Thomson and Townsend observed the rate of fall of a cloud in air and 210 THE STRUCTURE OF THE ATOM [Chap. 8 from this calculated an average size for the water droplets. The number of droplets in the cloud could then be estimated from the weight of water precipitated. The total charge of the cloud was measured by collecting the charged droplets on an electrometer. The conditions of cloud formation were such that condensations occurred only on negatively charged particles. Making the assumption that each droplet bore only one charge, it was now possible to estimate that the value of the elementary negative charge was e ~ 6.5 x 10~ 10 esu. This was of the same order of magnitude as the charge on the hydrogen ion, and thus further evidence was provided that the cathode particles themselves were "atoms" of negative electricity, with a mass ~^ that of the hydrogen atom. The exact proof of this hypothesis of the atomic nature of electricity and a careful measurement of the elementary electronic charge were obtained in 1909 by Robert A. Millikan in his beautiful oil-drop experiments. Millikan was able to isolate individual droplets of oil bearing an electric charge, and to observe their rate of fall under the combined influences of gravity and an electric field. A body falls in a viscous medium with an increasing velocity until the gravitational force is just balanced by the frictional resistance, after which it falls at a constant "terminal velocity," v. The frictional resistance to a spherical body is given by Stoke's equation of hydrodynamics as f-=-(mYirv (8.11) where rj is the coefficient of viscosity of the medium and r the radius of the sphere. The gravitational force (weight) is equal to this at terminal velocity, so that, if p is the density of the body, and /> that of the fluid medium, far*g(p - Po) ^ 67T *l rv ( 8 - 12 ) If a charged oil droplet falls in an electric field, it can be brought to rest when the upward electric force is adjusted to equal the downward gravita- tional force, eE = fri*g( P - ft) (8.13) Since r may be calculated from the terminal velocity in eq. (8.12), only the charge e remains unknown in eq. (8.13). Actually, somewhat better results were obtained in experiments in which the droplet was observed falling freely and then moving in an electric field. In all cases, the charge on the oil droplets was found to be an exact multiple of a fundamental unit charge. This is the charge on the electron, whose presently accepted value is 5 e = (4.8022 0.0001) x 10~ 10 esu = (1.6018 0.00004) x 10- 19 coulomb 5 Millikan's result of 4.774 x 10~ 10 esu was low, owing to his use of an erroneous value for the viscosity of air. J. D. Stranathan, The Particles of Modern Physics (Philadelphia: BJakiston, 1954), Chap. 2, gives a most interesting account of the measurements of e. Sec. 9] THE STRUCTURE OF THE ATOM 21 1 9. Radioactivity. The penetrating nature of the X rays emitted when cathode rays impinged upon solid substances was a matter of great wonder and interest for the early workers in the field, and many ingenious theories were advanced to explain the genesis of the radiation. It was thought at one time that it might be connected with the fluorescence observed from the irradiated walls of the tubes. Henri Becquerel therefore began to investigate a variety of fluorescent substances to find out whether they emitted pene- trating rays. All trials with various minerals, metal sulfides, and other com- pounds known to fluoresce or phosphoresce on exposure to visible light gave negative results, until he recalled the striking fluorescence of a sample of potassium uranyl sulfate that he had prepared 15 years previously. After exposure to an intense light, the uranium salt was placed in the dark-room under a photographic plate wrapped in "two sheets of thick black paper." The plate was darkened after several hours' exposure. Becquerel soon found that this amazing behavior had nothing to do with the fluorescence of the uranyl salt, since an equally intense darkening could be obtained from a sample of salt that had been kept for days in absolute darkness, or from other salts of uranium that were not fluorescent. The penetrating radiation had its source in the uranium itself, and Becquerel proposed to call this new phenomenon radioactivity* It was discovered that radioactive materials, like X rays, could render gases conducting so that charged bodies would be discharged, and the dis- charge rate of electroscopes could therefore be used as a measure of the intensity of the radiation. Marie Curie examined a number of uranium com- pounds and ores in this way, and found that the activity of crude pitchblende was considerably greater than would be expected from its uranium content. In 1898, Pierre and Marie Curie announced the separation from pitchblende of two extremely active new elements, polonium and radium. Three different types of rays have been recognized and described in the radiation from radioactive materials. The ft rays are high-velocity electrons, as evidenced by their deviation in electric and magnetic fields, and ratio of charge to mass. Their velocities range from 0.3 to 0.99 that of light. The a rays are made up of particles of mass 4 (O = 16 scale) bearing a positive charge of 2 (e 1 scale). They are much less penetrating than ft rays, by a factor of about 100. Their velocity is around 0.05 that of light. The y rays are an extremely penetrating (about 100 times ft rays) electromagnetic radia- tion, undeflected by either magnetic or electric fields. They are similar to X rays, but have a much shorter wave length. Owing to their large mass, the a particles travel through gases in essen- tially straight lines, producing a large amount of ionization along their paths. The paths of ft particles are longer than those of a's, but are much more irregular on account of the easy deflection of the lighter ft particle. The phenomena of radioactivity as well as the observations on the 6 Compt. rend., 127, 501, March 2 f 1896. 212 THE STRUCTURE OF THE ATOM [Chap. 8 electrical discharge in gases provided evidence that electrons and positive ions were component parts of the structure of atoms. 10. The nuclear atom. The problem of the number of electrons contained in an atom attracted the attention of Thomson and of C. G. Barkla. From measurements of the scattering of light, X rays, and beams of electrons, it was possible to estimate that this number was of the same order as the atomic weight. To preserve the electrical neutrality of the atom, an equal number of positive charges would then be necessary. Thomson proposed an atom model that consisted of discrete electrons embedded in a uniform sphere of positive charge. Lord Rutherford 7 has told the story of the next great development in the problem, at the University of Manchester in 1910. In the early days I had observed the scattering of a particles, and Dr. Geiger in my laboratory had examined it in detail. He found in thin pieces of heavy metal that the scattering was usually small, of the order of one degree. One day Geiger came to me and said, "Don't you think that young Marsden, whom I am training in radioactive methods, ought to begin a small research?" Now 1 had thought that too, so I said, "Why not let him see if any a particles can be scattered through a large angle?" I may tell you in confidence that 1 did not believe they would be, since we knew that the a particle was a very fast massive particle, with a great deal of energy, and you could show that if the scattering was due to the accumulated effect of a number of small scatterings, the chance of an a particle's being scattered backwards was very small. Then I remember two or three days later Geiger coming to me in great excitement and saying, "We have been able to get some of the a particles coming backwards. . . ." It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. On consideration I realized that this scattering backwards must be the result of a single collision and when I made calculations I saw it was impossible to get any- thing of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. . . . ~ In the experimental arrangement used by Marsden and Geiger, a pencil of a particles was passed through a thin metal foil and its deflection observed on a zinc sulfide screen, which scintillated whenever struck by a particle. Rutherford enunciated the nuclear model of the atom in a paper pub- lished 8 in 1911. The positive charge is concentrated in the massive center of the atom, with the electrons revolving in orbits around it, like planets around the sun. Further scattering experiments indicated that the number of elemen- tary positive charges in the nucleus of an atom is equal within the experi- mental uncertainty to one-half its atomic weight. Thus carbon, nitrogen, and oxygen would have 6, 7, and 8 electrons, respectively, revolving around a like positive charge. It follows that the charge on the nucleus or the number of orbital electrons may be set equal to the atomic number of the element, the ordinal number of the position that it occupies in the periodic table. 7 Ernest Rutherford, Lecture at Cambridge, 1936, in Background to Modern Science, ed. by J. Needham and W. Pagel (London: Cambridge, 1938). 8 Phil. Mag., 21, 669 (1911). Sec. ll] THE STRUCTURE OF THE ATOM 213 According to the nuclear hypothesis, the a particle is therefore the nucleus of the helium atom. It was, in fact, known that a particles became helium gas when they lost their energy. 11. X rays and atomic number. The significance of atomic number was strikingly confirmed by the work of H. G. J. Moseley. 9 Barkla had discovered that in addition to the general or white X radiation emitted by all the ele- ments, there were several series of characteristic X-ray lines peculiar to each element. Moseley found that the frequency v of a given line in the character- istic X radiation of an element depended on its atomic number Z in such a way that Vv-=fl(Z-6) (8.14) where, for each series, a and b are constant for all the elements. The method by which the wave length of X rays is measured by using the regular inter- atomic spacings in a crystal as a diffraction grating will be discussed in Chapter 13. When the Moseley relationship was plotted for the K* X-ray lines of the elements, discontinuities in the plot appeared corresponding te missing elements in the periodic table. These vacant spaces have since been filled. This work provided further convincing evidence that the atomic number and not the atomic weight governs the periodicity of the properties of the chemical elements. 12. The radioactive disintegration series. Rutherford in 1898, soon after the discovery of radioactivity, observed that the activity from thorium would diffuse through paper but not through a thin sheet of mica. The radioactivity could also be drawn into an ionization chamber by means of a current of air. It was therefore evident that radioactive thorium was continuously pro- ducing an "emanation" that was itself radioactive. Furthermore, this emana- tion left a deposit on the walls of containers, which was likewise active. Each of these activities could be quantitatively distinguished from the others by its time of decay. As a result of a large amount of careful research by Rutherford, Soddy, and others, it was gradually established that a whole series of different elements was formed by consecutive processes of radio- active change. The number of radioactive atoms that decomposes per second is directly proportional to the number of atoms present. Thus Jf =e ~" (8 - 15) if NQ is the number of radioactive atoms present at t - 0. The constant A is 9 Phil. Mag., 26, 1024 (1913); 27, 703 (1914). 214 THE STRUCTURE OF THE ATOM [Chap. 8 called the radioactive-decay constant', the larger the value of A, the more rapid the decay of the radioactivity. The exponential decay law of eq. (8.15) is plotted in curve A, Fig. 8.5, the experimental points being those obtained from uranium X l9 the first product in the uranium series. A sample of uranium or of any of its salts is found to emit both a and ft particles. If an iron salt is added to a solution 100 60 80 100 TIME, DAYS Fig. 8.5. Radioactive decay and regeneration of UX t . of a uranium salt, and the iron then precipitated as the hydroxide, it is found that the ft activity is removed from the uranium and coprecipitated with the ferric hydroxide. This ft activity then gradually decays according to the exponential curve A of Fig. 8.5. The original uranium sample gradually regains ft activity, according to curve B. It is apparent that the sum of the activities given by curves A and B is always a constant. The amount of UX 1 (the ft emitter) decomposing per second is just equal to the amount being formed from the parent uranium. Sec. 12] THE STRUCTURE OF THE ATOM 215 element is the half-life period r, the time required for the activity to be reduced to one-half its initial value. From eq. (8.15), therefore, In 2 r 0.693 (8.16) The half life of uranium is 4.4 x 10 9 years, whereas that of UX^ is 24.5 days. Because of the long life of uranium compared to UX t , the number of uranium atoms present in a sample is effectively constant over measurable experimental periods, and the recovery curve of Fig. 8.5 reaches effectively the same initial activity after repeated separations of daughter UX t from the parent uranium. Many careful researches of this sort by Rutherford, Soddy, A. S. Russell, K. Fajans, R. Hahn, and others, are summarized in the complete radioactive series, such as that for the uranium family shown in Table 8.1. Examination TABLE 8.1 RADIOACTIVE SERIES URANIUM FAMILY Name Symbol of At. No. Z Mass No. A Particle Emitted Half Life Element Uranium 1 . U 92 238 a 4.56 < I0 9 y Uranium Xj Th 90 234 ft 24.1 d Uranium Xo Pa 91 234 fi,y 1.14m Uranium II . U 92 234 a 2.7 x IC^y Ionium Th 90 230 a 8.3 x 10 4 y Radium Ra 88 226 a 1590y Radon Rn 86 222 a 3.825 d Radium A . Po 84 218 a 3.05m Radium B . Pb 82 214 ft.v 26.8m Radium C . Bi 83 214 *,ft,v 19.7m Radium C' (99.96%) . Po 84 214 a 1.5 x 10~ 4 s Radium C" (0.04%) Tl 81 210 ft 1.32m Radium D . Pb 82 210 ft,Y 22 y Radium E . Bi 83 210 ft,y 5.0 d Radium F . . Po 84 210 a 140d Radium G . Pb 82 206 of the properties of the elements in this table established two important general principles. When an atom emits an a particle, its position is shifted two places to the left in the periodic table; i.e., its atomic number is decreased by two. The emission of a ft particle shifts the position one place to the right, increasing the atomic number by one. It is evident, therefore, that the source of the ft particles is in the nucleus of the atom, and not in the orbital electrons. No marked change in atomic weight is associated with the ft emission, whereas a emission decreases the atomic weight by four units. 216 THE STRUCTURE OF THE ATOM [Chap. 8 13. Isotopes. An important consequence of the study of the radioactive series was the demonstration of the existence of elements having the same atomic number but different atomic weights. These elements were called isotopes by Soddy, from the Greek taos TOKOS, "the same place" (i.e., in the periodic table). It was soon found that the existence of isotopes was not confined to the radioactive elements. The end product of the uranium series is lead, which, from the number of intermediate a particle emissions, should have an atomic weight of 206, compared to 207.21 for ordinary lead. Lead from the mineral curite (containing 21.3 per cent lead oxide and 74.2 per cent uranium trioxide), which occurs at Katanga, Belgian Congo, was shown to have an atomic weight of 206.03. This fact provided confirmation of the existence of nonradioactive isotopes and indicated that substantially all the lead in curite had arisen from the radioactive decay of uranium. The time at which the uranium was originally deposited can therefore be calculated from the amount of lead that has been formed. The geologic age of the earth obtained in this way is of the order of 5 X 10 9 years. This is the time elapsed since the minerals crystallized from the magma. The existence of isotopes provided the solution to the discrepancies in the periodic table and to the problem of nonintegral atomic weights. The measured atomic weights are weighted averages of those of a number of isotopes, each having a weight that is nearly a whole number. The generality of this solution was soon shown by the work of Thomson on positive rays. 14. Positive-ray analysis. In 1886, Eugen Goldstein, using a discharge tube with a perforated cathode, discovered a new type of radiation streaming Fig. 8.6. Thomson's apparatus for positive- ray analysis. into the space behind the cathode, to which he gave the name Kanahtrahlen. Eleven years later the nature of these rays was elucidated by W. Wien, who showed that they were composed of positively charged particles with ratios e/m of the same magnitude as those occurring in electrolysis. It was reason- able to conclude that they were free positive ions. In 1912, Thomson took up, the problem of the behavior of positive rays in electric and magnetic fields, using the apparatus shown in Fig. 8.6. The Sec. 14] THE STRUCTURE OF THE ATOM 217 positive rays, generated by ionization of the gas in a discharge tube A, were drawn out as a thin pencil through the elongated hole in the cathode B. They were then subjected in the region EE' simultaneously to a magnetic and to an electric field. This was accomplished by inserting strips of mica insulation (D, D') in the soft iron pole pieces of the magnet. Then by connecting E and " to a bank of batteries, it was possible to supply an electric field that would act parallel to the magnetic field of the magnet. The trace of the deflected positive rays was recorded on the photographic plate P. The effect of the superimposed fields may be seen from Fig. 8.7. Consider a positive ion with charge e to be moving perpendicular to the plane of the paper so that, if undeflected, it would strike the origin O. If it is subjected somewhere along its path to the action of an electric field directed along the positive X direction, it will be deflected from O to P, the deflection being inversely proportional to the radius of curvature of the approximately circular path traveled in the electric field between the plates at EE' in Thomson's apparatus. The actual magnitude of the deflection depends on the dimensions of the apparatus. From eq. (8.5) and Fig. 8.3, the deflection may therefore be written, taking /q as a proportionality constant, x = ^ = ^jr (8.17) If instead of the electric field a magnetic field in the same direction acts on the moving ion, it will be deflected upwards from O to Q 9 the deflection being given from eq. (8.9) by Fig. 8.7. Thomson's parabola method. mv (8-18) ^ ' The constants k l are the same in eqs. (8.17) and (8.18) if the electric and magnetic fields act over the same length of the ion's path, as is the case in Thomson's apparatus. If the electric and magnetic fields act simultaneously, the ion will be deflected to a point R dependent on its velocity v, and its ratio of charge to mass. In general, the individual positive ions in a beam are traveling with different velocities, and the pattern they form on a viewing screen may.be calculated by eliminating v between eqs. (8.17) and (8.18). Thus if C *1 ~TT ' X Em (8.19) 218 THE STRUCTURE OF THE ATOM [Chap. 8 This is the equation of a parabola. The important conclusion is thereby established that all ions of given ratio of charge to mass will strike the screen along a certain parabolic curve. Since the charge e' must be an integral multiple of the fundamental electronic charge e, the position of the parabola will effectively be determined only by the mass of the positive ion. The first evidence that isotopes existed among the stable elements was found in Thomson's investigation of neon in 1912. He observed a weak parabola accompanying that of Neon 20, which could be ascribed only to a Neon 22. As a result of the work of A. J. Dempster, F. W. Aston, and others, positive-ray analysis has been developed into one of the most precise methods for measuring atomic masses. The existence of isotopes has been shown to be the rule rather than the exception among the chemical elements. Apparatus for measuring the masses of positive ions are known as mass spectrographs when a photographic record is obtained, and otherwise as mass spectrometers. 15. Mass spectra The Dempster method. The disadvantage of the para- bola method is that the ions of any given e/m are spread out along a curve so that the density of the pictures ob- tained is low at reasonable times of exposure. It was most desirable to make use of some method that would bring all I 1 ~ II y ions of the same e/m to a sharp focus. -L fi*t A ^ S\\ ^ ne wa y ^ doing tn * s devised by A. J. Dempster in 1918, is shown in Fig. 8.8. The positive ions are obtained by vaporizing atoms from a heated fila- ment A, and then ionizing them by means of a beam of electrons from an "electron gun" 10 at B. Alternatively, ions can be Fig. 8.8. Dempster's mass spectrom- eter (direction focusing). formed by passing the electron beam through samples of gas. A potential difference V between A and the slit C accelerates the ions uniformly, so that they issue from the slit with approxi- mately the same kinetic energies, V - e (8.20) The region D is a channel between two semicircular pieces of iron, through which is passed the field from a powerful electromagnet. The field direction is perpendicular to the plane of the paper. The ions emerge from the slit C in various directions, but since they all have about the same velocity, they 10 An electron gun is an arrangement by which electrons emitted from a filament are accelerated by an electric field and focused into a beam with an appropriate slit system. Sec. 16] THE STRUCTURE OF THE ATOM 219 are bent into circular paths of about the same radius, given by eq. (8.9) as R = mv/Be . Therefore, from eq. (8.20), m 7' 2V (8.21) It is apparent that for any fixed value of the magnetic field B, the accelera- ting potential can be adjusted to bring the ions of the same m/e' to a focus 80 60 I cr UJ > 20 X 124 AND SCALE MAGNIFIED 4O X 124 126 128 130 132 134 ATOMIC MASS Fig. 8.9. Isotopes of xenon. 136 138 at the second slit F, through which they pass to the electrometer G. The electrometer measures the charge collected or the current carried through the tube by the ions. This was the method used by Dempster in operating the apparatus; it is called "direction focusing." A typical curve of ion current vs. the mass number calculated from eq. (8.21) is shown in Fig. 8.9, the heights of the peaks corresponding to the relative abundances of the isotopes. 16. Mass spectra Aston's mass spectrograph. A different method of focusing was devised by F. W. Aston in 1919, and used by him in the first extensive investigations of the occurrence of stable isotopes. The principle of this method may be seen from Fig. 8.10. Positive ions are generated in a gas discharge tube (not shown) and drawn off through the very narrow parallel slits S l and S 2 . Thus, in contrast with 220 THE STRUCTURE OF THE ATOM [Chap. 8 Dempster's system, a thin ribbon of rays of closely defined direction is taken for analysis ; the velocities of the individual ions may vary considerably, since they have been accelerated through different potentials in the discharge tube. The thin beam of positive rays first passes through the electric field between parallel plates P l and P 2 . The slower ions experience a greater deflection, since they take longer to traverse the field ; the beam is accordingly spread out, as well as being deflected as shown. A group of these rays, selected by the diaphragm D, next passes between the parallel pole pieces of the magnet M. The slower ions again experience the greater deflection. If the magnetic deflection $ is more than twice the S,S 2 Fig. 8.10. Aston's mass spectrograph (velocity focusing). electric deflection 0, all the ions, regardless of velocity, will be brought to a sharp focus at some point on the photographic plate P. Aston's method is therefore called "velocity focusing." More recent developments in mass spectrometry have combined velocity and direction focusing in a single instrument. The design has been refined to such an extent that it is possible to determine atomic masses to an accuracy of one part in 100,000. The precise determination of atomic weights with the mass spectrometer is accomplished by carefully comparing sets of closely spaced peaks. Thus one may resolve doublets such as H 2 + and He++, 16 O+ and CH 4 +, C lf and D 3 +. n By working with such doublets, instrumental errors are minimized. Mass spectrometers are finding increasing application in the routine analysis of complex mixtures of compounds, especially of hydrocarbons. For example, a few tenths of a milliliter of a liquid mixture of isomeric hexanes and pentanes can be quantitatively analyzed with a modern mass spectrometer, a task of insuperable difficulty by any other method. Hydro- carbon isomers do not differ in mass, but each isomer ionizes and decom- poses in a different way as a result of electron impact. Therefore each isomer yields a characteristic pattern of mass peaks in the spectrometer. Most com- mercial mass spectrometers follow the Dempster-type design. 11 The symbol D stands for deuterium or heavy hydrogen, H 2 , which will be discussed in following sections. Sec. 17] THE STRUCTURE OF THE ATOM 221 It may be noted that mass-spectrometer chemistry often seems to have little respect for our preconceived notions of allowable ionic species. Thus Ha 4 " and D 3 + are observed, and benzene vapor yields some C 6 +, a benzene ring completely stripped of its hydrogens. Such ions have, of course, a less than ephemeral lifetime, since they take only about a microsecond (10~ 6 sec) to traverse the spectrometer tube. 17. Atomic weights and isotopes. A partial list of naturally occurring stable isotopes and their relative abundance is given in Table 8.2. Not all of these isotopes were first discovered by positive-ray analysis, one notable exception being heavy hydrogen or deuterium, whose existence was originally demonstrated from the optical spectrum of hydrogen. The isotopic weights in Table 8.2 are not exactly integral. Thus the old TABLE 8.2 Atomic Number Z Element Symbol Mass Number A Isotopic Physical Atomic Weight (O lfl - 16)Af Relative Abundance (per cent) 1 Hydrogen H D 1 2 1.008131 2.01473 99.985 0.015 2 Helium He 3 3.01711 10~ 6 4 4.00389 100 3 Lithium Li 6 6.01686 7.8 7 7.01818 92.1 4 5 Beryllium Boron Be B 9 10 9.01504 10.01631 100 20 11 11.01292 80 6 Carbon C 12 12.00398 98.9 13 13.00761 1.1 7 Nitrogen N 14 15 14.00750 15.00489 99.62 0.38 8 Oxygen O 16 17 16.000000 17.00450 99.76 0.04 18 18.00369 0.20 9 Fluorine F 19 19.00452 100 10 Neon Ne 20 19.99881 90.00 21 21.00018 0.27 22 21.99864 9.73 15 16 Phosphorus Sulfur P S 31 32 30.98457 31.98306 100 95.1 33 32.98260 0.74 34 33.97974 4.2 36 0.016 17 Chlorine Cl 35 34.98107 75.4 37 36.97829 24.6 82 Lead Pb 204 1.5 206 23.6 207 22.6 208 208.060 52.3 92 Uranium U 234 0.006 235 0.720 238 99.274 222 THE STRUCTURE OF THE ATOM [Chap. 8 hypothesis of Prout is nearly but not exactly confirmed. The nearest whole number to the atomic weight is called the mass number of an atomic species. A particular isotope is conventionally designated by writing the mass number 80 70 60 50 40 30 20 10 -10 CURVE FOR LIGHT ELEMENTS 4He 02468 10 12 14 16 18 20 22 24 MASS NUMBER (0) z O <r -4 u_ o -6 z 5 " 8 2 -10 -12 - CURVE FOR HEAVY ELEMENTS l 20 40 60 80 100 120 140 160 180 200 220 240 MASS NUMBER (b) Fig. 8.11. Packing fraction curves, (a) Curve for light elements, (b) Curve for heavy elements. as a left- or right-hand superscript to the symbol of the element; e.g., 2 H, U 235 , and so on. The packing fraction of an isotope is defined by atomic weight mass number packing fraction = mass number The curves in Fig. 8.11 show how the packing fraction varies with mass number, according to the latest atomic-weight data. The further discussion of these curves, whose explanation requires an enquiry into the structure of the atomic nucleus, will be postponed till the following chapter. Sec. 18] THE STRUCTURE OF THE ATOM 223 It will be noted that oxygen, the basic reference element for the calcula- tion of atomic weights, is itself composed of three isotopes, 16, 17, and 18. Chemists have been unable to abandon the convention by which the mixture of isotopes constituting ordinary oxygen is assigned the atomic weight O ~ 16. Weights calculated on this basis are called chemical atomic weights. The physicists prefer to call O 16 16, whence ordinary oxygen becomes O -^ 16.0043. This leads to a set of physical atomic weights. 18. Separation of isotopes. For a detailed discussion of separation methods, reference may be made to standard sources. 12 Several of the more important procedures will be briefly discussed. 1. Gaseous diffusion. This was the method used to separate 235 UF 6 from 238 UF 6 in the plant at Oak Ridge, Tennessee. The fundamental principle involved has been discussed in connection with Section 7-8 on the effusion of gases. The separation factor f of a process of isotope separation is defined as the ratio of the relative concentration of a given species after processing to its relative concentration before processing. Thus/ (fli7 AI 2 / )/( /7 i/ AI 2) where (n l9 A?/) an d (>*2> "2') are tne concentrations of species 1 and 2 before and after processing. Uranium 235 occurs in natural uranium to the extent of one part in 140 (njn^ -= 1/140). If it is desired to separate 90 per cent pure U 235 from U 238 , therefore, the over-all separation factor must be /- (9/l)/(l/140) - 1260. For a single stage of diffusion the separation factor cannot exceed the ideal value a, given from Graham's Law, as a VM 2 /M ly where M 2 and M 1 are the molecular weights of the heavy and light components, respec- tively. For the uranium hexafluorides, a = A/352/349 1.0043. Actually, the value of /for a single stage will be less than this, owing to diffusion in the reverse direction, nonideal mixing at the barrier surface, and partially nondiffusive flow through the barrier. It is therefore necessary to use several thousand stages in a cascade arrangement to effect a considerable concentration of 235 UF 6 . The theory of a cascade is very similar to that of a fractionating column with a large number of theoretical plates. The light fraction that passes through the barrier becomes the feed for the next stage, while the heavier fraction is sent back to an earlier stage. It may be noted that UF 6 has at least one advantage for use in a process for separating uranium isotopes, in that there are no isotopes of fluorine except 19 F. 2. Thermal diffusion. This method was first successfully employed by H. Clusius and G. Dickcl, 13 and the experimental arrangement is often 12 H. S. Taylor and S. Glasstone, Treatise on Physical Chemistry, 3rd ed. (New York: Van Nostrand, 1941); H. D. Smyth, Atomic Energy for Military Purposes (Princeton Univ. Press, 1945); F. W. Aston, Mass Spectra and Isotopes, 4th ed. (New York: Longmans, 1942). 13 Naturmssenschaften, 26, 546 (1938). For the theory of the thermal diffusion separa- tion see K. Schafer, Angew. Chem., 59, 83 (1947). The separation depends not only on mass but also on difference in intermodular forces. With isotopic molecules the mass effect predominates and the lighter molecules accumulate in the warmer regions. 224 THE STRUCTURE OF THE ATOM [Chap. 8 called a Clusius column. It consists of a long vertical cylindrical pipe with an electrically heated wire running down its axis. When a temperature gradient is maintained between the hot inner wire and the cold outer walls, the lighter isotope diffuses preferentially from the cold to the warmer regions. The separation is tremendously enhanced by the convection currents in the tube, which carry the molecules arriving near the warm inner wire upwards to the top of the column. The molecules at the cold outer wall are carried downwards by convection. With columns about 30 meters high and a temperature difference of about 600C, Clusius was able to effect a virtually complete separation of the isotopes of chlorine, Cl 35 and Cl 37 . The cascade principle can also be applied to batteries of thermal diffusion columns, but for mass production of isotopes this operation is in general less economical than pressure-diffusion methods. 3. Electromagnetic separators. This method employs large mass spectro- meters with split collectors, so that heavy and light ions are collected separ- ately. Its usefulness is greatest in applications in which the throughput of material is comparatively small. 4. Separation by exchange reactions. Different isotopic species of the same element differ significantly in chemical reactivity. These differences are evident in the equilibrium constants of the so-called isotopic exchange reactions. If isotopes did not differ in reactivity, the equilibrium constants of these reactions would all be equal to unity. Some actual examples follow: J S 16 2 + H 2 18 -= i S 18 2 + H 2 16 K =-- 1.028 at 25C i3 CO + i2 C Q 2 = 12CO + 13 CO 2 K = 1.086 at 25C 15 NH 3 (g) + 14 NH 4 +(aq.) = 14 NH 3 (g) + 15 NH 4 +(aq.) K - 1.023 at 25C Such differences in affinity are most marked for the lighter elements, for which the relative differences in isotopic masses are greater. Exchange reactions can be applied to the separation of isotopes. The possible separation factors in a single-stage process are necessarily very small, but the cascade principle is again applicable. H. C. Urey and H. G. Thode concentrated 15 N through the exchange between ammonium nitrate and ammonia. Gaseous ammonia was caused to flow countercurrently to a solution of NH 4 4 ions, which trickled down columns packed with glass helices or saddles. After equilibrium was attained in the exchange columns, 8.8 grams of 70.67 per cent 15 N could be removed from the system, as nitrate, every twelve hours. As a result of exchange reactions, the isotopic compositions of naturally occurring elements show small but significant variations depending on their sources . If we know the equilibrium constant of an exchange reaction over a range of temperatures, it should be possible to calculate the temperature at which a product was formed, from a measurement of the isotopic ratio in the product. Urey has applied this method, based on O 18 : O 16 ratios, to Sec. 19] THE STRUCTURE OF THE ATOM 225 the determination of the temperature of formation of calcium carbonate deposits. The exchange equilibrium is that between the oxygen in water and in bicarbonate ions. The temperature of the seas in remote geologic eras can be estimated to within 1C from the O 18 : O 16 ratio in deposits of the shells of prehistoric molluscs. 19. Heavy hydrogen. The discovery of the hydrogen isotope of mass 2," which is called deuterium, and the investigation of its properties comprise one of the most interesting chapters in physical chemistry. In 1931, Urey, Brickwedde, and Murphy proved the existence of the hydrogen isotope of mass 2 by a careful examination of the spectrum of a sample of hydrogen obtained as the residue from the evaporation of several hundred liters of liquid hydrogen. Deuterium is contained in hydrogen to the extent of one part in 4500. In 1932, Washburn and Urey discovered that an extraordinary concen- tration of heavy water, D 2 O, occurred in the residue from electrolysis of water. 14 The production of 99 per cent pure D 2 O in quantities of tons per day is now a feasible operation. Some of the properties of pure D 2 O as compared with ordinary H 2 O are collected in Table 8.3. TABLE 8.3 PROPERTIES OF HEAVY WATER AND ORDINARY WATER Property Units Ordinary Water Heavy Water g/cc C 0.997044 4.0 1.104625 11.6 C 0.000 3.802 c 100.00 101.42 cal/mole cal/mole 1436 10,484 81.5 1510 10,743 80.7 1.33300 1.32828 dynes/cm millipoise 72.75 13.10 72.8 16.85 Density at 25C .... Temperature of maximum density . Melting point .... Boiling point .... Heat of fusion .... Heat of vaporization at 25 . Dielectric constant Refractive index at 20 (Na D line) Surface tension (20C) . Viscosity (10C) . PROBLEMS 1. An Na 4 " ion is moving through an evacuated vessel in the positive x direction at a speed of 10 7 cm per sec. At x 0, y =~ 0, it enters an electric field of 500 volts per cm in the positive y direction. Calculate its position (;c, y) after 10~ 6 sec. 2. Make calculations as in Problem 1 except that the field is a magnetic field of 1000 gauss in the positive z direction. 14 The mechanism of the separation of H 1 from H 2 during electrolysis is still obscure. For discussions see Eyring, et aL, J. Chem. Phys., 7, 1053 (1939); Urey and Teal, Rev. Mod. Phys., 7, 34(1935). 226 THE STRUCTURE OF THE ATOM [Chap. 8 3. Calculate the final position of the Na+ ion in the above problems if the electric and magnetic fields act simultaneously. 4. Consider a Dempster mass spectrometer, as shown in Fig. 8.8, with a magnetic field of 3000 gauss and a path radius of 5.00 cm. At what accelera- ting voltage will (a) H+ ions, (b) Na^ ions be brought to focus at the ion collector ? 5. Radium-226 decays by a particle emission with a half life of 1590 years, the product being radon-222. Calculate the volume of radon evolved from 1 g of radium over a period of 50 years. 6. The half life of radon is 3.825 days. How long would it take for 90 per cent of a sample of radon to disintegrate? How many disintegrations per second are produced in a microgram ( 1 0~ 6 g) of radium ? 7. Derive an expression for the average life of a radioactive atom in terms of the half life r. 8. The half life of thorium-C is 60.5 minutes. How many disintegrations would occur in 15 minutes from a sample containing initially 1 mg of Th-C (at wt. 212)? 9. Radioactivity is frequently measured in terms of the curie (c) defined as the quantity of radioactive material producing 3.7 X 10 10 disintegrations per sec. The millicurie is 10~ 3 c, the microcurie, 10~ 6 c. How many grams of (a) radium, (b) radon are there in one curie? 10. It is found that in 10 days 1.07 x 10~ 3 cc of helium is formed from the a particles emitted by one gram of radium. Calculate a value for the half life of radium from this result. 11. The half life of U-238 is 4.56 x 10 9 years. The final decay product is Pb-206, the intermediate steps being fast compared with the uranium dis- integration. In Lower Pre-Cambrian minerals, lead and uranium are found associated in the ratio of approximately 1 g Pb to 3.5 g U. Assuming that all the Pb has come from the U, estimate the age of the mineral deposit. 12. A ft particle moving through a cloud chamber under a magnetic field of 10 oersteds traverses a circular path of 18 cm radius. What is the energy of the particle in ev? REFERENCES BOOKS 1. Born, M., Atomic Physics (London: Blackie, 1951). 2. Feather, N., Lord Rutherford (London: Blackie, 1940). 3. Finkelnburg, W., Atomic Physics (New York: McGraw-Hill, 1950). 4. Rayleigh, Lord, Life ofJ. J. Thomson (Cambridge University Press, 1942). 5. Richtmeyer, F. K., and E. A. Kennard, Introduction to Modern Physics (New York: McGraw-Hill, ,1947). 6. Semat, H., Introduction to Atomic Physics (New York: Rinehart, 1946). Chap. 8] THE STRUCTURE OF THE ATOM 227 7. Stranathan, J. D., The Particles of Modern Physics (Philadelphia: Blaki- ston, 1954). 8. Tolansky, S., Introduction to Atomic Physics (London: Longmans, 1949). 9. Van Name, F., Modern Physics (New York: Prentice-Hall, 1952). ARTICLES 1. Birge, R. T., Am. J. Phys., 13, 63-73 (1945), "Values of Atomic Con- stants." 2. Glasstone, S., /. Chem. Ed., 24, 478-81 (1947), "William Prout." 3. Hooykaas, R., Chymia, 2, 65-80 (1949), "Atomic and Molecular Theory before Boyle." 4. Jauncey, G. E., Am. J. Phys., 14, 226-41 (1946), "The Early Years of Radioactivity." 5. Lemay, P., and R. E. Oesper, Chymia, 1, 171-190 (1948), "Pierre Louis Dulong." 6. Mayne, K. I., Rep. Prog. Phys., 15, 24-48 (1952), "Mass Spectrometry." 7. Rayleigh, Lord, /. Chem. Soc., 467-75 (1942), "Sir Joseph J. Thomson." 8. Urey, H. C, Science in Progress, vol. I (New Haven: Yale University Press, 1939), 35-77, "Separation of Isotopes." 9. Winderlich, R., J. Chem. Ed., 26, 358-62 (1949), "Eilard Mitscherlich." CHAPTER 9 Nuclear Chemistry and Physics 1. Mass and energy. During the nineteenth century, two important prin- ciples became firmly established in physics : the conservation of mass and the conservation of energy. Mass was the measure of matter, the substance out of which the physical world was constructed. Energy seemed to be an independent entity that moved matter from place to place and changed it from one form to another. In a sense, matter contained energy, for heat was simply the kinetic energy of the smallest particles of matter, and potential energy was associated with the relative positions of material bodies. Yet there seemed to be one instance, at Jeast, in which energy existed independently of matter, namely in the form of radiation. The electromagnetic theory of Clerk Maxwell required an energy in the electromagnetic field and the field traversed empty space. Yet no experiments can be performed in empty space, so that actually this radiant energy was detected only when it impinged on matter. Now a very curious fact was observed when this immaterial entity, light energy, struck a material body. The observation was first made in 1628 by Johannes Kepler, who noted that the tails of comets always curved away from the sun. He correctly- assigned the cause of this curvature to a pressure exerted by the sun's rays. In 1901 this radiation pressure was experimentally demonstrated in the laboratory, by means of delicate torsion balances. Thus the supposedly immaterial light exerts a pressure. The pressure implies a momentum asso- ciated with the light ray, and a momentum implies a mass. If we return to Newton's picture of a light ray as made up of tiny particles, simple calcula- tions show that the energy of the particles E is related to their 'mass by the equation E - c 2 m (9.1) where c is the speed of light. As a result of Albert Einstein's special theory of relativity (1905) it appeared that the relation E c*m was applicable to masses and energies of any origin. He showed first of ail that no particle could have a speed greater than that of light. Thus the inertial resistance that a body offers to acceleration by an applied force must increase with the speed of the body. As the speed approaches that of light, the mass must approach infinity. The relation between mass and speed v is found to be (9.2) 228 Sec. 2] NUCLEAR CHEMISTRY AND PHYSICS 229 When v = the body has a rest mass, m Q . Only at speeds comparable with that of light does the variation of mass with speed become detectable. Equation (9.2) has been confirmed experimentally by measurements of e/m for electrons accelerated through large potential differences. If a rapidly moving particle has a larger mass than the same particle would have at rest, it follows that the larger the kinetic energy, the larger the mass, and once again it turns out that the increase in mass Am and the increase in kinetic energy AE are related by A" - c 2 AAW. As we shall see in the next section, the Einstein equation E = c 2 m has been conclusively checked by experiments on nuclear reactions. The situation today is therefore that mass and energy are not two distinct entities. They are simply two different names for the same thing, which for want of a better term is called mass-energy. We can measure mass-energy in mass units or in energy units. In the COS system, the relation between the two is: 1 gram ^ c 2 ergs = 9 x 10 20 ergs. One gram of energy is sufficient to convert 30,000 tons of water into steam. 2. Artificial disintegration of atomic nuclei. In 1919, Rutherford found that when a particles from Radium C were passed through nitrogen, protons were ejected from the nitrogen nuclei. This was the first example of the dis- integration of a normally stable nucleus. It was soon followed by the demon- stration of proton emission from other light elements bombarded with a particles. In 1923, P. M. S. Blackett obtained cloud-chamber photographs showing that these reactions occurred by capture of the a particle, a proton and a new nucleus then being formed. For example, 7 N 14 + 2 He 4 > ( 9 P) -> iH 1 + 8 17 This type of reaction does not occur with heavy elements because of the large electrostatic repulsion between the doubly charged alpha and the high positive charges of the heavier nuclei. It was realized that the singly charged proton, 1 H 1 , would be a much more effective nuclear projectile, but it was not available in the form of high-velocity particles from radioactive materials. J. D. Cockroft and E. T. S. Walton 1 therefore devised an electrostatic accelerator. This appara- tus was the forerunner of many and ever more elaborate machines for pro- ducing high-velocity particles. The protons produced by ionization of hydrogen in an electric discharge were admitted through slits to the accel- erating tube, accelerated across a high potential difference, and finally allowed to impinge on the target. The energy unit usually used in atomic and nuclear physics, the electron volt, is the energy acquired by an electron in falling through a potential difference of one volt. Thus 1 ev = eV ~ 1.602 x 10~ 19 volt coulomb 1 Proc. Roy. Soc., A 729, 477 (1930); 136, 619 (1932). 230 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 (joule) = 1.602 x 10~ 12 erg. The usual chemical unit is the kiiocalorie per mole. 1.602 x 10~ 19 x 6.02 x 10 23 , . , ! ev ____ ^ 23.05 kcal per mole One of the first reactions to be studied by Cockroft and Walton was iH 1 -f 3 Li 7 -> 2 2 He 4 The bombarding protons had energies of 0.3 million electron volt (mev.) From the range of the emergent a particles in the cloud chamber, 8.3 cm, their energy was calculated to be 8.6 mev each, or more than 17 mev for the pair. It is evident that the bombarding proton is merely the trigger that sets off a tremendously exothermic nuclear explosion. The energies involved in these nuclear reactions are several million times those in the most exothermic chemical changes. Thus an opportunity is pro- vided for the quantitative experimental testing of the E = c 2 m relation. The mass-spectrographic values for the rest masses of the reacting nuclei are found to be H + Li = 2 He 1.00812 + 7.01822 2 x 4.00391 Thus the reaction occurs with a decrease in rest mass Aw of 0.01852 g per mole. This is equivalent to an energy of 0.01852 x 9 x 10 20 - 1.664 x 10 19 erg per mole I 664 x 10 19 or -' - 2.763 x 10~ 5 erg per lithium nucleus O.v/^c X L\J or 2.763 x 10~ 5 x 6.242 x 10 11 - 17.25 x 10 6 - 17.25 mev This figure is in excellent agreement with the energy observed from the cloud-chamber experiments. Nor is this an isolated example, for hundreds of these nuclear reactions have been studied and completely convincing evidence for the validity of the equation E = c 2 m has been obtained. It has become rather common to say that a nuclear reaction like this illustrates the conversion of mass to energy, or even the annihilation of matter. This cannot be true in view of the fact that mass and energy are the same. It is better to explain what happens as follows: Rest mass is a par- ticularly concentrated variety of energy; Jeans once called it bottled energy. When the reaction 1 H 1 + 3 Li 7 -> 2 2 He 4 takes place, a small amount of this bottled energy is released ; it appears as kinetic energy of the particles re- acting, which is gradually degraded into the random kinetic energy or heat of the environment. As the molecules of the environment gain kinetic energy, they gain mass. The hotter a substance, the greater is its mass. Thus, in the nuclear explosion, the concentrated rest mass (energy) is degraded into the heat mass (energy) of the environment. There has been no over-all change in mass and no over-all change in energy; mass-energy is conserved. Sec. 3] NUCLEAR CHEMISTRY AND PHYSICS 231 The measurement of the large amounts of energy released in nuclear reactions now provides the most accurate known means of determining small mass differences. The reverse process of calculating atomic masses from the observed energies of nuclear reactions is therefore widely applied in the determination of precise atomic weights. A few of the values so obtained are collected in Table 9.1 and compared with mass-spectrometer data. The TABLE 9.1 ATOMIC WEIGHTS (O 16 - 16.0000) Atom Mass Spectrometer Value Mass-Energy Value H 1 1.008141 1.008142 H 2 2.014732 2.014735 * H 3 3.01700 He 3 3.01698 He 4 4.00386 4.00387 *He 6 6.02047 Li 6 6.0145 6.01686 Li 7 7.01818 7.01822 C 12 12.00381 12.00380 agreement between the two methods is exact, within the probable error of the experiments. It would be hard to imagine a more convincing proof of the equivalence of mass and energy. The starred isotopes are radioactive, and the only available mass values are those from the E = c 2 m relation. 3. Methods for obtaining nuclear projectiles. It was at about this point in its development that nuclear physics began to outgrow the limitations of small-scale laboratory equipment. The construction of machines for the pro- duction of enormously accelerated ions, capable of overcoming the repulsive forces of nuclei with large atomic numbers, demanded all the resources of large-scale engineering. One of the most generally useful of these atom-smashing machines has been the cyclotron, shown in the schematic drawing of Fig. 9.1, which was invented by E. O. Lawrence of the University of California. The charged particle is fed into the center of the "dees" where it is accelerated by a strong electric field. The magnetic field, however, constrains it to move in a circular path. The time required to traverse a semicircle is t ^R^v = (TT/B) (m/e) from eq. (8.9); this is a constant for all particles having the same ratio e/m. The electric field is an alternating one, chosen so that its polarity changes with a frequency twice that of the circular motion of the charged particle. On each passage across the dees, therefore, the particle receives a new for- ward impulse, and describes a trajectory of ever increasing radius until it is drawn from the accelerating chamber of the cyclotron. The 184-in. machine at Berkeley, California, will produce a beam of 100 mev deuterons (nuclei of deuterium atoms) having a range in air of 140 fe'et. 232 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 A limit to the energy of ions accelerated in the original type of cyclotron is the relativistic increase of mass with velocity; this eventually destroys the synchronization in phase between the revolving ions and the accelerating field across the dees. This problem has been overcome in the synchro- cyclotron, in which frequency modulation, applied to the alternating accel- erating potential, compensates for the relativistic defocusing. This modifica- tion of the original design of the Berkeley instrument has more than doubled the maximum ion energies obtained. Iniulotor Feed lines Oee Internal beam Electric deflector Vacuum can Lower pole Magnet pole piece Fig. 9.1. Schematic diagram of the cyclotron. (From Lapp and Andrews, Nuclear Radiation Physics, 2nd Ed. Prentice-Hall, 1954.) The synchrotron employs modulation of both the electric accelerating field and the magnetic focusing field. With this principle, it is possible to achieve the billion-volt range for protons. The cosmotron, a synchrotron completed in 1952 at Brookhaven National Laboratory, accelerates protons in a toroidal vacuum chamber with orbits 60 ft in diameter. The C-shaped magnets are placed around the vacuum chamber. Pulses of about 10 11 protons at 3.6 mev are fired into the chamber. After about 3 x 10 6 revolutions, the pulse of protons has reached 3 bev (3000 mev), and is brought to the target. A similar machine at Berkeley is designed to produce 10 bev protons. These particles are thus well within the energy range of cosmic rays. 4. The photon. The essential duality in the nature of radiation has already been remarked: sometimes it is appropriate to treat it as an electromagnetic wave, while at other times a corpuscular behavior is displayed. The particle of radiation is called the photon. A more detailed discussion of the relation between waves and particles will be given in the next chapter. One important result may be stated here. A homogeneous radiation of wave length A or frequency v = cjX may be considered to be composed of photons whose energy is given by the relation, = hv (9.3) Sec. 4] NUCLEAR CHEMISTRY AND PHYSICS 233 Here h is a universal constant, called Planck's constant, with the value 6.624 x 10~ 27 erg sec. A photon has no rest mass, but since e me 2 , its mass is m hv/c 2 . The corpuscular nature of light was first clearly indicated by the photo- electric effect, discovered by Hertz in 1887, and theoretically elucidated by Einstein in 1905. Many substances, but notably the metals, emit elections when illuminated with light of appropriate wave lengths. A simple linear relation is observed between the maximum kinetic energy of the photo- electrons emitted and the frequency of the incident radiation. The slope of the straight line is found to be Planck's constant h. Thus, \mv 2 = hv ~ < (9.4) Such an equation can be interpreted only in terms of light quanta, or photons, which in some way transmit their energy hv to electrons in the metal, driving them beyond the field of attraction of the metal ions. The term <f> represents the energy necessary to overcome the attractive force tending to hold the electron within the metal. If a photon (e.g., from X or y rays) strikes an electron, an interchange of energy may take place during the collision. The scattered photon will have a higher frequency if it gains energy, a lower fre- quency if it loses energy. This is called the Compton effect. Consider in Fig. 9.2 a photon, with Fig. 9.2. The Compton effect, initial energy hv, hitting an electron at rest at O. Let hv' be the energy of the scattered photon and let the scattered electron acquire a speed v. Then ifm is the mass of the electron, its momentum will be mv and its kinetic energy \mv 2 . The scattering angles are a and ft. The laws of conservation of energy and of momentum both apply to the collision. From the first, hv = hv' + \mv 2 From the second, for the jc and y components of the momentum, hv hv' r^ cos a + mv cos p c c hv' . = sin a mv sin p c Eliminating ft from the momentum equations by setting sin 2 ft + cos 2 ft = 1, and assuming that v' v <^ v, we find for the momentum imparted to the electron, 2hv . a -* (9 ' 5) 234 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 Then from the energy equation, by eliminating v, the change in frequency of the photon is hv* & v -- v _ v ' ~ ( 1 _ cos a ) me 2 The predicted angular dependence of the change in frequency has been confirmed experimentally in studies of the Compton scattering of X rays by electrons in crystals. The Compton effect has also been observed in cloud- chamber photographs. 5. The neutron. In 1930, W. Bothe and H. Becker discovered that a very penetrating secondary radiation was produced when a particles from polo- nium impinged on light elements such as beryllium, boron, or lithium. They believed this radiation to consist of 7 rays of very short wavelength, since no track was made in a cloud chamber, and therefore charged particles were not being formed. In 1932, Frederic and Irene Curie-Joliot found that this new radiation had much greater ionizing power after it had passed through paraffin, or some other substance having a high hydrogen content, and during its passage protons were emitted from these hydrogen-rich materials. James Chadwick 2 solved the problem of the new "radiation." He realized that it was made up of particles of a new kind, having a mass comparable with that of the proton, but bearing no electric charge. These particles were called neutrons. Because of its electrical neutrality, forces between the neutron and other particles become appreciable only at very close distances of approach. The neutron, therefore, loses energy only slowly as it passes through matter; in other words, it has a great penetrating power. The hydrogen nucleus is most effective in slowing a neutron, since it is of com- parable mass, and energy exchange is a maximum between particles of like mass, during actual collisions or close approaches. The reaction producing the neutron can now be written 2 He 4 4- 4 Be* - 6 C 12 + ^ Neutrons can be produced by similar reactions of other light elements with high energy a particles, protons, deuterons, or even y rays, for example : 1 H 2 + /n>-> 1 H l + /2 1 H l + 3 Li 7 ~> 4 Be 7 + o/? 1 Beams of neutrons can be formed by means of long pinholes or slits in thick blocks of paraffin, and methods are available for producing beams of uniform energy. 3 Because it can approach close to an atomic nucleus without being electro- statically repelled, the neutron is an extraordinarily potent reactant in nuclear processes. 2 Proc. Roy. Soc., A 136, 692 (1932). 3 E. Fermi, J. Marshall, and L. Marshall, Phys. Rev., 72, 193 (1947); W. Zinn, ibid., 71, 757 (1947). Sec. 6] NUCLEAR CHEMISTRY AND PHYSICS 235 6. Positron, meson, neutrino. The year 1932 was a successful one for nuclear physics, because two new fundamental particles were discovered, the neutron and the positron. The latter was detected by Carl D. Anderson in certain cloud-chamber tracks from cosmic rays. The positron is the positive electron e+. It had previously been predicted by the theoretical work of Dirac. In 1933 Frederic and Irene Curie-Joliot found that a shower of positrons was emitted when a rays from polonium impinged on a beryllium target. When targets of boron, magnesium, or aluminum were used, the emission of p6sitrons was observed to continue for some time after the particle bombardment was stopped. This was the first demonstration of artificial radioactivity. 4 A typical reaction sequence is the following: 5 B 10 + 2 He 4 -> O n l + 7 N 1:i ; 7 N 13 -> 6 C 13 + e+ More than a thousand artificially radioactive isotopes are now known, produced in a variety of nuclear reactions. 5 The positron escaped detection for so long because it can exist only until it happens to meet an electron. Then a reaction occurs that annihilates both of them, producing a y-ray photon: e -\ f e~~ -> hv The energy equivalent to the rest mass of an electron is: f - me 2 - 9.11 x 10- 28 x (3.00 x 10 10 ) 2 --- 8.20 x 10 7 erg If this is converted into a single y-ray photon, the wavelength would be A - n,c ~ 9.U X-IO The y radiation obtained in 'the annihilation of electron-positron pairs has either this wavelength or one-half of it. The latter case corresponds to the conversion of the masses of both e + and e~~ into a single y-ray photon. The reverse process, the production of an electron-positron pair from an energetic photon, has also been observed. In 1935, H. Yukawa proposed for the structure of the nucleus a theory that postulated the existence of a hitherto unknown kind of particle, which would be unstable and have a mass of about 150 (electron ~ 1). From 1936 to 1938 Anderson's work at Pasadena revealed the existence of particles, produced by cosmic rays, which seemed to have many of the properties predicted by Yukawa. These particles are the ^-mesons, which may be charged plus or minus, have a mass of 209 2, and a half life of 2.2 x 10~ 6 sec. The particles required by the theory, however, resemble more closely the 7r-mesons, discovered in 1947 by the Bristol cosmic-ray group headed by C. F. Powell. These have a mass of 275, and decay to /^-mesons, with a half life of 2.0 x 10~ 8 sec. Several other particles, with masses of 800 to 4 C. R. Acad. Set. Paris, 198, 254, 559 (1934). 5 G. Seaborg and I. Perlman, Rev. Mod. Phys., 20, 585 (1948). 236 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 1300, have also been discovered. The theoretical interpretation of the variety of particles now known will require some new great advance in fundamental theory. In order to satisfy the law of conservation of mass-energy in radioactive decays, decay of mesons, and similar processes, it is necessary to postulate the existence of neutral particles with rest masses smaller than that of the electron. These neutrinos have not yet been detected by physical methods, since their effects are necessarily small. 7. The structure of the nucleus. The discovery of the neutron led to an important revision in the previously accepted picture of nuclear structure. Instead of protons and electrons, it is now evident that protons and neutrons are the true building units. These are therefore called nucleons. Each nucleus contains a number of protons equal to its atomic number Z, plus a number of neutrons /?, sufficient to make up the observed mass number A. Thus, A n + Z. The binding energy E of the nucleus is the sum of the masses of the nucleons minus the actual nuclear mass M. Thus, E -=- Zm n + (A - Z)m n M (9.6) The proton mass m ir = 1.00815, the neutron mass m n 1.00893 in atomic mass units. To convert this energy from grams per mole to mev per nuclcon, it must be multiplied by C 2 . _ 934 W x 10 x 1.602 x 10 ~ 12 One of the convincing arguments against the existence of electrons as separate entities in the nucleus is based on the magnitude of the observed binding energies. For example, if the deuteron 1 H 2 were supposed to be made up of two protons and an electron, the binding energy would be 0.001 53 gram per mole. Yet the electron's mass is only 0.00055 gram per mole. For the electron to preserve its identity in the nucleus while creating a binding energy about three times its own mass would seem to be physically most unreasonable. We do not yet know the nature of the forces between nucleons. The nuclear diameter is given approximately by d 1.4 x 10~ 13 A m cm, A being the mass number. The forces therefore must be extremely short-range, unlike electrostatic or gravitational forces. The density of nuclear material is around 10 14 g per cc. A drop big enough to see would weigh 10 7 tons. There is an electrostatic repulsion between two protons, but this longer-range (inverse square) force is outweighed by the short-range attraction, so that at separa- tions around 10~ 13 cm the attraction between two protons is about the same as that between two neutrons or a neutron and a proton. According to Yukawa's theory, the attractive forces between nucleons are due to a new type of radiation field, in which the mesons play a role like that of the photons in an ordinary electromagnetic field. Sec. 7] NUCLEAR CHEMISTRY AND PHYSICS 237 A further insight into nuclear forces can be obtained by examining the composition of the stable (nonradioactive) nuclei. In Fig. 9.3 the number of neutrons in the nucleus is plotted against the number of protons. The line has an initial slope of unity, corresponding to a one-to-one ratio, but it curves upward at higher atomic numbers. The reason for this fact is that the electrostatic repulsion of the protons increases as the nucleus becomes larger, since it is a longer range force than the attraction between protons. To compensate for this repulsion more neutrons are necessary. Yet there is 140 130 120 MO 100 90 c/> i 70 t; 60 UJ 2 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 110 PROTONS, Z Fig. 9.3. Number of neutrons vs. number of protons in stable nuclei. a limit to the number of extra neutrons that can be accommodated and still produce added stability, so that the heavier nuclei become less stable. This effect is illustrated clearly in Fig. 9.4, which shows the binding energy per nucleon as a function of the mass number. Only the stable iso- topes lie on this reasonably smooth curve. Natural or artificial radioactive elements fall below the curve by an amount that is a measure of their in- stability relative to a stable isotope of the same mass number. The successive maxima in the early part of the curve occur at the following nuclei: He 4 , Be 8 , C 12 , O 16 , Ne 20 . These are all nuclei containing an equal number of protons and neutrons, and in fact they are all polymers of He 4 . It is possible to say, therefore, that the forces between nucleons become saturated, like the valence bonds between atoms. The unit He 4 , two protons and two neutrons, appears to be one of exceptional stability. The nuclear shell structure is also clearly indicated in the packing-fraction vs. mass number 238 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 curves of Fig. 8.1 1 . The lower the packing fraction, the greater is the binding energy per nucleon. Another viewpoint is to consider that there are certain allowed energy levels in the nucleus. Each level can hold either two neutrons or two protons. 6 The upper proton levels become raised in energy owing to the coulombic GY PER NUCLEON - MEV T) ->l 00 Q r \ FISSIO ENERG N ^ Y f \ 1 \ \ o: ~ UJ z UJ o z 5 \ z CD 4 50 200 250 100 150 MASS NUMBER Fig. 9.4. Binding energy per nucleon as a function of atomic mass number. repulsion. Of all the stable nuclei, 152 have both n and Z even; 52 have Z odd, n even; 55, Z even, n odd; and only 4 have both n and Z odd. The four odd-odd nuclei are H 2 , Li 6 , B 10 , N 14 . Not only are the even-even nuclei the most frequent, they also usually have the greatest relative abundance. It can be concluded that filled nuclear energy levels confer exceptional stability. 8. Neutrons and nuclei. Since the neutron is an uncharged particle, it is not repelled as it approaches a nucleus, even if its energy is very low. We often distinguish fast neutrons, with a kinetic energy of > 100 ev, and slow neutrons, with energies from 0.01 to 10 ev. If the energies have the same magnitude as those of ordinary gas molecules (kT), the neutrons are called thermal neutrons. At 300K, kT --= 0.026 ev. The interaction of a neutron and a nucleus can be represented by the intermediate formation of a compound nucleus which may then react in several ways. If the neutron is released again, with the reformation of the original nucleus, the process is called scattering. If the neutron is retained for some time, although there may be a subsequent decomposition of the compound nucleus into new products, the process is called capture or absorption . * See Section 10-25 and discussion of nuclear spin on p. 247. Sec. 8] NUCLEAR CHEMISTRY AND PHYSICS 239 A quantitative description of the interaction between a nucleus and a neutron is given in terms of the effective nuclear cross section, a. Consider a beam of neutrons in which the neutron flux is n per cnv 2 per sec. If the beam passes through matter in which there are c nuclei of a given kind per cc, the number of neutrons intercepted per sec in a thickness fix is given by - dn nac dx (9.7) An initial flux of n is therefore reduced after a distance .v to n x n c cax . The scattering cross section a s is distinguished from the absorption cross section cr rt , and a a s \ a (l . Nuclear cross sections are generally of the order 10~ 24 cm 2 , and the whimsical physicists have called this unit the barn. 500 CD J 100 o UJ o o: o j < i- o 50 10 I 500 Fig. 9.5. 5 10 5O 100 NEUTRON ENERGY-6V Nuclear cross section of silver. [From Rainwater, Havens, Wu, and Dunning, Phys. Rev. 71, 65 (1947),] The cross sections depend on the kinetic energy of the neutrons and may be quite different in the low- and high-velocity ranges. The dependence of a on energy yields important information about energy levels in the nucleus, for when the neutron energy is very close to a nuclear energy level, a "resonance" occurs that greatly facilitates capture of the neutron, and hence greatly increases the value of a u . For example, for thermal neutrons, a^H 1 ) = 0.31 barn, a^H 2 ) - 0.00065 barn. Both ^H 1 and X H 2 have high scattering cross sections, and are therefore effective in slowing fast neutrons, but many of the thermal neutrons produced would be lost by capture to jH 1 dH 1 + Q/2 1 -> 1 H 2 ). It is for this reason that heavy water is a much more efficient neutron moderator than light water. In H 2 O a thermal neutron 240 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 would have, on the average, 150 collisions before capture; in D 2 O, 10 4 ; in pure graphite, 10 3 . A particularly important scattering cross section is that of cadmium. The cadmium nucleus has a resonance level in the thermal neutron region, leading to the tremendously high a = 7500 at 0.17ev. Thus a few milli- meters of cadmium sheet is practically opaque to thermal neutrons. The cross section for silver is shown in Fig. 9.5 as a function of neutron energy. The peaks in the curve correspond to definite neutron energy levels in the nucleus. The task of the nuclear physicist is to explain these levels, as the extranuclear energy levels of the electrons have been explained by the Bohr theory and quantum mechanics. (See Chapter 10.) 9. Nuclear reactions. The different types of nuclear reactions are con- veniently designated by an abbreviated notation that shows the reactant particle and the particle emitted. Thus an (n, p) reaction is one in which a neutron reacts with a nucleus to yield a new nucleus and a proton, e.g., 7 N 14 + X -* 6 C 14 + iH 1 would be written 7 N 14 (/i,/7) 6 C 14 . In Table 9.2 the various nuclear reaction types are summarized. The TABLE 9.2 TYPES OF NUCLEAR REACTIONS Reaction Type n capture np no. n,2n p capture pn pen Normal Mass Change Slightly + si 4 light clem, si heavy Very - si 4- light elem. heavy r*~ OH ~' j si light elem. 4 heavy ap si 4- except -for light clem. dp Always + dn Always + * Always 4- yn Always yp Always - Dependence on Energy of Projectile Yield Type of Radio- activity Usually Produced Example Resonance 100 per cent ft- Ag 107 + ! = Ag 108 Smooth High for light elements ft- N 14 f n 1 - C 14 4- H l Smooth High for light elements fi- Mg* -f n 1 - Ne" 4- He* Smooth Low P P" 4- n 1 = P" + 2n Resonance High fi+ C 13 4 H 1 - N 18 Threshold then smooth High ^ Cu" 4- H 1 = Zn" f- n 1 Smooth High Stable F" 4- H 1 = O 16 4- He* Smooth Low Be* 4- H 1 = Be 8 4- H Smooth High for light elements P C 11 4- He* - O 18 + n 1 Smooth High for light elements Stable N 14 4- He 4 = 0" 4- H 1 Smooth High for light elements 0- Co" 4- H - Co" -f H 1 Smooth High for light elements fi+ C 11 4- H 1 - N 18 4- H 1 Smooth High for light elements Stable O" 4- H 1 = N 14 4- He* Sharp threshold Low f Be* 4- y = Be' + /i l Sharp threshold Low H* 4- Y - H 1 4 /> Sec. 10] NUCLEAR CHEMISTRY AND PHYSICS 241 second column gives the normal rest-mass change for the reaction. A positive mass change is equivalent to an endothermic reaction, a negative mass change to an exothermic reaction. The next column indicates how the yield depends on the energy of the bombarding particle. In most cases there is a smooth increase in yield with increasing energy, but for capture processes there is a marked resonance effect. 10. Nuclear fission. Perusal of the binding energy curve in Fig. 9.4 reveals that a large number of highly exothermic nuclear reactions are possible, since the heavy nuclei toward the end of the periodic table are all unstable relative to the nuclei lying around the maximum of the curve. In the January 1939 number of Naturwissenschaften, Otto Hahn and S. Strassman reported that when the uranium nucleus is bombarded with neutrons it may split into fragments, one of which they identified as an isotope of barium. About 200 mev of energy is released at each fission. It was immediately realized that secondary neutrons would very possibly be emitted as a result of uranium fission, making a chain reaction possible. The likelihood of this may be seen as follows: Consider the fission of a 92 U 235 nucleus to yield, typically, a 56 Ba 139 as one of the observed disintegra- tion products. If balance is to be achieved between the numbers of protons and neutrons before and after fission, the other product would have to be 36 Kr 98 . This product would be far heavier than any previously known krypton isotope, the heaviest of which was 36 Kr 87 , a ft" emitter of 4 hours half-life. Now the hypothetical 36 Kr 96 can get back to the proton-neutron curve of Fig. 9.3 by a series of ft" emissions, and in fact a large number of new ft" emitters have been identified among the fission products. The same result can be achieved, however, if a number of neutrons are set free in the fission process. Actually, both processes occur. The fission process usually consists, therefore, of a disintegration of uranium into two lighter nuclei, one of mass number from 82 to 100, and the other from 128 to 150, plus a number, perhaps about three, of rapidly moving neutrons. In only about one case in a thousand does symmetrical fission into two nuclei of approximately equal mass occur. To determine which isotope of uranium is principally responsible for fission, A. O. Nier and his coworkers separated small samples of U 235 (0.7 per cent abundance) and U 238 (99.3 per cent) with a mass spectrometer. It was found that U 235 undergoes fission even when it captures a slow thermal neutron, but U 238 is split only by fast neutrons with energies greater than 1 mev. As usual, the capture cross section for slow neutrons is much greater than that for fast neutrons, so that U 235 fission is a much more likely process than that of U 238 . The process of fission can be visualized by con- sidering the nucleus as a drop of liquid. When a neutron hits it, oscillations are set up. The positive charges of the protons acquire an unsymmetrical distribution, and the resulting repulsion can lead to splitting of the nuclear drop. Since U 235 contains an odd number of neutrons, when it gains a 242 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 neutron considerable energy is set free. This kinetic energy starts the dis- turbance within the nucleus that leads to fission. The isotope U 238 already contains an even number of neutrons, and the capture process is not so markedly exothermic. Therefore the neutron must be a fast one, bringing considerable kinetic energy into the nucleus, in order to initiate fission. Fission of other heavy elements, such as lead, has been produced by bom- bardment with 200 mev deuterons produced by the Berkeley cyclotron. Fission can also be induced by y rays with energies greater than about 5 mev (photofissiori). In Fig. 9.6 are shown the mass distributions of the fission products in three different cases that have been carefully studied. When fission is produced uj 6 u. (/) O < 50 IOO MASS NUMBER ISO Fig. 9.6. Mass distribution of products in three different fission reactions. The energies of the particles initiating the fission are: n, thermal; a, 38 mev; rf, 200 mev. Note how the distribution becomes more symmetrical as the energy of the incident particle increases. (From P. Morrison, "A Survey of Nuclear Reactions" in Experi- mental Nuclear Physics, ed. E. Segre, Wiley, 1953.) by highly energetic particles, the distribution of masses is quite symmetrical, and the most probable split is one that yields two nuclei of equal mass. This is the result that would be expected from the liquid-drop model. The un- symmetrical splitting that follows capture of slower particles has not yet received a satisfactory theoretical explanation, but it is undoubtedly related to the detailed shell structure inside the nucleus. A nuclear reaction of great interest is spontaneous fission, discovered in 1940 by Flerov and Petrzhak in the U.S.S.R. It cannot be attributed to cosmic radiation or to any other known external cause, and it must be considered to Sec. 11] NUCLEAR CHEMISTRY AND PHYSICS 243 be a new type of natural radioactivity. For example, when about 6 g of Th 232 were observed for 1000 hr, 178 spontaneous fissions were detected. Sponta- neous fission is usually a very rare reaction, but it becomes much more frequent in some of the transuranium elements. 11. The transuranium elements. In 1940, E. McMillan and P. H. Abelson 7 found that when U 238 is irradiated with neutrons, a resonance capture can occur that leads eventually to the formation of two new transuranium elements. U 238 f" 1 - 92 92 U 239 23inin -> 93 Np 239 -f The 94 Pu 239 is a weakly radioactive a emitter (r 2.4 x 10 4 years). Its most important property is that, like U 235 , it undergoes fission by slow neutrons. It was shown by G. T. Seaborg 8 and his coworkers that bombardment of U 238 with a particles leads by an (a, n) reaction to Pu 241 . This is a ft emitter and decays to give 95 Am 241 , an isotope of americium which is a-radioactive with 500 years half-life. By 1954, the last of the transuranium elements to have been prepared were curium (96), berkelium (97), californium (98), and elements (99) and (100). Curium can be made by an (a, ri) reaction on Pu 239 : 94 Pu 239 f 2 He 4 ^ ^ + 96 Cm 242 The preparation of new examples of the transuranium elements has been facilitated by the technique of using heavy ions accelerated in the cyclotron. Thus high energy beams of carbon ions, ( 6 C 12 ) 6 ! , can increase the atomic number of a target nucleus by six units in one step. For example, isotopes of californium have been synthesized as follows: 9 92 U 238 + 6 C 12 -> 98 Cf 244 + 6/1 Element (99) was prepared by: 92 U 238 + 7 N 14 -> 99 X 247 + 5/i 12. Nuclear chain reactions. Since absorption of one neutron can initiate fission, and more than one neutron is produced at each fission, a branching chain can occur in a mass of fissionable material. The rate of escape of neutrons from a mass of U 235 , for example, depends on the area of the mass, whereas the rate of production of neutrons depends on the volume. As the volume of the mass is increased, therefore, a critical point is finally reached at which neutrons are being produced more rapidly than they are being lost. 7 Phys. Rev., 57, 1185 (1940). 8 Science, 104, 379 (1946); Chem. Eng. News, 25, 358 (1947). 9 A. Ghiorso, S. G. Thompson, K. Street, and G. T. Seaborg, Phys. Rev. 81, 1954 (1951). 244 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 If two masses of U 235 of subcritical mass are suddenly brought together a nuclear explosion can take place. For the continuous production of power, a nuclear pile is used. The fissionable material is mixed with a moderator such as graphite or heavy water, to slow down the neutrons. Control of the rate of fission is effected by introducing rods of a material such as cadmium, which absorbs the thermal neutrons. The depth to which the cadmium rods are pushed into the pile controls the rate of fission. The pile also serves as a source of intense beams of neutrons for research purposes. As shown in Fig. 9.7, a diagram of the Brookhaven pile, these REMOVABLE PLUG ION CHAMBER FOR PILE CONTROL PNEUMATIC TUBE RABBIT" Fig. 9.7. Diagrammatic sketch of the Brookhaven pile showing the features of importance for pile neutron research. (From D. J. Hughes, Pile Neutron Research, Addison-Wesley, 1953.) beams can be either fast neutrons from the center of the pile, or thermal neutrons drawn out through a layer of moderator. 13. Energy production by the stars. The realization of the immense quantities of energy that are released in exothermic nuclear reactions has also provided an answer to one of the great problems of astrophysics the source of the energy of the stars. At the enormous temperatures prevailing in stellar interiors (e.g., around 10 million degrees in the case of our sun) the nuclei have been stripped of electrons and are moving with large kinetic- theory velocities. Thus the mean thermal kinetic energy of an a particle at room temperature is of the order of ^ ev, but at the temperature of the sun it has become 10 4 ev. In other words, at stellar temperatures many of the nuclei have attained energies comparable with those of the high-velocity particles produced on earth by means of the cyclotron and similar devices. Nuclei with these high energies will be able to overcome the strong electrostatic repulsion between their positive charges and approach one Sec. 14] NUCLEAR CHEMISTRY AND PHYSICS 245 another sufficiently closely to initiate various nuclear reactions. It is these so-called thermonuclear reactions that account for the energy production of the stars. In 1938, Carl von Weizsacker and Hans Bethe independently proposed a most ingenious mechanism for stellar-energy production. This is a cycle proceeding as follows: C 12 + H 1 ->N 13 -\- hv N 13 -C 13 I- <?- C 13 f H 1 -^ N 14 |- hv N u I H 1 ^O ir M hv O 15 ^N 15 4-e'- N 15 -h H 1 >C 12 -|- He 4 The net result is the conversion of four H nuclei into one He nucleus through the mediation of C 12 and N 14 as "catalysts" for the nuclear reaction; 30mev are liberated in each cycle. This carbon cycle appears to be the principal source of energy in very hot stars (T> 5 x 10 s K). The energy of somewhat cooler stars, like our sun (T ~ 10" K), appears to be generated by the proton-proton cycle : iH 1 + t H l -- jH- | H | 0.42 mev ^ + X H 2 -- 2 He 3 4- y f 5.5 mev 2 He 3 + 2 He 3 - 2 He 4 f 2 1 H 1 f 12.8 mev The net result is the conversion of 4 protons to one helium nucleus, with the liberation of 24.6 mev plus the annihilation energy of the positron. Gamow has estimated 10 that reactions between hydrogen nuclei ^H 1 -| t H 2 -> 2 He 3 4- y; 2 jH 2 -> 2 He 4 -1 y) would have an appreciable rate at temperatures below 10 6 degrees; reactions of protons with lithium nuclei ( X H 1 -f 3 Li 6 -^ 2 He 4 -f 2 He 3 ; X H 1 f 3 Li 7 -> 2 2 He 4 ) require about 6 x 10 6 degrees; reactions such as jH 1 + 5 B 10 - -> 6 C X1 h y require about 10 7 degrees. The temperatures attainable by means of uranium or plutonium fission are high enough to initiate thermonuclear reactions of the lighter elements. The fission reaction acts as a "match" to start the fusion reactions. Easiest of all to "ignite" should be mixtures containing tritium, the hydrogen isotope of mass 3. X H 3 + jH 2 -- 2 He 4 4 O n l f 17.6 mev (y) !H 3 f 1 H 1 - 2 He 4 + 19.6 mev (y) The tritium can be prepared by pile reactions such as 3 Li (J f A? 1 ^ 2 He 4 f 1 H 3 . The isotope Li 6 has an abundance of 7.52 atom per cent. 14. Tracers. The variety of radioactive isotopes now available has made possible many applications in tracer experiments, in which a given type of atom can often be followed through a sequence of chemical or physical 10 George Gamow, The Birth and Death of the Sun (frew York: Penguin, 1945), p. 128. 246 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 changes. Stable isotopes can also be used as tracers, but they are not so easily followed and are available for relatively few elements. Radioactive isotopes can be obtained from four principal sources: (1) natural radio- activity; (2) irradiation of stable elements with beams of ions or electrons obtained from accelerators such as cyclotrons, betatrons, etc.; (3) pile irradiation with neutrons; (4) fission products. In Table 9.3 are listed a few of the many available isotopes. TABLE 9.3 ARTIFICIAL RADIOACTIVE ELEMENTS Nucleus Activity Half Life C 11 p+, y 21 min c 14 p- 5700 yr N 13 /?+, y 9.9 min o 16 p 125 sec Na 22 ft*, y 3.0 yr Na 24 /?-, y 14.8 hr P 32 j j- 14.3 days S 35 jJ- 87.1 days Ca 45 /?- 152 days Fe 59 j3~, y 46 days Co 60 0~, y 5.3 yr Cu 64 ^ + , p~ 12.8hr One of the earliest studies with radioactive tracers used radioactive lead to follow the diffusion of lead ions in solid metals and salts. For example, a thin coating of radiolead can be plated onto the surface of a sample of metallic lead. After this is maintained at constant temperature for a definite time, thin slices are cut off and their radioactivity measured with a Geiger counter. The self-diffusion constant of Pb in the metal can readily be cal- culated from the observed distribution of activity. Many such diffusion studies have now been made in metals and in solid compounds. The results obtained are of fundamental importance in theories of the nature and prop- erties of the solid state. Diffusion in liquids, as well as the permeability of .natural and synthetic membranes, can also be conveniently followed by radioactive tracer methods. The solubility of water in pure hydrocarbons is so low that it is scarcely measurable by ordinary methods. If water containing radioactive hydrogen, or tritium, ^^ a ft~~ emitter of 12 years half life, is used, even minute amounts dissolved in the hydrocarbons are easily measured. 11 A useful tracer method is isotopic dilution analysis. An example is the determination of amino acids in the products of protein hydrolysis. The conventional method would require the complete isolation of each amino acid in pure form. Suppose, however, a known amount of an amino acid 11 C. Black, G. G. Joris, and H. S. Taylor, /. Chem. Phys., 16, 537 (1948). Sec. 15] NUCLEAR CHEMISTRY AND PHYSICS 247 labeled with deuterium or carbon-14 is added to the hydrolysate. After thorough mixing, a small amount of the given acid is isolated and its activity measured. From the decrease in activity, the total concentration of the acid in the hydrolysate can be calculated. Tracers are used to elucidate reaction mechanisms. One interesting problem was the mechanism of ester hydrolysis. Oxygen does not have a radioactive isotope of long enough half life to be a useful tracer, but the stable O 18 can be used. By using water enriched with heavy oxygen (O*) the reaction was $hown to proceed as follows: O O R C/ + HO*H -, R-C< I R'OH X OR' X O*H The tagged oxygen appeared only in the acid, showing that the OR group is substituted by O*H in the hydrolysis. 12 Radioactive isotopes of C, Na, S, P, etc., are of great use in investigations of metabolism. They supplement the stable isotopes of H, N, and O. For ^example it has been found that labeled phosphorus tends to accumulate preferentially in rapidly metabolizing tissues. This has led to its trial in cancer therapy. The results in this case have not been particularly encouraging, but it may be possible to find metabolites or dyes that are specifically concen- trated in tumor tissues, and then to render these compounds radioactive by inclusion of appropriate isotopic atoms. 13 15. Nuclear spin. In addition to its other properties, the nucleus may have an intrinsic angular momentum or spin. All elementary particles (i.e., neutrons, protons, and electrons) have a spin of one-half in units of h/27r. The spin of the electron will be considered in some detail in the next chapter. The spin of the elementary particles can be either plus or minus. If an axis is imagined passing through the particle, the sign corresponds to a clockwise or counterclockwise spin, although this picture is a very crude one. The spin of a nucleus is the algebraic sum of the spins of the protons and neutrons that it contains. The hydrogen nucleus, or proton, has a spin of one-half. If two hydrogen atoms are brought together to form H 2 , the nuclear spins can be either parallel ( 1f ) or antiparallel ( 11, ). Thus there are two nuclear spin isomers of H 2 . The molecule with parallel spins is called "orthohydrogen," the one with antiparallel spins is called "parahydrogen." Since spins almost never change their orientation spontaneously, these two isomers are quite stable. They have different heat capacities and different molecular spectra. Other molecules composed of two identical nuclei having nonzero spin behave similarly, but only in the cases of H 2 and D 2 are there marked differences in physical properties. 12 M. Polanyi and A. L. Szabo, Trans. Farad. Soc., 30, 508 (1934). 13 M. D. Kamen, Radioactive Tracers in Biology (New York: Academic, 1947). 248 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 PROBLEMS 1. What is the Am in g per mole for the reaction H 2 -f t O 2 H 2 O for which A// - - 57.8 kcal per mole? 2. From the atomic weights in Table 9.1, calculate the AE in kcal per mole for the following reactions: ^ + o/;' - ^2, t H 2 f- o* 1 = ^ 1 H 1 f e - X, 2 t H 2 - 2 He 4 3. Calculate the energies in (a) ev (b) kcal per mole of photons having wavelengths of 2.0 A, 1000 A, 6000 A, 1 mm, 1 m. 4. To a hydrolysate from 10 g of protein is added 100 mg of pure CD 3 CHNH 2 COOH (deuterium-substituted alanine). After thorough mixing, 100 mg of crystalline alanine is isolated which has a deuterium content of 1.03 per cent by weight. Calculate the per cent alanine in the protein. 5. A 10-g sample of iodobenzene is shaken with 100 ml of a 1 M KI solu- tion containing 2500 counts per min radio-iodine. The activity of the iodo- benzene layer at the end of 2 hours is 250 cpm. What per cent of the iodine atoms in the iodobenzene have exchanged with the iodide ions in solution? 6. Calculate the mass of an electron accelerated through a potential of 2 x 10 8 volts. What would the mass be if the relativity effect is ignored? 7. Naturally occurring oxygen consists of 99.76 per cent O 16 , 0.04 per cent O 17 , and 0.20 per cent O 18 . Calculate the ratio of atomic weights on the physical scale to those on the chemical scale. 8. The work function of a cesium surface is 1.81 volts. What is the longest wavelength of incident light that can eject a photoelectron from Cs? 9. The 77-meson has a mass about 285 times that of the electron; the /t-meson has a mass about 215 times that of the electron. The 7r-meson decays into a //-meson plus a neutrino. Estimate A for the reaction in ev. 10. Calculate the energy necessary to produce a pair of light mesons. This pair production has been accomplished with the 200-in. California cyclotron. 11. The scattering cross section, a, of lead is 5 barns for fast neutrons. How great a thickness of lead is required to reduce the intensity of a neutron beam to 5 per cent of its initial value? How great a thickness of magnesium with a = 2 barns? 12. According to W. F. Libby [Science, 109, 22V (1949)] it is probable that radioactive carbon- 14 (r = 5720 years) is produced in the upper atmo- sphere by the action of cosmic-ray neutrons on N 14 , being thereby main- tained at an approximately constant concentration of 12.5 cpm per g of carbon. A sample of wood from an ancient Egyptian tomb gave an activity of 7.04 cpm per g C. Estimate the age of the wood. 13. A normal male subject weighing 70.8 kg was injected with 5.09 ml of water containing tritium (9' x 10 9 cpm). Equilibrium with body water was Chap. 9] NUCLEAR CHEMISTRY AND PHYSICS 249 reached after 3 hr when a 1-ml sample of plasma water from the subject had an activity of 1.8 x 10 5 cpm. Estimate the weight per cent of water in the human body. 14. When 38 Sr 88 is bombarded with deuterons, 38 Sr 89 is formed. The cross section for the reaction is 0.1 barn. A SrSO 4 target 1.0 mm thick is exposed to a deuteron beam current of 100 microamperes. If scattering of deuterons is neglected, compute the number of Sr atoms transmuted in 1.0 hr. The Sr 88 is 82.6 per cent of Sr, and Sr 89 is a Remitter of 53-day half life. Compute the curies of Sr 89 produced. 15. When 79 Au 197 (capture cross section a c =-- 10~ 22 cm 2 ) is irradiated with slow neutrons it is converted into 79 Au 198 (r --= 2.8 days). Show that in general the number of unstable nuclei present after irradiation for a time / is o^ (1 _ e ^ A Here n Q is the number of target atoms and <f> is the slow neutron flux. For the case in question, calculate the activity in microcuries of a 100-mg gold sample exposed to a neutron flux of 200/cm 2 sec for 2 days. 16. The conventional unit of quantity of X radiation is the roentgen, r. It is the quantity of radiation that produces 1 esu of ions in 1 cc of air at STP (1 esu === 3.3 x 10~ 10 coulomb). If 32.5 ev are required to produce a single ion pair in air, calculate the energy absorbed in 1 liter of air per roentgen. 17. Potassium-40 constitutes 0.012 per cent of natural K, and K is 0.35 per cent of the weight of the body. K 40 emits /? and y rays and has r 4.5 x 10 8 yr. Estimate the number of disintegrations per day of the K 40 in each gram of body tissue. 18. The isotope 89 Ac 225 has r 10 days and emits an a with energy of 5.80 mev. Calculate the power generation in watts per 100 mg of the isotope. REFERENCES BOOKS 1. Baitsell, G. A. (editor), Science in Progress, vol. VI (New Haven: Yale Univ. Press, 1949). Articles by H. D. Smyth on Fission; J. A. Wheeler on Elementary Particles; E. O. Lawrence on High Energy Physics; G. T. Seaborg on Transuranium Elements. 2. Bethe, H., Elementary Nuclear Theory (New York: Wiley, 1947). 3. Friedlander, G., and J. W. Kennedy, Introduction to Radiochemistry (New York: Wiley, 1949). 4. Gamow, G., and C. L. Critchfield, Theory of Atomic Nucleus and Nuclear Energy Sources (New York: Oxford, 1949). 5. Goodman, C. (editor), The Science and Engineering of Nuclear Power (2 vols) (Boston: Addison-Wesley, 1947, 1949). 250 NUCLEAR CHEMISTRY AND PHYSICS [Chap. 9 6. Halliday, D., Introductory Nuclear Physics (New York: Wiley, 1950). 7. Hughes, D. J., Pile Neutron Research (Boston: Addison- Wesley, 1953). 8. Lapp, R. E., and H. L. Andrews, Nuclear Radiation Physics, 2nd ed. (New York: Prentice-Hall, 1954). 9. Libby, W. F., Radiocarbon Dating (Chicago: Univ. of Chicago Press, 1952). ARTICLES 1. Anderson, C. D., Science in Progress, 7, 236-249 (1951), 'The Elementary Particles of Physics." 2. Curtan, S. C., Quart. Rev., 7, 1-18 (1953), "Geological Age by Means of Radioactivity/' 3. Dunning, J. R., Science in Progress, 7, 291-355 (1951), "Atomic Structure and Energy." 4. Hevesy, G. C., J. Chem. Soc., 1618-1639 (1951), "Radioactive Indicators in Biochemistry." 5. Pryce, M. H. L., Rep. Prog. Phys., 17, 1-35 (1954), "Nuclear Shell Structure." 6. Wilkinson, M. K., Am. J. Phys., 22, 263-76 ( 1 954), "Neutron Diffraction." CHAPTER 10 Particles and Waves 1. The dual nature of light. It has already been noted that in the history of light two different theories were alternately in fashion, one based on the particle model and the other on the wave model. At the present time both must be regarded with equal respect. In some experiments light displays notably corpuscular properties: the photoelectric and Compton effects can be explained only by means of light particles, or photons, having an energy s hv. In other experiments, which appear to be just as convincing, the wave nature of light is manifest: polarization and interference phenomena require an undulatory theory. This unwillingness of light to fit neatly into a single picture frame has been one of the most perplexing problems of natural philosophy. The situa- tion recalls the impasse created by the "null result" of the Michelson-Morley experiment. This result led Einstein to examine anew one of the most basic of physical concepts, the idea of the simultaneity of events in space and time. The consequence of his searching analysis was the scientific revolution expressed in the relativity theories. An equally fundamental enquiry has been necessitated by the develop- ments arising from the dual nature of light. These have finally required a re-examination of the meaning and limitations of physical measurement when applied to systems of atomic dimensions or smaller. The results of this analysis are as revolutionary as the relativity theory; they are embodied in what is called quantum theory or wave mechanics. Before discussing the significant experiments that led inexorably to the new theories, we shall review briefly the nature of vibratory and wave motions. 2. Periodic and wave motion. The vibration of a simple harmonic oscilla- tor, discussed on page 190, is a good example of a motion that is periodic in time. The equation of motion (/= ma) is md 2 x/dt 2 --== KX. This is a simple linear differential equation. 1 It can be solved by first making the sub- stitution p dx/dt. Then d*x/dt* =-- dpjdt -= (dp/dx)(dx/dt) = p(dp/dx), and the equation becomes p(dp\dx) + (K/W)X 0. Integrating, p 2 +(t</m)x 2 = const. The integration constant can be evaluated from the fact that when the oscillator is at the extreme limit of its vibration, x = A, the kinetic energy is zero, and hence p 0. Thus the constant = (K/m)A 2 . Then 1 See, for example, Gran vi lie et al., Calculus, p. 383.. 252 PARTICLES AND WAVES [Chap. 10 ~~df V/,4 2 - * 2 . , X IK sin" 1 - / t -\ const A *< m This integration constant can be evaluated from the initial condition that at / =-- 0, A- ---=-- 0; therefore constant 0. The solution of the equation of motion of the simple harmonic oscillator is accordingly: x -= AsmJ-~t (10.1) If we set VK//W --- 2771-, this becomes x ^ A sin 2irvt (10.2) The simple harmonic vibration can be represented graphically by this sine function, as shown in Fig. 10.1. A cosine function would do just as well. The constant v is called the frequency of the motion; it is the number of vibrations in unit time. The reciprocal of the frequency, r -= ]/v, is called the period of the motion, the time required for a single vibration. Whenever t -- n(r/2), where n is an integer, the displace- ment x passes through zero. Fig. 10.1. Simple harmonic vibration. The quantity A, the maximum value of the displacement, is called the amplitude of the vibration. At the position x =- A, the oscillator reverses its direction of motion. At this point, therefore, the kinetic energy is zero, and all the energy is potential energy E p . At position x 0, all the energy is kinetic energy E k . Since the total energy, E = E p + E k , is always a constant, it must equal the potential energy at x -= A. On page 190 the potential energy of the oscillator was shown to be equal to } 2 KX 2 , so that the total energy is E - \KA 2 (10.3) The total energy is proportional to the square of the amplitude. This im- portant relation holds true for all periodic motions. The motion of a harmonic oscillator illustrates a displacement periodic with time, temporally periodic. If such an oscillator were immersed in a fluid medium it would set up a disturbance which would travel through the Sec. 3] PARTICLES AND WAVES 253 medium. Such a disturbance would be not only temporally periodic but also spatially periodic. It would constitute what is called a wave. For example, a tuning fork vibrating in air sets up sound waves. An oscillating electric dipole sets up electromagnetic waves in space. Let us consider a simple harmonic wave moving in one dimension, x. If one takes an instantaneous "snapshot" of the wave, it will have the form of a sine or cosine function. This snapshot is the profile of the wave. If at a point x = the magnitude of the disturbance <f> equals 0, then at some further point x = A, the magnitude will again be zero, and so on at 2A, 3A . . . A. This quantity A is called the wavelength. It is the measure of the wave's periodicity in space, just as the period T is the measure of its periodicity in time. The profile of the simple sine wave has the form: <-,4sin27r~ (10.4) X. Now consider the expression for the wave at some later time /. The idea of the velocity of the wave must then be introduced. If the disturbance is moving through the medium with a velocity c in the positive x direction, in a time t it will have moved a distance ct. The wave profile will have exactly the same form as before if the origin is shifted from ;c = to a new origin at x = ct. Referred to this moving origin, the wave profile always maintains the form of eq. (10.4). To refer the disturbance back to the stationary origin, it is necessary only to subtract the distance moved in time t from the value of x. Then the equation for the moving wave becomes f\ <f>^-Asm~(x-ct) (10.5) Note that the nature of the disturbance <f> need not be specified: in the case of a water wave it is the height of the undulation ; in the case of an electro- magnetic wave it is the strength of an electric or magnetic field. Now it is evident that c/A is simply the frequency: v = c/A. The number of wavelengths in unit distance is called the wave number, k I/A, so that eq. (10.5) can be written in the more convenient form: <f> = A sin 27r(kx - vt) (10.6) 3. Stationary waves. In Fig. 10.2, two waves, fa and fa, are shown that have the same amplitude, wavelength, and frequency. They differ only in that fa has been displaced along the X axis relative to fa by a distance d/2,7rk. Thus they may be written fa = A sin 2ir(kx vt) fa = A sin [2ir(kx vt) + d] The quantity d is called the phase of fa relative to fa. 254 PARTICLES AND WAVES [Chap. 10 When the displacement is exactly an integral number of wavelengths, the two waves are said to be in phase \ this occurs when d = 2?r, 4?r, or any even multiple of 77. When d --= 77, 3rr 9 or any odd multiple of n, the two waves are exactly out of phase. Interference phenomena are readily explained in terms of these phase relationships, for when two superimposed waves of equal amplitude are out of phase, the resultant disturbance is reduced to zero. Fig. 10.2. Waves differing in phase. The expression (10.6) is one solution of the general partial differential equation of wave motion, which governs all types of waves, from tidal waves to radio waves. In one dimension this equation is In three dimensions the equation becomes v a; 2 (10.7) (10.8) The operator V 2 (del squared) is called the Laplacian. One important property of the wave equation is apparent upon inspec- tion. The disturbance $ and all its partial derivatives appear only in terms of the first degree and there are no other terms. This is therefore a linear homogeneous differential equation. 2 It can be verified by substitution that if fa and <^ 2 are any two solutions of such an equation, then a new solution can be written having the form i i i i / 1 r\ r>\ <p - fli9i ~r #2r2 (10.9) where a l and a 2 are arbitrary constants. This is an illustration of the principle of superposition. Any number of solutions can be added together in this way to obtain new solutions. This is essentially what is done when a complicated vibratory motion is broken down into its normal modes (page 191), or when a periodic function is represented by a Fourier series. An important application of the superposition principle is found in the addition of two waves of the form of eq. (10.6) that are exactly the same 8 Granvilie, he. c//., pp. 372, 377. Sec. 4] PARTICLES AND WAVES 255 except that they are going in opposite directions. Then the new solution will be </> -^ A sin 2rr(kx vt) + A sin 2v(kx } vt) or ^ 2 A sin 27r&jc cos 27rvt (10.10) . x \ y xv since sin x + sin j 2 sin cos This new wave, which does not move either forward or backward, is a stationary wave. The waves of the original type [eq. (10.6)] are called pro- gressive waves. It will be noted that in the stationary wave represented by eq. (10.10), the disturbance <f> always vanishes, irrespective of the value of /, for points at which sin 2-nkx = or x 0, iAr, $A% S/v . . . (n/2)k. These points are called nodes. The distance between successive nodes is Ik or A/2, one-half a wavelength. Midway between the nodes are the positions of maximum amplitude, or antinodes. Solutions of the one-dimensional type, which have just been discussed, will apply to the problem of a vibrating string in the idealized case in which there is no damping of the vibrations. In a string of infinite length one can picture the occurrence of progressive waves. Consider, however, as in Fig. 10.2, a string having a certain finite length L. This limitation imposes certain boundary conditions on the permissible solutions of the wave equation. If the ends of the string are held fixed: at x and at x L, the displacement <f) must 0. Thus there must be an integral number of nodes between and L, so that the allowed wavelengths are restricted to those that obey the equation n Y L (10.11) where n is an integer. This occurrence of whole numbers is very typical of solutions of the wave equation under definite boundary conditions. In order to prevent destruction of the wave by interference, there must be an integral number of half wavelengths fitted within the boundary. This principle will be seen to have important consequences in quantum theory. 4. Interference and diffraction. The interference of light waves can be visualized with the aid of the familiar construction of Huygens. Consider, for example, in (a) Fig. 10.3, an effectively plane wave front from a single source, incident upon a set of slits. The latter is the prototype of the well known diffraction grating. Each slit can now be regarded as a new light source from which there spreads a semicircular wave (or hemispherical in the three-dimensional case). If the wavelength of the radiation is A, a series of concentric semicircles of radii A, 2A, 3A . . . may be drawn with these sources as centers. Points on these circles represent the consecutive maxima in amplitude of the new wavelets. Now, following Huygens, the new resultant wave fronts are the curves or surfaces that are simultaneously tangent to the 256 PARTICLES AND WAVES [Chap. 10 a cos oi secondary wavelets. These are called the "envelopes" of the wavelet curves and are shown in the illustration. The important result of this construction is that therfe are a number of possible envelopes. The one that moves straight ahead in the same direction as the original incident light is called the zero-order beam. On either side of this are first-, second-, third-, etc., order diffracted beams. The angles by which the diffracted beams deviate from the original direction evidently depend on the wavelength of the incident radiation. The longer the wave- length, the greater is the diffraction. This is, of course, the basis for the use of the diffraction grating in the measurement of the wavelength of radiation. l* f ORDER ORDER 2 nd ORDER (a) (b) Fig. 10.3. Diffraction: (a) Huygens* construction; (b) path difference. The condition for formation of a diffracted beam can be derived from a consideration of (b) Fig. 10.3, where attention is focused on two adjacent slits. If the two diffracted rays are to reinforce each other they must be in phase, otherwise the resultant amplitude will be cut down by interference. The condition for reinforcement is therefore that the difference in path for the two rays must be an integral number of wavelengths. If a is the angle of diffraction and a the separation of the slits, this path difference is a cos a and the condition becomes a cos a = /a (10.12) where h is an integer. This equation applies to a linear set of slits. For a two-dimensional plane grating, there are two similar equations to be satisfied. For the case of light incident normal to the grating, a cos a = AA b cos ft = A It will be noted that the diffraction is appreciable only when the spacings Sec. 5] PARTICLES AND WAVES 257 of the grating aj^rftt very much larger than the wavelength of the incident light. In order /to obtain diffraction effects with X rays, for example, the spacings should be of the order of a few Angstrom units. 3 Max von Laue, in 1912, realized that the interatomic spacings in crystals were probably of the order of magnitude of the wavelengths of X rays. Crystal structures should therefore serve as three-dimensional diffraction gratings for X rays. This prediction was immediately verified in the critical experiment of Friedrich, Knipping, and Laue. A typical X-ray diffraction picture is shown in Fig. 13.7 on page 375. The far-reaching consequences of Laue's discovery will be considered in some detail in a later chapter. It is mentioned here as a demonstration of the wave properties of X rays. 5. Black-body radiation. The first definite failure of the old wave theory of light was not found in the photoelectric effect, a particularly clear-cut case, but in the study of black-body radiation. All objects are continually absorbing and emitting radiation. Their properties as absorbers or emitters may be extremely diverse. Thus a pane of window glass will not absorb much of the radiation of visible light but will absorb most of the ultraviolet. A sheet of metal will absorb both the visible and the ultraviolet but may be reasonably transparent to X rays. In order for a body to be in equilibrium with its environment, the radia- tion it is emitting must be equivalent (in wavelength and amount) to the radiation it is absorbing. It is possible to conceive of objects that "are perfect absorbers of radiation, the so-called ideal black bodies. Actually, no sub- stances approach very closely to this ideal over an extended range of wave- lengths. The best laboratory approximation to an ideal black body is not a substance at all, but a cavity. This cavity, or hohlraum, is constructed with excellently insulating walls, in one of which a small orifice is made. When the cavity is heated, the radia- tion from the orifice will be a good sample of the equilibrium radiation within the heated enclosure, which is practically ideal black-body radiation. There is a definite analogy between the behavior of the radiation within such a hohlraum and that of gas molecules in a box. Both the molecules and the radiation are characterized by a density and both exert pressure on the confining walls. One difference is that the gas density is a function of the volume and the temperature, whereas the radiation density is a function of temperature alone. Analogous to the various velocities distributed among the gas molecules are the various frequencies distributed among the oscilla- tions that comprise the radiation. At any given temperature there is a characteristic distribution of the gas velocities given by Maxwell's equation. The corresponding problem of the spectral distribution of black-body radiation, that is, the fraction of the 8 It is also possible to use larger spacings and work with extremely small angles of incidence. The complete equation, corresponding to eq. (10.12), for incidence at an angle oto, is 0(cos a cos ao) = M. 258 PARTICLES AND WAVES [Chap. 10 total energy radiated that is within each range of wavelength, was first explored experimentally (1877-1900) by O. Lummer and E. Pringsheim. Some of their results are shown in Fig. 10.4. These curves indeed have a marked resemblance to those of the Maxwell distribution law. At high tem- peratures the position of the maximum is shifted to shorter wavelengths an iron rod glows first dull red, then orange, then white as its temperature is raised and higher frequencies become appreciable in the radiation. 140 12345 WAVE LENGTH, MICRONS Fig. 10.4. Data of Lummer and Pringsheim on spectral distribution of radiation from a black body at three different temperatures. When these data of Lummer and Pringsheim appeared, attempts were made to explain them theoretically by arguments based on the wave theory of light and the principle of equipartition of energy. Without going into the details of these efforts, which were uniformly unsuccessful, it is possible to see why they were foredoomed to failure. According to the principle of the equipartition of energy, an oscillator in thermal equilibrium with its environment should have an average energy equal to kT, \kT for its kinetic energy and \kT for its potential energy, where k is the Boltzmann constant. This classical theory states that the average energy depends in no way on the frequency of the oscillator. In a system containing 100 oscillators, 20 with a frequency v l of 10 10 cycles per sec and 80 with v 2 =-- 10 14 cycles per sec, the equipartition principle predicts that 20 per cent of the energy shall be in the low-frequency oscillators and 80 per cent in the. high-frequency oscillators. The radiation within a hohlraum can be considered to be made up of Sec. 6] PARTICLES AND WAVES 259 standing waves of various frequencies. The problem of the energy distribu- tion over the various frequencies (intensity / vs. v) apparently reduces to the determination of the number of allowed vibrations in any range of frequencies. The possible high-frequency vibrations greatly outnumber the low- frequency ones. The one-dimensional case of the vibrating string can be used to illustrate this fact. We have seen in eq. (10.11) that in a string of length L, standing waves can occur only for certain values of the wavelength given by X = 2L/n. It follows that the number of allowed wavelengths from any given value A to the maximum 2L is equal to n -= 2L/A. We wish to find the additional number of allowed wavelengths that arise if the limiting wave- length value is decreased from X to A dX. The result is obtained by differentiation 4 as dn = ^dl (10.13) A This indicates that the number of allowed vibrations in a region from A to A dk increases rapidly as the wavelength decreases (or the frequency increases). There are many more high-frequency than low-frequency vibra- tions. The calculation in three dimensions is more involved 5 but it yields essentially the same answer. For the distribution of standing waves in an enclosure of volume V, the proper formula is dn (Sir F/A 4 )c/A, or dn^%7T-v*dv (10.14) c 3 Since there are many more permissible high frequencies than low fre- quencies, and since by the equipartition principle all frequencies have the same average energy, it follows that the intensity / of black-body radiation should rise continuously with increasing frequency. This conclusion follows inescapably from classical Newtonian mechanics, yet it is in complete dis- agreement with the experimental data of Lummer and Pringsheim, which show that the intensity of the radiation rises to a maximum and then falls off sharply with increasing frequency. This abject failure of classical mechani- cal principles when applied to radiation was viewed with unconcealed dismay by the physicists of the time. They called it the "ultraviolet catastrophe." 6. Planck's distribution law. The man who first dared to discard classical mechanics and the equipartition of energy was Max Planck. Taking this step in 1900, he was able to derive a new distribution law, which explained the experimental data on black-body radiation. Newtonian mechanics (and relativity mechanics too) was founded upon the ancient maxim that natura non facit saltum ("nature does not make a jump"). Thus an oscillator could be presumed to take up energy continuously 4 It is assumed that in a region of large L and small A, n is so large that it can be con- sidered to be a continuous function of A. 5 R. H. Fowler, Statistical Mechanics (London: Cambridge, 1936), p. 112. 260 PARTICLES AND WAVES [Chap. 10 in arbitrarily small increments. Although matter was believed to be atomic in its constitution, energy was assumed to be strictly continuous. Planck discarded this precept and suggested that an oscillator, for example, could acquire energy only in discrete units, called quanta. The quantum theory began therefore as an atomic theory of energy. The magni- tude of the quantum or atom of energy was not fixed, however, but depended on the oscillator frequency according to s - hv (10.15) Planck's constant h has the dimensions of energy times time (e.g., 6.62 x 10-27 er g sec ^ a q uan tity known as action. According to this hypothesis it is easy to see qualitatively why the in- tensity of black-body radiation always falls off at high frequencies. At fre- quencies such that hv ^> kT, the size of the quantum becomes much larger than the mean kinetic energy of the atoms comprising the radiator. The larger the quantum, the smaller is the chance of an oscillator having the necessary energy, since this chance depends on an e~ h ' f * T Boltzmann factor. Thus oscillators of high frequency have a mean energy considerably less than the kT of the classical case. Consider a collection of N oscillators having a fundamental vibration frequency v. If these can take up energy only in increments of hv, the allowed energies are 0, hv, 2hv, 3hv, etc. Now according to the Boltzmann formula, eq. (7.31), if N Q is the number of systems in the lowest energy state, the number N { having an energy e { above this ground state is given by AT, = TV/** 1 (10.16) In the collection of oscillators, for example, NI = N e~ hv l kT N* = Ne-~ 2hvlkT N 3 = N Q e~* hv l kT The total number of oscillators in all energy states is therefore -tfo 2 t-O The total energy of all the oscillators equals the energy of each level times the number in that level. E = # Sec. 7] PARTICLES AND WAVES 261 The average energy of an oscillator is therefore . E _ e ~~ ~N ~ According to this expression, the mean energy of an oscillator whose fundamental frequency is v approaches the classical value of kT when hv becomes much less than kT. 1 Using this equation in place of the classical equipartition of energy, Planck derived an energy-distribution formula in excellent agreement with the experimental data for black-body radiation. The energy density E(v) dv is simply the number of oscillations per unit volume between v and v + dv [eq. (10.14)] times the average energy of an oscillation [eq. (10.17)]. Hence Planck's Law is STT/Z r 3 dv E(v) dv - - hv/kT -- (10.18) 7. Atomic spectra. Planck's quantum theory of energy appeared in 1901. Strong confirmation was provided by the theory of the photoelectric effect proposed by Einstein in 1905. Another most important application of the theory was soon made, in the study of atomic spectra. An incandescent gas emits a spectrum composed of lines at definite wave- lengths. Similarly if white light is transmitted through a gas, certain wave- lengths are absorbed, causing a pattern of dark lines on a bright background when the emergent light is analyzed with a spectrograph. These emission and absorption spectra must be characteristic of certain preferred frequencies in the gaseous atoms and molecules. A sharply defined line spectrum is typical of atoms. Molecules give rise to spectra made up of bands, which can often be analyzed further into closely packed lines. For example, the spectra of atomic hydrogen (H) and of molecular nitrogen (N 2 ) are shown in Fig. 10.5a and b. In 1885, J. J. Balmer discovered a regular relationship between the fre- quencies of the atomic hydrogen lines in the visible region of the spectrum. The wave numbers v' are given by with Wj = 3, 4, 5 . . . etc. The constant ^ is called the Rydberg constant, and has the value 109,677.581 cm" 1 . It is one of the most accurately known physical constants. 6 In eq. (10.17) let e~ x = y, then the denominator S/ = 1 + y 4- y* -f - . . . = 1/(1 ~ y\ (y< 1). The numerator, Zi>' = y(\ + 2v + 3/ +...)= yl(\ - y) 2 , (y < 1) so that eq. (10.17) becomes hvy/(l - y) = Ar/fr*"/** 1 - 1). 7 When hv < kT, e^/** 1 1 -f (hv/kT). 262 PARTICLES AND WAVES [Chap. 10 Other hydrogen series were discovered later, which obeyed the more general formula, JL.-JL) !/ (10.19) Lyman found the series with 2 = 1 in the far ultraviolet, and others were found in the infrared by Paschen (/ 2 = 3), Bracket! (n a 4), and Pfund H, A. Fig. 10.5a. Spectra of atomic hydrogen. (From Herzberg, Atomic Spectra and Atomic Structure, Dover, 1944.) fcjJK-ftfVl^ MIBffP 8 ''' vCTP Fig. 10.5b. Spectra of molecular nitrogen. (From Harrison, Lord, and Loofbourow, Practical Spectroscopy, Prentice-Hall, 1948.) (/? 2 = 5). A great number of similar series have been observed in the atomic spectra of other elements. 8. The Bohr theory. These characteristic atomic line spectra could not be explained on the basis of the Rutherford atom. According to this model, electrons are revolving around a positively charged nucleus, the coulombic attraction balancing the force due to the centripetal acceleration. The classical theory of electromagnetic radiation demands that an accelerated electric charge must continuously emit radiation. If this continuous emission of energy actually occurred, the electrons would rapidly execute a descending spiral and fall into the nucleus. The Rutherford atom is therefore inherently unstable according to classical mechanics, but the predicted continuous radiation does not in fact occur. The fact that the electrons in atoms do not follow classical mechanics is also clearly shown by the heat-capacity values of gases. The C v for monatomic gases equals f R, which is simply the amount expected for the translation of the atom as a whole. It is evident that the electrons in the atoms do not take up energy as the gas is heated. Niels Bohr, in 1913, suggested that the electrons can revolve around the nucleus only in certain definite orbits, corresponding to certain allowed Sec. 8] PARTICLES AND WAVES 263 energy states. Radiation is emitted in discrete quanta whenever an electron falls from an orbit of high to one of lower energy, and is absorbed whenever an electron is raised from a low to a higher energy orbit. If E ni and E nt are the energies of two allowed states of the electron, the frequency of the spectral line arising from a transition is A 1 v = = -(E ni -E nt ) (10.20) A separate and arbitrary hypothesis is needed to specify which orbits are allowed. The simplest orbits of one electron moving in the field of force of a positively charged nucleus are the circular ones. For these orbits, Bohr postulated the following frequency condition: 8 only those orbits occur for which the angular momentum mvr is an integral multiple of h\2-n. mvr = n~> n - 1, 2, 3 . . . (10.21) ITT The integer n is called a quantum number. The mechanics of motion of the electron in its circular orbit of radius r can be analyzed starting with Newton's equation,/ ma. The force is the coulombic attraction between nucleus, with charge Ze, and electron, i.e., Ze 2 /r 2 . The acceleration is the centripetal acceleration, v 2 /r. Therefore Ze 2 /r 2 mv 2 /r, and (10.22) mv* h 2 Then, from eq. (10.21) r - n 2 - (10.23) In the case of a hydrogen atom Z ==- 1, and the smallest orbit, n 1, would have a radius, "o = TT~2 - ' 529 A (10 ' 24 ) 4n 2 me 2 This radius is of the same order of magnitude as that obtained from the kinetic theory of gases. It may be noted that the radii of the circular Bohr orbits depend on the square of the quantum number. It can now be demonstrated that the Balmer series arises from transitions between the orbit with n = 2 and outer orbits; in the Lyman series, the lower term is the orbit with n = 1 ; the other series are explained similarly. These results are obtained by calculating the energies corresponding to the different orbits and applying eq. (10.20). The energy level diagram for the hydrogen atom is shown in Fig. 10.6. 8 It will be seen a little later that this condition is simply another form of Planck's hypothesis that h is the quantum of action. 264 PARTICLES AND WAVES [Chap. 10 Volts n cm l 14 13.53 13 1 1A 12 1 1 !"l^ 10,000- 23.23 Ti io - n & ^ "T w o> eo r- t- m 0> o t- o^ ^ o > S ^ 11 l!r ss "f *i* J" 10 ^ 30,000- 9 _ 1 | 40,000- 8 - S 50,000- 7 - 6 (A 60,000- "C -^ to eo *! 5 c 10 H5 CS 5 c * *" CO 04<=> <T **7 oo IS 70.000- $ 4 - 80,000- 3 - 90,000- 2 - 100,000- 1 Fig. 10.6. Energy levels of the H atom. (After G. Herzberg, Atomic Spectra, Dover, 1944.) The total energy E of any state is the sum of the kinetic and potential energies: ^ ^ * p r Ze* Ze 2 From eq. (10.22), E = / Therefore from eq. (10.23), E == The frequency of a spectral line is then 2r Z 2 l 2 """^ 2 (10.25) (10.26) Sec. 9] PARTICLES AND WAVES 265 Comparison with the experimental eq. (10.19) yields a theoretical value of the Rydberg constant for atomic hydrogen of .^-~ 109,737 cm- ch 3 This is in excellent agreement with the experimental value. This pleasing state of affairs represented a great triumph for the Bohr theory and lent some solid support to the admittedly ad hoc hypothesis on which it is based. Several improvements in the original Bohr theory were made by Arnold Sommerfeld. He considered the possible elliptical orbits of an electron around the nucleus as one focus. Such orbits are known to be stable con- figurations in dynamical systems such as the planets revolving around the sun. For a circular orbit, the radius r is constant so that only angular momen- tum, associated with the variable 6, need be considered. For elliptical orbits two quantum numbers are needed, for the two variables r and 0. The azimutha! quantum number k was introduced to give the angular momentum in units of h\1-n. The principal quantum number n was defined 9 so that the ratio of the major axis to the minor axis of the elliptical orbit was n\k. Then k can take any value from 1 to /?, the case n k corresponding to a circular orbit. 9. Spectra of the alkali metals. An electron moving about a positively charged nucleus is moving in a spherically symmetrical coulombic field of force. Besides the hydrogen atom, a series of hydrogenlike ions satisfy this condition. These ions include He^, Li++, and Be l+ +, each of which has a single electron. Their spectra are observable when electric sparks discharge through the vapour of the element (spark spectra). They are very similar in structure to the hydrogen spectrum, but the different series are displaced to shorter wavelengths, as a consequence of the dependence of frequency on the square of the nuclear charge, given by eq. (10.26). If an electron is moving in a spherically symmetrical field, the energy level is the same for all elliptical orbits of major axis a as it is for the circular orbit of radius a. In other words, the energy is a function only of the principal quantum number n. All energy levels with the same n are the same, irrespec- tive of the value of k, the azimuthal quantum number. For example, if n = 3, there are three superimposed levels or terms of identical energy, having k = 1, 2, or 3. Such an energy level is said to have a threefold de- generacy. Actually, even in hydrogen, a very slight splitting of these degener- ate levels is found in the fine structure of the spectra, revealed by spectro- graphs of high resolving power. 9 Derivations and detailed discussions of these aspects of the old quantum theory may be found in S. Dushman's article in Taylor's Treatise on Physical Chemistry, 2nd ed., p. 1 170. 266 PARTICLES AND WAVES [Chap. 10 EV 5.37 5 UL For most of the atoms and ions that may give rise to spectra the electrons concerned in the transitions are not moving in spherically symmetric fields. Consider, for example, the case of the lithium atom, which is typical of the alkali metals. The electron whose transitions are responsible for the observed spectrum is the outer, valence, or optical electron. This electron does not move in a spherical field, since its position at any instant is influenced by the positions of the two inner electrons. If the outer electron is on one side of the nucleus, it is less likely that the other two will be there also, because of the electrostatic repulsions. Thus the field is no longer spherical, and the elliptical orbits can no longer have the same energy as a circular orbit of the same n value. The elliptical orbits will have different energy levels depending on their ellipticity, which is governed by the allowed values of the azimuthal quantum number k. For each n, there will be n different energy levels characterized by different k's. The lowest ojr ground state is that for which n = 1 and k ^ 1. States with k ~ 1 are called s states. This is therefore a Is state. When n = 2, k can be either 1 or 2. States with k = 2 are called p states. We therefore have a 2s state and a 2p state. Similarly, when n = 3, we have 3s, 3p, and 3d (k = 3) states; when n = 4, we have 4s, 4/?, 4J, and 4f(k = 4) states. In this discussion there has been a tacit assumption that the energy levels of the atom are determined solely by the quan- tum states of the valence electrons. This is actually not true, and all the electrons and even the nucleus should be considered in discussing the allowed energy states. Then, instead of the quantum number k, which gives the angular momentum of the single electron, a new quantum number L must be used that gives the resultant angular momentum of all the electrons. According as L ~ 0, 1, 2, 3 . etc., we refer to the atomic states as S, P, A F . . . etc. In the case of atoms like the alkali metals, which have only one valence electron, it turns out that the resultant angular momenta of the inner electrons add vectorially to zero. Therefore in this case only ttie single electron need be considered after all. 10 Nevertheless we shall use the more proper notation, 5, />, D, F, to refer to the energy levels. The energy-level diagram for lithium is shown in Fig. 10.7. The observed 10 The situation becomes more complicated when there are two or more optical electrons. An excellent discussion is given by G. H. Herzberg, Atomic Spectra and Atomic Structure (New York: Dover Publications, 1944). Fig. 10.7. Energy levels and spec- tral transitions in the lithium atom. Sec. 10] PARTICLES AND WAVES 267 spectral series arise from the combinations of these terms, as shown in the diagram. It will be noted that only certain transitions are allowed; others are forbidden. Certain selection rules must be obeyed, as for example in this case the rule that AL must be + 1 or 1 . Experimentally four distinct series have been observed in the atomic spectra of the alkalis. The principal series is the only one found in absorption spectra and arises from transitions between the ground state 1*9 and fc the various P states. It may be written symbolically : v^ 15 - mP Absorption spectra almost always arise from transitions from the ground state only, since at ordinary temperatures the proportion of atoms in excited states is usually vanishingly small, being governed by the exponential Boltz- mann factor e ~^ ElkT . At the much higher temperatures required to excite emission spectra, some of the higher states are sufficiently populated by atoms to give rise to a greater variety of lines. Thus in the emission spectra of the alkali metals, in addition to the prin- cipal series, three other series appear. These may be written symbolically as v 2P - mS the sharp series v 2P - mD the diffuse series v --= 3D mF the fundamental series The names are not notably descriptive, although the lines in the sharp series are indeed somewhat narrower than the others. 10. Space quantization. So far in the discussion of allowed Bohr orbits, we have not considered the question of how the orbits can be oriented in space. This is because in the absence of an external electric or magnetic field there is no way of distinguishing between different orientations, since there is no physically established axis of reference. If an atom is placed in a magnetic field, however, one can ask how the orbits will be oriented relative to the field direction. The answer given by the Bohr theory is that only certain orientations are allowed. These are determined by the condition that the component of angular momentum in the direction of the magnetic field, e.g., in the Z direction, must be an integral multiple of h/27r. Thus P. - % (10.27) where m is the magnetic quantum number. This behavior is called space quantization. The allowed values of m are 1, 2, 3, etc., up to &, k being the azimuthal quantum number, which gives the magnitude of the total angular 268 PARTICLES AND WAVES [Chap. 10 k3 Fig. 10.8. Spatial quantiza- tion of angular momentum in a magnetic field H. momentum in units of h/2ir. An example of space quantization for the case k = 3 is illustrated in Fig. 10.8. For any value of k, there are 2k allowed orientations corresponding to the different values of m. In the absence of an external field, the correspond- ing energy level will be 2/r-fold degenerate. In the presence of an electric or magnetic field this energy level will be split into its individual components. This splitting gives rise to a splitting of the corresponding spec- tral lines. In a magnetic field this is called the Zeeman effect', in an electric field, the Stark effect. This observed splitting of the spectral lines is the experimental basis for the introduction into the Bohr theory of space quantization and the quantum number m. 11. Dissociation as series limit. It will be noted in the term diagram for lithium that the energy levels become more closely packed as the height above the ground state increases. They finally converge to a common limit whose height above the ground level corresponds to the energy necessary to remove the electron completely from the field of the nucleus. In the observed spectrum, the lines become more and more densely packed and finally merge into a continuum at the onset of dissociation. The reason for the continuous absorption or emission is that the free electron no longer has quantized energy states but can take up kinetic energy of translation continuously. The energy difference between the series limit and the ground level represents the ionization potential I of the atom or ion. Thus the fast ionization potential of Li is the energy of the reaction Li+ 4- e -> Li. The second ionization potential is the energy of Li+ f + e -> Li+. Examples of ionization potentials are given in Table 10.1. The way in which the values of / vary with position in the periodic table should be noted. This periodicity is very closely related to the periodic character of the chemical properties of the elements, for it is the outer electrons of an atom that enter into its chemical reactions. Thus the alkali metals have low ionization potentials; the inert gases, high ionization potentials. 12. The origin of X-ray spectra. The origin of the characteristic X-ray line series studied by Moseley (see Chapter 8) is readily understood in terms of the Bohr theory. The optical spectra are caused by transitions of outer or valence electrons, but the X-ray spectra are caused by transitions of the inner electrons. X rays are generated when high-velocity particles such as electrons impinge upon a suitable target. As the result of such a collision, an electron may be driven completely from its orbit, leaving a "hole" in the Sec. 13] PARTICLES AND WAVES 269 TABLE 10.1 IONIZATION POTENTIALS OF CHEMICAL ELEMENTS (IN ELECTRON VOLTS) Element First lonizatlon Potential Second lonization Potential H 13.60 He 24.58 54.41 Li 5.39 75.62 Be 9.32 18.21 B 8.30 25.12 C 11.27 24.38 N 14.55 29.61 O 13.62 35.08 F 17.42 34.98 Ne 21.56 40.96 Na 5.14 47.29 Mg 7.65 15.03 Al 5.99 18.82 Si 8.15 16.34 P 10.98 19.65 S 10.36 23.41 Cl 12.96 23.80 A 15.76 27.62 K 4.34 31.81 Ca 6.11 11.87 target atom. When electrons in outer shells, having larger values of the principal quantum number n, drop into this hole, a quantum of X radiation is emitted. 13. Particles and waves. One might go on from here to describe the further application of the Bohr theory to more complex problems in atomic structure and spectra. Many other quite successful results were obtained, but there were also a number of troublesome failures. Attempts to treat cases in which more than one outer electron is excited, as in the helium spectrum, were in general rather discouraging. The Bohr method is essentially nothing more than the application of a diminutive celestial mechanics, with coulombic rather than gravitational forces, to tiny solar-system models of the atom. Certain quantum conditions have been arbitrarily superimposed on this classical foundation. The rather capricious way in which the quantum numbers were introduced and adjusted always detracted seriously from the completeness of the theory. Now there is one branch of physics in which, as we have seen, integral numbers occur very naturally, namely in the stationary-state solutions of the equation for wave motion. This fact suggested the next great advance in physical theory: the idea that electrons, and in fact all material particles, must possess wavelike properties. It was already known that radiation exhibited both corpuscular and undulatory aspects. Now it was to be shown, first theoretically and soon afterwards experimentally, that the same must be true of matter. 270 PARTICLES AND WAVES [Chap. 10 This new way of thinking was first proposed in 1923 by Due Louis de Broglie. In his Nobel Prize Address he has described his approach as follows. 11 . . . When 1 began to consider these difficulties [of contemporary physics] I was chiefly struck by two facts. On the one hand the quantum theory of light cannot be considered satisfactory, since it defines the energy of a light corpuscle by the equation E ---= hv, containing the frequency v. Now a purely corpuscular theory contains nothing that enables us to define a frequency; for this reason alone, there- fore, we are compelled, in the case of light, to introduce the idea of a corpuscle and that of periodicity simultaneously. On the other hand, determination of the stable motion of electrons in the atom introduces integers; and up to this point the only phenomena involving integers in Physics were those of interference and of normal modes of vibration. This fact suggested to me the idea that electrons too could not be regarded simply as corpus- cles, but that periodicity must be assigned to them also. A simple two-dimensional illustration of this viewpoint may be seen in Fig. 10.9. There are shown two possible electron waves of different wave- lengths for the case of an electron revolving around an atomic nucleus. In one case, the circumference of the electron orbit is an integral multiple of the wavelength of the electron wave. In the other case, this condi- tion is not fulfilled and as a result the wave is destroyed by interference, and the supposed state is nonexistent. The introduction of in- tegers associated with the permissible states Fig. 10.9. Schematic drawing Qf e|ectronic motjon therefore occurs quite of an electron wave constrained ~ to move around nucleus. The naturally once the electron is given wave solid line represents a possible properties. The situation is exactly analogous stationary wave. The dashed line with the occurrence of stationary waves on shows how a wave of somewhat a vibrating string. The necessary condition different wavelength would be for a staWc ^ Qf radius . fc destroyed by interference. e 27rr e =-. nX (10.28) A free electron is associated with a progressive wave so that any energy is allowable. A bound electron is represented by a standing wave, which can have only certain definite frequencies. In the case of a photon there are two fundamental equations to be obeyed : e ~ hv, and e = me 2 . When these are combined, one obtains hv = me 2 or X ---- c/v -= h/mc hip, where p is the momentum of the photon. Broglie considered that a similar equation governed the wave- length of the electron wave. Thus, * * A = = - (10.29) mv p 11 L. de Broglie, Matter and Light^ (New York: Dover Publications [1st ed., W. W. Norton Co.], 1946). Sec. 14] PARTICLES AND WAVES 271 The original Bohr condition for a stable orbit was given by eq. (10.21) as 27Ttnvr e = nh. By combination with eq. (10.28), one again obtains eq. (10.29) so that the Broglie formulation gives the Bohr condition directly. The Broglie relation, eq. (10.29), is the fundamental one between the momentum of the electron considered as a particle and the wavelength of its associated wave. Consider, for example, an electron that has been accelerated through a potential difference V of 10 kilovolts. Then Ve ~ -i/w 2 , and its velocity would be 5.9 x 10 9 cm per sec, about one-fifth that of light. The wavelength of such an electron would be h 6.62 x 10~ 27 ~ mv ~ (9AI x \Q r )(53~ 0.12 A This is about the same wavelength as that of rather hard X rays. H 2 a F 22- Golf ball Baseball TABLE 10.2 WAVELENGTHS OF VARIOUS PARTICLES Particle Mass (g) Velocity (cm/sec) Wavelength (A) ilectron .... 9.1 x 10- 28 5.9 x 10 7 12.0 It electron 9.1 X 10~ 28 5.9 x 10 8 1.2 volt electron . 9.1 X 10~ 28 5.9 x 10 9 0.12 t proton t a particle lecule at 200C 1.67 x 10- 24 6.6 x 10- 24 3.3 x 10~ 24 1.38 x JO 7 6.9 x 10 6 2.4 x 10 5 0.029 0.015 0.82 cle from radium 6.6 x 10- 24 1.51 x 10 9 6.6 X 10~ 5 bullet .... 1.9 3.2 x 10 4 1.1 x 10~ 23 ill .... 45 3 x 10 3 4.9 x 10~ 24 11 .... 140 2.5 x 10 3 1.9 x 10~ 24 Table 10.2 lists the theoretical wavelengths associated with various particles. 12 The wavelengths of macroscopic bodies are exceedingly short, so that any wave properties will escape our observation. Only in the atomic world does the wave nature of matter become manifest. 14. Electron diffraction. If any physical reality is to be attached to the idea that electrons have wave properties, a 1 .0 A electron wave should be diffracted by a crystal lattice in very much the same way as an X-ray wave. Experiments along this line were first carried out by two groups of workers, who shared a Nobel prize for their efforts. C. Davisson and L. H. Germer worked at the Bell Telephone Laboratories in New York, and G. P. Thom- son, the son of J. J. Thomson, and A. Reid were at the University of Aber- deen. Diffraction diagrams obtained by Thomson by passing beams of 12 After J. D. Stranathan, The Particles of Modern physics (Philadelphia: Blakiston, 1942), p. 540. 272 PARTICLES AND WAVES [Chap. 10 electrons through thin gold foils are shown in Fig. 10.10. The wave nature of the electron was unequivocally demonstrated by these researches. More recently, excellent diffraction patterns have been obtained from crystals placed in beams of neutrons. Electron beams, owing to their negative charge, have one advantage not possessed by X rays as a means of investigating the fine structure of matter. Appropriate arrangements of electric and magnetic fields can be designed to act as "lenses" for electrons. These arrangements have been applied in the [The photograph below was one of the first ob- tained. The one at the right is a recent example.] Fig. 10.10. Diffraction diagrams obtained by passing beams of electrons through thin gold foils. (Courtesy Professor Sir George Thomson.) development of electron microscopes capable of resolving images as small as 20 A in diameter. We could wish for no clearer illustrations of the wave properties of electrons than the beautiful electron micrographs of viruses, fibers, and colloidal particles that have been obtained with these instruments. 15. The uncertainty principle. In the development of atomic physics we have noted the repeated tendency toward the construction of models of the atom and its constituents from building blocks that possess all the normal properties of the sticks and stones of everyday life. One fundamental axiom of the classical mechanics developed for commonplace occurrences was the possibility of simultaneously measuring different events at different places. Such measurement appears at first to be perfectly possible because to a first approximation the speed of light is infinitely large, and it takes practically no time to signal from place to place. More refined measurements must consider the fact that this speed is really not infinite, but only 3 x 10 10 cm per sec. This speed is indeed large compared with that of a rocket, but not compared with that of an accelerated electron. As a result, attempts to apply the old mechanics to moving electrons were a failure, and the new relativisitic mechanics of Einstein was needed to correct the situation. Sec. 15] PARTICLES AND WAVES 273 In a similar way, in our ordinary macroscopic world, the value of the Planck constant h may be considered to be effectively zero. The Broglie wavelengths of ordinary objects are vanishingly small, and a batter need not consider diffraction phenomena when he swings at an inside curve. If we enter into the subatomic world, h is no longer so small as to be negligible. The Broglie wavelengths of electrons are of such a magnitude that diffraction effects occur in crystal structures. One of the fundamental tenets of classical mechanics is that it is possible to specify simultaneously the position and momentum of any body. The strict determinism of mechanics rested upon this basic assumption. Knowing the position and velocity of a particle at any instant, Victorian mechanics would venture to predict its position and velocity at any other time, past or future. Systems were completely reversible in time, past configurations being obtained simply by substituting / for t in the dynamical equations. But, is it really possible to measure simultaneously the position and momentum of any particle? The possible methods of measurement must be analyzed in detail before an answer can be given. To measure with precision the position of a very small object, a micro- scope of high resolving power is required. With visible light one cannot expect to locate objects much smaller than a tenth of a micron. The size of the smallest body that can be observed is limited by diffraction effects, which begin to create a fuzziness in the image when the object is of the same order of magnitude as the wavelength of the incident light. The limit of resolution is given according to the well known formula of Abbe as R A/2/4, and the maximum value of the numerical aperture A is unity. In order to determine the position of an electron to within a few per cent uncertainty, radiation of wavelength around 10~ 10 cm or 10~ 2 A would have to be used. We shall conveniently evade the technical problems involved in the design and manufacture of a microscope using these y rays. With such very short rays, there will be a very large Compton effect, and the y ray will impart considerable momentum to the electron under observation. This momentum is given by eq. (9.5) as mv = 2(hv/c) sin a/2. Since the range in scattering angle is from to 7T-/2, corresponding to the aperture of the micro- scope (A = 1), the momentum is determined only to within an uncertainty of A/? = mv & A/A. On account of diffraction, the error A^ in the determina- tion of position is of the order of the wavelength A. The product of the uncertainty in momentum times the uncertainty in position is therefore of the order of h, &p-&q~h (10.30) This is the famous uncertainty principle of Werner Heisenberg (1926). It is impossible to specify simultaneously the exact position and momentum of a particle because our measuring instruments necessarily disturb the object being measured. This disturbance is negligible with man-sized objects, but 274 PARTICLES AND WAVES [Chap. 10 the disturbance of atom-sized particles cannot be neglected. Herein is the essential meaning of the failure of classical mechanics and the success of wave mechanics. 13 16. Waves and the uncertainty principle. Some kind of uncertainty prin- ciple is always associated with a wave motion. This fact can be seen very clearly in the case of sound waves. Consider the case of an organ pipe, set into vibration by depressing a key, whose vibration is stopped as soon as the key is released. The vibrating pipe sets up a train of sound waves in the air, which we hear as a note of definite frequency. Now suppose the time between the depression and the release of the key is gradually shortened. As a result, the length of the train of waves is shortened also. Finally the time will come when the period during which the key is depressed is actually less than the period r of the sound wave, the time required for one complete vibration. Once this happens, the frequency of the wave is no longer precisely deter- mined, for at least one complete vibration must take place to define the frequency. It appears, therefore, that the time and the frequency cannot both be fixed at any arbitrary value. If a very small time is chosen, the frequency becomes indeterminate. When waves are associated with particles, a similar uncertainty principle is a necessary consequence. If the wavelength or frequency of an electron wave, for example, is to be a definitely fixed quantity, the wave must be infinite in extent. Any attempt to confine a wave within boundaries requires destructive interference at these boundaries in order to reduce the resultant amplitudes there to zero. This interference can be secured only by super- imposing waves of different frequencies. It follows that an electron wave of perfectly definite frequency, or momentum, must be infinitely extended and therefore must have a completely indeterminate position. In order to fix the position, superimposed waves of different frequency are required, and as the position becomes more closely defined the momentum becomes fuzzier. The uncertainty relation eq. (10.30) can be expressed not only in terms of position and momentum but also for energy and time. Thus, A/? A? - AF A/ ?v h (10.31) This equation is used to estimate the sharpness of spectral lines. In general, lines arising from transitions from the ground state of an atom are sharp. This is because the optical electron spends a long time in the ground state and thus A", the uncertainty in the energy level, is very small. On the other hand, the lifetime of excited states may sometimes be very short, and trans- itions between such excited energy levels may give rise to diffuse or broad- ened lines as a result of the uncertainty A in the energy levels, which is 18 Many natural philosophers would, not agree with this statement. See H. Margenau, Physics Today, 7, 6 (1954). Sec. 17] PARTICLES AND WAVES 275 reflected in an uncertainty, Av -= A//z, in the frequency of the observed line. 14 17. Zero-point energy. According to the old quantum theory, the energy levels of a harmonic oscillator were given by E n nhv. If this were true the lowest energy level would be that with n 0, and would therefore have zero energy. This would be a state of complete rest, represented by the minimum in the potential energy curve in Fig. 7.15. The uncertainty principle does not allow such a state of completely defined position and completely defined (in this case, zero) momentum. As a result, the wave treatment shows that the energy levels of the oscillators are given by * = ( + I)'"' 0- 32 ) Now, even when n 0, the ground state, there is a residual zero-point energy amounting to =-, \hv (10.33) This must be added to the Planck expression for the mean energy of an oscillator, which was derived in eq. (10.17). 18. Wave mechanics the Schrodinger equation. In 1926, Erwin Schr5- dinger and W. Heisenberg independently laid the foundations for a distinctly new sort of mechanics which was expressive of the wave-particle duality of matter. This is called wave or quantum mechanics. The starting point for most quantum mechanical discussions is the Schrodinger wave equation. We may recall that the general differential equation of wave motion in one dimension is given by eq. (10.7) as _ dx* ~ & ' a/ 2 where <f> is the displacement and v the velocity. In order to separate the variables, let <f> =--- y(x) sin 2-nvt. On substitution in the original equation, this yields d*W 47T 2 V 2 T? + ^T-V = (10-34) dx* v 2 This is the wave equation with the time dependence removed. In order to apply this equation to a "matter wave," the Broglie relation is introduced, as follows: The total energy E is the sum of the potential energy U and the kinetic energy p 2 j2m. E = p*/2m + U. Thus, p = [2m(E - (7)] 1/2 , or X = hip = h[2m(E U)]~ m . Substituting this in eq. (10.34), one obtains: (,0.35) 14 This is not the only cause of broadening of spectral lines. There is in addition a pressure broadening due to interaction with the electric fields of neighboring atoms or molecules, and a Doppler broadening, due to motion of the radiating atom or molecule with respect to the observer. 276 PARTICLES AND WAVES [Chap. 10 This is the famous Schrftdinger equation in one dimension. For three dimensions it takes the form o2.*| W + -r^-(E- U)y> = (10.36) rr Although the equation has been obtained in this way from the ordinary wave equation and Broglie's relation, it is actually so fundamental that it is now more usual simply to postulate the equation as the starting point of quantum mechanics, just as Newton's/^ ma is postulated as the starting point of ordinary mechanics. As is usual with differential equations, the solutions of eq. (10.36) for any particular set of physical conditions are determined by the particular boundary conditions imposed upon the system. Just as the simple wave equation for a vibrating string yields a discrete set of stationary-state solutions when the ends of the string are held fixed, so in general solutions are obtained for the SchrOdinger equation only for certain energy values E. In many cases the allowed energy values are discrete and separated, but in certain other cases they form a continuous spectrum of values. The allowed energy values are called the characteristic, proper, or eigen- values for the system. The corresponding wave functions y am called the characteristic functions or eigenf unctions. 19. Interpretation of the y functions. The eigenfunction ip is by nature a sort of amplitude function. In the case of a light wave, the intensity of the light or energy of the electromagnetic field at any point is proportional to the square of the amplitude of the wave at that point. From the point of view of the photon picture, the more intense the light at any place, the more photons are falling on that place. This fact can be expressed in another way by saying that the greater the value of y>, the amplitude of a light wave in any region, the greater is improbability of a photon being within that region. It is this interpretation that is most useful when applied to the eigen- functions of Schrftdinger's equation. They are therefore sometimes called probability amplitude functions. If y(x) is a solution of the wave equation for an electron, then the probability of finding the electron within the range from x to x + dx is given 16 by y> 2 (x)dx. The physical interpretation of the eigenfunction as a probability ampli- tude function is reflected in certain mathematical conditions that it must obey. It is required that y>(x) be single T valued, finite, and continuous for all physically possible values of x. It must be single-valued, since the probability of finding the electron at any point x must have one and only one value. It cannot be infinite at any point, for then the electron would be fixed at exactly that point, which would be inconsistent with the wave properties. The require- ment of continuity is helpful in the selection of physically reasonable solutions for the wave equation. 15 Since the function y may be a complex quantity, the probability is written more generally as ^v>, where $ is the complex conjugate of y>. Thus, e.g., if y> = e~ ix t y> = *'* Sec. 20] PARTICLES AND WAVES 277 20. Solution of wave equation the particle in a box. The problem of finding the solution of the wave equation in any particular case may be an extremely difficult one. Sometimes a solution can be devised in principle that in practice would involve several decades of calculations. The recent develop- ment of high-speed calculating machines has greatly extended the range of problems for which numerical solutions can be obtained. The simplest case to which the wave equation can be applied is that of a free particle; i.e., one moving in the absence of any potential field. In this case we may set U = and the one-dimensional equation becomes 87T 2 /M -- E V = A solution of this equation is readily found 16 to be y A sin ( -^ V2mE x \ (10.37) (10.38) where A is an arbitrary constant. This is a perfectly allowable solution as long as E is positive, since the sine of a real quantity is everywhere single- valued, finite, and continuous. Thus all positive values of E are allowable w (a) (b) (0 Fig. 10.11. Electron in a one-dimensional box. (a) the potential function, (b) allowed electron waves, (c) tunnel effect. and the free particle has a continuous spectrum of energy states. This con- clusion is in accord with the picture previously given of the onset of the continuum in atomic spectra as the result of dissociation of an electron from the atom. What is the effect of imposing a constraint upon the free particle by requiring that its motion be confined within fixed boundaries? In three 16 See, for example, Granville et-aL, op. ciY., p. 390. The solution can be verified by substitution into the equation. 278 PARTICLES AND WAVES [Chap. 10 dimensions this is the problem of a particle enclosed in a box. The one- dimensional problem is that of a particle required to move between set points on a straight line. The potential function that corresponds to such a condition is shown in (a), Fig. 10.11. For values of x between and a the particle is completely free, and U =- 0. At the boundaries, however, the particle is constrained by an infinite potential wall over which there is no escape; thus U --= oo when x ^ 0, x - a. The situation now is similar to thcit of the vibrating string considered at the beginning of the chapter. Restricting the electron wave within fixed boundaries corresponds to seizing hold of the ends of the string. In order to obtain stable standing waves, it is again necessary to restrict the allowed wavelengths so that there is an integral number of half wavelengths between and a; i.e., n(A/2) a. Some of the allowed electron waves are shown in (b), Fig. 10.11, superimposed upon the potential-energy diagram. The permissible values of the kinetic energy E n of the electron in a box can be obtained from the Broglie relation X - hjmv. 2 i = \m (10.39) From this equation, two important consequences can be deduced which will hold true for the energy of electrons, not only in this special case, but quite generally. First of all, it is apparent that as the value of a increases, the energy decreases. Other factors being the same, the more room the electron has to move about in, the lower will be its energy. The more localized is its motion, the higher will be its energy. Remember that the lower the energy, the greater the stability of a system. Secondly, the integer n is a typical quantum number, which now appears quite naturally and without any ad hoc hypotheses. It determines the number of nodes in the electron wave. When n 1 there are no nodes. When n 2 there is a node in the center of the box; when n -- 3 there are two nodes, and so on. The value of the energy depends directly on 2 , and therefore rises rapidly as the number of nodes increases. The extension of the one-dimensional result to a three-dimensional box of sides a, b, and c is very simple. The allowed energy levels for the three- dimensional case depend on a set of three integers (n l9 2 , %): since there are three dimensions, there are three quantum numbers. h* In* TV* 3 2 \ =o-(-V + 7J + -r) 00.40) 8/w \ a 2 b* c* / This result shows that according to wave mechanics even the trans- lational motion of a particle in a box is quantized. Because of the extremely Sec. 21] PARTICLES AND WAVES 279 small value of h 2 these levels lie very closely packed together except in cases where the dimensions of the box are vanishingly small. If electron waves in one dimension are comparable with vibrations of a violin string, those in two dimensions are like the pulsations of a drumhead, whereas those in three dimensions are like the vibrations of a block of steel. The waves can then have nodes along three directions, and the three quantum numbers determine the number of nodes. 21. The tunnel effect. Let us take a baseball, place it in a well constructed box, and nail the lid down tightly. Now any proper Newtonian will assure us that the ball is in the box and is going to stay there until someone takes it out. There is no probability that the ball will be found on Monday inside the box and on Tuesday rolling along outside it. Yet if we transfer our attention from a baseball in a box to an electron in a box, quantum mechanics predicts exactly this unlikely behavior. To be more precise, consider in (c), Fig. 10.11, a particle moving in a "one-dimensional box" with a kinetic energy E k . It is confined by a potential- energy wall of thickness d and height U Q . Classical mechanics indicates that the particle can simply move back and forth in its potential energy well; since the potential-energy barrier is higher than the available kinetic energy, the possibility of escape is absolutely nil. Quantum mechanics tells a different story. The wave equation (10.35) for the region of constant potential energy U Q is *) 4 (87T 2 w/7* 2 ) (E - U )y> - This equation has the general solution W -- ^ e ^^!/i)^2m(E'-U n )x In the region within the box E ^ U (} and this solution is simply the familiar sine or cosine wave of eq. (10.38) written in the complex exponential form. 17 In the region within the potential-energy barrier, however, U Q > ", so that the expression under the square root sign is, negative. One can therefore multiply out a V 1 term, obtaining the following result: y> -= Ae-V* 1 ^***^'* (10.41) This exponential function describes the behavior of the wave function within the barrier. It is evident that according to wave mechanics the prob- ability of finding an electron in the region of negative energy is not zero, but is a certain finite number that falls off exponentially with the distance of penetration within the barrier . The behavior of the wave function is shown in (c), Fig. 10.11. So long as the barrier is not infinitely high nor infinitely wide there is always a certain probability that electrons (or particles in general) will leak through. This is called the tunnel effect. 17 See, for example, Courant and Robbins, What Is Mathematics (New York: Oxford, 1941), p. 92, for a description of this notation: e io =- cos -f / sin 0. 280 PARTICLES AND WAVES [Chap. 10 The phenomenon is not observed with baseballs in boxes or with cars in garages, 18 being rendered extremely improbable by the various parameters in the exponential. In the world of atoms, however, the effect is a common one. One of the best examples is the emission of an a particle in a radio- active disintegration. The random nature of this emission is a reflection of the fact that the position of the particle is subject to probability laws. 22. The hydrogen atom. If the translational motion of the atom as a whole and the motion of the atomic nucleus are neglected, the problem of the hydrogen atom can be reduced to that of a single electron in a coulombic field. This is in a sense a modification of the problem of a particle in a three- dimensional box, except that now the box is spherical. Also, instead of steep walls and zero potential energy within, there is now a gradual rise in potential with distance from the nucleus: at r = oo, U 0; at r = 0, U = oo. The potential energy of the electron in the field of the nucleus is given by U -= e 2 /r. The Schrddinger equation therefore becomes In view of the spherical symmetry of the potential field, it is convenient to transform this expression into spherical coordinates, i a / a^A i a 2 ^ i a / a^A r* Or V Or / + r 2 sin 2 6 * M>* + ^sinO ' 00 \ Sm OO/ + The polar coordinates r, 0, and <f> have their usual significance (Fig. 7.2, page 168). The coordinate r measures the radial distance from the origin; is a "latitude"; and <f> a "longitude." Since the electron is moving in three dimensions, three coordinates obviously suffice to describe its position at any time. In this equation, the variables can be separated, since the potential is a function of r alone. Let us substitute That is, the wave function is a product of three functions, one of which depends only on r, one only on 6, and the last only on <f>. We shall skip the intervening steps in the solution and the application of the boundary con- ditions that permit only certain allowed eigenfunctions to be physically meaningful. 19 From our previous experience, however, we shall not be sur- prised to find that the final solutions represent a set of discrete stationary 18 This extreme example is described by G. Gamow in Mr. Tompkins in Wonderland (New York: Macmillan, 1940), which is recommended as an introduction to this chapter in Physical Chemistry. 19 For the steps in the solution see, for example, L. Pauling and E. B. Wilson, Introduc- tion to Quantum Mechanics (New York: McGraw-Hill, 1935), Chap. V. Sec. 22] PARTICLES AND WAVES 281 energy states for the hydrogen atom, characterized by certain quantum numbers, n, /, and m. Nor is it surprising that exactly three quantum numbers are required for this three-dimensional motion, just as one sufficed for the waves on a string, whereas three were needed for the particle in a box. The allowed eigenfunctions are certain polynomials whose properties had been extensively studied by mathematicians well before the advent of quan- tum mechanics. In order to give them a measure of concreteness, some examples of these hydrogen wave functions are tabulated in Table 10.3 for the lower values of the quantum numbers n, /, and m. TABLE 10.3 THE HYDROGENLIKE WAVE FUNCTIONS K Shell n = i, / = o, m = 0: /2T\ 3 / 2 1 /2T\ 3 --^ ~ V-rr \<V L Shell n = 2, / = 0, m = 0: -!=(?)''' (2 -*)*-* 4X/27T W V <*J Y> 2P . - -4 4V 2- = 2, /= 1,/n = 1: v = L^ (? ) 3/2 5: e -Zr/2 sin cos 4V/27T ^o 7 flb _ 4V27 These quantum numbers can be assigned a significance purely in terms of the wave-mechanical picture, but they are also the logical successors to the numbers of the old quantum theory. Thus n is still called the principal quantum number. It determines the total number of nodes in the wave function, which is equal to n 1 . These nodes may be either in the radial function R(r), or in the azimuthal function 0(0). When the quantum number / is zero, there are no nodes in the function. In this case the number of nodes in R(r) equals n 1 . The azimuthal quantum number /replaces the k(= I + 1) of old quantum theory. The angular momentum is given by Vl(l + 1) H/2iT. Now / can take any value from to n 1 ; then / is the number of nodal surfaces passing through the origin. The magnetic quantum number m still gives the value of the components 282 PARTICLES AND WAVES [Chap. 10 of angular momentum along the r axis, since p 0tZ mh/27T 9 exactly as in eq. (10.27). The allowed values of m now run from / to {-/, including zero. The great advantage of the new theory is that these numbers all arise quite naturally from Schrodinger's equation. n = 3 (a) mo 6 Fig. 10.12. (a) Radial part of wave functions for hydrogen atom, (b) Radial distribution functions giving probability of finding electrons at a given distance from nucleus. (After G. Herzberg, Atomic Spectra, Dover, 1944.) 23. The radial wave functions. In (a), Fig. 10.12, the radial wave functions have been plotted for various choices of n and /. In case / 0, all the nodes appear in the radial function. The value of ^ 2 (r) is proportional to the probability of finding the electron at any particular distance r in some definite direction from the nucleus. More important physically is the radial distribution function, 47rr 2 ^ 2 (r), which gives the probability of finding the electron within a spherical shell of thickness dr Sec. 23] PARTICLES AND WAVES 283 at a distance r from the nucleus, irrespective of direction. (Compare the problem of gas-velocity distribution on page 187.) The radial distribution functions are shown in (b), Fig. 10.12. In place of the sharply defined electron orbits of the Bohr theory, there is a more diffuse distribution of electric charge. The maxima in these distribution curves, however, correspond closely with the radii of the old Bohr orbits. Yet there is always a definite probability n -2 / -o > 2, m - 1 3, m - n = 3, m 2 n = 3, m = db 1 n = 3, w = n = 4, m Fig. 10.13. Electron clouds of the H atom. (From Herzberg, Atomic Spectra and Atomic Structure, Dover, 1944.) of finding the electrons much closer to or much farther from the nucleus. The strict determinism of position in the classical description has been replaced by the probability language of wave mechanics. A particularly clear illustration of the wave mechanical representation of the hydrogen atom can be obtained from the illustrations in Fig. 10.13. Here the intensity of the shading is proportional to the value of y> 2 , the probability distribution function. There is a greater probability of an electron being in a light-colored region. It should be clearly understood that quantum mechanics does not say that the electron itself is smeared out into a cloud. 284 PARTICLES AND WAVES [Chap. 10 It is still to be regarded as a point charge. Its position and momentum cannot be simultaneously fixed, and all that the theory can predict that has physical meaning is the probability that the electron is in any given region. A wave function for an electron is sometimes called an orbital. When / = we have an 5 orbital, which is always spherically symmetrical. When / = 1 we have a p orbital. The p orbitals can have various orientations in space corresponding to the allowed values of w, which may be 1, 0, or +L In Fig. 10.14, the angular parts of the wave functions are represented for s z Fig. 10.14. Polar representation of absolute values of angular part of wave function for the H atom. The j-type function (/ == 0) is spherically symmetrical. There are three possible />-type functions, directed along mutually perpendicular axes (x, y. z). and p orbitals, and the directional 1 character of the p orbitals is very evident. It will be shown later that the directional character of certain chemical bonds is closely related to the directed orientations of these orbitals. 24. The spinning electron. There is one aspect of atomic spectra that cannot be explained on the basis of either the old quantum theory or the newer wave mechanics. This is the multiplicity or multiple! structure of spectral lines. Typical of this multiplicity are the doublets occurring in the spectra of the alkali metals: for example, in the principal series each line is in reality a closely spaced double line. This splitting is revealed immediately with a spectroscope of good resolving power. The occurrence of double lines indicates that each term or energy level for the optical electron must also be split into two. Sec. 25] PARTICLES AND WAVES 285 A satisfactory explanation for the occurrence of multiple energy levels was first proposed in 1925 by G. E. Uhlenbeck and S. Goudsmit. They postulated that an electron itself may be considered to be spinning on its axis. 20 As a result of spin the electron has an inherent angular momentum. Along any prescribed axis in space, for example, the direction of a magnetic field, the components of the spin angular momentum are restricted to values given by sh/2-rr, where s can have only a value of + J or -J. In effect, the electron spin adds a new quantum number s to those re- quired to describe completely the state of an electron. We now have, therefore, the following quantum numbers: n the principal quantum number; allowed values 1, 2, 3, . . . / the azimuthal quantum number, which gives the orbital angular momentum of the electron; allowed values 0, 1, 2, . . ., n I. m the magnetic quantum number, which gives the allowed orientation of the "orbits" in an external field; allowed values /, / + 1 -/ + 2, . . ., + /. s the spin quantum number; allowed values \-\ or J. 25. The Pauli Exclusion Principle. An exact solution of the wave equation for an atom has been obtained only in the case of hydrogen; i.e., for the motion of a single electron in a spherically symmetric coulombic field. Nevertheless, in more complex atoms the energy levels can still be specified in terms of the four quantum numbers n, /, m, s 9 although in many cases the physical picture of the significance of the numbers will be lost. This is especially true of electrons in inner shells, for which a spherically symmetric field would be a very poor approximation. On the other hand, the behavior of an outer or valence electron is sometimes strikingly similar to that of the electron in the hydrogen atom. In any case, the important fact is that the four quantum numbers still suffice to specify completely the state of an electron even in a complex atom. There is a most important principle that determines the allowable quan- tum numbers for an electron in an atom and consequently has the most profound consequences for chemistry. It is the Exclusion Principle, first enunciated by Wolfgang Pauli. In a single atom no two electrons can have the same set of four quantum numbers, n, /, m, s. At present this principle cannot be derived from fundamental concepts, but it may have its ultimate origin in relativity theory. It is suggestive that relativity theory introduces a "fourth dimension," so that a fourth quantum number becomes necessary. 26. Structure of the periodic table. The general structure of the periodic table is immediately clarified by the Exclusion Principle. We recall that even in a complex atom the energy levels of the electrons can be specified by 80 No attempt will be made to reconcile this statement with the idea that an electron is x>int charge. It is merely properties of electrons, ab .a point charge. It is merely a convenient pictorial way of speaking of one of the fundamental about which the complete story is not yet written. 286 PARTICLES AND WAVES [Chap. 10 means of four quantum numbers : , /, m, s. The Exclusion Principle requires that no two electrons in an atom can have the same values for all four quan- tum numbers. The most stable state, or ground state, of an atom will be that in which the electrons are in the lowest possible energy levels that are con- sistent with the Exclusion Principle. The structure of the periodic table is a direct consequence of this requirement. The lowest atomic energy state is that for which the principal quantum number n is 1, and the azimuthal quantum number / is 0. This is a Is state. The hydrogen atom has one electron and this goes therefore into the \s level. The helium atom has two electrons, which may both be accommodated in the \s state if they have opposing spins. With two electrons in the \s state, there is an inert gas configuration since the shell n 1 or K shell is com- pleted. The completed shell cannot add electrons and a large energy would be needed to remove an electron. Continuing to feed electrons into the lowest lying energy levels, we come to lithium with 3 electrons. The first two go into the Is levels, and the third electron must occupy a 2s level. The 2s electron is much less tightly bound than the Is electrons. The first ionization potential of Li is 5.39 ev, the second 75.62 ev. This is true because the 2s electron is usually much farther from the nucleus than the Is, and besides it is partially shielded from the +3 nuclear charge by the two Is electrons. A Is electron, on the other hand, is held by the almost unshielded +3 nuclear charge. The L shell, with n --= 2, can hold 8 electrons two 2s and six 2p electrons, the quantum numbers being as follows : n I 2 m -1 + 1 s 4 4 4 4 When the L shell is filled, the next electron must enter the higher-lying M shell of principal quantum number n = 3. A qualitative picture of the stability of the complete octet is obtained by considering the elements on either side of neon. Z Is 2s IP 3s o . 8 2 2 4 F . 9 2 2 5 Ne . 10 2 2 6 Na . 11 2 2 6 1 Mg . . 12 2 2 6 2 The attraction of an electron by the positively charged nucleus is governed by Coulomb's Law, but for electrons outside the innermost shell the shielding effect of the other electrons must be taken into consideration. For a given Sec. 27] PARTICLES AND WAVES 287 electron, the shielding effect of other electrons is pronounced only if they lie in a shell between the given electron and the nucleus. Electrons in the same shell as the given electron have little shielding effect. Thus in fluorine, the nuclear charge is -f 9; each of the five 2p electrons is attracted by this +9 charge minus the shielding of the Is and 2s electrons, four in all, resulting in an effective nuclear charge of about 4 5. The 2p electrons in fluorine are therefore tightly held, the first ionization potential being about 18 volts. If an extra electron is added to the 2p level in fluorine, forming the fluoride ion F~, the added electron is also tightly held by the effective +5 nuclear charge. The electron affinity of F is 4.12ev; that is, F + e->F- + 4.12ev. Now suppose one attempted to add another electron to F~ to form F 53 . This electron would have to go into the 3s state. In this case, all ten of the inner electrons would be effective in shielding the -f 9 nucleus, and indeed the hypothetical eleventh electron would be repelled rather than attracted. Thus the fluoride ion is by far the most stable configuration and the - 1 valence of fluorine is explained. If the tendency of one atom to add an electron (electron affinity) is of the same magnitude as the tendency of another atom to lose an electron (ionization potential) a stable electrovalent bond is possible. Considering now the sodium atom, we can see that its eleventh electron, 3s 1 , is held loosely (/ ~ 5.11 ev). It is shielded from the +11 nucleus by 10 inner electrons. If we continue to feed electrons into the allowed levels, we find that the 3/? level is complete at argon (Is 2 2s 2 2/? 6 3s 2 3/? 6 ), which has the stable s 2 /? 6 octet associated with inert gas properties. 27. Atomic energy levels. In Table 10.4 the assignment of electrons to levels is shown for all the elements, in accordance with our best present knowledge as derived from chemical and spectroscopic data. In the element following argon, potassium with Z = 19, the last electron enters the 4s orbital. This is required by its properties as an alkali metal, and the fact that its spectral ground state is *S as in Li and Na. We may well ask, however, why the 4s orbitals are lower than the 3d orbitals, which pro- vide 10 vacant places. The answer to this question should help to clarify the structure of the remainder of the periodic table and the properties of the elements in the transition series. It may be noted that in this section we are speaking of orbitals, or quantum mechanical wave functions y> for the elec- trons. The Bohr picture was useful in dealing with the lighter elements (up to A) but it gives an inadequate picture of the remainder of the periodic table. The reason why the 4s orbital for potassium has a lower energy than a 3d orbital arises from the fundamental difference in form of s, p, and d orbitals. The electron distributions in. the 3s, 3/?, and 3d orbitals for the hydrogen atom were shown in (b), Fig. 10.12. The ordinates of the curves are proportional to the radial distribution functions, and therefore to the TABLE 10.4 ELECTRON CONFIGURATIONS OF THE ELEMENTS Shell: K L A/ AT Element Is 2s 2p 3* 3/7 3rf 45 4p 4d 4f 1. H 1 2. He 2 3. Li 2 1 4. Be 2 2 5. B 2 2 1 6. C 2 2 2 7. N 2 2 3 8. 2 2 4 9. F 2 2 5 10. Ne 2 2 6 11. Na 2 2 6 1 12. Mg 2 2 6 2 13. Al 2 2 6 2 1 14. Si 2 2 6 2 2 15. P 2 2 6 2 3 16. S 2 2 6 2 4 17. Cl 2 2 6 2 5 18. A 2 2 6 2 6 19. K 2 2 6 2 6 1 20. Ca 2 2 6 2 6 2 21. Sc 2 2 6 2 6 1 2 22. Ti 2 2 6 262 2 23. V 2 2 6 263 2 24. Cr 2 2 6 265 1 25. Mn 2 2 6 265 2 26. Fe 2 2 6 266 2 27. Co 2 2 6 267 2 28. Ni 2 2 6 268 2 29. Cu 2 2 6 2 6 10 1 30. Zn 2 2 6 2 6 10 2 31. Ga 2 2 6 2 6 10 2 1 32. Ge 2 2 6 2 6 10 2 2 33. As 2 2 6 2 6 10 2 3 34. Se 2 2 6 2 6 10 2 4 35. Br 2 2 6 2 6 10 2 5 36. Kr 2 2 6 2 6 10 2 6 Shell: K L Af AT P Q Element 45 4p 4d 4f 5s 5p 5d 5f 5g 6s 6p 6d Is 37. Rb 38. Sr 2 2 2 2 2 2 2 2 2 2 8 8 18 18 2 6 2 6 1 2 6 2 39. Y 40. Zr 41. Nb 42. Mo 43. Tc 44. Ru 45. Rh 46. Pd 8 8 8 8 8 8 8 8 18 18 18 18 18 18 18 18 2 6 1 262 264 265 2 6 (5) 267 268 2 6 10 2 2 1 1 (2) 1 1 TABLE 10.4 (Cont.) Shell: K L M N O /> (2 Element 4s 4j> 4d 4f 5s 5p 5d . V% 6s 6p 6d 7* 47. Ag 2 8 18 2 6 10 1 48. Cd 2 8 18 2 6 10 2 49. In 2 8 18 2 6 10 2 1 50. Sn 2 8 18 2 6 10 2 2 51. Sb 2 8 18 2 6 10 2 3 52. Te 2 8 18 2 6 10 2 4 53. I 2 8 18 2 6 10 2 5 54. Xe 2 8 18 2 6 10 2 6 55. Cs 2 8 18 2 6 10 2 6 1 56. Ba 2 8 18 2 6 10 2 6 2 57. La 2 8 18 2 6 10 2 6 1 2 58. Ce 2 8 18 2 6 10 2 2 6 2 59. Pr 2 8 18 2 6 10 3 2 6 2 60. Nd 2 8 18 2 6 10 4 2 6 2 61. Pm 2 8 18 2 6 10 5 2 6 2 62. Sm 2 8 18 2 6 10 6 2 6 2 63. Eu 2 8 18 2 6 10 7 2 6 2 64. Gd 2 8 18 2 6 10 7 2 6 2 65. Tb 2 8 18 2 6 10 8 2 6 2 66. Dy 2 8 18 2 6 10 9 2 6 2 67. Ho 2 8 18 2 6 10 10 2 6 2 68. Er 2 8 18 2 6 10 11 2 6 2 69. Tu 2 8 18 2 6 10 13 2 6 2 70. Yb 2 8 18 2 6 10 14 2 6 2 71. Lu 2 8 18 2 6 10 14 2 6 1 2 72. Hf 2 8 18 2 6 10 14 262 2 73. Ta 2 8 18 2 6 10 14 263 2 74. W 2 8 18 2 6 10 14 264 2 75. Re 2 8 18 2 6 10 14 265 2 76. Os 2 8 18 2 6 10 14 266 2 77. Ir 2 8 18 2 6 10 14 267 2 78. Pt 2 8 18 2 6 10 14 269 1 79. Au 2 8 18 2 6 10 14 2 6 10 1 80. Hg 2 8 18 2 6 10 14 2 6 10 2 81. Tl 2 8 18 2 6 10 14 2 6 10 2 1 82. Pb 2 8 18 2 6 10 14 2 6 10 2 2 83. Bi 2 8 18 2 6 10 14 2 6 10 2 3 84. Po 2 8 18 2 6 10 14 2 6 10 2 4 85. At 2 8 18 2 6 10 14 2 6 10 2 5 86. Rn 2 8 18 2 6 10 14 2 6 10 2 6 87. Fr 2 8 18 2 6 10 14 2 6 10 2 6 1 88. Ra 2 8 18 2 6 10 14 2 6 10 2 6 2 89. Ac 2 8 18 2 6 10 14 2 6 10 2 6 1 2 90. Th 2 8 18 2 6 10 14 2 6 10 2 6 2 2 91. Pa 2 8 18 2 6 10 14 2 6 10 2 2 6 1 2 92. U 2 8 18 2 6 10 14 2 6 10 3 2 6 1 2 93. Np 2 8 18 2 6 10 14 2 6 10 5 2 6 2 94. Pu 2 8 18 2 6 10 14 2 6 10 6 2 6 2 95. Am 2 8 18 2 6 10 14 2 6 10 7 2 6 2 96. Cm 2 8 18 2 6 10 14 2 6 10 1 2 6 1 2 97. Bk 2 8 18 2 6 10 14 2 6 10 8 2 6 1 2 98. Cf 2 8 18 2 6 10 14 2 6 10 9 2 6 1 2 290 PARTICLES AND WAVES [Chap. 10 probability of finding an electron within a given region. Now, of course, these hydrogen wave functions are not a completely accurate picture of the orbitals in a more complex atom with many electrons. The approximation is satisfactory, however, for valence electrons, which move in the hydrogen- like field of a nucleus shielded by inner electrons. The 4s and 3p orbitals predict a considerable concentration of the charge cloud closely around the nucleus, 21 whereas the 3d orbital predicts an ex- tremely low probability of finding the electron close to the nucleus. As a result of this penetration of the 4s orbital inward towards the nucleus, a 4s electron will be more tightly bound by the positive nuclear charge, and will therefore be in a lower energy state than a 3d electron, whose orbital does not penetrate, and which is therefore more shielded from the nucleus by the inner shells. It is true that the most probable position for a 4s electron is farther from the nucleus than that for a 3d electron; the penetration effect more than makes up for this, since the coulombic attraction decreases as the square of r, the distance of the electron from the nucleus. Since 4s lies lower than 3d, the nineteenth electron in potassium enters the 4s rather than the 3d level, and potassium is a typical alkali metal. In Fig. 10.15 the relative energies of the orbitals are plotted as functions of the atomic number (nuclear charge). This graph is not quantitatively exact, but is designed to show roughly how the relative energy levels of the various orbitals change with increasing nuclear charge. The energies are obtained from atomic spectra. Although the effect is not shown in the figure, it should be noted that the energy levels of the s and p orbitals fall steadily with increasing atomic number, since the increasing nuclear charge draws the penetrating s and p orbitals closer and closer to the nucleus. At low atomic numbers, up to Z ^ 20 (Ca), the 3d levels are not lowered, since there are not yet sufficient electrons present for the d's to penetrate the electron cloud that surrounds and shields the nucleus. As more electrons are added, however, the 3d orbitals eventually penetrate the shielding electrons and begin to fall with increasing Z. This phenomenon is repeated later with the 4rfand 4f orbitals. At high Z, therefore, orbitals with the same principal quantum number tend to lie together; at low Z they may be widely separated because of different penetration effects. Following calcium, the 3d orbitals begin to be filled rather than the 4p. One obtains the first transition series of metals, Sc, Ti, V, Cr, Mn, Fe, Co, Ni. These are characterized by variable valence and strongly colored compounds. Both these properties are associated with the closeness of the 4s and 3d levels, 21 The distinct difference between this quantum-mechanical picture and the classical Bohr orbits should be carefully noted. There are four successive maxima in the y function for the 4s orbital, at different distances from the nucleus. The quantum mechanical picture of an atom is a nucleus surrounded by a cloud of negative charge. There are differences in density of the cloud at different distances from the nucleus. The cloud is the superposition of the v> functions for all the orbitals occupied by electrons. Sec. 27] PARTICLES AND WAVES 291 which provide a variable number of electrons for bond formation, and possible excited levels at separations corresponding with the energy available in visible light (~ 2 ev). The filling of the 3d shell is completed with copper, which has the con- figuration \s*2s*2p*3s*3p*3d l 4s l . Copper is not an alkali metal despite the outer 4s electron, since the 3d level is only slightly below the 4s and Cu++ ions are readily formed. 2s Fig. 10.15. Dependence of energies of orbitals on the nuclear charge Z. The next electrons gradually fill the 4s and 4p levels, the process being completed with krypton. The next element, rubidium, is a typical alkali with one 5s electron outside the 4s 2 4p B octet. Strontium, with two 5s electrons, is a typical alkaline earth of the Mg, Ca, Sr, Ba series. Now, however, the 4d levels become lowered sufficiently to be filled before the 5p. This causes the second transition series, which is completed with palladium. Silver follows with the copper type structure, and the filling of the 5,y and 5p levels is completed with xenon. A typical alkali (Cs) and alkaline earth (Ba) follow with one and two 6s electrons. The next electron, in lanthanum, enters the 5d level, and one might suspect that a new transition series is underway. Meanwhile, however, with 292 PARTICLES AND WAVES [Chap. 10 increasing nuclear charge, the 4forbitals have been drastically lowered. The 4/ levels can hold exactly 14 electrons. 22 As these levels are filled, we obtain the 14 rare earths with their remarkably similar chemical properties, deter- mined by the common 5s 2 5p 6 outer configuration of their ions. This process is complete with lutecium. The next element is hafnium, with 5d 2 6s 2 . Its properties are very similar to those of zirconium with 4d 2 5s 2 . This similarity in electronic structures was predicted before the discovery of hafnium, and led Coster and Hevesy to look for the missing element in zirconium minerals, where they found it in 1923. Following Hf the 5d shell is filled, and then the filling of the 6p levels is completed, the next s 2 p* octet being attained with radon. The long missing halogen (85) and alkali (87) below and above radon have been found as artificial products from nuclear reactions. 23 They are called "astatine" and "francium." Radium is a typical alkaline earth metal with two Is electrons. In the next element, actinium, the extra electron enters the 6d level, so that the outer configuration is 6d l ls 2 ; this is to be compared with lanthanum with 5d*6s 2 . It was formerly thought that the filling of the 6d levels continued in the elements following actinium. As i result of studies of the properties of the new transuranium elements it now appears more likely that actinium marks the beginning of a new rare-earth group, successive electrons entering the 5f shell. Thus the trivalent state becomes more stable compared to the quadrivalent state as one proceeds through Ac, Th, Pa, U, Np, Pu, Am, Cm, just as it does in the series La, Ce, Pr, Nd, Pm, Sm, Eu, etc. This is true because successive electrons added to the/shell are more tightly bound as the nuclear charge increases. The actinide "rare earths" therefore resemble the lanthanide rare earths rather than the elements immediately above themselves in the periodic table. PROBLEMS 1. What is the average energy, , of a harmonic oscillator of frequency 10 13 sec- 1 at 0, 200, 1000C? What is the ratio ejkT at each temperature? 2. The K* X-ray line of iron has a wavelength of 1 .932 A. A photon of this wavelength is emitted when an electron falls from the L shell into a vacancy in the K shell. Write down the electronic configuration of the ions before and after emission of this line. What is the energy difference in kcal per mole between these two configurations? " As follows: n 4 / 3 m 3, -2, -1, 0, 4-1, +2, -fl * i, i, i, i, t, t, * 23 For an excellent account, see Glenn T. Seaborg, "The Eight New Synthetic Elements," American Scientist, 36, 361 (1948). Chap. 10] PARTICLES AND WAVES 293 3. The fundamental vibration frequency of N 2 corresponds to a wave number of 2360 cm" 1 . What fraction of N 2 molecules possess no vibrational energy (except their zero-point energy) at 25C ? 4. The first line in the Lyman series lies at 1216 A, in the Balmer series, at 6563 A. In the absorption spectra of a certain star, the Balmer line appears to have one-fourth the intensity of the Lyman line. Estimate the temperature of the star. 5. Calculate the ionization potential of hydrogen as the energy required to remove the electron from r r - 0.53A to infinity against the coulombic attraction of the proton. 6. An excited energy level has a lifetime of 10~ 10 sec. What is the mini- mum width of the spectral line arising in a transition from the ground state to this level ? 7. Calculate the wavelength of a proton accelerated through a potential difference of 1 mev. 8. For a particle of mass 9 x 10~ 28 g confined to a one-dimensional box 100 A long, calculate the number of energy levels lying between 9 and 10 ev. 9. Consider an electron moving in a circular path around the lines of force in a magnetic field. Apply the Bohr quantum condition eq. (10.21) to this rotation. What is the radius of the orbit of quantum number n 1 in a magnetic field of 10 5 gauss? 10. The K al X-ray line is emitted when an electron falls from an L level to a hole in the K level. Assume that the Rydberg formula holds for the energy levels in a complex atom, with an effective nuclear charge 7! equal to the atomic number minus the number of electrons in shells between the given electron and the nucleus. On this basis, estimate the wavelength of the Af al X-ray line in chromium. The experimental value is 2.285 A. 11. The wave function for the electron in the ground state of the hydrogen atom is y ls = (7ra 3 )~ 1/2 e~ r/a , where a is the radius of the Bohr orbit. Calculate the probability that an electron will be found somewhere between 0.9 and 1.1 a . What is the probability that the electron will be beyond 2 a ? 12. Write an account of the probable inorganic chemistry of Np, Pu, Am, Cm, in view of their probable electron configurations. Compare the chemistry of astatine and iodine, francium and cesium. REFERENCES BOOKS 1. de Broglie, L., Matter and Light (New York: Dover, 1946). 2. Heitler, W., Elementary Wave Mechanics (New York: Oxford, 1945). 3. Herzberg, G., Atomic Spectra and Atomic Structure (New York: Dover Publications, 1944). 294 PARTICLES AND WAVES [Chap. 10 4. Mott, N. F., Elements of Wave Mechanics (Cambridge: Cambridge Univ. Press, 1952). 5. Pauling, L., and E. B. Wilson, Introduction to Quantum Mechanics (New York: McGraw-Hill, 1935). 6. Pitzer, K. S., Quantum Chemistry (New York: Prentice-Hall, 1953). 7. Slater, J. C, Quantum Theory of Matter (New York: McGraw-Hill, 1951). 8. Whittaker, E. T., From Euclid to Eddington, A Study of Conceptions of the External World (London: Cambridge, 1949). ARTICLES 1. Compton, A. H., Am. J. Phys. 9 14, 80-84 (1946), "Scattering of X-Ray Photons." 2. de Vault, D., /. Chem. Ed., 21, 526-34, 575-81 (1944), "The Electronic Structure of the Atom." 3. Glockler, G.,J. Chem. Ed., 18, 418-23 (1941), "Teaching the Introduction to Wave Mechanics." 4. Margenau, H., Am. J. Phys., 13, 73-95 (1945); 12, 119-30, 247-68 (1944), "Atomic and Molecular Theory Since Bohr." 5. Meggers, W. F., /. Opt. Soc. Am., 41, 143-8 (1951), "Fundamental Research in Atomic Spectra." 6. Zworykin, V. K., Science in Progress, vol. Ill (New Haven: Yale Univ. Press, 1942), pp. 69-107, "Image Formation by Electrons." CHAPTER 11 The Structure of Molecules 1. The development of valence theory. The electrical discoveries at the beginning of the nineteenth century strongly influenced the concept of the chemical bond. Indeed, Berzelius proposed in 1812 that all chemical com- bination was caused by electrostatic attraction. As it turned out 115 years later, this theory happened to be true, though not in the sense supposed by its originator. It did much to postpone the acceptance of diatomic structures for the common gaseous elements, such as H^ N 2 , and O 2 . It was admitted that most organic compounds fitted very poorly into the electrostatic scheme, but until 1828 it was widely believed that these compounds were held together by "vital forces," arising by virtue of their formation from living things. In that year, Wohler's synthesis of urea from ammonium cyanate destroyed this distinction between organic and inorganic compounds, and the vital forces gradually retreated to their present refuge in living cells. Two general classes of compounds came to be distinguished, with an assortment of uncomfortably intermediate specimens. The polar compounds, of which NaCl was a prime example, could be adequately explained as being composed of positive and negative ions held together by coulombic attrac- tion. The nature of the chemical bond in the nonpolar compounds, such as CH 4 , was completely obscure. Nevertheless, the relations of valence with the periodic table, which were demonstrated by Mendeleev, emphasized the remarkable fact that the valence of an element in a definitely polar compound was usually the same as that in a definitely nonpolar compound, e.g., O in K 2 O and (C 2 H 5 ) 2 O. In 1904 Abegg pointed out the rule of eight: To many elements in the periodic table there could be assigned a negative valence and a positive valence the sum of which was eight, for example, Cl in LiCl and C1 2 O 7 , N in NH 3 and N 2 O 5 . Drude suggested that the positive valence was the number of loosely bound electrons that an atom could give away, and the negative valence was the number of electrons that an atom could accept. Once the concept of atomic number was clearly established by Moseley (1913), further progress was possible, for then the number of electrons in an atom became known. The special stability of a complete outer octet of electrons was soon noticed. For example: He, 2 electrons; Ne, 2 + 8 elec- trons; A, 2 4- 8 + 8 electrons. In 1916, W. Kossel made an important con- tribution to the theory of the electrovalent bond, and in the same year G. N. Lewis proposed a theory for the nonpolar bond. Kossel explained the formation of stable ions by a tendency of the atoms 295 296 THE STRUCTURE OF MOLECULES [Chap. 11 to gain or lose electrons until they achieve an inert-gas configuration. Thus argon has a completed octet of electrons. Potassium has 2 + 8 + 8 + 1, and it tends to lose the outer electron, becoming the positively charged K+ ion having the argon configuration. Chlorine has 2 + 8 + 7 electrons and tends to gain an electron, becoming Cl with the argon configuration. If an atom of Cl approaches one of K, the K donates an electron to Cl, and the resulting ions combine as K f Cl:, the atoms displaying their valences of one. The extension to other ionic compounds is familiar. G. N. Lewis proposed that the links in nonpolar compounds resulted from the sharing of pairs of electrons between atoms in such a way as to form stable octets to the greatest possible extent. Thus carbon has an atomic number of 6; />., 6 outer electrons, or 4 less than the stable neon configura- tion. It can share electrons with hydrogen as follows: H xo XO H Each pair of shared electrons constitutes a single covalent bond. The Lewis theory explained why the covalence and electrovalence of an atom are usually identical, for an atom usually accepts one electron for each covalent bond that it forms. The development of the Bohr theory led to the idea that the electrons were contained in shells or energy levels at various distances from the nucleus. These shells were specified by the quantum numbers. By about 1925, a systematic picture of electron shells was available that represented very well the structure of the periodic table and the valence properties of the elements. The reason why the electrons are arranged in this way was unknown. The reason why a shared electron pair constitutes a stable chemical bond was also unknown. An answer to both these fundamental chemical problems was provided by the Pauli Exclusion Principle. Its application to the problem of the periodic table was shown in the previous chapter. Its success in explaining the nature of the chemical bond has been equally remarkable. 2. The ionic bond. The simplest type of molecular structure to understand is that formed from two atoms, one of which is strongly electropositive (low ionization potential) and the other, strongly electronegative (high electron affinity). Such, for example, would be sodium and chlorine. In crystalline sodium chloride, one cannot speak of an NaCl molecule since the stable arrangement is a three-dimensional crystal structure of Na+ and Cl~ ions. In the vapor, however, a true NaCl molecule exists, in which the binding is almost entirely ionic. The attractive force between two ions with charges q and q 2 can be represented at moderate distances of separation r by the coulombic force Sec. 3] THE STRUCTURE OF MOLECULES 297 or ty a potential V ~q\q^r. If the ions are brought so close to- gether that their electron clouds begin to overlap, a mutual repulsion between the positively charged nuclei becomes evident. Born and Mayer have sug- gested a repulsive potential having the form U be~ r/a , where a and b are constants. The net potential for two ions is therefore + be r/a (11-1) This potential-energy function is plotted in Fig. 11.1 for NaCl, the minimum in the curve representing the stable internuclear separation for a Na+Cl~ 5 10 15 20 INTERNUCLEAR SEPARATION, r, A Fig. 11.1. Potential energy of Na + }- Cl . (The internuclear distance in the stable molecule is 2.51 A. Note the long range of the coulombic attraction.) molecule. Spectra of this molecule are observed in the vapor of sodium chloride. 3. The covalent bond. One of the most important of all the applications of quantum mechanics to chemistry has been the explanation of the nature of the covalent bond. The simplest example of such a bond is found in the H 2 molecule. Although Lewis, in 1918, declared that this bond consists of a shared pair of electrons, it was in 1927 that a real understanding of the nature of the binding was provided by the work of W. Heitler and F. London. If two H atoms are brought together there results a moderately com- plicated system consisting of two 4- 1 charged nuclei and two electrons. If the atoms are very far apart their mutual interaction is effectively nil. In 298 THE STRUCTURE OF MOLECULES [Chap. 1 1 other words, the potential energy of interaction V ~ when the internuclear distance r oo. At the other extreme, if the two atoms are forced very closely together, there is a large repulsive force between the two positively charged nuclei, so that as r -> 0, U > oo. Experimentally we know that two hydrogen atoms can unite to form a stable hydrogen molecule, whose dis- sociation energy is 4.48 ev, or 103.2 kcal per mole. The internuclear separation in the molecule is 0.74 A. -5 - 0.5 2.5 I 1.5 2 ANGSTROMS Fig. 11.2. Potential energy curve for hydrogen molecule. (Note the shorter range of the valence forces in H 2 , as compared with the ionic molecule NaCl shown in Fig. 11.1.) These facts about the interaction of two H atoms are summarized in the potential-energy curve of Fig. 11.2. The problem before us is to explain the minimum in the curve. This is simply another way of asking why a stable molecule is formed, or what is the essential nature of the covalent bond in H 2 . The quantum-mechanical problem is to solve the Schrodinger equation for the system of two electrons and two protons. Consider the situation in Fig. 11.3, where the outer electron orbits overlap somewhat. According to quantum mechanics, of course, these orbits are not sharp. There are eigen- functions ^(1) for electron (1) and y(2) for electron (2), which determine the probability of finding the electrons at any point in space. As long as the atoms are far apart, the eigenfunction for electron (1) on nucleus (a) will be simply that found on page 281 for the ground state of a hydrogen atom namely, y ls (\) - (7ra *)- l/2 e~ r/a : Sec. 3] THE STRUCTURE OF MOLECULES 299 For the two electrons, a wave function is required that expresses the probability of simultaneously finding electron (1) on nucleus (a) and electron (2) on nucleus (b). Since the combined probability is the product of the two individual probabilities, such a function would be a(\)b(2). Here a(\) and b(2) represent eigenfunctions for electron (1) on nucleus (a) and electron (2) on nucleus (b). A very important principle must now be considered. There are no physical differences and no way of distinguishing between a system with (1) on (a) and (2) on (b) and a system with (2) on (a) and (1) on (b). The electrons cannot be labeled. The proper wave function for the system must contain in itself an expression of this fundamental truth. To help solve this problem we need only recall from page 254 that if ^i and y> 2 are two solutions of the wave equation, then any linear combination Fig< u 3 , nte ract,on of two h>drogen of these solutions is also a solution, e.g., atoms. c iVi + C 2 1 /V There are two particular linear combinations that inherently express the principle that the electrons are indistinguishable. These are Vf = fl(l)A(2) + a(2)b(\) V_ -a(\)b(2) -a( If the electrons are interchanged in these functions, y> + is not changed at all; it is called a symmetric function. y_ is changed to y>_, but this in itself does not change the electron distribution since it is y>* which gives the probability of finding an electron in a given region, and ( 1/>) 2 = y> 2 . The function ^_ is called antisymmetric. So far the spin properties of the electrons have not been included, and this must be done in order to obtain a correct wave function. The electron- spin quantum number s, with allowed values of either + 1 or I, determines the magnitude and orientation of the spin. We introduce two spin functions a and ft corresponding to s - +A and s = \. For the two-electron system there are then four possible complete spin functions: Spin Function Electron 1 Electron 2 a(l)a(2) -ft +i 0)0(2) +J -1 00 M2) -J -fj 00)0(2) -t -* When the spins have the same direction they are said to be parallel', when they have opposite directions, antiparallel. Once again, however, the fact that the electrons are indistinguishable 300 THE STRUCTURE OF MOLECULES [Chap. 11 forces us to choose linear combinations for the two-electron system which are either symmetric or antisymmetric. There are three possible symmetric spin functions: a(l)a(2) \ sym There is one antisymmetric spin function : <x(l)/3(2) oc(2)/?(l) antisym The possible complete wave functions for the H-H system are obtained by combining these four possible spin functions with the two possible orbital wave functions. This leads to eight functions in all. At this point in the argument the Pauli Principle enters in an important way. The Principle is stated in a more general form than was used before: "Every allowable eigenfunction for a system of two or more electrons must be antisymmetric for the interchange of the coordinates of any pair of electrons." It will be shown a little later that the prohibition against four identical quantum numbers is a special case of this statement. As a consequence of the exclusion principle, the only allowable eigen- functions are those made up either of symmetric orbitals and antisymmetric spins or of antisymmetric orbitals and symmetric spins. There are four such combinations for the H-H system: rbital Spin Total Spin Term + 0(2 W) a( 1)0(2) - a(2)0(l) (singlet) tZ - a(2)b(\) 1 1 (triplet) 3 D a( 1)0(2) 1 a(2)0(l)J ' The term symbol S expresses the fact that the molecular state has a total angular momentum of zero, since it is made up of two atomic S terms. The multipliu.y of the term, or number of eigenfunctions corresponding with it, is added as a left-hand superscript. This multiplicity is always 2f? + 1 where & is the total spin. The way in which the general statement of the exclusion principle reduces to that in terms of quantum numbers can readily be seen in a typical example. Multiplying out the *X function gives y> = aa(l)6(2) aa(2)6(l). If the quantum numbers n, /, m are the same for both electrons, their orbital func- tions are identical, a = b, so that y = a<x.(\)afi(2) aa(2)00(l). If the fourth quantum number s is also the same for both, either or , the spin functions must be either both a or both /?. Then y = 0, that is, the proba- bility of such a system is zero, In other words, eigenfunctions that assign Sec. 4] THE STRUCTURE OF MOLECULES 301 identical values of n, /, m, and s to two electrons are outlawed. This result was shown in a special case, but it is in fact a completely general consequence of the requirement of antisymmetry. 4. Calculation of the energy in H-H molecule. The next step is to calculate the energy for the interaction of two hydrogen atoms using the allowed wave functions. The different electrostatic interactions are shown in Fig. 1 1 .3: i. electron (1) with electron (2), potential, (7, 2 / r i2 ii. electron (1) with nucleus (/>), / 2 e 2 /r lb iii. electron (2) with nucleus (a\ t/ 3 ^ e z /r 2a iv. nucleus (a) with nucleus (/?), (7 4 ^ e 2 /r att Note that the interactions of electron (1) with its own nucleus (a) and of electron (2) with nucleus (b) are already taken into account by the fact that we are starting with two hydrogen atoms. The potential for the interaction of two electrons a distance r 12 apart is /! -e 2 /r l2 . In order to find the energy of interaction, we must multiply this by the probability of finding an electron in a given element of volume dv, and then integrate over all of space. Since the required probability is ifdv, this gives E --- J U^dv. Since the total potential is U U -f t/ 2 + U 3 + t/ 4 , the total energy of interaction of the two hydrogen atoms becomes --= J t/yVr (11.2) This energy must now be calculated for both the symmetric and the anti- symmetric orbital wave functions. Squaring these functions, one obtains y 2 -0 2 (l)/>*(2) f <P(2)b\ I) 2a( 1)6(2X2)6(1) The f sign is for the X S function, the sign for the 3 2 function. The integral in eq. (1 1.2) can therefore be written E^2C2A (11.3) where C = J Ua\\}b*(2)dv A - J Ua(\)b(\)a(2)b(2)dv ( ' ' C is called the coulombic energy , and A is called the exchange energy. The coulombic energy is the result of the ordinary electrostatic interaction between the charges of the electrons and the nuclei. The behavior of this coulombic energy as the two hydrogen atoms approach each other can be estimated qualitatively as follows, although the actual integration is not too difficult if we use the simple Is orbitals for a and b. At large intern uclear distances, C is zero. At very small distances C approaches infinity owing to the strong repulsions between the nuclei. At intermediate distances where the electron clouds overlap there is a net attractive potential since portions of the diffuse electron clouds are close to the nuclei and the resulting attrac- tion more than compensates for the repulsions between different parts of tL diffuse clouds and between the still relatively distant nuclei. The resulting 302 THE STRUCTURE OF MOLECULES [Chap. 1 1 dependence of the coulombic energy on the internuclear distance r is shown as curve C in Fig. 1 1 .4. The depth of the minimum in the coulombic potential energy curve is only about 0.6 ev compared to the observed 4.75 ev for the H-H bond. The classical electrostatic interaction between two hydrogen atoms is thus com- pletely inadequate to explain the strong covalent bond. The solution to the problem must be in the specifically quantum mechanical phenomenon of the exchange energy A. The exchange energy arises from the fact that the electrons are indis- tinguishable, and besides considering the interaction of electron 1 on nucleus a, we have to consider interactions occurring as if electron 1 were on nucleus b. Since quantum mechanics is expressed in the language of y> functions, we even have to consider interactions arising between charge densities that represent electron 1 on both a and b simultaneously. Even to try to express the phenomenon in terms of artificially labeled electrons involves us in difficulties, but it is clear qualitatively that "exchange" may increase the density of electronic charge around the positive nuclei and so increase the binding energy. Like the coulombic energy, the exchange energy is zero when there is no overlap. At a position of large overlap it may lead to a large attrac- tive force and large negative potential energy. The exact demonstration of this fact would require the evaluation of the integral. When this is done we obtain a curve for the variation of A with r. The total energy of interaction 2C 2A can now be plotted. It is clear that 2C I 2 A leads to a deep minimum in the potential energy curve. This is the solution for the symmetric orbital wave function; i.e., the anti- symmetric spin function. It is the case, therefore, in which the electron spins are antiparallel. The spin of one electron is -| i, and that of the other is i. The other curve, 2C-2/4, corresponds to the antisymmetric orbital wave function, which requires symmetric spin functions, or parallel spins. The two curves are drawn as *S and 3 X in Fig. 11.4. The deep minimum in the 1 S curve indicates that the Heitler-London theory has successfully explained the covalent chemical bond in the hydrogen molecule. The binding energy is about 10 per cent coulombic, and 90 per cent exchange energy. o -I Ul _i u -2 -3 -4 /EXPERIMENTAL " CURVE 05 10 15 20 A 25 30 35 Fig. 11.4. Heitler-London treatment of the H 2 molecule. Sec. 5] THE STRUCTURE OF MOLECULES 303 Since the covalent bond is formed between atoms that share a pair of electrons with opposite spins, covalence is often called spin valence. Only when the spins are opposed is there an attractive interaction due to the exchange phenomenon. If the spins are parallel, there is a net repulsion between two approaching hydrogen atoms. It is interesting to note that if two H atoms are brought together, there is only one chance in four that they will attract each other, since the stable state is a singlet and the repulsive state is a triplet. The Heitler-London theory is an example of the valence-bond (V.B.) approach to molecular structure. 5. Molecular orbitals. An alternative to the Heitler-London method of applying quantum mechanics to molecular problems is the method of mole- cular orbitals, developed by Hund, Mulliken, and Lennard-Jones. Instead of starting with definite atoms, it assumes the nuclei in a molecule to be held fixed at their equilibrium separations, and considers the effect of gradually feeding the electrons into the resulting field of force. Just as the electrons in an atom have definite orbitals characterized by quantum numbers, n, /, m, and occupy the lowest levels consistent with the Pauli Principle, so the elec- trons in a molecule have definite molecular orbitals and quantum numbers, and only two electrons having opposite spins can occupy any particular molecular orbital. In our description of the molecular orbital (M.O.) method we shall follow an excellent review by C. A. Coulson. 1 For diatomic molecules, the molecular quantum numbers include a prin- cipal quantum number n, and a quantum number A, which gives the com- ponents of angular momentum in the direction of the internuclear axis. This A takes the place of the atomic quantum number /. We may have states designated <r, TT, <5 . . . as A 0, 1 , 2 . . . . 6. Homonuclear diatomic molecules. Homonuclear diatomic molecules are those that are formed from two identical atoms, like H 2 , N 2 , and O 2 . Such molecules provide the simplest cases for application of the M.O. method. If a hydrogen molecule, H 2 , is pulled apart, it gradually separates into two hydrogen atoms, H a and H 6 , each with a single \s atomic orbital. If the process is reversed and the hydrogen atoms are squeezed together, these atomic orbitals coalesce into the molecular orbital occupied by the electrons in H 2 . We therefore adopt the principle that the molecular orbital can be constructed from a linear combination of atomic orbitals (L.C.A.O.). Thus y y(A : Is) -}- yy(B : Is) Since the molecules are completely symmetrical, y must be 1. Then there are two possible molecular orbitals : v , - y(A : Is) + y(B : Is) ^ M = y(A : Is) ~ y(B : Is) 1 Quarterly Reviews, 1, 144 (1947). 304 THE STRUCTURE OF MOLECULES [Chap. 11 These molecular orbitals are given a pictorial representation in (a), Fig. 11.5. The Is A.O.'s are spherically symmetrical (see page 283). If two of these are brought together until they overlap, the M.O. resulting can be represented as shown. The additive one, vv leads to a building up of charge (o) J V"(A U)t^(B'U) B A^(A 2Py 2Py 2P) F'ig. 11.5. Formation of molecular orbitals by linear combinations of atomic orbitals. density between the nuclei. The subtractive one, y u , has an empty space free of charge between the nuclei. Both these M.O.'s are completely symmetrical about the internuclear axis; the angular momentum about the axis is zero, and they are called a orbitals. The first one is designated as a a\s orbital. It is called a bonding orbital, for the piling up of charge between the nuclei tends to bind them together. The second one is written as a* Is, and is an antibonding orbital, corresponding to a net repulsion, since there is no Sec. 6] THE STRUCTURE OF MOLECULES 305 shielding between the positively charged nuclei. Antibonding orbitals will be designated with a star. A further insight into the nature of these orbitals is obtained if we imagine the H nuclei squeezed so tightly together that they coalesce into the united nucleus of helium. Then the bonding orbital a\s merges into the Is atomic orbital of helium. The antibonding o*\s must merge into the next lowest A.O. in helium, the 2s. This 2s level is 19.7 ev above the Lv, and this energy difference is further evidence of the antibonding nature of the a* Is. The electron configurations of the molecules are built up just as in the atomic case, by feeding electrons one by one into the available orbitals. In accordance with the Pauli Principle, each M.O. can hold two electrons with opposite spins. In the case of H 2 , the two electrons enter the o\s orbital. The configura- tion is (als) 2 and corresponds to a single electron pair bond between the H atoms. The next possible molecule would be one with three electrons, He 2 +. This has the configuration (orl,s) 2 (cr*l,s) 1 . There are two bonding electrons and one antibonding electron, so that a net bonding is to be expected. The molecule has, in fact, been observed spectroscopically and has a dissociation energy of 3.0 ev. If two helium atoms are brought together, the result is (crls) 2 (tf* Is) 2 . Since there are two bonding and two antibonding electrons, there is no ten- dency to form a stable He 2 molecule. We have now used up all of our avail- able M.O.'s and must make some more in order to continue the discussion. The next possible A.O.'s are the 2s, and these behave just like the Is providing a2s and a*2s M.O.'s with accommodations for four more elec- trons. If we bring together two lithium atoms with three electrons each, the molecule Li 2 is formed. Thus Li[]s*2s l ] 4 Li[\s*2s l ] ->Li 2 [(a\s)*(o*\s)*(a2s)*] Actually, only the outer-shell or valence electrons need be considered, and the M.O.'s of inner #-shell electrons need not be explicitly designated. The Li 2 configuration is therefore written as [KK(a2s)' 2 ] . The molecule has a dissociation energy of 1.14ev. The hypothetical molecule Be 2 , with eight electrons, does not occur, since the configuration would have to be [KK(a2s) 2 (a*2s)*\. The next atomic orbitals are the 2/?'s shown in Fig. 10.14. There are three of these, p X9 p v , p Z9 mutually perpendicular and with a characteristic wasp-waisted appearance. The most stable M.O. that can be formed from the atomic p orbitals is one with the maximum overlap along the inter- nuclear axis. This M.O. is shown in (b), Fig. 1 1.5, and with the corresponding antibonding orbital can be written y> = ip(A : 2p x ) + y( B : 2 Px) <*lp : 2p x ) - y(B : 2p x ) o*2p 306 THE STRUCTURE OF MOLECULES [Chap. II These orbitals have the same symmetry around the internuclear axis as the a orbitals formed from atomic s orbitals. They also have a zero angular momentum around the axis. The M.O.'s formed from the p v and p z A.O.'s have a distinctly different form, as shown in (c), Fig. 1 1.5. As the nuclei are brought together, the sides of the p y or p z orbitals coalesce, and finally form two streamers of charge density, one above and one below the internuclear axis. These are called TT orbitals; they have an angular momentum of one unit. We can summarize the available M.O.'s as follows, in order of increasing energy: crhy < cr*l s < o2s < o*2s < o2p < 7T y 2p 7r z 2p < Tr y *2p rr z *2p < o*2p With the good supply of M.O.'s now available, the configurations of other homonuclear molecules can be determined, by feeding pairs of electrons with opposite spins into the orbitals. The formation of N 2 proceeds as follows : There are six net bonding electrons, so that it can be said that there is a triple bond between the two N's. One of these bonds is a a bond; the other two are TT bonds at right angles to each other. Molecular oxygen is an interesting case: O[\s 2 2s*2p*] f- O(\s 2 2s 2 2p 4 ] -> O 2 [KK(a2s) 2 (o*2s) 2 (o2p) 2 (7r2p)*(TT*2p) 2 ] There are four net bonding electrons, or a double bond consisting of a a and a TT bond. Note that a single bond is usually a a bond, but a double bond is not just two equivalent single bonds, but a a plus a TT. In O 2 , the (n*2p) orbital, which can hold 4 electrons, is only half filled. Because of electrostatic repulsion between the electrons, the most stable state will be that in which the electrons occupy separate orbitals and have parallel spins. Thus these two electrons are assigned as (TT y *2p) l (7r.*2p) 1 . The total spin of O 2 is then if = 1, and its multiplicity, 2^ + 1 ~ 3. The ground state of oxygen is 3 2. TABLE 11.1 PROPERTIES OF HOMONUCLEAR DIATOMIC MOLECULES Molecule Binding Energy (ev) Internuclear Separation (A) Vibration Frequency (sec- 1 ) Bonding Antibonding Electrons 3.6 .59 3.15 x 10 13 2 5.5 .31 4.92 x 10 13 4 7.4 .09 7.08 x 10 13 6 5.1 .20 4.74 x 10 13 4 3.0 .30 3.40 x 10 13 2 Sec. 7] THE STRUCTURE OF MOLECULES 307 In the M.O. method, all the electrons outside closed shells make a con- tribution to the binding energy between the atoms. The shared electron pair bond is not particularly emphasized. The way in which the excess of bonding over antibonding orbitals determines the tightness of binding may be seen by reference to the simple diatomic molecules in Table 11.1. 7. Heteronuclear diatomic molecules. If the two nuclei in a diatomic molecule are different, it is still possible to build up molecular orbitals by an L.C.A.O., but now the symmetry of the homonuclear case is lost. Con- sider, for example, the molecule HC1. The bond between the atoms is un- doubtedly caused mainly by electrons in an M.O. formed from the \s A.O. of H and a 3/7 A.O. of Cl. The M.O. can be written as :\s) + yy(Q\ : 3/7) Now y is no longer 1, but there are still a bonding orbital for f-y and an antibonding orbital for y. Actually, the chlorine has a greater tendency than the hydrogen to hold electrons, and thus the resulting M.O. partakes more of the chlorine A.O. than of the hydrogen A.O. The larger y, the more unsymmetrical is the orbital, or the more polar the bond. Thus in the series HI, HBr, HC1, HF, the value of y increases as the halogen becomes more electronegative. 8. Comparison of M.O. and V.B. methods. Since the M.O. and the V.B. methods are the two basic approaches to the quantum theory of molecules, it is worth while to summarize the distinctions between them. The V.B. treatment starts with individual atoms and considers the inter- action between them. Consider two atoms a and b with two electrons (I) and (2). A possible wave function is ^, a(\)b(2). Equally possible is i/> 2 ^ b(\)a(2), since the electrons are indistinguishable. Then the valence bond (Heitler-London) wave function is The M.O. treatment of the molecule starts with the two nuclei. If a(\) is a wave function for electron (I) on nucleus (a), and b(\) is that for electron (I) on nucleus (b\ the wave function for the single electron moving in the field of the two nuclei is y>i = c v a(\) + c 2 b(\) (L.C.A.O.). Similarly for the second electron, y 2 = ^0(2) + c 2 b(2). The combined wave function is the product of these two, or VMO ViVa --= c^a(\)a(2) + c* b(\)b(2) + Cl c 2 [a(\)b(2) f- a(2)b(\)} Comparing the y VB with the ^MO> we see that VMO g ives a Iar g e wei g ht to configurations that place both electrons on one nucleus. In a molecule AB, these are the ionic structures A+Br and A~B + . The ^ vn neglects these ionic terms. Actually, for most molecules, M.O. considerably overestimates the ionic terms, whereas V.B. considerably underestimates them. The true 308 THE STRUCTURE OF MOLECULES [Chap. 1 1 structure is usually some compromise between these two extremes, but the mathematical treatment of such a compromise is much more difficult. 9. Directed valence. In the case of polyatomic molecules, a rigorous M.O. treatment would simply set up the nuclei in their equilibrium positions and pour in the electrons. It is, however, more desirable to preserve the idea of definite chemical bonds, and to do this we utilize bond orbitals, or localized molecular orbitals. For example, in the water molecule, the A.O.'s that take part in bond formation are the \s orbitals of hydrogen, and the 2p x and 2p y of oxygen. The stable structure will be that in which there is maximum overlap of these orbitals. Since p x and p v are at right angles to each other, the situation in Fig. 11.6 is obtained. The observed valence angle in H 2 O is not exactly 90 y Fig. 11.6. Formation of a molecular orbital for H 2 O. but actually 105. The difference can be ascribed in part 2 to the polar nature of the bond; the electrons are drawn toward the oxygen, and the residual positive charge on the hydrogens causes their mutual repulsion. In H 2 S the bond is less polar and the angle is 92. The important point is the straight- forward fashion in which the directed valence is explained in terms of the shapes of the atomic orbitals. The most striking example of directed valence is the tetrahedral orienta- tion of the bonds formed by carbon in aliphatic compounds. To explain these bonds, it is necessary to introduce a new principle, the formation of hybrid orbitals. The ground state of the carbon atom is \s 2 2s 2 2p*. There are two uncoupled electrons 2p x , 2p u , and one would therefore expect the carbon to be bivalent. In order to display a valence of four, the carbon atom must have four electrons with uncoupled spins. The simplest way to attain this condition is to excite or promote o,ne of the 2s electrons into the/? state, and to have all the resulting p electrons with uncoupled spins. Then the outer configuration would be 2s2/? 3 , with 2,?f 2/? J .J2/? 1/ f 2/^j. This excitation requires the investment of about 65 kcal per mole of energy, but the extra binding energy of the four bonds that are formed more than compensates for the promotion energy, and carbon is normally quadrivalent rather than bivalent. If these four 2s2p* orbitals of carbon were coupled with the Is orbitals 2 A more detailed theory shows that the 2s electrons of the oxygen also take part in the bonding, forming hybrid orbitals like those discussed below for carbon. Sec. 9] THE STRUCTURE OF MOLECULES 309 of hydrogen to yield the methane molecule, it might at first be thought that three of the bonds would be different from the remaining one. Actually, of course, the symmetry of the molecule is such that all the bonds must be exactly the same. Pauling 3 showed that in a case like this it is possible to form four identical hybrid orbitals that are a linear combination of the s and p orbitals. These are called tetrahedral orbitals, t l9 / 2 > 'a *4 since they are spatially directed to the corners of a regular tetrahedron. One of them is shown in (a), Fig. 11.7. In terms of the 2s and 2p orbitals it has the form: y^) = \\p(2s) + (V3/2)y(2p x ). The hybrid / orbitals then combine with the Is orbitals of hydrogen to form a set of localized molecular orbitals for methane. (0) Fig. 11.7. Hybrid atomic orbitals for carbon: (a) a single tetrahedral orbital; (b) three trigonal orbitals. The tetrahedral orbitals are exceptionally stable since they allow the electron pairs to avoid one another to the greatest possible extent. In addition to the tetrahedral hybrids, the four sp 3 orbitals of carbon can be hybridized in other ways. The so-called trigonal hybrids mix the 2s, 2p x , and 2p y to form three orbitals at angle of 120. These hybrids are shown in (b), Fig. 11.7. For example, y - Viy<2s) -f V|y<2/7 x ). The fourth A.O., 2/? z , is perpendicular to the plane of the others. This kind of hybridization is that used in the aromatic carbon compounds like benzene, and also in ethylene, which are treated separately in the next section. Hybrid orbitals are not restricted to carbon compounds. An interesting instance of their occurrence is in the compounds of the transition elements. It will be recalled that these elements have a d level that is only slightly lower than the outer s level. Cobalt, for example, has an outer configuration of 3d 7 4s 2 . The cobaltic ion, Co+++, having lost three electrons, has 3*/ 8 . It is noted for its ability to form complexes, such as the hexamminocobaltic ion, H.N^ /NH 3 1 H 3 N Co NH 3 \NH a 8 L. Pauling, Nature of the Chemical Bond (Ithaca, N.Y.: Cornell Univ. Press, 1940), p. 85. 310 THE STRUCTURE OF MOLECULES [Chap. 11 This characteristic can be explained by the fact that there are six low-lying empty orbitals, each of which can hold a pair of electrons: O OOOi Is 2s 2p 3s 3p 3d 4s 4p These cPspP orbitals can be filled by taking twelve electrons from six NH 3 groups, forming the hexamminocobaltic ion with the stable rare gas con- figuration. Once again, hybridization takes place, and six identical orbitals are formed. Pauling's calculation showed that these orbitals should be oriented toward the vertices of an octahedron, and the octahedral arrange- ment is confirmed by the crystal structures of the compounds. 10. Nonlocalized molecular orbitals. It is not always possible to assign the electrons in molecules to molecular orbitals localized between two nuclei. The most interesting examples of delocalization are found in conjugated and aromatic hydrocarbons. Consider, for example, the structure of butadiene, usually written CH 2 =CH~-CH=CH 2 . The molecule is coplanar, and the C C C bond angles are close to 120. The M.O.'s are evidently formed from hybrid carbon A.O.'s of the trigonal type. Three of these trigonal orbitals lie in a plane and are used to form localized bonds with C and H as follows: CH 2 CH CH CH 2 . The fourth orbital is a /?-shaped one, perpendicular (a) (b) Fig. 11.8. Nonlocalized IT orbital in butadiene. to the others. These orbitals line up as shown in (a), Fig. 11.8, for the in- dividual atoms. When the atoms are pushed together, the orbitals overlap to form a continuous sheet above and below the carbon nuclei as in (b). This typical nonlocalized orbital is called a n orbital, and it can hold four electrons. It is important to note that the four TT electrons are not localized in particular bonds, but are free to move anywhere within the region in the figure. Since a larger volume is available for the motion of the electrons, their energy levels are lowered, just as in the case of the particle in a box. Thus delocalization results in an extra binding energy, greater than would be achieved in the classical structure of alternating double and single bonds. In the case of butadiene, this delocalization energy, often called the resonance energy, amounts to about 7 kcal per mole. Sec. 11] THE STRUCTURE OF MOLECULES 311 Benzene and other aromatic molecules provide the most remarkable instances of nonlocalized orbitals. The discussion of benzene proceeds very similarly to that of butadiene. First the carbon A.O.'s are prepared as trigonal hybrids and then brought together with the hydrogens. The localized orbitals formed lie in a plane, as shown in (a), Fig. 1 1.9. The p orbitals extend their sausage-shaped sections above and below the plane, (b), and when they overlap they form two continuous bands, (c), the TT orbitals, above and below the plane of the ring. These orbitals hold six mobile electrons, which are Fig. 11.9. Localized trigonal orbitals (a) and nonlocalized -n orbitals (c) in benzene. completely delocalized. The resulting resonance energy is about 40 kcal per mole. The properties of benzene bear out the existence of these mobile -n elec- trons. All the C C bonds in benzene have the same length, 1.39 A compared to 1.54 in ethane and 1.30 in ethylene. The benzene ring is like a little loop of metal wire containing electrons; if a magnetic field is applied normal to the planes of the rings in solid benzene, the electrons are set in motion, and experimental measurements show that an induced magnetic field is caused that opposes the applied field. 11. Resonance between valence-bond structures. Instead of the M.O. method it is often convenient to imagine that the structure of a molecule is made up by the superposition of various distinct valence-bond structures. Applying this viewpoint to the case of benzene, one would say that the actual structure is formed principally by resonance between the two Kekule" structures, and 312 THE STRUCTURE OF MOLECULES with smaller contributions from the three Dewar structures, [Chap. 11 According to the resonance theory, the eigenfunction ^ describing the actual molecular structure is a linear combination of the functions for possible valence bond structures, This is an application of the general superposition principle for wave func- tions. Each eigenfunction y corresponds to some definite value E for the energy of the system. The problem is to determine the values of a l9 a 2 , # 3 , etc., in such a way as to make E a minimum. The relative magnitude of these coefficients when E is a minimum is then a measure of the contribution to the over-all structure of the different special structures represented by Vi ^2 Y>3 1* must b e clearly understood that the resonance description does not mean that some molecules have one structure and some another. The structure of each molecule can only be described as a sort of weighted average of the resonance structures. Two rules must be obeyed by possible resonating structures: (1) The structures can differ only in the position of electrons. Substances that differ in the arrangement of the atoms are ordinary isomers and are chemically and physically distinguishable as dis^jjact* compounds. (2) The resonating structures must have the same number of paired and unpaired electrons, otherwise they would have different total spins and be physically distinguish- able by their magnetic properties. In substituted benzene compounds, the contributions of various ionic structures must be included. For example, aniline has the following resonance structures : H H H N H H H H covalent ionic The ionic structures give aniline an additional resonance energy of 7 kcal, compared with benzene. The increased negative charge at the ortho and para positions in aniline accounts for the fact that the NH 2 group in aniline directs positively charged approaching substituents (NO a +, Br+) to these positions. The way in which the V.B. method would treat the hydrogen halides is Sec. 12] THE STRUCTURE OF MOLECULES 313 instructive. Two important structures are postulated, one purely covalent and one purely ionic : H+ :C1:~ and H:C1: The actual structure is visualized as a resonance hybrid somewhere between these two extremes. Its wave function is V ~~ ^covalent + The value of a is adjusted until the minimum energy is obtained. Then (a 2 /! + a 2 ) 100 is called the per cent ionic character of the bond. For the various halides the following results are found : Molecule % Ionic Character HF 60 HC1 17 HBr 11 HI 5 The bond in HI is predominantly covalent; in HF, it is largely ionic. The distinction between these different bond types is thus seldom clearcut, and most bonds are of an intermediate nature. The tendency of a pair of atoms to form an ionic bond is measured by the difference in their power to attract an electron, or in their electronegativity. Fluorine is the most, and the alkali metals are the least, electronegative of the elements. The fractional ionic character of a bond then depends upon the difference in electronegativity of its constituent atoms. 12. The hydrogen bond. It has been found that in many instances a hydrogen atom can act as if it formed a bond to two other atoms instead of to only one. A typical example is the dimer of formic acid, which has the structure O H O / \ H C C H \ / O H O This hydrogen bond is not very strong, usually having a dissociation energy of about 5 kcal, but it is extremely important in many structures, such as the proteins. It occurs in general between hydrogen and the electronegative elements N, O, F, of small atomic volumes. We know that hydrogen can form only one covalent bond, since it has only the single Is orbital available for bond formation. Therefore the hydro- gen bond is essentially an ionic bond. Since the proton is extremely small, its electrostatic field is very intense. A typical hydrogen-bonded structure is the ion (HF 2 )~, which occurs in hydrofluoric acid and in crystals such as KHF 2 . It can be represented as a resonance hybrid of three structures, :F: H F- F~ H :F: F- H+ F- 314 THE STRUCTURE OF MOLECULES [Chap. 11 The ionic F H f F~ structure is the most important. It is noteworthy that electroneg^Hve elements with large ionic radii, e.g., Cl, have little or no tendenc^o form hydrogen bonds, presumably owing to their less concen- trates electrostatic fields. 13. Dipole moments. If a bond is formed between two atoms that differ in electronegativity, there is an accumulation of negative charge on the more electronegative atom, leaving a positive charge on the more electro- positive atom. The bond then constitutes an electric dipole, which is by definition an equal positive and negative charge, _q, separated by a distance r. A dipole, as in (a), Fig. 11.10, is characterized by its dipole moment, a vector having the magnitude qr and the direction of the line joining the positive to the negative charge. The dimensions of a dipole moment are charge times length. Two charges with the magnitude of e(4. 80 x 10~ 10 esu) separated by a distance of I A would have a dipole moment of 4.80 x 10~~ 18 csu cm. The unit 10~ 18 esu cm is called the debye, (d). If a polyatomic molecule contains two or more dipoles in different bonds, the net dipole moment of the mole- cule is the resultant of the vector addition of the individual bond moments. An example of this is shown in (b), Fig. 11.10. The measurement of tha dipole moments of molecules provides an insight into their geometric structure and also into the character of their valence bonds. Before we can discuss the determination of dipole moments, however, it is necessary to review some aspects of the theory of dielectrics. 14. Polarization of dielectrics. Consider a parallel-plate capacitor with the region between the plates evacuated, and let the charge on one plate be -for and on the other a per square centimeter. The electric field within the capacitor is then directed perpendicular to the plates and has the magnitude 4 EQ -- 47TCT. The capacitance is q aA A Fig. 11.10. (a) Definition of dipole moment; (b) vector addition of dipole moments in orthodichlorobenzene. where A is the area of the plates, rfthe distance, and (/the potential difference between them. Now consider the space between the plates to be filled with some material substance. In general, this substance falls rather definitely into one of two classes, the conductors or the insulators. Under the influence of small fields, electrons move quite freely through conductors, whereas in insulators or 4 See, for example, G. P. Harnwell, Electricity and Magnetism (New York: McGraw- Hill, 1949), p. 26. Sec. 14] THE STRUCTURE OF MOLECULES 315 dielectrics these fields displace the electrons only slightly from their equi- librium positions. An electric field acting on a dielectric thus causes a separation of positive and negative charges. The field is said to polarize the dielectric. This polarizo- tion is shown pictorially in (a), Fig. 11.11. The polarization can occur in twa ways: the induction effect and the orientation effect. An electric field always induces dipoles in molecules on which it is acting, whether or not they contain dipoles to begin with. If the dielectric does contain molecules that are per- manent dipoles, the field tends to align these dipoles along its own direction. The random thermal motions of the molecules oppose this orienting action. I CM (a) (b) Fig. 11.11. (a) Polarization of a dielectric; (b) definition of the polarization vector, P. Our main interest is in the permanent dipoles, but before these can be studied, effects due to the induced dipoles must be clearly distinguished. It is found experimentally that when a dielectric is introduced between the plates of a capacitor the capacitance is increased by a factor e, called the dielectric constant. Thus if C is the capacitance with a vacuum, the capaci- tance with a dielectric is C eC . Since the charges on the capacitor plates are unchanged, this must mean that the field between the plates is reduced by the factor e, so that E = E Q /e. The reason why the field is reduced is clear from the picture of the polarized dielectric, for all the induced dipoles are aligned so as to produce an over-all dipole moment that cuts down the field strength. Consider in (b), Fig. 11.11, a unit cube of dielectric between the capacitor plates, and define a vector quantity P called the polarization, which is the dipole moment per unit volume. Then the effect of the polarization is equivalent to that which would be produced by a charge of -f P on one face and ~P on the other face (1 cm 2 ) of the cube. The field in the dielectric is now determined by the net charge on the plates, so that J-47r(er-P) (11.5) A new vector has been defined, called the displacement D, which depends only on the charge or, according to D = 47ror. It follows that D-J5+4rrP, and DIE = e (11.6) It is apparent that in a vacuum, where e = 1, D = E. 316 THE STRUCTURE OF MOLECULES [Chap. 11 15. The induced polarization. Let us consider the induced or distortion polarization, P I} , produced by an electric field acting on a dielectric that does not contain permanent dipoles. The first problem to be solved is the magnitude of the dipole moment m induced in a molecule by the field acting on it. It may be assumed that this induced moment is proportional to the intensity of the field 5 F, so that m-aoF (11.7) The proportionality constant OQ is called the distortion polarizability of the molecule. It is the induced moment per unit field strength, and has the dimensions of a volume, since q r/(q/r 2 ) r 3 v. At first it might seem that the field acting on a molecule should be simply the field E of eq. (1 1.5). This would be incorrect, however, for the field that polarizes a molecule is the local field immediately surrounding it, and this is different from the average field E throughout the dielectric. For an isotropic substance this local field can be calculated 6 to be F-B + *?-* (I I J) In the absence of permanent dipoles, the polarization or dipole moment per unit volume is the number of molecules per cc, , times the average moment induced in a molecule, m. Thus, from eqs. (11.7) and (11.8), - /7<x ( E + Since, from eq. (11.6), E(e 1) ^-- 4*rP D , 477/700 3 (H.9) This is the Clausius-Mossotti equation. Multiplying both sides by the ratio of molecular weight to density M/p, 4 B + 2 p 3p 3 The quantity P M is called the molar polarization. So far it includes only the contribution from induced dipoles, and in order to obtain the complete molar polarization, a term due to permanent dipoles must be added. -> 16. Determination of the dipole moment. Having examined the effect of induced dipoles on the dielectric constant, we are in a position to consider 5 This is true only for isotropic substances; otherwise, for example in nonisotropic crystals, the direction of the moment may not coincide with the field direction. This dis- cussion therefore applies only to gases, liquids, and cubic crystals. 6 A good derivation is given by Slater and Frank, Introduction to Theoretical Physics (New York: McGraw-Hill, 1933), p. 278; also, Syrkin and Dyatkina, The Structure of Molecules (New York: Interscience, 1950), p. 471. Sec. 16] THE STRUCTURE OF MOLECULES 317 the influence of permanent dipoles. If the bonds in a molecule are ionic or partially ionic, the molecule has a net dipole moment, unless the individual bond moments add vectorially to zero. It is now possible to distinguish an orientation polarization of a dielectric, which is that caused by permanent dipoles, from the distortion polarization, caused by induced dipoles. There will always be an induced moment. It is evoked almost instanta- neously in the direction of the electric field. It is independent of the tempera- ture, since if the molecule's position is disturbed by thermal collisions, the dipole is at once induced again in the field direction. The contribution to the polarization caused by permanent dipoles, however, is less at higher tem- peratures, since the random thermal collisions of the molecules oppose the tendency of their dipoles to line up in the electric field. It is necessary to calculate the average component of a permanent dipole in the field direction as a function of the temperature. Consider a dipole with random orientation. If there is no field, all orientations are equally probable. This fact can be expressed by saying that the number of dipole moments directed within a solid angle da) is simply Adw, where A is a constant depend- ing on the number of molecules under observation. If a dipole moment // is oriented at angle to a field of strength F its potential energy 7 is U - //Fcos 0. According to the Boltzmann equation, the number of molecules oriented within the solid angle da} is then Ae-' ulkT dco = A The average value of the dipole moment in the direction of the field, by analogy with eq. (7.39), can be written A* cos (>lkT 1 cos Oda> To evaluate this expression, let [iFjkT x, cos = y; then dw -- 2-n- sin 9 dO - 277 dy. Thus * i (e x e~ x ) Since e**dy ~- ---------- _ s m e x -- = coth x -- = L(x) p e x e~ x x x Here L(x) is called the "Langevin function," in honor of the inventor of this treatment. 7 Harnwell, op. cit., p. 64. M 318 THE STRUCTURE OF MOLECULES [Chap. 11 In most cases x = [iF/kTis a very small number 8 so that on expanding L(x) in a power series, only the first term need be retained, leavingL(x) = x/3, or /|2 The total polarizability of a dielectric is found by adding this contribution due to permanent dipoles to the distortion polarizability, and may be written a = a o 4- CM 2 /3*r). Instead of eq. (1 1.10), the total polarization is therefore This equation was first derived by P. Debye. 40 i 30 O NJ 5 20 .j o a. 3 .0 1.0 5.0 2.0 3.0 4.0 1/TXlO 3 Fig. 11.12. Application of the Debye equation to the polarizations of the hydrogen ha 1 ides. When the Clausius-Mossotti treatment is valid, 9 e - 1 M / PU - - - - ^ e + 2 P For gases, e is not much greater than 1, so that E - 1 M 4n = "=- ^ a o (11.13) 8 Values of n range around 10" 18 (esu) (cm). If a capacitor with 1 cm between plates is (3 x 10^\ airiov " 10 " 17 erg com P ared witn kT = 10 ~ 14 er at room temperature. 9 This is the case only for gases or for dilute solutions of dipolar molecules in non-polar solvents. If there is a high concentration of dipolar molecules, as in aqueous solutions, there are localized polarization fields that cannot be treated by the Clausius-Mossotti method. In other words, the permanent dipoles tend to influence the induced polarization. Sec. 17] THE STRUCTURE OF MOLECULES 319 It is now possible to evaluate both OQ and // from the intercept and slope of P M vs. l/T^plots, as shown in Fig. 11.12. The necessary experimental data are values of the dielectric constant over a range of temperatures. They are obtained by measuring the capacitance of a capacitor using the vapor or solution under investigation as the dielectric between the plates. A number of dipole-moment values are collected in Table 11.2. TABLE 11.2 DIPOLE MOMENTS Compound Moment (debyes) Compound HC1 1.03 CH 3 Br HBr 0.78 CH 3 C1 HI 0.38 CH 3 I H 2 1.85 CH 3 OH H 2 S 0.95 C 2 H 5 C1 NH 3 1.49 (C 2 H 5 ) 2 S0 2 1.61 C 6 H 5 OH C0 2 0.0 QH 5 N0 2 CO 0.11 C 6 H 5 .CH 2 C1 Moment (debyes) .45 .85 .35 .68 ; 2.02 .14 .70 4.08 1.85 17. Dipole moments and molecular structure. Two kinds of information about molecular structure are provided by dipole moments: (1) The extent to which a bond is permanently polarized, or its per cent ionic character; and (2) an insight into the geometry of the molecule, especially the angles be- tween its bonds. Only a few examples of the applications will be mentioned. 10 The H Cl distance in HC1 is 1.26 A (found by methods described on page 334). If the structure were H+C1 , the dipole moment would be H - (1.26)(4.80) - 6.05d The actual moment of 1.03 suggests therefore that the ionic character of the bond is equivalent to a separation of charges of about \e. Carbon dioxide has no dipole moment, despite the difference in electro- negativity between carbon and oxygen. It may be concluded that the molecule is linear, O C O; the moments due to the two C O bonds, which are surely present owing to the difference in electronegativity of the atoms, exactly cancel each other on vector addition. On the other hand, water has a moment of 1.85d, and must have a triangular structure (see Fig. 1 1 .6). It has been estimated that each O H bond has a moment of 1.60d and the bond angle is therefore about 105, as shown by a vector diagram. 10 R. J. W. LeFevre, Dipole Moments (London: Methiien, 1948) gives many interesting examples. 320 THE STRUCTURE OF MOLECULES [Chap. 11 A final simple example is found in the substituted benzene derivatives: OH OH - 1.55 1.70 The zero moments of /?-dichloro- and sym-trichlorobenzene indicate that benzene is planar and that the C -Cl bond moments are directed in the plane of the ring, thereby adding to zero. The moment of />di-OH benzene, on the other hand, shows that the O H bonds are not in the plane of the ring, but directed at an angle to it, thus providing a net moment. 18. Polarization and refractivity. It may be recalled that one of the most interesting results of Clerk Maxwell's electromagnetic theory of light 11 was the relationship f /r^ 2 , where n R is the index of refraction. Thus the refractive index is related through eq. (11.10) to the molar polarization. The physical reason for this relationship can be understood without going into the details of the electromagnetic theory. The refractive index of a medium is the ratio of the speed of light in a vacuum to its speed in the medium, n R - c/c m . Light always travels more slowly through a material substance than it does through a vacuum. A light wave is a rapidly alternating electric and magnetic field. This field, as any other, acts to polarize the dielectric through which it passes, pulling the electrons back and forth in rapid alternation. The greater the polarizability of the molecules, the greater is the field induced in opposition to the applied field, and the greater therefore is the "resistance" to the transmission of the light wave. Thus high polariz ability means low c m and high refractive index. We have already seen that increasing the polarization increases the dielectric constant. The detailed theory leads to the Maxwell relation, e -= n n 2 . This relation is experimentally confirmed only under certain conditions: (1) The substance contains no permanent dipoles. (2) The measurement is made with radiation of very long wavelength, in the infrared region. (3) The refractive index is not measured in the neighborhood of a wave- length where the radiation is absorbed. The first restriction arises from the fact that dielectric constants are measured at low frequencies (500 to 5000 kc), whereas refractive indices are measured with radiation of frequency about 10 12 kc. A permanent dipole cannot line up quickly enough to follow an electric field alternating this rapidly. Permanent dipoles therefore contribute to the dielectric constant but not to the refractive index. The second restriction is a result of the effect of high frequencies on the 11 G. P. Harnwell, opt cit., p. 579. Sec. 19] THE STRUCTURE OF MOLECULES 321 induced polarization. With high-frequency radiation (in the visible) only the electrons in molecules can adjust themselves to the rapidly alternating electric fields; the more sluggish nuclei stay practically in their equilibrium positions. With the lower-frequency infrared radiation the nuclei are also displaced. It is customary, therefore, to distinguish, in the absence of permanent dipoles, an electronic polarization P K and an atomic polarization P A . The total polarization, P A \ P K , is obtained from dielectric-constant measure- ments or infrared determinations of the refractive index. The latter are hard to make, but sometimes results with visible light can be successfully extra- polated. The electronic polarization P K can be calculated from refractive index measurements with visible light. Usually P A is only about 10 per cent of P E , and may often be neglected. When the Maxwell relation is satisfied, we obtain from eq. (11.10) the Lorenz-Lorentz equation: n 2 t AY vri'7 = />A/ (1L14) The quantity at the left of eq. (11.14) is often called the molar refraction R M . When the Maxwell relation holds, R M - P M . It will be noted that the molar refraction R M has the dimensions of volume. It can indeed be shown from simple electrostatic theory 12 that a sphere of conducting material of radius r, in an electric field F, has an induced electric moment of m = r 3 / 7 . According to this simple picture, the molar refraction should be equal to the true volume of the molecules contained in one mole. A comparison of some values of molecular volume obtained in this way from refractive index measurements with those obtained from van der Waal's b was shown in Table 7.5. 19. Dipole moments by combining dielectric constant and refractive index measurements. The Lorenz-Lorentz equation also provides an alternative method of separating the orientation and the distortion polarizations, and thereby determining the dipole moment. A solution of the dipolar compound in a nonpolar solvent e.g., nitrobenzene in benzene-- is prepared at various concentrations. The dielectric constant is measured and the apparent molar polarization calculated from eq. (1J.10). This quantity is made up of the distortion polarizations of both solute and solvent plus the orientation polarization of the polar solute. The molar polarizations due to distortion can be set equal to the molar refractions R M , calculated from the refractive indices of the pure liquids. When these R M are subtracted from the total apparent P M , the remainder is the apparent molar orientation polarisation for the solute alone. This polarization is plotted against the concentration in the solution and extrapolated to zero concentration. 13 A value is obtained in 12 Slater and Frank, op. r/7., p. 275. 13 E. A. Guggenheim, Trans. Faraday Soc., 47, 573 (1951), gives an improved method for extrapolation. 322 THE STRUCTURE OF MOLECULES [Chap. 1 i this way from which the effect of dipole interaction has been eliminated. From eq. (11.13), therefore, it is equal to (4n/3)N([i*/3kT) and the dipole moment of the polar solute can be calculated. 20. Magnetism and molecular structure. The theory for the magnetic properties of molecules resembles in many ways that for the electric polariza- tion. Thus a molecule can have a permanent magnetic moment and also a moment induced by a magnetic field. Corresponding to eq. (11.6), we have B H + 4nI (11.15) where B is the magnetic induction, H is the field strength, and / is the in- tensity of magnetization or magnetic moment per unit volume. These quantities are the magnetic counterparts of the electrical D, /?, and P. In a vacuum B H, but otherwise B -= e'H, where t', the permeability, is the magnetic counterpart of the dielectric constant F. Usually, however, mag- netic properties are discussed in terms of ~ X (H.16) where % is called the magnetic susceptibility per unit volume of the medium. (Electric susceptibility would be P/E.) The susceptibility per mole is % M - (M/p)x- The magnetig^fffialogue of eq. (11. 13) is (1U7) where a is the induced moment and JU M is the permanent magnetic dipole moment. Just as before, the two effects can be experimentally separated by temperature-dependence measurements. An important difference from the electrical case now appears, in that > or XM> can b e either positive or negative. If % M is negative, the medium is called diamagnetic; if % M is positive, it is called paramagnetic. For iron, nickel, and certain alloys, % M is positive and much larger than usual, by a factor of about a million. Such substances are called ferromagnetic. From eq. (11.15) it can be seen that the magnetic field in diamagnetic substances is weaker than in a vacuum, whereas in paramagnetic substances it is stronger. An experimental measurement of susceptibility can be made with the magnetic balance. The specimen is suspended so that it is partly inside and partly outside a strong magnetic field. When the magnet is turned on, a paramagnetic substance tends to be drawn into the field region, a dia- magnetic tends to be pushed out of the field. From the weight required to restore the original balance point, the susceptibility is calculated. Sec. 20] THE STRUCTURE OF MOLECULES 323 The phenomenon of diamagnetism is the counterpart of the distortion polarization in the electrical case. The effect is exhibited by all substances and is independent of the temperature. A simple interpretation is obtained if one imagines the electrons to be revolving around the nucleus. If a mag- netic field is applied, the velocity of the moving electrons is changed, pro- ducing a magnetic field that, in accordance with Lenz's Law, is opposed in direction to the applied field. The diamagnetic susceptibility is therefore always negative. When paramagnetism occurs, the diamagnetic effect is usually quite over- shadowed, amounting to only about 10 per cent of the total susceptibility. Paramagnetism is associated with the orbital angular momentum and the spin of uncoupled electrons, i.e., those that are not paired with others having equal but opposite angular momentum and spin. An electron revolving in an orbit about the nucleus is like an electric current in a loop of wire, or a turn in a solenoid. The resultant magnetic moment is a vector normal to the plane of the orbit, and proportional to the angular momentum p of the revolving electron. In the MKS system of units (charge in coulombs) the magnetic moment is (e/2m)p (weber meters). 14 Since p can have only quantized values, m^l-n, where m l is an integer, the allowed values of the magnetic moment are m^eh^nm). It is evident, there- fore, that there is a natural unit of magnetic moment, eh/47rm. It is called the Bohr magneton. The ratio of magnetic moment to angular momentum is called the gyro- magnetic ratio, R . For the orbital motion of an electron, R g e/2m. The spinning electron also acts as a little magnet. For electron spin, however, R g = e/m. Since the intrinsic angular momentum of an electron can have only quantized values %(h/2ir), the magnetic moment of an unpaired electron is eh/fam, or one Bohr magneton. In the case of molecules, only the contributions due to spin are very important. This is true because there is a strong internal field within a mole- cule. In a diatomic molecule, for example, this field is directed along the internuclear axis. This internal field holds the orbital angular momenta of the electrons in a fixed orientation. They cannot line up with an external magnetic field, and thus the contribution they would normally make to the susceptibility is ineffective. It is said to be quenched. There remains only the effect due to the electron spin, which is not affected by the internal field. Thus a measurement of the permanent magnetic moment of a molecule tells us how many unpaired spins there are in its structure. There have been many applications of this useful method, 15 of which only one can be mentioned here. Let us consider two complexes of cobalt, 14 A derivation is given by C. A. Coulson, Electricity (New York: Interscience, 1951), p. 91. In electrostatic units the magnetic moment is (e/2mc)p t where c is the speed of light i/i vacuo. 16 P. W. Selwood, Magnetochemistry (New York: Interscience, 1943). 324 THE STRUCTURE OF MOLECULES [Chap. 11 {Co(NH 3 ) 6 } C1 3 and K 3 {CoF 6 }. Two possible structures may be suggested for such complexes, one covalent and one ionic, as follows: 3d 4s 4p Unpaired Spins Covalent . 11 11 11 11 ft 11 ft ft ft Ionic . . 11 t t t t .. 4 The hexammino complex is obviously covalent, but the structure of the hexafluoro complex is open to question. It is found that the hexammino complex has zero magnetic moment, whereas the {CoF 6 } r ^ complex has a moment of 5.3 magnetons. The structures can thus be assigned as follows: NR, H 3 N Co NFi, NFL C1= and 21. Nuclear paramagnetism. In addition to the magnetism due to the electrons in an atom there is also magnetism due to the nuclei. We may consider a nucleus to be composed of protons and neutrons, and both these nucleons have intrinsic angular momenta or spins, and hence act as ele- mentary magnets. In most nuclei these spins add to give a nonzero resultant nuclear spin. It was first predicted that the magnetic moment of the proton would be 1 nuclear magneton, ehl^-nM, where M is the proton mass. Actually, however, the proton has a magnetic moment of 2.79245 nuclear magnetons, and the neutron moment is -1.9135. The minus sign indicates that the moment behaves like that of a negatively charged particle. Since M is almost 2000 times the electronic mass m, nuclear magnetic moments are less than electronic magnetic moments by a factor of about 1000. The existence of nuclear magnetism was first revealed in the hyperfine structure of spectral lines. As an example consider the hydrogen atom, a proton with one orbital electron. The nucleus can have a spin / i4, and the electron can have a spin S = i. The nuclear and the electron spins can be either parallel or antiparallel to each other, and these two different align- ments will differ slightly in energy, the parallel state being higher. Thus the ground state of the hydrogen atom will in fact be a closely spaced doublet, and this splitting is observed in the atomic spectra of hydrogen, if a spectro- graph of high resolving power is employed. The spacing between the two levels, A -- hv, corresponds to a frequency v of 1420 megacycles. After the prediction of the astrophysicist van der Hulst, an intense emission of radia- tion at this frequency was observed from clouds of interstellar dust. The study of this phenomenon is an important part of the rapidly developing subject of radioastronomy, which is providing much information about hitherto uncharted regions of our universe. Sec. 21] THE STRUCTURE OF MOLECULES 325 If a nucleus with a certain magnetic moment is placed in a magnetic field, we can observe the phenomenon of space quantization (see page 267). The component of the moment in the direction of the field is quantized, and for each allowed direction there will be a slightly different energy level. For readily accessible magnetic fields, the* frequencies v A//i for transitions between two such levels also lie in the microwave range of radio frequencies. TEST TUBE WITH SAMPLE TRANSMITTER COIL Fig. 11.13. Simplified apparatus for basic nuclear magnetic resonance experiment. (Drawing courtesy R. H. Varian.) For example, at a field of 7050 gauss, the frequency for protons is 30 mega- cycles. The earlier attempts to detect these transitions were unsuccessful, but in 1946 E. M. Purcell and Felix Bloch independently developed the method of nuclear magnetic resonance. The principle of this method is shown in Fig. 11.13. The field H of the magnet is variable from to 10,000 gauss. This field produces an equi- distant splitting of the nuclear energy levels which arise as a result of space quantization. The low-power radio-frequency transmitter operates at, for example, 30 megacycles. It causes a small oscillating magnetic field to be applied to the sample. This field induces transitions between the energy 326 THE STRUCTURE OF MOLECULES [Chap. 1 1 levels, by a resonance effect, when the frequency of the oscillating field equals that of the transitions. When such transitions occur in the sample, the resultant oscillation in magnetic field induces a voltage oscillation in the receiver coil, which can be amplified and detected. Figure 11.14 shows an oscillographic trace of these voltage fluctuations over a very small range of magnetic fields (38 milligauss) around 7050 gauss, with ethyl alcohol as the sample. Note that each different kind of proton in the molecule CH 3 -CH 2 -OH appears at a distinct value of H. The reason for this splitting is that the different protons in the molecule have a slightly different magnetic environment, and hence a slightly different CH 2 ETHYL ALCOHOL OH Fig. 11.14. Proton resonance under high resolution at 30 me and 7050 gauss. Total sweep width 38 milligauss. Field decreases linearly from left to right. resonant frequency. The areas under the peaks are in the ratio 3:2: 1, corresponding to the relative number of protons in the different environments. Each peak also has a fine structure. The structural information that can be provided by this method is thus almost unbelievably detailed, and a new and deep insight into the nature of the chemical bond is provided. Applications have been made to problems ranging from isotope analysis to structure determinations. 22. Electron diffraction of gases. One of the most generally useful methods for measuring bond distances and bond angles has been the study of the diffraction of electrons by gases and vapors. The wavelength of 40,000 volt electrons is 0.06 A, about one-tenth the order of magnitude of interatomic distances in molecules, so that diffraction effects are to be expected. The fact that the electron beam and the electrons in the scattering atoms both are negatively charged greatly enhances the diffraction. On page 256 diffraction by a set of slits was discussed in terms of the Huygens construction. In the same way, if a collection of atoms at fixed distances apart (i.e., a molecule) is placed in a beam of radiation, each atom can be regarded as a new source of spherical wavelets. From the interference pattern produced by these wavelets, the spatial arrangement of the scatter- ing centers can be determined. The experimental apparatus for electron Sec. 22] THE STRUCTURE OF MOLECULES 327 diffraction is illustrated in Fig. 11.15. The type of pattern found is a series of rings similar to those in Fig. 10.10 but somewhat more diffuse. The electron beam traverses a collection of many gas molecules, oriented at random to its direction. It is most interesting that maxima and minima HOT FILAMENT- ELECTRON" SOURCE 2 VAPOR SUPPLY | -| ,-T ,'?..-_- <*--PHUIUUKAHMIU PLATE 7^* j | v^J J ~~~-~- ACCELERATING 1] VOLTAGE HTO VACUUM ''PUMPS Fig. 11.15. Schematic diagram of electron diffraction apparatus. are observed in the diffraction pattern despite the random orientation of the molecules. This is because the scattering centers occur as groups of atoms with the same definite fixed arrangement within every molecule. A collection of individual atoms, e.g., argon gas, would give no diffraction rings. Diffrac- tion by gases was treated theoretically (for X rays) by Debye in 1915, but electron-diffraction experiments were not carried out till the work of Wierl in 1930. We can show the essential features of the diffraction theory by considering the simplest case, that of a diatomic molecule. 16 The molecule is represented in Fig. 11.16 with one atom A, at the origin, and the other B, a distance r away. The electron beam enters along Y'A and the diffracted beam, scattered through an angle 0, is picked up at P on a photo- graphic film, a distance R from the origin. The angles a and <f> give the orientation of AB to the primary beam. The interference between the waves scattered from A and B depends on the difference between the lengths of the paths which they traverse. This path difference is 6 - AP CB - BP. The difference in Fig . n 16 Scattering of electrons phase between the two scattered waves is by a diatomic molecule. In order to add waves that differ in phase and amplitude, it is convenient to represent them in the complex plane and to add vectorially. 17 In our case we shall assume for simplicity that the atoms A and B are identical. Then the resultant amplitude at P is A --= A -f /V"' 2 ^. A^ called the atomic scatter- ing factor, depends on the number of electrons in the atom. The intensity of 18 The treatment follows that given by M. H. Pirenne, The Diffraction of X rays and Electrons by Free Molecules (London: Cambridge, 1946), p. 7. 17 See Courant and Robbins, What Is Mathematics ? (New York: Oxford, 1941), p. 94. 328 THE STRUCTURE OF MOLECULES [Chap. 11 radiation is proportional to the square of the amplitude, or in this case to AA, the amplitude times its complex conjugate. Thus A 2 /O I ZirioJ* i ,.27n<5/A\ ^o v^ ~r c -re ; - 2/* 2 ( 1 + cos -^- j - 4^ 2 cos 2 It is now necessary to express 6 in terms of r, /, 0, a, and (/>. This can be done by referring to Fig. 11.16. We see first of all that CB -~ r sin a sin <f>. Then BP VR 2 + r 2 2rR sin a sin (6 + </). Since r is a few Angstroms while R is several cm, r <; /?, so that r 2 is negligible and the square root can be expanded 18 to yield BP R r sin <x sin (0 -f- <). Then we have 6 = AP CB #P r sin a [sin (0 + </>) sin $ - 2r sin 0/2 sin a cos [< + (0/2)]. In order to obtain the required formula for the intensity of scattering of a randomly oriented group of molecules, it is necessary to average the expression for the intensity at one particular orientation (a, <f>) over all possible orientations. The differential element of solid angle is sin a den d<f>, and the total solid angle of the sphere around AB is 4-rr. Hence the required average intensity becomes , 4 ^o 2 r r jo r L^ 2 \] , M lav ~ cos 2 2-rr - sin - sin a cos + - I sin a da. dd> 4n Jo Jo L A 2 \ 07 J On integration, 19 f av - 2 A 2 i\ + A n "' ' ( 1L18 ) 47T . where x = sin - 18 From the binomial theorem, (1 f x) 1 / 2 = 1 + x Jx 2 + . . . . 19 Let /<, = L L cos2 (A cos ft) dp sin a da. IT > > where A = ~. - sin - sin a and p $ -f 0/2 Then since cos 2 p = (1 f cos 2/?)/2, we obtain 7 - = VJo Jo (y + cos ( 2 where / is the Bessel function of order zero (see Woods, Advanced Calculus, p. 282). This can now be integrated by introducing the series expansion of 2 ( " I Sec. 23] THE STRUCTURE OF MOLECULES 329 In Fig. 11.17, }\A is plotted against x, and the maxima and minima in the intensity are clearly evident. In a more complex molecule with atoms j, k (having scattering factors Aj, A k ) a distance r )k apart, the resultant intensity would be (11.19) This is called the "Wierl equation." The summation must be carried out over all pairs of atoms in the molecule. 5 4 C\j O "5 \ ~ 2 I 27T 67T 87T I07T Fig. 11.17. Scattering curve for diatomic molecule plot of eq. (11.18). In the case of the homonuclear diatomic molecule already considered, eq. (1 1.19) becomes sin AT 22 - A A A A A si i/i j - -|- /i i/l 2 A A ^ 4- A A ' ^12 Sm Since r u =-- r 22 =- 0, and (sin x)/x -> 1 as x > 0, and r l2 =- r 21 =^ r, this reduces to eq. (1 1.18). 23. Application of Wierl equation to experimental data. The scattering angles of maximum intensity are calculated from the positions of the dark rings on the picture and the geometry of the apparatus and camera. This gives an experimental scattering curve, whose general form resembles that of the theoretical curve shown in Fig. 11.17, although the positions of the maxima depend, of course, on the molecule being studied. Then a particular molecular structure is assumed and the theoretical scattering curve corre- sponding to it is calculated from eq. (11.19). For example, in the benzene structure there are three different carbon-carbon distances, six between ortho positions, six between meta positions, and three between para positions. Therefore the r }k terms consist of 6r cc , 6(V3 r cc ), and 3(2 r cc ). The positions where in our case x ~ 2B sin a, with B = (2wr/A) sin 6/2. The required integral is given in Pierce's tables (No. 483) as C* - r/i f : The series that results is that for (sin x)/x. (Pierce No. 772.) 330 THE STRUCTURE OF MOLECULES [Chap. 11 of hydrogen atoms are generally ignored because of their low scattering power. It is often sufficiently accurate to substitute the atomic number Z for the atomic scattering factor A. For benzene, the Wierl equation would then become 7(61) 6 sin xr 3 sin 2xr 6 sin A/3 xr Z* = ~~x7~ ~ f ~~^x7~ + ^~Vlxr~~ This function is plotted for various choices of the parameter r, the inter- atomic distance, until the best agreement with the experimental curve is obtained. In other cases bond angles also enter as parameters to be adjusted to obtain the best fit between the observed and calculated curves. It may be noted that only the positions of the maxima and not their heights are used. TABLE 11.3 THE ELECTRON DIFFRACTION OF GAS MOLECULES Molecule NaCl NaBr Nal Bond Distance (A) Diatomic Molecules Molecule 2.51 - 0.03 2.64 i_ 0.01 2.90 0.02 C1 2 Br 2 Bond Distance (A) 2.01 -b 0.03 2.28 0.02 2.65 0.10 Polyatomic Molecules Molecule Configuration Bond Bond Distance 1 CdI 2 Linear Cd I 2.60 0.02 HgCl 2 Linear Hg-Cl 2.34 0.01 BC1 3 Planar B Cl .73 0.02 SiF 4 Tetrahedral Si F .54 0.02 SiCl 4 Tetrahedral | Si Cl 2.00 0.02 P 4 Tetrahedral i P P 2.21 0.02 C1 2 O Bent, 115 JL 4 1 Cl O .68 0.03 S0 2 Bent, 124 -h 15 i s-o .45 0.02 CH 2 Planar C .15 0.05 C0 2 Linear c o .13 0.04 QH 6 _ _ _ i Planar - - - A c c 1.390 0.005 Some results of electron diffraction studies are collected in Table 11.3. As molecules become more complieated, it becomes increasingly difficult to determine an exact structure, since usually only a dozen or so maxima are visible, which obviously will not permit the exact calculation of more than five or six parameters. Each distinct interatomic distance or bond angle Sec. 24] THE STRUCTURE OF MOLECULES 331 constitutes a parameter. It is possible, however, from measurements on simple compounds, to obtain quite reliable values of bond distances and angles, which may be used to estimate the structures of more complex molecules. Some interesting effects of resonance on bond distances have been observed. For example, the C Cl distance in CH 3 C1 is 1.76 A but in CH 2 =CHCi it is only 1.69 A. The shortening of the bond is ascribed to resonance between the following structures: Cl: Cl^ /' .. / H 2 O=C and H 2 C C \ \ H H The C Cl bond in ethylene chloride is said to have about 18 per cent double bond character. 24. Molecular spectra. Perhaps the most widely useful of all methods for investigating molecular architecture is the study of molecular spectra. It affords information about not only the dimensions of molecules but also the possible molecular energy levels. Thus, other methods pertain to the ground state of the molecule alone, but the analysis of spectra also elucidates the nature of excited states. It has been mentioned that the spectra of atoms consist of sharp lines, and those of molecules appear to be made up of bands in which a densely packed line structure is sometimes revealed under high resolving power. Spectra arise from the emission or absorption of definite quanta of radia- tion when transitions occur between certain energy levels. In an atom the energy levels represent different allowed states for the orbital electrons. A molecule too can absorb or emit energy in transitions between different electronic energy levels. Such levels would be associated, for example, with the different 'molecular orbitals discussed on pages 303-311. In addition there are two other possible ways in which a molecule can change its energy level, which do not occur in atoms. These are by changes in the vibrations of the atoms within the molecule and by changes in the rotational energy of the molecule. These energies, like the electronic, are quantized, so that only certain distinct levels of vibrational and rotational energy are permissible. In the theory of molecular spectra it is customary, as a good first approxi- mation, to consider that the energy of a molecule can be expressed simply as the sum of electronic, vibrational, and rotational contributions. Thus, E - Zf elee -} vib + rot (11.20) This complete separation of the energy into three distinct categories is not strictly correct. For example, the atoms in a rapidly rotating molecule are separated by centrifugal forces, which thus affect the character of the 332 THE STRUCTURE OF MOLECULES [Chap. 11 vibrations. Nevertheless, the approximation of eq. (11.20) suffices to explain many of the observed characteristics of molecular spectra. It will be seen in the following discussions that the separations between electronic energy levels are usually much larger than those between vibra- tional energy levels, which in turn are much larger than those between rotational levels. The type of energy-level diagram that results is shown in Fig. 1 1.18. Associated with each electronic level there is a series of vibrattonal J' J' J'- Fig. 11.18. Energy-level diagram for a molecule. Two electronic levels A and B, with their vibrational levels (v) and rotational levels (J) levels, each of which is in turn associated with a series of rotational levels. The close packing of the rotational levels is responsible for the banded structure of molecular spectra. Transitions between different electronic levels give rise to spectra in the visible or ultraviolet region; these are called electronic spectra. Transitions between vibrational levels within the same electronic state are responsible for spectra in the near infrared (< 20/^), called vibration-rotation spectra. Finally, spectra are observed in the far infrared (> 20^) arising from transi- tions between rotational levels belonging to the same vibrational level; these are called pure rotation spectra. Sec. 25] THE STRUCTURE OF MOLECULES 333 25. Rotational levels far-infrared spectra. The model of the rigid rotator, described on page 189, may be used for the interpretation of pure rotation spectra. The calculation of the allowed energy levels for such a system is a straightforward problem in quantum mechanics. The SchrCdinger equation in this case is very similar to that for the motion of the electron about the nucleus in the hydrogen atom, except that for a diatomic molecule it is a question of the rotation of two nuclei about their center of mass. We recall that the rotation of a dumbbell model is equivalent to the rotation of the reduced mass // at a distance r from the rotation axis. For a rigid rotator the potential energy U is zero, so that the wave equation becomes V ^0 (11.21) Without too great difficulty this equation can be solved exactly. 20 It is then found that the eigenfunction y is single valued, continuous, and finite, as is required for physical meaning, only for certain values of the energy E y the allowed eigenvalues. These are WA./ 10 _/(/+!) trai ' fcrV* ~~"M*r ( } Here / is the moment of inertia of the molecule and the rotational quantum number J can have only integral values, 0, 1,2, 3, etc. The value of J gives the allowed values of the rotational angular momen- tum /?, in units of h/2n: p =-- (h/27r)Vj(J + 1) ^ (h/2n) J. This is exactly similar to the way in which the quantum number / in the hydrogen-atom system, and the corresponding A in molecules, determine the orbital angular momenta of electrons. The selection rule for rotational levels is found to be A/ = or 1. Thus an expression for AE for the rigid-rotator model is readily derived from eq. (11.22). Writing B =-- h/Kir'*!, we obtain for two levels with quantum numbers J and J': A ^ hv hB[J(J f 1) -J \J' + I)]. Since v -= (A//0, and7 -J' = 1, v = 2fl/ (11.23) The spacing between energy levels increases linearly with 7, as shown in Fig. 11.18. The absorption spectra due to pure rotation arise from transitions from each of these levels to the next higher one. By means of a spectrograph of good resolving power, the absorption band will be seen to consist of a series of lines spaced an equal distance apart. From eq. (11.23) this spacing is AT --- v v ----- 2B. Pure rotation spectra occur only when the m9lecule possesses a permanent 20 K. S. Pitzer, Quantum Chemistry (New York: Prentice-Hall, 1953), p. 53. An approximate formula is obtained directly from the Bohr hypothesis that the angular momentum is quantized in units of h/2ir. Thus /co = Jh/2*, and the kinetic energy rV. 334 THE STRUCTURE OF MOLECULES [Chap. 1 1 dipole moment. This behavior has been elucidated by quantum mechanical arguments, but it can be understood also in. terms of the classical picture that radiation is produced when a rotating dipole sends out into space a train of electromagnetic waves. If a molecule has no dipole, its rotation cannot produce an alternating electric field. We have discussed only the problem of the diatomic rotator. The rota- tional energy levels of polyatomic molecules are considerably more complex, but do not differ much from the diatomic case in the principles involved. 26. Internuclear distances from rotation spectra. The analysis of rotation spectra can give accurate values of the moments of inertia, and hence inter- nuclear distances and shapes of molecules. Let us consider the example of HC1. Absorption by HC1 has been observed in the far infrared, around A = 50 microns or v --= 200cm" 1 . The spacing between successive lines is A/ 20.1 to 20.7 cm" 1 . Analysis shows that the transition from / = to J 1 corresponds to a wave number of v' I/A = 20.6 cm" 1 . The frequency is therefore v - ~ - (3.00 x 10 10 )(20.6) - 6.20 x 10 11 sec' 1 A The first rotational level, / = 1, lies at an energy of hv - (6.20 x 10 U )(6.62 x 10~ 27 ) - 4.10 x 10~ 15 erg Fromeq. (11.22), = 1^=4.10X10- so that /= 2.72 x 10- 40 gcm 2 Since / --- jur 2 , where // is the reduced mass, we can now determine the inter- nuclear distance r. For HC1, 72 x i()- 40 \ 1/2 (2 l 27. Vibrational energy levels. Investigations in the far infrared are difficult to make, and a much greater amount of useful information has been obtained from the near-infrared spectra, arising from transitions between different vibrational energy levels. The simplest model for a vibrating molecule is that of the harmonic oscillator, whose potential energy is given by U -= J/cjt 2 , the equation of a parabola. The Schrftdinger equation is therefore: = (11.24) Sec. 27] THE STRUCTURE OF MOLECULES 335 The solution to this equation can be obtained exactly by quite simple methods. 21 The result has already been mentioned as a consequence of uncertainty-principle arguments (page 275), being vib = ( + i)** (11.25) The energy levels are equally spaced, and the existence of a zero point energy, EQ ~= \hv$ when v = 0, will be noted. The selection rule for transitions between vibrationai energy levels is found to be Ai? i I . Actually, the harmonic oscillator is not a very good model for molecular vibrations except at low energy levels, near the bottom of the potential- energy curve. It fails, for example, to represent the fact that a molecule may dissociate if the amplitude of vibration becomes sufficiently large. The sort of potential-energy curve that should be used is one like that pictured for the hydrpgen molecule in Fig. 1 1.2 on page 298. Two heats of dissociation may be defined by reference to this curve. The xpectroscopic heat of dissociation, D e , is the height from the asymptote to the minimum. The chemical heat of dissociation, Z) , is measured from the ground state of the molecule, at v = 0, to the onset of dissociation. Therefore, D e = + !Av (11-26) In harmonic vibration the restoring force is directly proportional to the displacement r. The potential-energy curve is parabolic and dissociation can never take place. Actual potential-energy curves, like that in Fig. 1 1.2, corre- spond to anharmonic vibrations. The restoring force is no longer directly proportional to the displacement. The force is given by dU/dr, the slope of the potential curve, and this decreases to zero at large values of r, so that dissociation can occur as the result of vibrations of large amplitude. The energy levels corresponding to an anharmonic potential-energy curve can be expressed as a power series in (v f- i), v ib = hv[(v \ 1) - x e (v + i) 2 + y f (v + I) 3 - . . .] (1 1.27) Considering only the first anharmonic term, with anharmonicitv constant, x e : v ib MM |) -/iwt,(r f i) 2 (11.28) The energy levels are not evenly spaced, but lie more closely together as the quantum number increases. This fact is illustrated in the levels superimposed on the curve in Fig. 1 1 .2. Since a set of closely packed rotational levels is associated with each of these vibrationai levels, it is sometimes possible to determine with great precision the energy level just before the onset of the continuum, and so to calculate the heat of dissociation from the vibration- rotation spectra. As an example of near-infrared spectra, let us consider some observations with hydrogen chloride. There is an intense absorption band at 2886cm" 1 . 21 Pauling and Wilson, he. cit., p. 68. A student might well study this as a typical quantum mechanical problem, since it is about the simplest one available. 336 THE STRUCTURE OF MOLECULES [Chap, n This arises from transitions from the state with v = to that with v = 1, or Ay = +1. In addition, there are very much weaker bands at higher fre- quencies, corresponding to Ai? =-= +2, +3, . . . etc., which are not com- pletely ruled out for an anharmonic oscillator. For the v = 1 band in HCl, we have, therefore, v = (2886) x 3 x 10 10 - 8.65 x 10 13 sec" 1 as the fundamental vibration frequency. This is about one hundred times the rotation frequency found from the far-infrared spectra. The force constant of a harmonic oscillator with this frequency, from eq. (10.2), would be K --=-- 4n*v 2 p = 4.81 x 10 5 dynes per cm. If the chemical bond is thought of as a spring, the force constant is a measure of its tightness. Potential-energy curves of the type shown in Fig. 11.2 are so generally useful in chemical discussions that it is most convenient to have an analytical expression for them. An empirical function that fits very well is that suggested by P. M. Morse: I/- D,(l *-'-'>)* (11.29) Here /? is a constant that can be evaluated in terms of molecular parameters as ft - v V2n*[i/D e . 28. Microwave spectroscopy. Microwaves are those with a wavelength in or around the range from 1 mm to 1 cm. Their applications were rapidly advanced as a result of wartime radar research. In recent years, radar tech- niques have been applied to spectroscopy, greatly extending the accuracy with which we can measure small energy jumps within molecules. In ordinary absorption spectroscopy, the source of radiation is usually a hot filament or high-pressure gaseous-discharge tube, giving in either case a wide distribution of wavelengths. This radiation is passed through the absorber and the intensity of the transmitted portion at diffeient wavelengths is measured after analysis by means of a grating or prism. In microwave spectroscopy, the source is monochromatic, at a well defined single wave- length which can, however, be rapidly varied (fiequency modulation). It is provided by an electronically controlled oscillator employing the recently developed klystron or magnetron tubes. After passage through the cell con- taining the substance under investigation, the microwave beam is picked up by a receiver, often of the crystal type, and after suitable amplification is fed to a cathode-ray oscillograph acting as detector or recorder. The resolving power of this arrangement is 100,000 times that of the best infrared grating spectrometer, so that wavelength measurements can be made to seven significant figures. One of the most thoroughly investigated of microwave spectra has been that of the "umbrella" inversion of the ammonia molecule, the vibration in which the nitrogen atom passes back and forth through the plane of the three hydrogen atoms. The rotational fine structure of this transition has been beautifully resolved, over 40 lines having been catalogued for 14 NH 3 Sec. 29] THE STRUCTURE OF MOLECULES 337 and about 20 for 15 NH 3 . Such measurements provide an almost embarrassing wealth of experimental data, permitting the construction of extremely detailed theories for the molecular energy levels. Pure rotational transitions in heavier molecules are inaccessible to ordi- nary infrared spectroscopy because, in accord with eq. (11.22), the large moments of inertia would correspond to energy levels at excessively long, wavelengths. Microwave techniques have made this region readily accessible. From the moments of inertia so obtained, it is possible to calculate inter- nuclear distances to better than _t0.002 A. A few examples are shown in Table 11.4. TABLE 11.4 INTERNUCIEAR DISTANCES FROM MICROWAVE SPECTRA Molecule Distance (A) Molecule i Distance (A) C1CN C Cl 1.630 I ocs C .161 C N 1.163 C S .560 BrCN C Br 1 .789 N. 2 O N N .126 C N 1.160 N O .191 S0 2 S .433 By observing the spectra under the influence of an electric field (Stark effect) the dipole moments of gas molecules can be accurately determined. Microwave measurements also afford one of the best methods for finding nuclear spins. 29. Electronic band spectra. The energy differences A between electronic states in a molecule are in general much larger than those between successive vibrational levels. Thus the corresponding electronic band spectra are observed in the visible or ultraviolet region. The A's between molecular electronic levels ape usually of the same order of magnitude as those between atomic energy levels, ranging therefore from 1 to 10 ev. In Fig. 11.19 are shown the ground state of a molecule (Curve A), and two distinctly different possibilities for an excited state. In one (Curve B), there is a minimum in the potential energy curve, so that the state is a stable configuration for the molecule. In the other (Curve C), there is no minimum, and the state is unstable for all internuclear separations. A transition from ground state to unstable state would be followed immediately by dissociation of the molecule. Such transitions give rise to a continuous absorption band in the observed spectra. Transitions between different vibrational levels of two stable electronic states also lead to a band in ihe spectra, but in this case the band can be analyzed into closely packed lines corresponding to the different upper and lower vibrational and rota- tional levels. The task of the spectroscopist is to measure the wavelengths of the various lines and interpret them in terms of the energy levels from which 338 THE STRUCTURE OF MOLECULES [Chap. 1 1 they arise. There is obviously a wealth of experimental data here, which should make possible a profound knowledge of the structure of molecules. There is a general rule, known as the Franck-Condon principle, which is helpful in understanding electronic transitions. An electron jump takes place very quickly, much more quickly than the period of vibration of the atomic nuclei (~ 10~ 13 sec), which are heavy and sluggish compared with electrons. It can therefore be assumed that the positions and velocities of the nuclei are virtually unchanged during transitions, 22 which can thus be represented by vertical lines drawn on the potential energy curves, Fig. 11.19. Fig. 11.19. Transitions between electronic levels in molecules. By applying the Franck-Condon principle it is possible to visualize how transitions between stable electronic states may sometimes give rise to dis- sociation. For example, in Curve A of Fig. 11.19, the transition XX' leads to a vibrational level in the upper state that lies above the asymptote to the potential energy curve. Such a transition will lead to dissociation of the molecule. If a molecule dissociates from an excited electronic state, the fragments formed, atoms in the diatomic case, are not always in their ground states. In order to obtain the heat of dissociation into atoms in their ground states, it is therefore necessary to subtract the excitation energy of the atoms. For 22 It may be noted that the vertical line for an electronic transition is drawn from a point on the lower curve corresponding with the midpoint in the internuclear vibration. This is done because according to quantum mechanics the maximum in \p in the ground state lies at the mid-point of the vibration. This is not true in higher vibrational states, for which the maximum probability lies closer to the extremes of the vibration. Classical theory predicts a maximum probability at the extremes of the vibration. Sec. 30] THE STRUCTURE OF MOLECULES 339 example, in the ultraviolet absorption spectrum of oxygen there is a series of bands corresponding to transitions from the ground state to an excited state. These bands converge to the onset of a continuum at 1759 A, equiva- lent to 7.05 ev. The two atoms formed by the dissociation are found to be a normal atom (3P state) and an excited atom (1 D state). The atomic spec- trum of oxygen reveals that this 1 D state lies 1 .97 ev above the ground state. Thus the heat of dissociation of molecular oxygen into two normal atoms (O 2 - 2 O (3P) ) is 7.05 - 1 .97 - 5.08 ev or 1 17 kcal per mole. 30, Color and resonance. The range of wavelengths from the red end of the visible spectrum at 8000 A to the near ultraviolet at 2600 A corresponds with a range of energy jumps from 34 to 1 14 kcal per mole/ A compound with an absorption band in the visible or near ultraviolet must therefore possess at least one electronic energy level from 34 to 114 kcal above the ground level. This is not a large energy jump compared with the energy of binding of electrons in an electron pair bond. It is therefore not surprising that most stable chemical compounds are actually colorless. In fact, the appearance of color indicates that one of the electrons in the structure is loosely held and can readily be raised from the ground molecular orbital to an excited orbital. For example, molecules containing an unpaired electron (odd molecules and free radicals) are usually colored (NO 2 , CIO 2 , triphenylmethyl, etc.). Groups such as NO 2 , C==O, or N N often confer color on a mole- cule since they contain electrons, in 7r-type orbitais, that are readily raised to excited orbitais. In other cases, resonance gives rise to a series of low-lying excited levels. The ground state in the benzene molecule can be assigned an orbital written as y A + y Ry where A and B denote the two Kekule structures shown on page 311. The first excited state is then y> A y B . This state lies 115 kcal above the ground level, and the excitation of an electron into this state is responsible for the near-ultraviolet absorption band of benzene around 2600 A. In a series of similar molecules such as benzene, naphthalene, anthracene, etc., the absorption shifts toward longer wavelength as the molecule becomes longer. The same effect is observed in the conjugated polyenes; butadiene is colorless but by the time the chain contains about twelve carbon atoms, the compounds are deeply colored. This behavior can be explained in terms of the increasing delocalization of the ^-electrons as the length of the mole- cule increases. Let us recall the simple expression for the energy levels of an electron in a box, eq. (10.39), E n = /zV/8m/ 2 , where /is the length of the box. In a transition from n^ to n 2 the energy jump is (/i 2 /8m/ 2 ) (nf w 2 2 ). Thus not only the value of the energy but also the size of the energy jump falls markedly with increasing /. Now the molecular orbitals in organic molecules are of course not simple potential boxes, but the situation is physically very similar. Anything that increases the space in which the 7r-electron is free to 340 THE STRUCTURE OF MOLECULES [Chap. 11 move tends to decrease the energy gap between the ground state and excited states, and shifts the absorption toward the red. Most dyes have structures that consist of two resonating forms. For instance, the phenylene blue ion is / vw v +NH NH 2 In this and similar cases, the transition responsible for the color can be ascribed to an electron jump between a y A + y> B anc * a y> A y B orbital. 31. Raman spectra. If a beam of light is passed through a medium, a certain amount is absorbed, a certain amount transmitted, and a certain N Fig. 11.20. Raman spectrum of O 2 excited by Hg 2537-A line. (From Herzberg, Molecular Spectra and Molecular Structure, Van Nostrand, 1950.) amount scattered. The scattered light can be studied by observations per- pendicular to the direction of the incident beam. Most of the light is scattered without change in wavelength (Rayleigh scattering); but there is in addition a small amount of scattered light whose wavelength has been altered. If the incident light is monochromatic, e.g., the Na D line, the scattered spectrum will exhibit a number of faint lines displaced from the original wavelength. An example is shown in Fig. 11.20. This effect was first observed by C. V. Raman and K. S. Krishnan in 1928. It is found that the Raman displacements, Av, are multiples of vibra- tional and rotational quanta characteristic of the scattering substance. There are therefore rotational and vibration-rotational Raman spectra, which are the counterparts of the ordinary absorption spectra observed in the far and near infrared. Since the Raman spectra are studied with light sources in the visible or ultraviolet, they provide a convenient means of obtaining the same sort of information about molecular structure as is given by the infrared spectra. In many cases, the two methods supplement each other, since vibra- tions and rotations that are not observable in the infrared (e.g., from mole- cules without permanent dipoles) may be active in the Raman. Sec. 32] THE STRUCTURE OF MOLECULES 341 32. Molecular data from spectroscopy. Table 1 1.5 is a collection of data derived from spectroscopic observations on a number of molecules. TABLE 11.5 SPECTROSCOPIC DATA ON THE PROPERTIES OF MOLECULES* Diatomic Molecules Molecule Equilibrium Inter nuclear Separation, r. (A) Heat of Dissociation, Do (ev) Fundamental Vibration Frequency (a>, cm" 1 ) Moment of Inertia, (g cm* x 10-") Cl, 1.989 2.481 564.9 114.8 CO 1.1284 9.144 2168 14.48 H, 0.7414 4 777 4405 0.459 HD 0.7413 4.513 3817 0.611 D, 0.7417 4556 3119 0.918 HBr 1.414 3.60 2650 3.30 HC1" 1.275 4.431 2989 2.71 I. 2.667 1.542 214.4 748 Li, 2.672 1 14 351.3 41.6 N, 1.095 7 384 2360 13.94 NaCl 251 425 380 NH 1.038 34 3300 1.68 0, 1 2076 5.082 1580 19.14 OH 0.971 4.3 3728 1 48 Triatomic Molecules Molecule. X-Y-Z Internuclear Separation (A) Bond Angle Moments of Inertia (g cm* x 10 *) Fundamental Vibration Frequencies (cm^ 1 ) (deg) ffv r v* '.i '71 J c (U l >i W| o~~c o 1.162 1 162 180 71 67 1320 668 2350 H O H 096 096 105 1 024 1 920 2947 3652 1595 3756 D O D 0.96 0.96 105 1.790 3812 5752 2666 1179 2784 H S H 1.35 1.35 92 2667 3076 5.845 2611 1290 2684 S -0 1.40 1.40 120 123 732 85.5 1151 524 1361 N -N- O 1.15 1 23 180 66.9 1285 589 2224 * From G. Herzberg, Molecular Spectra and Molecular Structure, Vols. I and II (New York: D. Van Nostrand Co., 1950). In this chapter we have not discussed the spectra of polyatomic molecules, one of the most active branches of modern spectroscopy. It is possible, how- ever, to evaluate moments of inertia and vibration frequencies for polyatomic molecules by extensions of the methods described for diatomic molecules. Generally the high-frequency vibrations are those that stretch the bonds, and the lower frequencies are bond-bending vibrations. It is often possible to characterize a given type of chemical bond by a bond vibration frequency, which is effectively constant in a large number of different compounds. For example, the stretching frequency of the O-O bond is 1706 in acetone, 1715 in acetaldehyde, 1663 in actetic acid, and 1736 in methyl acetate. The approximate constancy of these bond or group frequencies is the basis for the widespread application of infrared spectroscopy to the structure H ^termination r\f* n<\i/ /"\ranir rrmnnnHc anrl th* HftnilpH cnPftrilTTI nrOV1flP.<5 342 THE STRUCTURE OF MOLECULES [Chap. 11 a method for characterizing a new compound which is as reliable as the finger-printing of a suspect citizen. Some typical bond frequencies are summarized in Table 1 1 .6. TABLE 11.6 BOND-FREQUENCY INTERVALS FOR INFRARED SPECTRA OF GASES* Group H-- < H N< H C=C< I H S N^C c-c Frequency Interval 3500-3700 3300-3500 3300-3400 3000-3100 2550-2650 2200-2300 2170-2270 Group Frequency Interval cm" 1 O=-C< 1700-1850 >c=c< 1550-1650 s-=c< 1500-1600 F C 1100-1300 Cl C 700-800 Br C 500-600 I C 400-500 * After B. Bak, Elementary Introduction to Molecular Spectra (Amsterdam: North Holland Publ. Co., 1954). 33. Bond energies. In discussions of structure, thermodynamics, and chemical kinetics, it is often necessary to have some quantitative information about the strength of a certain chemical bond. The measure of this strength is the energy necessary to break the bond, the so-called bond energy. The energy of a bond between two atoms, A B, depends on the nature of the rest of the molecule in which the bond occurs. There is no such thing as a strictly constant bond energy for A B that persists through a varied series TABLE 11.7 BOND ENERGIES (KCAL/MOLE) Elements Hydrides Chlorides H H 103.2 Li H 58 C Cl 78 Li Li 26 C H 98.2 Cl 49 C C 80 N H 92.2 Si Cl 87 N N 37 O H 109.4 P Cl 77 O O 34 P H 77 I Cl 49.6 Cl Cl 57.1 S H 87(?) I I 35.6 Cl H 102.1 Br H 86.7 Single Double Triple C C 80 145 198 N N 37 225 O O 34 117 C N 66 209 Sec. 33] THE STRUCTURE OF MOLECULES 343 of compounds. Nevertheless, it is possible to strike an average from which actual A B bonds do not deviate too widely. Pauling has reduced a large amount of experimental data to a list of normal covalent single-bond energies. 23 If the actual bond is markedly polarized (partial ionic character), or if through resonance it acquires some double-bond character, its energy may be considerably higher than the norm. Values from a recent compilation 24 are given in Table 1 1 .7. These values are obtained by a combination of various methods: (1) spectroscopy, (2) thermochemistry, and (3) electron impact. The electron impact method employs a mass spectrometer and gradually increases the energy of the electrons from the ion gun until the molecule is broken into fragments. An instance of the application of thermochemical data is the following determination of the O H bond strength : H 2 ~ 2 H AH = 103.4 kcal (spectroscopic) O 2 = 2 O AH 1 18.2 (spectroscopic) H 2 f- * O 2 -= H 2 AH - - 57.8 (calorimetric) 2 H + O = H 2 O AH - 220.3 This is A// for the formation of 2 O H bonds, so that the bond strength is taken as 220.3/2 - 110 kcal. PROBLEMS 1. Write down possible resonance forms contributing to the structures of the following: CO 2 , CH 3 COO-, CH 2 .CH-CH:CH 2 , CH 3 NO 2 , C 6 H 5 C1, C 6 H 5 NH 2 , naphthalene. 2. On the basis of molecular orbital theory, how would you explain the following? The binding energy of N 2 4 is 6.35 and that of N 2 7.38 ev, whereas the binding energy of O 2 + is 6.48 and that of O 2 , 5.08 ev. 3. The following results are found for the dielectric constant e of gaseous sulfur dioxide at 1 atm as a function of temperature: K 267.6 297.2 336.9 443.8 e . 1.009918 1.008120 1.005477 1.003911 Estimate the dipole moment of SO 2 , assuming ideal gas behavior. 4. M. T. Rogers 25 found the following values for the dielectric constant e and density p of isopropyl cyanide at various mole fractions X in benzene solution at 25C: X . . . 0.00301 0.00523 0.00956 0.01301 0.01834 0.02517 e . . . 2.326 2.366 2.442 2.502 2.598 2.718 P . . . 0.87326 0.87301 0.87260 0.87226 0.87121 0.87108 For pure C 3 H 7 NC, p = 0.7 '6572, refractive index n D = 1.3712; for pure 23 For a full discussion: L. Pauling, op. cit., p. 53. 24 K. S. Pitzer, J. Am. Chem. Soc., 70, 2140 (1948). 26 J. Am. Chem. Soc.> 69, 457 (1947). 344 THE STRUCTURE OF MOLECULES [Chap. 11 benzene, p 0.87345, n D -=- 1.5016. Calculate the dipole moment /i of isopropyl cyanide. 5. Chlorobenzene has /t -- 1.55 d, nitrobenzene // = 3.80 d. Estimate the dipole moments of: metadinitrobenzene, orthodichlorobenzene, metachloro- nitrobenzene. The observed moments are 3.90, 2.25, 3.40 d. How would you explain any discrepancies? 6. The angular velocity of rotation o> 27rv rot where v roi is the rotation frequency of a diatomic rotor. The angular momentum is (h/27r)Vj(J | 1). Calculate the rotation frequency of the HC1 molecule for the state with / --=--- 9. Calculate the frequency of the spectral line corresponding to the transition J ^ 9 to / - 8. 7. In the far infrared spectrum of HBr is a series of lines having a separa- tion of 16.94cm *. Calculate the moment of inertia and the internuclear separation in HBr from this datum. 8. In the near infrared spectrum of carbon monoxide there is an intense band at 2144cm" 1 . Calculate (a) the fundamental vibration frequency of CO; (b) the period of the vibration; (c) the force constant; (d) the zero-point energy of CO in cal per mole. 9. Sketch the potential-energy curve for the molecule Li 2 according to the Morse function, given D - 1.14 ev, v ~- 351.35 cm" 1 , r f 2.672 A. 10. The Schumann-Runge bands in the ultraviolet spectrum of oxygen converge to a well defined limit at 1759 A. The products of the dissociation are an oxygen atom in the ground state and an excited atom. There are two low-lying excited states of oxygen, 1 D and 1 S at 1.967 and 4.190 volts above the ground state. By referring to the dissociation data in Table 4.4, page 81, decide which excited state is formed, and then calculate the spectroscopic dissociation energy of O 2 into two O atoms in the ground state. 11. In a diffraction investigation of the structure of CS 2 with 40-kv electrons, Cross and Brockway 26 found four sharp maxima ( f+) each followed by a weak maximum ( 4 ) and a deep minimum ( ), at the following values of 4-77/A (sin 0/2) 4.713 6.312 7.623 8.698 10.63 11.63 12.65 14.58 15.54 16.81 I f- -I -I \ \- -f - -}-+ 4- + + CS 2 is a linear molecule. Calculate the C S distance from these data, using the approximation that the scattering factor is equal to the atomic number Z. 12. With data from Table 11.5, draw to scale the first five rotational levels in the molecule NaCl. At what frequency would the transition J = 4 to 75 be observed? In NaCl vapor at 1000C what would be the relative numbers of molecules in the states with J = 0, J = 1, and J = 2. * J. Chem. Phys., 3, 821 (1935). Chap. 11] THE STRUCTURE OF MOLECULES 345 13. In ions of the first transition series, the paramagnetism is due almost entirely to the unpaired spins, being approximately equal to /* 2v / S(S -f- 1 ) magnetons where S is the total spin. On this basis, estimate // for K 13 , Mn f 2 , Co+ 2 , and Cu+. REFERENCES BOOKS 1. Bates, L. F., Modern Magnetism (London: Cambridge, 1951). 2. Bottcher, C. J. F.., Theory of Electric Polarisation (Amsterdam: Elsevier, 1952). 3. Burk, R. E., and O. Grummitt (editors), Chemical Architecture (New York: Interscience, 1948). 4. Coulson, C. A., Valence (New York: Oxford, 1952). 5. Debye, P., Polar Molecules (New York: Dover, 1945). 6. Gaydon, A. G., Dissociation Energies (London: Chapman and Hall, 1952). 7. Gordy, W., W. V. Smith, and R. F. Trambarulo, Microwave Spectra- scopy (New York: Wiley, 1953). 8. Herzberg, G., Infrared and Raman Spectra (New York: Van Nostrand, 1945). 9. Herzberg, G., Molecular Spectra and Molecular Structure (New York: Van Nostrand, 1950). 10. Ketelaar, J. A. A., Chemical Constitution (Amsterdam: Elsevier, 1953). 11. Palmer, W. G., Valency, Classical and Modern (Cambridge, 1944). 12. Pauling, L., The Nature of the Chemical Bond (Ithaca: Cornell Press, 1940). 13. Pitzer, K. S., Quantum Chemistry (New York: Prentice-Hall, 1953). 14. Rice, F. O., and E. Teller, The Structure of Matter (New York: Wiley, 1949). ARTICLES 1. Condon, E. U., Am. J. Phys., 75, 365-74 (1947), "The Franck-Condon Principle and Related Topics." 2. Klotz, I. M., /. Chem. Ed., 22, 328-36 (1945), "Ultraviolet Absorption Spectroscopy." 3. Mills, W. H., J. Chem. Soc., 1942, 457-66 (1942), "The Basis of Stereo- chemistry." 4. Pake, G. E., Am. J. Phys., 18, 438-73 (1950), "Nuclear Magnetic Resonance." 5. Pauling, L., /. Chem. Soc., 1461-67 (1948), "The Modern Theory of Valency." 6. Purcell, E. M., Science, 118, 431-36 (1953), "Nuclear Magnetic Resonance." 346 THE STRUCTURE OF MOLECULES [Chap. 11 7. Selwood, P. W., /. Chem. Ed., 79, 181-88 (1942), "Magnetism and Molecular Structure." 8. Spurr, R., and L. Pauling, J. Chem. Ed., 18, 458-65 (1941), "Electron Diffraction of Gases." 9. Sugden, S., J. Chem. Soc., 328-33 (1943), "Magnetochemistry." 10. Thompson, H. W., J. Chem. Soc. 9 183-92 (1944), "Infrared Measure- ments in Chemistry." 11. Wilson, E. B., Ann. Rev. Phys. Chem., 2, 151-76 (1951), "Microwave Spectroscopy of Gases". CHAPTER 12 Chemical Statistics 1. The statistical method. If you take a deck of cards, shuffle it well, and draw a single card at random, it is not possible to predict what the card will be, unless you happen to be a magician. Nevertheless, a -number of significant statements can be made about the result of the drawing. For example: the probability of drawing an ace is one in thirteen ; the probability of drawing a spade is one in four; the probability of drawing the ace of spades is one in fifty-two. Similarly, if you were to ask an insurance company whether a certain one of its policyholders was going to be alive 10 years from now, the answer might be: "We cannot predict the individual fate of John Jones, but our actuarial tables indicate that the chances are nine out of ten that he will survive." We are familiar with many statements of this kind and call them "statisti- cal predictions." In many instances it is impossible to foretell the outcome of an individual event, but if a large number of similar events are considered, a statement based on probability laws becomes possible. An example from physics is found in the disintegration of radioactive elements. No one can determine a priori whether an isolated radium atom will disintegrate within the next 10 minutes, the next 10 days, or the next 10 centuries. If a milligram of radium is studied, however, we know that very close to 2.23 x 10 10 atoms will explode in any 10-minute period. Some applications of statistical principles to chemical systems were dis- cussed in Chapter 7. It was pointed out that since the atoms and molecules of which matter is composed are extremely small, any large-scale body con- tains an enormous number of elementary particles. It is impossible to keep track of so many individual particles. Any theory that attempts to interpret the behavior of macroscopic systems in terms of atoms and molecules must therefore rely heavily on statistical considerations. But just because a system does contain so very many particles, its actual behavior will be practically indistinguishable from.that predicted by statistics. If a man tossed 10 coins, the result might deviate widely from 50 per cent heads; if he tossed a thous- and, the percentage deviation would be fairly small; but if some tireless player were to toss 10 23 coins, the result would be to all intents and purposes exactly 50 per cent heads. We have seen already that from the molecular-kinetic point of view the Second Law of Thermodynamics is a statistical law. It expresses the drive toward randomness or disorder in a system containing a large number of particles. Applied to an individual molecule it has no meaning, for in this 347 348 CHEMICAL STATISTICS [Chap. 12 case any distinction between heat (disordered energy) and work (ordered energy) disappears. Even for intermediate cases, such as colloidal particles in Brownian motion, the Second Law is inapplicable, since the particles contain only about 10 6 to 10 9 atoms. Now that the structures and energy levels of atoms and molecules have been considered, in Chapters 8 through 11, it is possible to see how the behavior of macroscopic systems is determined by these atomic and mole- cular parameters. We shall confine our attention to systems in equilibrium, which are usually treated by thermodynamics. This is not, however, a necessary restriction for the statistical method, which is competent to handle also situations in which the system is changing with time. These are some- times called "rate processes," and include transport phenomena, such as diffusion and thermal conductivity, as well as the kinetics of chemical reactions. Statistical thermodynamics is still a very young science, and many funda- mental problems remain to be solved. Thus the only systems that have been treated at all accurately are ideal gases and perfect crystals. Imperfect gases and liquids present unsurmounted difficulties. 2. Probability of a distribution. The discussion of statistical thermo- dynamics upon which we are embarking will not be distinguished for its mathematical precision, nor will any attempt be made to delve into the logical foundations of the subject. 1 The general question to be answered is this: given a macroscopic physical system, composed of molecules (and/or atoms), and knowing from quantum mechanics the allowed energy states for these molecules, how will we dis- tribute the large number of molecules among the allowed energy levels? The problem has already been discussed for certain special cases, the answers being expressed in the form of "distribution laws," for example, the Maxwell distribution law for the kinetic energies of molecules, the Planck distribution law for the energies of harmonic oscillators. We wish now to obtain a more general formulation. The statistical treatment is based on an important principle: the most probable distribution in a system can be taken to be the equilibrium dis- tribution. In a system containing a very large number of particles, deviations from the most probable distribution need not be considered in defining the equilibrium condition. 2 We first require an expression for the probability P of a distribution. Then the expression for the maximum probability is obtained by setting the variation of P equal to zero, subject to certain restraining conditions imposed on the system. 1 For such treatments, see R. H. Fowler and E. A. Guggenheim, Statistical Thermo- dynamics (London: Cambridge, 1939); and R. C. Tolman, Statistical Mechanics (New York: Oxford, 1938). a See J. E. Mayer and M. Mayer, Statistical Mechanics (New York: Wiley, 1940), for a good discussion of this point. Sec. 3] CHEMICAL STATISTICS 349 The method of defining the probability may be illustrated by an example that is possibly familiar to some students, the rolling of dice. The probability of rolling a certain number n will be defined as the number of different ways in which n can be obtained, divided by the total number of combinations that can possibly occur. There are six faces on each of two dice so that the total number of combinations is 6 2 36. There is only one way of rolling a twelve; if the dice are distinguished as a and b y this way can be designated as a(6) b(6). Its probability P(\2) is equal to one in 36. For a seven, there are six possibilities: a(6)-b(\) a(l)-b(6) a(5)-b(2) a(2)~b(5) a(4)-b(3) Therefore, P(7) = H \ =- J. Just as with the dice, the probability of a given distribution of molecules among energy levels could be defined as the number of ways of realizing the particular distribution divided by the total number of possible arrangements. For a given system, this total number is some constant, and it is convenient to omit it from the definition of the probability of the system. The new definition therefore is: the probability of a distribution is equal to the number of ways of realizing the distribution. 3. The Boltzmann distribution. Let us consider a system that has a total energy E and contains n identical particles. Let us assume that the allowed energy levels for the particles (atoms, molecules, etc.) are known from quantum mechanics and are specified as e l9 2 , % " " ' K> ' ' ' etc - How will the total energy E be distributed among the energy levels of the n particles? For the time being, we shall assume that each particle is distinguishable from all the others and that there are no restrictions on how the particles may be assigned to the various energy levels. These assumptions lead to the "classical" or Boltzmann distribution law. It will be seen later that this law is only an approximation to the correct quantum mechanical distribution laws, but the approximation is often completely satisfactory. Now the n distinguishable particles are assigned to the energy levels in such a way that there are n t in level e l9 n 2 in 2 , or in general n K in level e K . The probability of any particular distribution, characterized by a particular set of occupation numbers , is by definition equal to the number of ways of realizing that distribution. Since permuting the particles within a given energy level does not produce a new distribution, the number of ways of realizing a distribution is the total number of permutations !, divided by the number of permutations of the particles within each level, ^ ! n 2 \ . . .n K ! . . . The required probability is therefore (.2..) . n K \ N 350 CHEMICAL STATISTICS [Chap. 12 As an example of this formula, consider four particles a, b, c, d distributed so that two are in e l9 none in 2 an d one ea h i n a an d 4- The possible arrangements are as follows : l % *3 4 ab C d ab d c ac b d ac d b ad . b c ad c b be a d be d a bd a c bd c a cd a b cd b a There are twelve arrangements as given by the formula [0! = 1]: ,2 2!0! 1! 1! 2- 1 1 1 Note that interchanges of the two particles within level s l are not significant. Returning to eq. (12.1), the equilibrium distribution is the one for which this probability is a maximum. The maximum is subject to two conditions, the constancy of the number of particles and the constancy of the total energy. These conditions can be written = n Y F I, n K e K - E 2 ' 2) By taking the logarithm of both sides of eq. (12.1), the continued product is reduced to a summation. In In n\ n K l The condition for a maximum in P is that the variation of P, and hence of In P, be zero. Since In A?! is a constant, Stirling's formula 3 for the factorials of large numbers is In n\ = n In n n (12.3) (12.4) 3 For derivation see D. Widder, Advanced Calculus (New York: Prentice-Hall, 1947), f>. 317. Sec. 3] CHEMICAL STATISTICS 351 Therefore eq. (12.3) becomes ^ 2 n K ^ n n K ~ ^ ^ w^ = or 2 In 77^^ - (12.5) The two restraints in eq. (12.2), since n and E are constants, can be written (5/2^2 (5 i,- = AIT v ji n < 12 - 6 ) oE ^= Z, G K on K These two equations are multiplied by two arbitrary constants, 4 a and /?, and added to eq. (12.5), yielding S a a/i^ + S /?e A , a/ijr + Z\nn K Sn K - (12.7) The variations 6n K may now be considered to be perfectly arbitrary (the restraining conditions having been removed) so that for eq. (12.7) to hold, each term in the summation must vanish. As a result, In n K + a f fte K ~ or n K -=e~*e-*** (12.8) This equation has the same form as the Boltzmann distribution law previously obtained and suggests that the constant ft equals \jkT. It could have been calculated anew. Thus n K ^ e -*e~*K lkT (12.9) It is convenient at this point to make one extension of this distribution law. It is possible that there may be more than one state corresponding with the energy level e K . If this is so, the level is said to be degenerate and should be assigned a statistical weight g K , equal to the number of superimposed levels. The distribution law in this more general form is accordingly e-**t kT (12.10) The constant a is evaluated from the condition Zn K = n whence S e~ *g K e~ ** lkT = n Therefore eq. (12.10) becomes -e K lkT " * This is an application of Lagrange*s method of undetermined multipliers, the stan- dard treatment of constrained maxima problems. See, for example, D. Widder, Advanced Calculus, p. 113 352 CHEMICAL STATISTICS [Chap. 12 This is the Boltzmann distribution law in its most general form. The expression 2 gK e ~'* lkT * n ^ e denominator of eq. (12. 1 1) is very important in statistical mechanics. It is called the partition function, and will be denoted by the symbol -*** (12-12) The average energy e of a particle is given by (see eq. 7.38) = or g = kT* (12.13) oT 4. Internal energy and heat capacity. It is now possible to make use of the distribution law to calculate the various functions of thermodynamics. Thermodynamics deals not with individual particles, but with large-scale systems containing very many particles. The usual thermodynamic measure is the mole, 6.02 x 10 23 molecules. Instead of considering a large number of individual particles, let us consider a large number of systems, each containing a mole of the substance being studied. The average energy of these systems will be the ordinary internal energy E. We again use eq. (12.13), except that now a whole system takes the place of each particle. If the allowed energies of the whole system are E l9 2 , . . . E K , the average energy will be Writing Z-S&t*-**'* 71 (12.14) then, E = kT*--- (12.15) oT We may call Z the molar partition function to distinguish it from the molecular partition function/. It is also called the sum-over-states From eq. (12.15) the heat capacity at constant volume is 5T 5. Entropy and the Third Law. Equation (12.16) can be employed to calculate the entropy in terms of the molar partition function Z. Thus : Sec. 5] CHEMICAL STATISTICS 353 Integrating by parts, we find 5 = : ) +*r / v *>o T /a In Z dT E T (12.17) In this equation, only 5 and \k lnZ| T=0 are temperature-independent terms. The constant term, 5 , the entropy at the absolute zero, is therefore 5 = *lnZ| T . = *In ft (12.18) Here g Q is the statistical weight of the Jowest possible energy state of the system. Equation (12.18) is the statistical-mechanical formulation of the Third Law of Thermodynamics. If we consider, for example, a perfect crystal at the absolute zero, there will usually be one and only one equilibrium arrangement of its constituent atoms, ions, or molecules. In other words, the statistical weight of the lowest energy state is unity: the entropy at 0K becomes zero. This formulation ignores the possible multiplicity of the ground state due to nuclear spin. If the nuclei have different nuclear-spin orientations, there will be a residual entropy at 0K. In chemical problems such effects are of no importance, since in any chemical reaction the nuclear-spin entropy would be the same on both sides of the reaction equation. It is thus conventional to set 5 -= for the crystalline elements and hence for all crystalline solids. Many statistical calculations on this basis have been quantitatively checked by experimental Third-Law values based on heat-capacity data. Examples are given in Table 12.1. TABLE 12.1 COMPARISON OF STATISTICAL (SPECTROSCOPIC) AND THIRD-LAW (HEAT-CAPACITY) ENTROPIES Entropy as Ideal Gas at 1 atm, 298.2K Gas Statistical Third Law N 2 45.78 45.9 2 49.03 49.1 a, 53.31 53.32 H 2 31.23 29.74 HCl 44.64 44.5 HBr 47.48 47.6 HI 49.4 49.5 H 2 O 45.10 44.28 N 2 O 52.58 51.44 NH 8 45.94 45.91 CH 4 44.35 44.30 C 2 H 4 52.47 52.48 $54 CHEMICAL STATISTICS [Chap. 12 In certain cases, however, it appears that even at absolute zero the particles in a crystal may persist in more than one geometrical arrangement. An example is crystalline nitrous oxide. Two adjacent molecules of N 2 O can be oriented either as (ONN NNO) or as (NNO NNO). The energy difference A between these alternative configurations is so slight that their relative probability e * EIRT is practically unity even at low temperatures. By the time the crystal has been cooled to the extremely low temperature at which even a minute A might produce a reorientation, the rate of rotation of the molecules within the crystal has become vanishingly slow. Thus the random orientations are effectively "frozen." As a result, heat-capacity measure- ments will not include a residual entropy S Q equal to the entropy of mixing of the two arrangements. From eq. (3.42) this would amount to 5 - -R S X, In X, - R(\ In J + In i) - R In 2 - 1.38 eu It is found that the entropy calculated from statistics is actually larger by 1.14eu than the Third-Law value, which is within the experimental uncer- tainty of iO.25 eu in S Q . A number of examples of this type have been carefully studied. 5 If the substance at temperatures close to 0K is not crystalline, but a glass, there is also a residual entropy owing to the randomness characteristic of vitreous structures. Another instance of a residual entropy of mixing at 0K arises from the isotopic constitution of the elements. This effect can usually be ignored since in most chemical reactions the isotopic ratios change very slightly. As a result of this discussion, we shall set S Q ~- in eq. (12.18), obtaining S ~+*lnZ (12.19) 6. Free energy and pressure. From the relation A = E TS and eqs. (12.15) and (12.19), the work function becomes A = -kT\nZ (12.20) The pressure, @A/dV) T , is then P = kT*^ (12.2.) The Gibbs free energy is simply F = A -\- PV, and from AF the equi- librium constants for a reaction can be calculated. Expressions have now been obtained that enable us to calculate all thermodynamic properties of interest, once we know how to evaluate the molar partition function Z. 7. Evaluation of molar partition functions. The evaluation of the molar partition function Z has not yet been accomplished for all types of systems, which is of course hardly surprising, for the function Z contains in itself the 5 For the interesting case of ice, see L. Pauling, /. Am. Chem. Soc., 57, 2680 (1935). Sec. 7] CHEMICAL STATISTICS 355 answer to all the equilibrium properties of matter. If we could calculate Z from the properties of individual particles, we could then readily calculate all the energies, entropies, free energies, specific heats, and so forth, that might be desired. In many cases, it is a good approximation to consider that E K , an energy of the system, can be represented simply as the sum of energies E K of non- interacting individual particles. This would be the case, for example, of a crystal composed of independent oscillators, or of an almost perfect gas in which the intermolecular forces were negligible. In such instances we can write EK ^ i(0 + 2 (2) -I- * 3 (3) f . . . e N (N) (12.22) This expression indicates that particle (1) occupies an energy level e l9 particle (2) an energy level F 2 , etc. Each different way of assigning the particles to the energy levels determines <* distinct state of the system E K . The molar partition function, or sum over the states E K , then becomes (The statistical weights g K are omitted for convenience in writing the ex- pressions.) The second summation must be taken over all the different ways of assigning the particles to the energy levels E K . It can be rewritten as e Since each particle has the same set of allowed energy levels, this sum is equal 6 to (2 e - 8 * lkT ) N K Thus we find that Z =--/* The relation Z /' v applies to the case in which rearranging the particles among the energy levels in eq. (12.22) actually gives rise to different states that must be included in the summation for Z. This is the situation in a perfect crystal, the different particles (oscillators) occupying distinct localized positions in the crystal structure. In the case of a gas, on the other hand, each particle is free to move throughout the whole available volume. States in the gas that differ merely * It may be rather hard to see this equality at first. Consider therefore a simple case in which there are only two particles (1) and (2) and two energy levels f t and e 2 . The ways of assigning the particles to the levels are: i = i (0 Ma (2), 2 = *i (2) + 2 (1), E* = i (1) t- FI (2), The sum over states is: Z = e-W* -f e- E *l* T 4- e ~ which is equal to *l kT -f e~* Now it is evident that this is identical with f n = (S*-**/** 1 ) 1 = (e~*il kT -f e- 356 CHEMICAL STATISTICS [Chap. 12 by the interchange of two particles are not distinguishable and should be counted only once. If each level in eq. (12.22) contains only one particle, 7 the number of permutations of the particles among the levels is AH We there- fore divide the expression for Z by this factor, obtaining for the ideal gas case, Z -(!/#!)/*. Thus the relations between /and Z in the two extreme cases are Ideal crystals Z -/* 1 (12.23) Ideal gases Z = N J N Intermediate kinds of systems, such as imperfect gases and liquids, are much more difficult to evaluate. In proceeding to calculate the partition functions for an ideal gas, it is convenient to make use of a simplifying assumption. The energy of a mole- cule will be expressed as the sum of translational, rotational, vibrational, and electronic terms. Thus = ? trans + *rot +" f vlb + *elec (12.24) It follows that the partition function is the product of corresponding terms, /" ftr&nsfiotf \lbfelec (12.25) The simplest case to be considered is that of the monatomic gas, in which there are no rotational or vibrational degrees of freedom; except at very high temperatures the electronic excitation is usually negligible. 8. Monatomic gases translational partition function. In Section 10-20 it was shown that the translational energy levels for a particle in a one- dimensional box are given by The statistical weight of each level is unity, g n = 1. Therefore the molecular partition function becomes ~*!* ml *} The energy levels are so closely packed together that they can be considered to be continuous, and the summation can be replaced by an integration, 7 When the volume is large and the temperature not very low, there will be many more energy levels than there are particles. This will be evident on examination of eq. (10.39) for the levels of a particle in a box. Since there is no housing shortage, there is no reason for the particles to "double-up" and hence the assumption of single occupancy is a good one. For a further discussion, see Tolman, he. cit., pp. 569-572. Sec. 8] CHEMICAL STATISTICS 357 (12.26) For three degrees of translational freedom this expression is cubed, and since / 3 K, we obtain (,2.27) This is the molecular partition function for translation. The molar partition function is (12 28) \ ( } The energy is therefore This is, of course, the simple result to be expected from the equipartition principle. The entropy is evaluated from eq. (12.19), using the Stirling formula, AM = (N/e) N . It follows that Nlf The entropy is therefore (12.29) ATT This is the famous equation that was first obtained by somewhat un- satisfactory arguments by Sackur and Tetrode (1913). As an example, let us apply it to calculate the entropy of argon at 273.2K and at one atmosphere pressure. Then R = 1.98 cal per C 77 = 3.1416 * = 2.718 m-6.63 X 10" 23 g V = 22,414 cc k - 1.38 x 10~ 18 erg per C # = 6.02 x 10 23 7- 273.2 h = 6.62 x 10~ 27 ergsec 358 CHEMICAL STATISTICS [Chap. 12 On substituting these quantities into eq. (12.29), the entropy is found to be 36.2 cal per deg mole. 9. Diatomic molecules rotational partition function. The energy levels for diatomic molecules, according to the rigid-rotator model, were given by eq. (11. 22) as J(J_+ l)/r fr<)t ^ " ~87T 2 / If the moment of inertia / is sufficiently high, these energy levels become so closely spaced as to be practically continuous. This condition is, in fact, realized for all diatomic molecules except H 2 , HD, and D 2 . Thus for F 2 , / - 25.3 x 10~ 40 gcm 2 ; for N 2 , 13.8 x lO^ 40 ; but for H 2 , / - 0.47 x 1Q- 40 . These values are calculated from the interatomic distances and the masses of the molecules, since / = //r 2 . Now the multiplicity of the rotational levels requires some consideration. The number of ways of distributing J quanta of rotational energy between two axes of rotation equals 2J -f 1, for in every case except J there are two possible alternatives for each added quantum. The statistical weight of a rotational level J is therefore 2J + 1 . The rotational partition function now becomes / rot -= E (27 4 \)e /<>+ !>*//"' (12 .30) Replacing the summation by an* integration, since the levels are closely spaced, we obtain One further complication remains. In homonuclear diatomic molecules (N 14 N 14 , C1 35 C1 35 , etc.) only all odd or all even /'s are allowed, depending on the symmetry properties of the molecular eigenfunctions. If the nuclei are different (N 14 N 15 , HC1, NO, etc.) there are no restrictions on the allowed 7's. A symmetry number a is therefore introduced, which is either a = 1 (heteronuclear) or a = 2 (homonuclear). Then ^ot ~ - ah2 (12.32) As an example of the application of this equation, consider the calcula- tion of the entropy of F 2 at 298.2K, assuming translational and rotational contributions only. From eq. (12.29), the translational entropy is found to be 36.88 eu. Then the rotational part is Sec. 10] CHEMICAL STATISTICS 359 Note that the rotational energy is simply RT in accordance with the equi- partition principle. Substituting / 25.3 x lO' 40 , S rot 8.74 eu. Adding the translational term, we have S =-- S mt f 5 tran8 - 8.74 h 36.88 - 45.62 eu This compares with a total entropy of 5 I 298 - 48.48 eu. The vibrational contribution at 25C is therefore small. 10. Polyatomic molecules rotational partition function. The partition function in eq. (12.32) holds also for linear polyatomic molecules, with a - 2 if the molecule has a plane of symmetry (such as O C O), and a - 1 if it has not (such as N-^N-- O). For a nonlinear molecule, the classical rotational partition function has been found to be /r rot In this equation A, /?, C are the three principal moments of inertia of the molecule. The symmetry number a is equal to the number of equivalent ways of orienting the molecule in space. For example: H 2 O, a 2; NH 3 , a-3;CH 4 , o- 12;C 6 H 6 , a == 12. 11. Vibrational partition function. In evaluating a partition function for the vibrational degrees of freedom of a molecule, it is often sufficient to use the energy levels of the harmonic oscillator, which from eq. (11.25) are fvib -~ 0' f i)** (12.34) At low temperatures vibrational contributions are usually small and this approximation is adequate. For reasonably exact calculations at higher tem- peratures the anharmonicity of the vibrations must be considered. Some- times the summation for f can be made by using energy levels obtained directly from molecular spectra. The partition function corresponding to eq. (12.34) would be, for each vibrational degree of freedom, f = J e -( p +W' v i kT -__- e -i' v W' y e -rWkT V V / vil) -f-"" m '(l -<,-*'/)-! (12.35) The total vibrational partition function is the product of terms such as eq. (12.35), one for each of the normal modes of vibration of the molecule, Aib-'TFAvib < 12 - 36 > i For the purposes of tabulation and facility in calculations, the vibrational contributions can be put into more convenient forms. The vibrational energy, from eqs. (12.15), (12.23), and (12.35), is 360 CHEMICAL STATISTICS [Chap. 12 Now Nhv/2 is the zero point energy per mole , whence, writing hvjkT = x, ^-^4 (12-37) (12.38) Then the heat capacity JRx* 2(cosh x 1) From eq. (12.20), since for the vibrational contribution 8 A F, ( I --.; C,, v Finally the contribution to the entropy is * ^0 * ''0 o (12.39) (12.40) T T An excellent tabulation of these functions has been given by J. G. Aston. 9 A much less complete set of values is given in Table 12.2. If the vibration TABLE 12.2 THERMODYNAMIC FUNCTIONS OF A HARMONIC OSCILLATOR hv (E - o) (F-E ) hv (E - ) -(F-Ej * kf v T T X kT V T T 0.10 .985 .891 4.674 1.70 .571 0.7551 0.4008 0.15 .983 .842 3.917 1.80 .528 0.7070 0.3591 0.20 .981 .795 3.394 1.90 .484 0.6640 0.3219 0.25 .977 .749 2.999 2.00 .439 0.6221 0.2889 0.30 .972 .704 2.683 2.20 .348 0.5448 0.2333 0.35 .967 .660 2.424 2.40 .256 0.4758 0.1890 0.40 .961 .616 2.206 2.60 .164 0.4145 0.1534 0.45 .954 .574 2.017 2.80 .074 0.3603 0.1246 0.50 .946 .532 1.853 3.00 0.9860 0.3124 0.1015 0.60 .929 .450 1.581 3.50 0.7815 0.2166 0.0610 0.70 .908 .372 1.364 4.00 0.6042 0.1483 0.0367 0.80 .884 .297 1.186 4.50 0.4571 0.1005 0.0223 0.90 .858 .225 1.037 5.00 0.3393 0.0674 0.0133 .00 .830 .157 0.9120 5.50 0.2477 0.0449 0.0081 .10 .798 .091 0.8044 6.00 0.1782 0.0296 0.0050 .20 .765 .028 0.7128 6.50 0.1266 0.0195 0.0030 .30 .729 0.9678 0.6321 7.00 0.0890 0.0127 0.0018 .40 .692 0.9106 0.5628 8.00 0.0427 0.0053 0.0006 .50 .653 0.8561 0.5016 9.00 0.0199 0.0022 0.0004 .60 .612 0.8043 0.4481 10.00 0.0090 0.0009 0.0001 8 This is evident from eq. (12.21) since /vib is not a function of K, P =-- 0, F = A + py=A. 9 H. S. Taylor and S. Glass tone, Treatise on Physical Chemistry, 3rd ed., vol. 1 , p. 655 (New York: Van Nostrand, 1942). Sec. 12] CHEMICAL STATISTICS 361 frequency is obtainable from spectroscopic observations, these tables can be used to calculate the vibrational contributions to the energy, entropy, free energy, and heat capacity. 12. Equilibrium constant for ideal gas reactions. From the relation AF RTln K v , the equilibrium constant can be calculated in terms of the partition functions. From eqs. (12.20) and (12.23), A = ~Ar7'lirZ = kTln(f y /N\). From the Stirling formula, N! - (N/e) N , and since for an ideal gas,F - A+PY= A f RT, we find that F = - RT In (f/N). Let us write J yint ~ ,3 -J Y where / int denotes the internal partition functions, / rot / vib / elcc , and /' is the partition function per unit volume; i.e., f/V. Then, the free energy ' The standard free energy F is the F at unit pressure of one atmosphere. The volume of a mole of ideal gas under standard conditions of 1 atm pressure is V RT/l. The standard free energy is accordingly 10 F - RT \nfkT For a typical reaction aA + bB ^-- cC \ dD, J AJ Therefore, K, - v fi o i*f b J AJ B Fromeq. (4.12), ** rrr: /C c (/v/ ) If the concentration terms in K c are expressed in units of molecules per cc rather than the more usual moles per cc, we obtain the more concise expression, ft c f d /'A/'B This equation can easily be given a simple physical interpretation. Con- sider a reaction A -> B, then K c ' ^ /B'//A'- The partition function is the sum of the 'probabilities e~ efkT of all the different possible states of the molecules (/= e" elkT ). The equilibrium constant is therefore the ratio of the total probability of the occurrence of the final state to the total probability of the occurrence of the initial state. 13. The heat capacity of gases. The statistical theory that has now been outlined provides a very satisfactory interpretation of the temperature dependence of the heat capacity of gases. The translational energy is effectively nonquantized. It makes a constant contribution C v = $/?, for all types of molecules. 10 Note that k is in units of cc atm/C. 362 CHEMICAL STATISTICS [Chap. 12 Except in the molecules H 2 , HD, and D 2 , the rotational energy quanta are small compared to kT at temperatures greater than about 80K. There is therefore a constant rotational contribution of C v = R for diatomic and linear polyatomic molecules or C r $R for nonlinear polyatomic molecules. For example, with nitrogen at 0C, Af lot =-- 8 x 10~ 16 erg compared to AT - 377 x 10 16 erg. At temperatures below 80K the rotational heat 2.00 LJ O o 1.50 >* _i < o >? o o i.OO i o o 2 O O V 1.0 2.0 3.0 "/hi, Fig. 12.1. Heat capacity contribution of a harmonic oscillator. capacity can be calculated from the partition function in eq. (12.30) and the general formula, eq. (12.16). The magnitude of the quantum of vibrational energy hv is usually quite large compared to kT at room temperatures. For example, the fundamental vibration frequency in N 2 is 2360 cm" 1 , corresponding to f vib of 46.7 x 10~ 14 erg, whereas at 0C kT 3.77 x 10~ 14 . Such values are quite usual and the vibrations therefore make relatively small contributions to low-temperature energies, entropies, and specific heats. The data in Table 7.6 (page 192) confirm this conclusion. In Fig. 12.1, the heat-capacity curve for a typical Sec. 14] CHEMICAL STATISTICS 363 harmonic oscillator is shown as a function of 7/0,,, where O v ---- hvfk is called the characteristic temperature of the vibration. As the temperature is raised, vibrational excitation becomes more and more appreciable. If we know the fundamental vibration frequencies of a molecule, we can determine from Fig. 12.1 or Table 12.2 the corresponding contribution to C r at any temperature. The sum of these contributions is the total vibrational heat capacity. 14. The electronic partition function. The electronic term in the partition function is calculated directly from eq. (12.12) and the observed spectro- scopic data for the energy levels. Often the smallest quantum of electronic energy is so large compared to kT that at moderate temperatures the elec- tronic energy acquired by the gas is negligible. In other cases, the ground state may be a multiplet, but have energy differences so slight that it may be considered simply as a degenerate single level. There are, however, certain intermediate cases in which the multiplet splitting is of the order of kT at moderate temperatures. A notable example is NO, with a doublet splitting of around 120 crn^ 1 or 2.38 x 10~ 14 erg. An electronic contribution to the heat capacity is well marked in NO. Complica- tions arise in these cases, however, owing to an interaction between the rotational angular momentum of the nuclei (quantum number J) and the electronic angular momentum (quantum number A). The detailed analysis is therefore more involved than a simple separation of the internal energy into vibrational, rotational, and electronic contributions would indicate. 11 15. Internal rotation. When certain polyatomic molecules are studied, it is found that the strict separation of the internal degrees of freedom into vibration and rotation is not valid. Let us compare, for example, ethyfene and ethane, CH 2 CH 2 and CH 3 CH 3 . The orientation of the two methylene groups in C 2 H 4 is fixed by the double bond, so that there is a torsional or twisting vibration about the bond but no complete rotation. In ethane, however, there is an internal rotation of the methyl groups about the single bond. Thus one of the vibra- tional degrees of freedom is lost, becoming an internal rotation. This rotation would not be difficult to treat if it were completely free and un- restricted, but such is not the case. There are potential-energy barriers, amounting to about 3000 calories per mole, which must be overcome before rotation occurs. The maxima in energy occur at positions where the hydrogen atoms on the two methyl groups are directly opposite to one another, the minima at positions where the hydrogens are "staggered." The theoretical treatment of the problems of restricted internal rotation is still incomplete, but good progress is being made. 12 16. The hydrogen molecules. Since the moment of inertia of the hydrogen molecule, H 2 , is only 0.47 x 10~ 40 gcm 2 , the quantum of rotational energy 11 Fowler and Guggenheim, op. cif. t p. 102. 12 J. G. Aston, loc. cit., p. 590. 364 CHEMICAL STATISTICS [Chap. 12 is too large for a classical treatment. To evaluate the partition function, the complete summation must her carried out. When this was first done, using eq. (12.30), modified with a symmetry number a = 2, the calculated specific heats were in poor agreement with the experimental values. It was later realized that the discrepancy must be a result of the existence of the two nuclear-spin isomers for H 2 . The proton (nucleus of the H atom) has a nuclear spin / -= % in units of /J/27T. The spins of the two protons in the H 2 molecule may either parallel or oppose each other. These two spin orientations give rise to the two spin isomers: ortho H 2 spins parallel resultant spin -= 1 para H 2 spins antiparallel resultant spin Spontaneous transitions between the ortho and para states are strictly prohibited. The ortho states are associated with only odd rotational levels (J I, 3, 5 . . .), and para states have only even rotational levels (J = 0, 2, 4 . . .). The nuclear-spin weights are g NS - 3 for ortho, corresponding to allowed directions 4-1,0, -1, andg NS 1 for para, whose resultant spin 13 is 0. At quite high temperatures (~ 0C), therefore, an equilibrium mixture of hydrogen consists of three parts ortho and one part para. At quite low temperatures (around 80K, liquid-air temperature) the equilibrium con- dition is almost pure para hydrogen, with the molecules in the lowest rota- tional state, J = 0. The equilibrium is attained very slowly in the absence of a suitable catalyst, such as oxygen adsorbed on charcoal, or other paramagnetic sub- stance. It is thus possible to prepare almost pure/?-H 2 by adsorbing hydrogen on oxygenated charcoal at liquid-air temperatures, and then warming the gas in the absence of catalyst. The calculated heat capacities of pure />-H 2 , pure o-H 2 and of the 1:3 normal H 2 , are plotted in Fig. 12.2. Mixtures of o- and/?-H 2 are conveniently analyzed by measuring their thermal conductivities, since these are pro- portional to their heat capacities. A similar situation arises with deuterium, D 2 . The nuclear spin of the D atom is 1. The possible resultant values for D 2 are therefore 0, 1, and 2. Of these, / = and 2 belong to the ortho modification and / ~ 1 is the para. The weights (2/ } 1) are 1 + 5 = 6, and 3, respectively. The high-tempera- ture equilibrium mixture therefore contains two parts ortho to one part para. In the molecule HD, which is not homonuclear, there are no restrictions on the allowed rotational energy levels. The partition function of eq. (12.30) is directly applicable. Other diatomic molecules composed of like nuclei with nonzero nuclear spins may also be expected to exist in both para and ortho modifications. 13 Compare the spatial quantization of the orbital angular momentum of an electron, page 268. Sec. 17] CHEMICAL STATISTICS 365 Any thermodynamic evidence for such isomers would be confined to ex- tremely low temperatures, because their rotational energy quanta are small. The energy levels are so close together that in calculating heat capacities it is unimportant whether all odds or all evens are taken. It is necessary only < 3.00 Fig. 12.2. 100 200 300 DEGREES KELVIN Heat capacities of pure para-hydrogen, pure ortho-hydrogen, and 3-o to \~p normal hydrogen. to divide the total number of levels by a =- 2. Spectroscopic observations, however, will often reveal an alternating intensity in rotational lines caused by the different nuclear-spin statistical weights. 17. Quantum statistics. In deriving the Boltzmann statistics, we assumed that the individual particles were distinguishable and that any number of particles could be assigned to one energy level. We know from quantum mechanics that the first of these assumptions is invalid. The second assump- tion is also incorrect if one is dealing with elementary particles or particles composed of an odd number of elementary particles. In such cases, the Pauli Exclusion Principle requires that no more than one particle can go into each energy level. If the particles considered are composed of an even number of elementary particles, any number can be accommodated in a single energy level. Two different quantum statistics therefore arise, which are characterized as follows: Name (1) Fermi-Dirac (2) Bose-Einstein Obeyed by Odd number of elementary particles (e.g., electrons, protons) Even number of elementary particles (e.g., deuterons, photons) Restrictions on n K Only one particle per state, n K < g K Any number of particles per state It is interesting to note that photons follow the Bose-Einstein statistics, indicating that they are complex particles and recalling the formation of electron-positron pairs from X-ray photons. 366 CHEMICAL STATISTICS [Chap. 12 A schematic illustration of the two types of distribution would be O00O O O O F.D. B.E. Distribution laws are calculated for these two cases by exactly the same sort of procedure as was used for the Boltzmann statistics. 14 The results are found to be very similar, < 12 - 42 > F.D. case + B.E. case Now in almost every case the exponential term is very large compared to unity, and the Boltzmann statistics are a perfectly good approximation for almost all practical systems. This can be seen by using the value of e* =f/n from eq. (12.10). The condition for the Boltzmann approximation is then e/kT f ---- ^ 1, or n Using the translational partition function /in eq. (12.27), we have e tl1fT (27rmkTj^V ^ ->' (12 - 43) This condition is obviously realized for a gas at room temperature. It is interesting to note, however, the circumstances under which it would fail. If n/V, proportional to the density, became very high, the classical statistics would eventually become inapplicable. This is the situation in the interior of the stars, and forms the basis of R. H. Fowler's brilliant contribution to astrophysics. A more mundane case also arises, namely in the electron gas in metals. We shall consider this in the next chapter, with only a brief mention here. A metallic crystal, to a first approximation, may be considered as a regular array of positive ions, permeated by a gas of Mobile electrons. In this case the density term in eq. (12.43) is exceptionally high and in addition the mass term m is lower by about 2 x 10 3 than in any molecular case. Thus the electron gas will not obey Boltzmann statistics; it must indeed follow the Fermi-Dirac statistics since electrons obey the Pauli Principle. PROBLEMS 1. In the far infrared spectrum of HC1, there is a series of lines with a spacing of 20.7 cm" 1 . In the near infrared spectrum, there is an intense band at 3.46 microns. Use these data to calculate the entropy of HC1 as an ideal gas at 1 atm and 298 K. 14 For these calculations, see, for example, Tolman, op. cit,, p. 388. Chap. 12] CHEMICAL STATISTICS 367 2. Estimate the equilibrium constant of the reaction C1 2 --- 2 Cl at 1000K. The fundamental vibration frequency of C1 2 is 565 cm" 1 and the equilibrium C1-C1 distance is 1.99 A. Compare with the experimental value in Table 4.5. 3. The isotopic composition of zinc is: 64 Zn 50.9 per cent; 68 Zn 27.3 per cent; 67 Zn 3.9 per cent; 68 Zn 17.4 per cent; 70 Zn 0.5 per cent. Calculate the entropy of mixing per mole of zinc at 0K. 4. Thallium forms a monatomic vapor. The normal electronic state of the atom is 2 P 1/2 but there is a 2 P^/ 2 state lying only 0.96 ev. above the ground state. The statistical weights of the state.s are 2 and 4, respectively. Plot a curve showing the variation with temperature of the contribution to the specific heat of the vapor caused by the electronic excitation. 5. In a star whose temperature is 10 6 K, calculate the density of material at which the classical statistics would begin to fail. 6. Calculate the equilibrium constant of the reaction H 2 f D 2 2 HD at 300K given: H 2 HD D 2 4371 3786 3092 0.5038 0.6715 1.0065 0.458 0.613 0.919 ) e , cm" 1 ..... Reduced mass, /i, at. wt. units Moment of inertia, /, g cm 2 x 10 40 7. In Problem 4.10, heat-capacity data were listed for a calculation of the Third-Law entropy of nitromethane. From the following molecular data, calculate the statistical entropy S 298 . Bond distances (A): N O 1.21; CN, 1.46; C H, 1.09. Bond angles: O N O 127; H C N 109J. From these distances, calculate the principal moments of inertia, / = 67.2, 76.0, 137.9 x 10~ 40 gcm 2 . The fundamental vibration frequencies 15 in cm" 1 are: 476, 599, 647, 921, 1097, 1153, 1384, 1413, 1449, 1488, 1582, 2905, 3048 (2). One of the torsional vibrations has become a free rotation around the CN bond with / = 4.86 x 10 40 . 8. Calculate the equilibrium constant K p at 25C for O 2 1H + O 2 16 - 2 O 16 O 1H . The nuclear spins of O 18 and O 16 are both zero. The vibration fre- quencies are given by v = (l/27r)(/c/ 1/2 , where K is the same for all three molecules. For O 2 10 , v 4.741 x 10 13 sec" 1 . The equilibrium internuclear distance, 1.2074 A, does not depend on the isotopic species. 9. The ionization potential of Na is 5.14 ev. Calculate the degree of dis- sociation, Na -= Na+ + e, at 10 4 K and 1 atm. 15 A. J. Wells and E. B. Wilson, /. Chem. Phys., 9, 314 (1941). 368 CHEMICAL STATISTICS [Chap. 12 REFERENCES BOOKS 1. Born, M., Natural Philosophy of Cause and Chance (New York: Oxford, 1949). 2. Dole, M., Introduction to Statistical Thermodynamics (New York: Prentice-Hall, 1954). 3. Gurney, R. W., Introduction to Statistical Mechanics (New York: McGraw-Hill, 1949). 4. Khinchin, A. I., Statistical Mechanics (New York: Dover, 1949). 5. Lindsay, R. B., Physical Statistics (New York: Wiley, 1941). 6. Rushbrooke, G. S., Introduction to Statistical Mechanics (New York: Oxford, 1949). 7. Schrddinger, E., Statistical Thermodynamics (Cambridge, 1946). 8. Ter Haar, D., Elements of Statistical Mechanics (New York: Rinehart, 1954). ARTICLES 1. Bacon, R. H., Am. J. Phys., 14, 84-98 (1946), "Practical Statistics for Practical Physicists/' 2. Eyring, H., and J. Walter, /. Chem. Ed., 18, 73-78 (1941), "Elementary Formulation of Statistical Mechanics." CHAPTER 13 Crystals 1. The growth and form of crystals. The symmetry of crystalline forms, striking a responsive chord in our aesthetic nature, has fascinated many men, from the lapidary polishing gems for a royal crown to the natural philosopher studying the structure of matter. Someone once said that the beauty of crystals lies in the planeness of their faces. It was also the measure- ment and explanation of these plane faces that first demanded scientific attention. In 1669, Niels Stensen (Steno), Professor of Anatomy at Copenhagen and Vicar Apostolic of the North, compared the interfacial angles in various specimens of quartz rock crystals. An interfacial angle may be defined as the angle between lines drawn perpendicular to two faces. Steno found that the corresponding angles (in different crystals) were always equal. After the invention of the contact goniometer in 1780, this conclusion was checked and extended to other substances, and the constancy of interfacial angles has been called the "first law of crystallography." It was a most important principle, for out of a great number of crystalline properties it isolated one that was constant and unchanging. Different crystals of the same substance may differ greatly in appearance, since corresponding faces may have developed to diverse extents as the crystals were growing. The interfacial angles, nevertheless, remain the same. We can consider that a crystal grows from solution or melt by the de- position onto its faces of molecules or ions from the liquid. If molecules are deposited preferentially on a certain face, this face will not extend rapidly in area, compared with faces at angles to it on which deposition is less frequent. The faces with the largest area are therefore those on which added molecules are deposited most slowly. Sometimes an altered rate of deposition can completely change the form, or habit, of a crystal. A well known case is sodium chloride, which grows from pure water solution as cubes, but from 15 per cent aqueous urea solution as octahedra. It is believed that urea is preferentially adsorbed on the octahedral faces, preventing deposition of sodium and chloride ions, and therefore causing these faces to develop rapidly in area. The real foundations of crystallography may be said to date from the work of the Abbe Rene Just Haiiy, Professor of the Humanities at the University of Paris. In 1784, he proposed that the regular external form of crystals was a reflection of an inner regularity in the arrangement of their constituent building units. These units were believed to be little cubes or 369 370 CRYSTALS [Chap. 13 polyhedra, which he called the molecules integrates of the substance This picture also helped to explain the cleavage of crystals along uniform planes. The Haiiy model was essentially confirmed, 128 years later, by the work of Max von Laue with X-ray diffraction, the only difference being in a more advanced knowledge of the elementary building blocks. 2. The crystal systems. The faces of crystals, and also planes within crystals, can be characterized by means of a set of three noncoplanar axes. Consider in Fig. 13.1 three axes having lengths a, b, and c, which are cut by the plane ABC, making intercepts OA, OB, and OC. If a, b, c, are chosen as unit lengths, the lengths of the intercepts may be expressed as OAja, OBjh, OC/c. The reciprocals of these Fig. 13.1. Crystal axes. lengths will then be a/OA, b/OB, c/OC. Now it has been established that it is always possible to find a set of axes on which the reciprocal intercepts of crystal faces are small whole numbers. Thus, if //, k, /are small integers: OA oc This is equivalent to the law of rational intercepts, first enunciated by Haiiy. The use of the reciprocal intercepts (hkl) as indices defining the crystal faces was first proposed by W. H. Miller in 1839. If a face is parallel to an axis, (001) (III) (211) Fig. 13.2. Miller indices. the intercept is at oo, and the Miller index becomes l/oo or 0. The notation is also applicable to planes drawn within the crystal. As an illustration of the Miller indices, some of the planes in a cubic crystal are shown in Fig. 13.2. Sec. 3] CRYSTALS 371 According to the set of axes used to represent their faces, crystals may be divided into seven systems. These are summarized in Table 13.1. They range from the completely general set of three unequal axes (a, b, c) at three unequal angles (a, /?, y) of the triclinic system, to the highly symmetrical set of three equal axes at right angles of the cubic system. TABLE 13.1 THE SEVEN CRYSTAL SYSTEMS System Axes Angles Example Cubic a - b = c a = ft = y = 90 Rock salt Tetragonal Orthorhombic Monoclinic a -- b\ c a\b\ c a\ b\ c OL-ft=y-- 90 a _ ft =- y =, 90 White tin Rhombic sulfur Monoclinic sulfur Rhombohedral Hexagonal Triclinic a b c a = b\ c a', b\ c a - ft y I 90 a -_= ft = 90 ;y 120 a y= ^ ^ y ^ 90 Calcite Graphite Potassium dichromate 3. Lattices and crystal structures. Instead of considering, as Haiiy did, that a crystal is made of elementary material units, it is helpful to introduce a geometrical idealization, consisting only of a regular array of points in space, called a lattice. An example in two dimensions is shown in Fig. 13.3. o 1 Fig. 13.3. Two-dimensional lattice with unit cells. The lattice points can be connected by a regular network of lines in various ways. Thus the lattice is broken up into a number of unit cells. Some examples are shown in the figure. Each cell requires two vectors, a and b, for its description. A three-dimensional space lattice can be similarly divided into unit cells that require three vectors for their description. If each point in a space lattice is replaced by an identical atom or group of atoms there is obtained a crystal structure. The lattice is an array of points; in the crystal structure each point is replaced by a material unit. In 1848, A. Bravais showed that all possible space lattices could be 372 CRYSTALS [Chap. 13 assigned to one of only 14 classes. 1 The 14 Bravais lattices are shown in Fig. 13.4. They give the allowed different translational relations between points in an infinitely extended regular three-dimensional array. The choice of the 14 lattices is somewhat arbitrary, since in certain cases alternative descriptions are possible. TRICLINIC A 2 SIMPLE 3 SIDE-CENTERED MONOCLINIC MONOCLINIC \ 4. SIMPLE 5 END-CENTERED ORTHORHOMBIC ORTHORHOMBIC 6. FACE-CENTERED 7 BODY- ORTHORHOMBIC CENTERED ORTHORHOMBIC 9 RHOMBOHEDRAL 10 SIMPLE II BODY-CENTERED 8. HEXAGONAL TETRAGONAL TETRAGONAL Fig. 13.4. The fourteen Bravais lattices. 4. Symmetry properties. The word "symmetry" has been used in referring to the arrangement of crystal faces. It is now desirable to consider the nature of this symmetry in more detail. If an actual crystal of a substance is studied, some of the faces may be so poorly developed that it is difficult or impossible to see its full symmetry just by looking at it. It is necessary therefore to 1 A lattice that contains body-, face-, or end-centered points can always be reduced to one that does not (primitive lattice). Thus the face-centered cubic can be reduced to a primitive rhombohedral. The centered lattices are chosen when possible because of their higher symmetry. Sec. 4] CRYSTALS 373 consider an ideal crystal in which all the faces of the same kind are developed to the same extent. It is not only in face development that the symmetry of the crystal is evident but also in all of its physical properties, e.g., electric and thermal conductivity, piezoelectric effect, and refractive index. Symmetry is described in terms of certain symmetry operations, which are those that transform the crystal into an image of itself. The symmetry operations are imagined to be the result of certain symmetry elements: axes of rotation, mirror planes, and centers of inversion. The possible symmetry elements of finite figures, i.e., actual crystals, are shown in Fig. 13.5 with schematic illustrations. (a) T (b) MM/ v- i' Fig. 13.5. Examples of symmetry elements: (a) mirror plane m; (b) rotation axes; (c) symmetry center 1 ; (d) twofold rotary inversion axis 2. The possible combinations of these symmetry elements that can occur in crystals have been shown to number exactly 32. These define the 32 crystallo- graphic point groups* which determine the 32 crystal classes. The symbols devised by Hermann and Mauguin are used to represent the symmetry elements. An axis of symmetry is denoted by a number equal to its multiplicity. The combination of a rotation about an axis with reflection through a center of symmetry is called an "axis of rotary inversion"; it is denoted by placing a bar above the symbol for the axis, e.g., 2, 3. The center of symmetry alone is then T. A mirror plane is given the symbol m. All crystals necessarily fall into one of the seven systems* but there are several classes in each system. Only one of these, called the holohedral class, possesses the complete symmetry of the system. For example, consider two crystals belonging to the cubic system, rock salt (NaCl) and iron pyrites (FeS 2 ). Crystalline rock salt, Fig. 13.6, possesses the full symmetry of the cube: three 4-fold axes, four 3-fold axes, six 2-fold axes, three mirror planes perpendicular to the 4-fold axes, six mirror planes perpendicular to the 2-fold axes, and a center of inversion. The cubic crystals of pyrites might at first seem to possess all these symmetry elements too. Closer examination reveals, 2 A set of symmetry operations forms a. group when the consecutive application of any two operations in the set is equivalent to an operation belonging to the set (law of multi- plication). It is understood that the identity operation, leaving the crystal unchanged, is included in each set; that the operations are reversible; and that the associative law holds, A(BC) = (AB)C. 374 CRYSTALS [Chap. 13 however, that the pyrites crystals have characteristic striations on their faces, as shown in the picture, so that all the faces are not equivalent. These crystals therefore do not possess the six 2-fold axes with the six planes normal to them, and the 4-fold axes have been reduced to 2-fold axes. In other cases, such departures from full symmetry are only revealed, as far as external appearance goes, by the orientation of etch figures formed by treating the surfaces with acids. Sometimes the phenomenon of pyro- electricity provides a useful symmetry test. When & crystal that contains no center of symmetry is heated, a difference in potential is developed across its faces. This can be observed by the resultant electrostatic attraction between individual crystals. CD (a) (b) Fig. 13.6. (a) Rock salt, (b) Pyrites. All these differences in symmetry are caused by the fact that the full symmetry of the point lattice has been modified in the crystal struc- ture, as a result of replacing the geometrical points by groups of atoms. Since these groups need not have so high a symmetry as the original lattice, classes of lower than holohedral symmetry can arise within each system. 5. Space groups. The crystal classes are the various groups of symmetry operations of finite figures, i.e., actual crystals. They are made up of opera- tions by symmetry elements that leave at least one point in the crystal invariant. This is why they are called point groups. In a crystal structure, considered as an infinitely extended pattern in space, new types of symmetry operation are admissible, which leave no point invariant. These are called space operations. The new symmetry opera- tions involve translations in addition to rotations and reflections. Clearly only an infinitely extended pattern can have a space operation (translation) as a symmetry operation. The possible groups of symmetry operations of infinite figures are called space groups. They may be considered to arise from combinations of the Sec. 6] CRYSTALS 375 14 Bravais lattices with the 32 point groups. 3 A space group may be visualized as a sort of crystallographic kaleidoscope. If one structural unit is introduced into the unit cell, the operations of the space group immediately generate the entire crystal structure, just as the mirrors of the kaleidoscope produce a symmetrical pattern from a few bits of colored paper. The space group expresses the sum total of the symmetry properties of a crystal structure, and mere external form or bulk properties do not suffice for its determination. The inner structure of the crystal must be studied and this is made possible by the methods of X-ray diffraction. 6. X-ray crystallography. At the University of Munich in 1912, there was gathered a group of physicists interested in both crystallography and the Fig. 13.7. A Laue photograph taken with X-rays. (From Lapp and Andrews, Nuclear Radiation Physics, 2nd Ed., Prentice-Hall, 1953.) behavior of X rays. P. P. Ewaid and A. Sommerfeld were studying the passage of light waves through crystals. At a colloquium discussing some of this work, Max von Laue pointed out that if the wavelength of the radia- tion became as small as the distance between atoms in the crystals, a diffrac- tion pattern should result. There was some evidence that X rays should have the right wavelength, and W. Friedrich agreed to make the experimental test. On passing an X-ray beam through a crystal of copper sulfate, there was obtained a diffraction pattern like that in Fig. 13.7, though not nearly so 3 A good example of the construction of space groups is given by Sir Lawrence Bragg, The Crystalline State (London: G. Bell & Sons, 1933), p. 82. The spjice-group notation is described in International Tables for the Determination of Crystal Structures, Vol. I. There are exactly 230 possible crystallographic space groups. 376 CRYSTALS [Chap. 13 distinct in these first trials. The wave properties of X rays were thus definitely established and the new science of X-ray crystallography began. Some of the consequences of Laue's great discovery have already been mentioned, and on page 257 the conditions for diffraction maxima from a regular three-dimensional array of scattering centers were found to be cos (a 00) hh cos08-/? )-*A (13.1) cos (y - y ) - tt If monochromatic X rays are used, there is only a slim chance that the orientation of the crystal is fixed in such a way as to yield diffraction maxima. The Laue method, however, uses a continuous spectrum of X radiation with a wide range of wavelengths. This is the so-called white radiation, conveniently obtained from a tungsten target at high voltages. In this case, at least some of the radiation is at the proper wavelength to experience interference effects, no matter what the orientation of crystal to beam. 7. The Bragg treatment. When the news of the Munich work reached England, it was immediately taken up by W. H. Bragg and his son W. L. Fig. 13.8. Bragg scattering condition. Bragg who had been working on a corpuscular theory of X rays. W. L. Bragg, using Laue-type photographs, analyzed the structures of NaCl, KC1, and ZnS (1912, 1913). In the meantime (1913), the elder Bragg devised a spectrom- eter that measured the intensity of an X-ray beam by the amount of ioniza- tion it produced, and he found that the characteristic X-ray line spectrum could be isolated and used for crystallographic work. Thus the Bragg method uses a monochromatic (single wavelength) beam of X rays. The Braggs developed a treatment of X-ray scattering by a crystal that was much easier to apply than Laue's theory, although the two are essentially equivalent. It was shown that the scattering of X rays could be represented as a "reflection" by successive planes of atoms in the crystal. Consider, in Fig. 13.8, a set of parallel planes in the crystal structure and a beam of X rays incident at an angle 0. Some of the rays will be "reflected" from the upper layer of atoms, the angle of reflection being equal to the angle of inci- dence. Some of the rays will be absorbed, and some will be "reflected" from Sec. 8] CRYSTALS 377 the second layer, and so on with successive layers. All the waves "reflected" by a single crystal plane will be in phase. Only under certain strict conditions will the Waves "reflected" by different underlying planes be in phase with one another. The condition is that the path difference between the waves scattered from successive planes must be an integral number of wavelengths, nk. If we consider the "reflected" waves at the point P, this path distance Tor the first two planes is 6 = "AB + ~BC. Since triangles AOB and COB are congruent, AB BC and d 2 AB. Therefore d 2d sin 0. The condition for reinforcement or Bragg "reflection" is thus /7A-2</sin0 (13.2) According to this viewpoint, there are different orders of "reflection" specified by the values n = 1 , 2, 3 . . . The second order diffraction maxima from (100) planes may then be regarded as a "reflection" due to a set of planes (200) with half the spacing of the (100) planes. The Bragg equation indicates that for any given wavelength of X rays there is a lower limit to the spacings that can give observable diffraction spectra. Since the maximum value of sin is 1, this limit is given by " A -- ~ 2 sin max " 2 8. The structures of NaCl and KC1. Among the first crystals to be studied by the Bragg method were sodium and potassium chlorides. A single crystal was mounted on the spectrometer, as shown in Fig. 13.9, so that the X-ray IX-RAY BEAM L-V/V//J SLIT SYSTEM IONIZATION CHAMBER **" ^^J^^ DIVIDED SCALE /TO ELECTROMETER Fig. 13.9. Bragg X-ray spectrometer. beam was incident on one of the important crystal faces, (100), (1 10), or (1 1 1). The apparatus was so arranged that the "reflected" beam entered the ioniza- tion chamber, which was filled with methyl bromide. Its intensity was measured by the charge built up on an electrometer. The experimental data are shown plotted in Fig. 13.10 as "intensity of scattered beam" vs. "twice the angle of incidence of beam to crystal." As the crystal is rotated, successive maxima "flash out" as the angles are passed 378 CRYSTALS [Chap. 13 conforming to the Bragg condition, eq. (13.2). In these first experiments the monochromatic X radiation was obtained from a palladium target. Both the wavelength of the X rays and the structure of the crystals were unknown to begin with. It was known, of course, from external form, that both NaCl and KC1 could be based on a cubic lattice, simple, body-centered, or face-centered. By comparing the spacings calculated from X-ray data with those expected for these lattices, a decision could be made as to the proper assignment. (100) (MO) (IN) / I A KCl / /\ A A (100) (HO) (Ml) L A /^^ NoCl A. ^\ > \ A 0* 5 10 15 20 25 30 35 40 45 Fig. 13.10. Bragg spectrometer data, / vs. 20. The general expression for the spacing of the planes (hkl) in a cubic lattice is -" Vh*-+k*n i When this is combined with the Bragg equation, we obtain sin 2 6 = (A 2 /4a 2 )(// 2 + k 2 + I 2 ) Thus each observed value of sin can be indexed by assigning to it the value of (hkl) for the set of planes responsible for the "reflection." For a simple cubic lattice, the following spacings are allowed: (hkl) . . . 100 110 111 200 210 211 220 221,300 etc. h 2 + k 2 + 1 2 . .1 2 3 4 5 6 8 9 etc. If the observed X-ray pattern from a simple cubic crystal was plotted as intensity vs. sin 2 we would obtain a series of six equidistant maxima, with the seventh missing, since there is no set of integers hkl such that h 2 + k 2 + I 2 7. There would then follow seven more equidistant maxima, with the 15th missing; seven more, the 23rd missing; four more, the 28th missing; and so on. Sec. 8] CRYSTALS 379 In Fig. 13.11 (a) we see the (100), (110), and (111) planes for a simple cubic lattice. A structure may be based on this lattice by replacing each lattice point by an atom. If an X-ray beam strikes such a structure at the Bragg angle, sin" 1 (A/20), the rays scattered from one (100) plane will be exactly in phase with the rays from successive (100) planes. The strong scattered beam may be called the "first-order reflection from the (100) planes." A similar result is obtained for the (1 10) and (111) planes. We shall -*- a ^ Fig. 13.11. Spacings in cubic lattices: (a) simple cubic; (b) body-centered cubic; (c) face-centered cubic. obtain a diffraction maximum from each set of planes (hkl), since for any given (hkl) all the atoms will be included in the planes. Fig. 13.11 (b) shows a structure based on a body-centered cubic lattice. The (110) planes, as in the simple-cubic case, pass through all the lattice points, and a strong first-order (1 10) reflection will occur. In the case of the (100) planes, however, we find a different situation. Exactly midway between any two (100) planes, there lies another layer of atoms. When X rays scattered from the (100) planes are in phase and reinforce one another, the rays scattered by the interleaved atomic planes will be retarded by half a wave- length, and hence will be exactly out of phase with the others. The observed intensity will therefore be the difference between the scattering from the two sets of planes. If the atoms all have identical scattering powers, the resultant intensity will be reduced to zero by the destructive interference, and no 380 CRYSTALS [Chap. 13 first-order (100) reflection will appear. If, however, the atoms are different, the first-order (100) will still appear, but with a reduced intensity given by the difference between the scatterings from the two interleaved sets of planes. The second-order diffraction from the (100) planes, occurring at the Bragg angle with n ^ 2 in eq. (13.2), can equally well be expressed as the scattering from a set of planes, called the (200) planes, with just half the spacing of the (100) planes. In the body-centered cubic structure, all the atoms lie in these (200) planes, so that all the scattering is in phase, and a strong scattered beam is obtained. The same situation holds for the (111) planes: the first-order (111) will be weak or extinguished, but the second-order (111), i.e. the (222) planes, will give strong scattering. If we examine successive planes (hkl) in this way, we find for the body-centered cubic structure the results shown in Table 13.2, in which planes missing due to extinction are indicated by dotted lines. TABLE 13.2 CALCULATED AND OBSERVED DIFFRACTION MAXIMA 300 (hkl) . ... 100 110 111 200 210 211 220 211 310 /,2 + p 4. 72 ! 2 3 4 5 6 8 9 10 simple cubic . | | | | | | | | | body-centered cubic : | | | III face-centered cubic Sodium Chloride 200 220 222 400 420 422 440 600 620 Potassium Chloride .1 I I I I I I 422 I In the case of the face-centered cubic structure, Fig. 13.1 1 (c), reflections from the (100) and (110) planes are weak or missing, and the (111) planes give intense reflection. The results for subsequent planes are included in Table 13.2. In the first work on NaCl and KC1, the X-ray wavelength was not known, so that the spacings corresponding to the diffraction maxima could not be calculated. The values of sin 2 0, however, can be used directly. The observed maxima are compared in Table 13.2 with those calculated for the different cubic lattices. The curious result is now not^d that apparently NaCl is face centered Sec. 8] CRYSTALS 381 Fig. 13.12. Sodium chloride structure. while KC1 is simple cubic. The reason why the KC1 structure behaves toward X rays like a simple cubic array is that the scattering powers of K + and Cl~ ions are indistinguishable since they both have an argon configuration with 18 electrons. In the NaCl structure the difference in scattering power of the Na+ and Cl~ ions is responsible for the deviation from the simple cubic pattern. The observed maxima from the (111) face of NaCl include a weak peak at an angle of about 10, in addition to the stronger peak at about 20, corresponding to that observed with KC1. These results are all explained by the NaCl structure shown in Fig. 13.12, which consists of a face-centered cubic array of Na+ ions and an interpenetrating face-centered cubic array of Cl~ ions. Each Na+ ion is surrounded by six equidistant Cl~ ions and each Cl~ ion by six equidistant Na+ ions. The (100) and (1 10) planes contain an equal number of both kinds of ions, but the (111) planes consist of either all Na f or all Cl~ ions. Now if X rays are scattered from the (111) planes in NaCl, whenever scattered rays from successive Na+ planes are exactly in phase, the rays scattered from the interleaved Cl~ planes are retarded by half a wavelength and are therefore exactly out of phase. The first-order (111) reflection is therefore weak in NaCl since it represents the difference between these two scatterings. In the case of KC1, where the scattering powers are the same, the first-order reflections are altogether extinguished by interference. Thus the postulated structure is in complete agreement with the experimental X-ray evidence. Once the NaCl structure was well established, it was possible to calculate the wavelength of the X rays used. From the density of crystalline NaCl, p = 2.163 g per cm 3 , the molar volume is M/p =--- 58.45/2.163 = 27.02 cc per mole. Then the volume occupied by each NaCl unit is 27.02 : (6.02 x 10 23 ) = 44.88 x 10~ 24 cc. In the unit cell of NaCl, there are eight Na+ ions at the corners of the cube, each shared between eight cubes, and six Na+ ions at the face centers, each shared between two cells. Thus, per unit cell, there are 8/8 + 6/2 = 4 Na + ions. There is an equal number of Cl~ ions, and there- fore four NaCl units per unit cell. The volume of the unit cell is there- fore 4 x 44.88 x 10" 24 = 179.52 (A) 3 . The interplanar spacing for the (200) planes is \a =- J179.52 173 =r 2.82 A. Substituting this value and the observed diffraction angle into the Bragg equation, A = 2(2.82) sin 5 58'; A - 0.586 A. Once the wavelength has been measured in this way, it can be used to determine the interplanar spacings in other crystal structures. Conversely, crystals with known spacings can be used to measure the wavelengths of other X-ray lines. The most generally useful target material is copper, with 382 CRYSTALS [Chap. 13 A -- 1.537 A (A^), a convenient length relative to interatomic distances. When short spacings are of interest, molybdenum (0.708) is useful, and chromium (2.285) is often employed for study of longer spacings. The Bragg spectrometer method is generally applicable but is quite time consuming. Most crystal structure investigations have used photographic methods to record the diffraction patterns. Improved spectrometers have been developed recently in which a Geiger-counter tube replaces the electrom- eter and ionization chamber. 9. The powder method. The simplest technique for obtaining X-ray diffrac- tion data is the powder method, first used by P. Debye and P. Scherrer. Instead of a single crystal with a definite orientation to the X-ray beam, a CYLINDRICAL " CAMERA POWDER SPECIMEN X-RAY BEAM FILM Fig. 13.13. The powder method. Powder picture of sodium chloride, Cu-K a radiation, (c). (Courtesy Dr. Arthur Lessor, Indiana University.) mass of finely divided crystals with random orientations is used. The experi- mental arrangement is illustrated in (a), Fig. 13.13. The powder is contained in a thin-walled glass capillary, or deposited on a fiber. Polycrystalline metals are studied in the form of fine wires. The sample is rotated in the beam to average as well as possible the orientations of the crystallites. Out of the many random orientations of the little crystals, there will be some at the proper angle for X-ray reflection from each set of planes. The direction of the reflected beam is limited only by the requirement that the angle of reflection equal the angle of incidence. Thus if the incident angle is 0, the reflected beam makes an angle 20 with the direction of the incident beam, (b), Fig. 13.13. This angle 26 may itself be oriented in various directions around the central beam direction, corresponding to the various orientations of the individual crystallites. For each set of planes, therefore, the reflected beams outline a cone of scattered radiation. This cone, intersecting a Sec. 10] CRYSTALS 383 cylindrical film surrounding the specimen, gives rise to the observed dark lines. On a flat plate film, the observed pattern consists of a series of con- centric circles. A typical X-ray powder picture is shown in (c), Fig. 13.13. It may be compared with the electron-diffraction picture obtained by G. P. Thomson from a polycrystalline gold foil (page 272). After obtaining a powder diagram, the next step is to index the lines, assigning each to the responsible set of planes. The distance x of each line from the central spot is measured carefully, usually by halving the distance between the two reflections on either side of the center. If the film radius is r, the circumference 2nr corresponds to a scattering angle of 360. Then, x/2irr = 2(9/360. Thus is calculated and, from eq. (13.2), the interplanar spacing. The spacing data are often used, without further calculation, to identify solids or analyze solid mixtures. Extensive tables are available 4 that facilitate the rapid identification of unknowns. To index the reflections, one must know the crystal system to which the specimen belongs. This system can sometimes be determined by microscopic examination. Powder diagrams of monoclinic, orthorhombic, and triclinic crystals may be almost impossible to index. For the other systems straight- forward methods are available. Once the unit-cell size is found, by calculation from a few large spacings (100, 110, 111, etc.), all the interplanar spacings can be calculated and compared with those observed, thus completing the indexing. Then more precise unit-cell dimensions can be calculated from high-index spacings. The general formulae giving the interplanar spacings are straightforward derivations from analytical geometry. 5 10. Rotating-crystal method. The rotating-single-crystal method, with photographic recording of the diffraction pattern, was developed by E. Schiebold around 1919. It has been, in one form or another, the most widely used technique for precise structure investigations. The crystal, which is preferably small and well formed, perhaps a needle a millimeter long and a half-millimeter wide, is mounted with a well defined axis perpendicular to the beam which bathes the crystal in X radiation. The film may be held in a cylindrical camera, and the crystal is rotated slowly during the course of the exposure. In this way, successive planes pass through the orientation necessary for Bragg reflection, each producing a dark spot on the film. Sometimes only part of the data is recorded on a single film, by oscillating through some smaller angle rather than rotating through 360. An especially useful method employs a camera that moves the film back and forth with a period synchronized with the rotation of the crystal. Thus the position of a spot on the film immediately indicates the orientation of the crystal at which the spot was formed (Weissenberg method). We cannot give here a detailed interpretation of these several varieties 4 J. D. Hanawalt, Ind. Eng. Chem. Anal., 10, 457 (1938). 6 C. W. Bunn, Chemical Crystallography (New York: Oxford, 1946), p. 376. 384 CRYSTALS [Chap. 13 , Fig. 13.14. Rotation picture of zinc oxine dihydrate Weisscnberg method. (Courtesy Prof. L. L. Merritt, Indiana University.) of rotation pictures. 6 A typical example is shown in Fig. 13.14. Methods have been developed for indexing the various spots and also for measuring their intensities. These data are the raw material for crystal-structure determinations. 11. Crystal-structure determinations: the structure factor. The problem of reconstructing a crystal structure from the intensities of the various X-ray diffraction maxima is analogous in some ways to the problem of the forma- tion of an image by a microscope. According to Abbe's theory of the micro- scope, the objective gathers various orders of light rays diffracted by the specimen and resynthesizes them into an image. This synthesis is possible because two conditions are fulfilled in the optical case: the phase relation- ships between the various orders of diffracted light waves are preserved at all times, and optical glass is available to focus and form an image with radiation having the wavelength of visible light. We have no such lenses for forming X-ray images (compare, however, the electron microscope), and the way in which the diffraction data are necessarily obtained (one by one) means that all the phase relationships are lost. The essential problem in determining a crystal structure is to regain this lost information in some way or other, and to resynthesize the structure from the amplitudes and phases of the diffracted waves. We shall return to this problem in a little while, but first let us see how the intensities of the various spots on an X-ray picture are governed by the crystal structure. 7 The Bragg relation fixes the angle of scattering in terms of ' See Bragg, he. cit., p. 30. Also Bunn, he. c//., p. 137. 7 This treatment follows that given by M. J. Buerger in X-Ray Crystallography (New York: Wiley, 1942), which.should be consulted for more details. Sec. 11] CRYSTALS 385 the interplanar spacings, which are determined by the arrangement of points in the crystal lattice. In an actual structure, each lattice point is replaced by a group of atoms. It is primarily the arrangement and composition of this group that controls the intensity of the scattered X rays, once the Bragg condition has been satisfied. As an example, consider in (a), Fig. 13.15, a lattice in which each point has been replaced by two atoms (e.g., a diatomic molecule). Then if a set of Fig. 13.15. X-ray scattering from a typical structure. lattice planes is drawn through the black atoms, another parallel but slightly displaced set can be drawn through the white atoms. When the Bragg con- dition is met, as in (b), Fig. 13.15, the reflections from all the black atoms are in phase, and the reflections from all the white atoms are in phase. The radiation scattered from the blacks is slightly out of phase with that from the whites, so that the resultant amplitude, and therefore intensity, is diminished by interference. The problem now is to obtain a general expression for the phase 386 CRYSTALS [Chap. 13 difference. An enlarged view of the structure (two-dimensional) is shown in (c), Fig. 13.15, with the black atoms at the corners of a unit cell with sides a and /?, and the whites at displaced positions. The coordinates of a black atom may be taken as (0, 0) and those of a white as (x, y). A set of planes (hk) is shown, for which it is assumed the Bragg condition is being fulfilled; these are actually the (32) planes in the figure. Now the spacings a/h along a and b/k along b correspond to positions from which scattering differs in phase by exactly 360 or 2rr radians, i.e., scattering from these positions is exactly in phase. The phase difference between these planes and those going through the white atoms is proportional to the displacement of the white atoms. The phase difference P x for displacement v in the a direction is given by x/(a/h) --= PJ2ir, or P x 2irh(x/a). The total phase difference for dis- placement in both a and b directions becomes />, -f Py - 2* By extension to three dimensions, the total phase change that an atom at (xyz) in the unit cell contributes to the plane (hkl) is We may recall (page 327) that the superposition of waves of different amplitude and phase can be accomplished by vectorial addition. If /j and / 2 are the amplitudes of the waves scattered by atoms (1) and (2), and P l and P 2 are the phases, the resultant amplitude is F f\? lPl 4-/ 2 ^ /J ". For any number of atoms, ^' (13-4) When this is combined with eq. (13.3), there is obtained an expression for the resultant amplitude of the waves scattered from the (hkl) planes by all the atoms in a unit cell: F(hkl) = ZJ K <?****!*+ w* * '*/') (13.5) The expression F(hkl) is called the structure factor of the crystal. Its value is determined by the exponential terms, which depend on the positions of the atoms, and by the atomic scattering factors f K , which depend on the number and distribution of the electrons in the atom, and on the scattering angle 0. The intensity of scattered radiation is proportional to the absolute value of the amplitude squared, \F(hkl)\ 2 . The crystal structure problem now becomes that of obtaining agreement between the observed intensities and those calculated from a postulated structure. Structure-factor expressions have been tabulated for all the space groups. 8 8 International Tables for the Determination of Crystal Structures (1952). It is usually possible to narrow the choice of space groups to two or three by means of study of missing reflections (hkl) and comparison with the tables. Sec. 12] CRYSTALS 387 As an example of the use of the structure factor let us calculate F(hkl) for the 100 planes in a face-centered cubic structure, eg., metallic gold. In this structure there are four atoms in the unit cell (Z 4), which may be assigned coordinates (xja, v/b, z/c) as follows: (000), (J i 0), (i J), and (0 i i). Therefore, from eq."(13.5) -- f Au (2 h 2^) since e" 1 cos TT + / sin TT --- 1 Thus the structure factor vanishes and there is therefore zero intensity of scattering from the (100) set of planes. This is almost a trivial case, since inspection of the face-centered cubic structure immediately reveals that there is an equivalent set of planes interleaved midway between the 100 planes, so that the resultant amplitude of the scattered X rays must be reduced to zero by interference. In more complicated instances, however, it is essential to use the structure factor to obtain a quantitative estimation of the scattering intensity expected from any set of planes (hkl) in any postulated crystal structure. 12. Fourier syntheses. An extremely useful way of looking at a crystal structure was proposed by Sir William Bragg when he pointed out that it may be regarded as a periodic three-dimensional distribution of electron density, since it is the electrons that scatter the X rays. Any such density function may be expressed as a Fourier series, a summation of sine and cosine terms. 9 It is more concisely written in the complex exponential form. Thus the electron density in a crystal may be represented as p(xyz) - \ \ A pQr e **P*!* ^ '////ft i /r) ( { 3 6) p _ oo q uj r or It is not hard to show 10 that the Fourier coefficients A wr are equal to the structure factors divided by the volume of the unit cell. Thus p(xyz) 1 SSX F(hkl)e"^^ rla ^ vlb ^ lf} (13.7) This equation expresses the fact that the only Fourier term that contributes to the X-ray scattering by the set of planes (hkl) is the one with the coefficient F(hkl), which appears intuitively to be the correct formulation. Equation (13.7) summarizes the whole problem involved in structure determinations, since in a very real sense the crystal structure is simply p(xyz). Positions of individual atoms are peaks in the electron density function />, and interatomic regions are valleys in the plot of p. Thus if we knew the F(hkiy$ we could immediately plot the crystal structure. All we know, however, are the intensities, which are proportional to \F(hkl)\ 2 . As 9 See, for example, Widder, Advanced Calculus, p. 324. 10 Bragg, op. cit., p. 221. 388 CRYSTALS [Chap. 13 stated earlier, we know the amplitudes but we have necessarily lost the phases in taking the X-ray pattern. A trial structure is now assumed and the intensities are calculated. If the assumed arrangement is even approximately correct, the most intense observed reflections should have large calculated intensities. The observed F's for these reflections may be put into the Fourier series with the calculated signs. 11 The graph of the Fourier summation will give new positions for the VAAX \ \ \ Fig. 13.16. Fourier map of electron density in glycylglycine projected on base of unit cell: (a) 40 terms; (b) 100 terms; (c) 200 terms. atoms, from which new f's can be calculated, which may allow more of the signs to be determined. Gradually the structure is refined as more and more terms are included in the synthesis. In Fig. 13.16 are shown three Fourier summations for the structure of glycylglycine. As additional terms are in- cluded in the summation, the resolution of the structure improves, just as the resolution of a microscope increases with objectives that catch more and more orders of diffracted light. Sometimes a heavy atom can be introduced into the structure, whose position is known from symmetry arguments. The large contribution of the heavy atom makes it possible to determine the phases of many of the F's. 11 The complete Fourier series is rarely used; instead, various two-dimensional pro- jections are preferred. Sec. 13] CRYSTALS 389 This was the method used with striking success by J. M. Robertson in his work on the phthalocyanine structures, 12 and in the determination of the structure of penicillin. This last was one of the great triumphs of X-ray crystallography, since it was achieved before the organic chemists knew the structural formula. 13. Neutron diffraction. Not only X-ray and electron beams, but also beams of heavier particles may exhibit diffraction patterns when scattered from the regular array of atoms in a crystal. Neutron beams have proved to be especially useful for such studies. The wavelength is related to the mass and velocity by the Broglie equation, X -- hjmv. Thus a neutron with a speed of 3.9 x 10 5 cm sec" 1 (kinetic energy 0.08 ev) would have a wave- length of 1 .0 A. The diffraction of electron rays or X rays is caused by their interaction with the orbital electrons of the atoms in the material through which they pass; the atomic nuclei contribute practically nothing to the scattering. The diffraction of neutrons, on the other hand, is primarily caused by two other effects: (a) nuclear scattering due to interaction of the neutrons with the atomic nuclei, (b) magnetic scattering due to interaction of the magnetic moments of the neutrons with permanent magnetic moments of atoms or ions. In the absence of an external magnetic field, the magnetic moments of atoms in a paramagnetic crystal are arranged at random, so that the magnetic scattering of neutrons by such a crystal is also random. It contributes only a diffuse background to the sharp maxima occurring when the Bragg con- dition is satisfied for the nuclear scattering. In ferromagnetic materials, however, the magnetic moments are regularly aligned so that the resultant spins of adjacent atoms are parallel, even in the absence of an external field. In antiferromagnetic materials, the magnetic moments are also regu- larly aligned, but in such a way that adjacent spins are always opposed. The neutron diffraction patterns dis- tinguish experimentally between these different magnetic structures, and indi- cate the direction of alignment of spins within the crystal. For example, manganous oxide, MnO, has the rock-salt structure (Fig. 13.12), and is antiferromagnetic. The detailed magnetic structure as revealed by neutron diffraction is shown in Fig. 13.17. The manganous ion, Mn+ 2 , has the electronic structure 3s 2 3p B 3d*. 12 J. Chem. Soc. (London), 1940, 36. For an account of the work on penicillin, see Research, 2, 202 (1949). CHEMICAL UNIT CELL Fig. 13.17. Magnetic structure of MnO as found by neutron diffraction. Note that the "magnetic unit cell" has twice the length of the "chemical unit cell." [From C. G. Shull, E. O. Wollan, and W. A. Strauser, Phys. Rev., 81, 483 (1951).] 390 CRYSTALS [Chap. 13 The five 3c/ electrons are all unpaired, and the resultant magnetic moment is 2V%(jf I 1) = 5.91 Bohr magnetons. If we consider Mn +2 ions in successive (111) planes in the crystal, the resultant spins are oriented so that they are alternately positively and negatively directed along the [100] direction. Another useful application of neutron diffraction has been the location of hydrogen atoms in crystal structures. It is usually impossible to locate hydrogen atoms by means of X-ray or electron diffraction, because the small scattering power of the hydrogen is completely overshadowed by that of heavier atoms. The hydrogen nucleus, however, is a strong scatterer of neutrons. Thus it has been possible to work out the structures of such com- pounds as UH 3 and KHF 2 neutron-diffraction analysis. 13 14. Closest packing of spheres. Quite a while before the first X-ray struc- ture analyses, some shrewd theories about the arrangement of atoms and (a) (b) / / (c; (c) (d) Fig. 13.18. (a) Hexagonal closest packing; (b) cubic closest packing (edge cut away to show closest packing normal to cube diagonals); (c) plan of hexagonal closest packing; (d) plan of cubic closest packing. molecules in crystals were developed from purely geometrical considerations. From 1883 to 1897, W. Barlow proposed a number of structures based on the packing of spheres. There are two different ways in which spheres of the same size can be packed together so as to leave a minimum of unoccupied volume, in each case 26 per cent voids. They are the hexagonal-closest-packed (hep) and the 13 S. W. Peterson and H. A. Levy, /. Chem. Phys., 20, 704 (1952). Sec. 14] CRYSTALS 391 cubic-closest-packed (ccp) arrangements depicted in Fig. 13.18. In ccp the layers repeat as ABC ABC ABC . . ., and in hep the order is AB AB AB ... It will be noted that the ccp structure may be referred to a face-centered- cubic unit cell, the (ill) planes being the layers of closest packing. The ccp structure is found in the solid state of the inert gases, in crystal- line methane, etc. symmetrical atoms or molecules held together by van der Waals forces. The high-temperature forms of solid H 2 , N 2 , and O 2 occur in hep structures. The great majority of the typical metals crystallize in the ccp, the hep, or a body-centered-cubic structure. Some examples are collected in Table 13.3. Other structures include the following: 14 the diamond-type cubic of TABLE 13.3 STRUCTURES OF THE METALS Cubic Closest Packed (fee) or (ccp) Hexagonal Closest Packed (hep) Body-Centered Cubic (bcc) Ag yFe Al Ni Au Pb ocCa Pt aBe Os yCa aRu Cd flSc aCe aTi Ba Mo aCr Na Cs Ta aFe Ti PCO Sr aCo aTl ^Fe V Cu Th /?Cr Zn Mg aZr K ^W Li pZr grey tin and germanium; the face-centered tetragonal, a distorted fee, of y-manganese and indium; the rhombohedral layered structures of bismuth, arsenic, and antimony; and the body-centered tetragonal of white tin. It will be noted that many of the metals are polymorphic (allotropic), with two or more structures depending on conditions of temperature and pressure. The nature of the binding in metal crystals will be discussed later. For the present, we may think of them as a network of positive metal ions packed primarily according to geometrical requirements, and permeated by mobile electrons. This so-called electron gas is responsible for the high conductivity and for the cohesion of the metal. The ccp metals, such as Cu, Ag, Au, Ni, are all very ductile and malle- able. The other metals, such as V, Cr, Mo, W, are harder and more brittle. This distinction in physical properties reflects a difference between the struc- ture types. When a metal is hammered, rolled, or drawn, it deforms by the gliding of planes of atoms past one another. These slip planes are those that contain the most densely packed layers of atoms. In the ccp structure, the slip planes are therefore usually the (111), which occur in sheets normal to 14 For descriptions see R. W. G. Wyckoff, Crystal Structures (New York: Interscience, 1948). 392 CRYSTALS [Chap. 13 all four of the cube diagonals. In the hep and other structures there is only one set of slip planes, e.g., those perpendicular to the hexagonal axis. Thus the ccp metals are characteristically more ductile than the others, since they have many more glide ways. 15. Binding in crystals. The geometrical factors, seen in their simplest form in the closest packed structures of identical spheres, are always very important in determining the crystal structure of a substance. Once they are satisfied, other types of interaction must also be considered. Thus, for example, when directed binding appears, closest packing cannot be achieved. Two different theoretical approaches to the nature of the chemical bond in molecules have been described in Chapter 11. In the method of atomic orbitals, the point of departure is the individual atom. Atoms are brought together, each with the electrons that "belong to it," and one considers the effect of an electron in one atomic orbital upon that in another. In the second approach, the electrons in a molecule are no longer assigned possessively to the individual atoms. A set of nuclei is arranged at the proper final distances and the electrons are gradually fed into the available molecular orbitals. For studying the nature of binding in crystals, these two different treat- ments are again available. In one case, the crystal structure is pictured as an array of regularly spaced atoms, each possessing electrons used to form bonds with neighboring atoms. These bonds may be ionic, covalent, or intermediate in type. Extending throughout three dimensions, they hold the crystal together. The alternative approach is once again to consider the nuclei at fixed positions in space and then gradually to pour the electron cement into the periodic array of nuclear bricks. Both these methods yield useful and distinctive results, displaying com- plementary aspects of the nature of the crystalline state. We shall call the first treatment, growing out of the atomic-orbital theory, the bond model of the solid state. The second treatment, an extension of the method of mole- cular orbitals, we shall call, for reasons to appear later, the band model of the solid state. 16. The bond model. If we consider that a solid is held together by chemical bonds, it is useful to classify the bond types. Even though the available classifications are as usual somewhat frayed at the edges, the following categories may be distinguished: (1) The van der Waals bonds. These bonds are the result of forces between inert atoms or essentially saturated molecules. These forces are the same as those responsible for the a term in the van der Waals equation. Crystals held together in this way are sometimes called molecular crystals. Examples are nitrogen, carbon tetrachloride, benzene. The molecules tend to pack together as closely as their geometry allows. The binding between the mole- cules in van der Waals structures represents a combination of factors such as dipole-dipole and dipole-polarization interactions, and the quantum Sec. 16] CRYSTALS 393 mechanical dispersion forces, first elucidated by F. London, which are often the principal component. 15 (2) The ionic bonds. These bonds are familiar from the case of the NaCl molecule in the vapor state (page 297). In a crystal, the coulombic interaction between oppositely charged ions leads to a regular three-dimensional struc- ture. In rock salt, each positively charged Na f ion is surrounded by six negatively charged Cl ions, and each Cl is surrounded by six Na j . There are no sodium-chloride molecules unless one wishes to regard the. entire crystal as a giant molecule. The ionic bond is spherically symmetrical and undirected; an ion will be surrounded by as many oppositely charged ions as can be accommodated ID (o) (b) Fig. 13.19. (a) Diamond structure; (b) graphite structure. geometrically, provided that the requirement of over-all electrical neutrality is satisfied. (3) The covalent bonds. These bonds, we recall, are the result of spin valence (page 303), the sharing between atoms of two electrons with anti- parallel spins. When extended through three dimensions, they may lead to a variety of crystal structures, depending on the valence of the constituent atoms, or the number of electrons available for bond formation. A good example is the diamond structure in (a), Fig. 13.19. The structure can be based on two interpenetrating face-centered cubic lattices. Each point in one lattice is surrounded tetrahedrally by four equidistant points in the other lattice. This arrangement constitutes a three-dimensional polymer of carbon atoms joined together by tetrahedrally oriented sp 3 bonds. Thus the configuration of the carbon bonds in diamond is similar to that in the aliphatic compounds such as ethane. The C C bond distance is 1.54 A in both diamond and ethane. Germanium, silicon, and grey tin also crystallize in the diamond structure. The same structure is assumed by compounds such as ZnS (zinc blende), Agl, A1P, and SiC. In all these structures, each atom is surrounded by four unlike atoms oriented at the corners of a regular tetrahedron. In every case the binding is primarily covalent. It is interesting to note that it is not neces- sary that each atom provide the same number of valence electrons; the 15 See Chapter 14, Sect. 10. 394 CRYSTALS [Chap. 13 Fig. 13.20. Structure of selenium structure can occur whenever the total number of outer-shell electrons is just four times the total number of atoms. There is also a form of carbon, actually the more stable allotrope, in which the carbon bonds resemble those in the aromatic series of compounds. This is graphite, whose structure is shown in (b), Fig. 13.19. Strong bonds operate within each layer of carbon atoms, whereas much weaker binding joins the layers; hence the slippery and flaky nature of graphite. The C C distance within the layers of graphite is 1.34 A, identical with that in anthracene. Just as in the discussion of the nature of binding in aromatic hydrocarbons (page 311), we can distinguish two types of electrons within the graphite struc- ture. The a electrons are paired to form localized-pair (sp 2 ) bonds, and the 77 electrons are free to move throughout the planes of the C 6 rings. Atoms with a spin valence of only 2 cannot form regular three-dimen- sional structures. Thus we have the interesting structures of selenium (Fig. 13.20), and tellurium, which consist of endless chains of atoms extending through the crystal, the individual chains being held together by much weaker forces. Another way of solving the problem is illustrated by the struc- ture of rhombic sulfur, Fig. 13.21. Here there are well defined, puckered, eight-membered rings of sulfur atoms. The bivalence of sulfur is maintained and the S 8 "molecules" are held together by van der Waals attractions. Elements like arsenic and antimony that in their compounds display a covalence of 3 tend to crystallize in structures that contain well defined layers of atoms. (4) 77?^ intermediate-type bonds. Just as in individual molecules, these bonds arise from resonance between covalent and ionic contributions. Alternatively, one may consider the polarization of one ion by an oppositely charged ion. An ion is said to be polarized when its electron "cloud" is distorted by the presence of the oppositely charged ion. The larger an ion the more readily is it polarized, and the smaller an ion the rhore intense is its electric field and the greater Fig. 13.21. Structure of rhombic sulfur. Sec. 17] CRYSTALS 395 Hg. 13.22, Structure of ice. its polarizing power. Thus in general the larger anions are polarized by the smaller cations. Even apart from the size effect, cations are less polarizable than anions because their net positive charge tends to hold their electrons in place. The structure of the ion is also important: rare-gas cations such as K+ have less polarizing power than transition cations such as Ag+, since their positive nuclei are more effectively shielded. The effect of polarization may be seen in the structures of the silver halides. AgF, AgCl, and AgBr have the-rock-salt structure, but as the anion becomes larger it becomes more strongly polarized by the small Ag+ ion. Finally, in Agl the binding has very little ionic character and the crystal has the zinc-blende structure. It has been confirmed spectroscopically that crystal- line silver iodide is composed of atoms and not ions. (5) The hydrogen bond. The hydrogen bond, discussed on page 313, plays an important role in many crystal struc- tures, e.g., inorganic and organic acids, salt hydrates, ice. The structure of ice is shown in Fig. 13.22. The coordination is similar to that in wurtzite, the hexago- nal form of zinc sulfide. Each oxygen is surrounded tetrahedrally by four nearest neighbors at a distance of 2.76 A. The hydrogen bonds hold the oxygens together, leading to a very open structure. By way of contrast, hydrogen sulfide, H 2 S, has a ccp structure, each molecule having twelve nearest neighbors. (6) 77?? metallic bond. The bond model has also been extended to metals. According to this picture, the metallic bond is closely related to the ordinary covalent electron-pair bond. Each atom in a metal forms covalent bonds by sharing electrons with its nearest neighbors. It is found that there are more orbitals available for bond formation than there are electrons to fill them. As a result the covalent bonds resonate among the available interatomic positions. In the case of a crystal this resonance extends throughout the entire structure, thereby producing great stability. The empty orbitals permit a ready flow of electrons under the influence of an applied electric field, leading to metallic conductivity. Structures such as those of selenium and tellurium, and of arsenic and antimony, represent transitional forms in which the electrons are much more localized because the available orbitals are more completely filled. In a covalent crystal like diamond the four .s/; 3 tetrahedral orbitals are completely filled. 17. The band model. It was in an attempt to devise an adequate theory for metals that the band model had its origin. The high thermal and electrical conductivities of metals focused attention on the electrons as the important 396 CRYSTALS [Chap. 13 entities in their structures. If we use as a criterion the behavior of the elec- trons, three classes of solids may be distinguished: (1) Conductors or metals, which offer a low resistance to the flow of electrons, an electric current, when a potential difference is applied. The resistivity of metals increases with the temperature. (2) Insulators, which have a high electric resistivity. (3) Semiconductors, whose resistivity is intermediate between that of typical metals and that of typical insulators, and decreases, usually ex- ponentially, with the temperature. The starting point of the band theory is a collection of nuclei arrayed in space at their final crystalline internuclear separations. The total number 3s 2p - - 2s Is - - 3s 2p 2 I -a - (a) (b) Fig. 13.23. Energy levels in sodium: (a) isolated atoms; (b) section of crystal. of available electrons is poured into the resultant field of force, a regularly periodic field. What happens? Consider in Fig. 13.23 the simplified model of a one-dimensional struc- ture. For concreteness, let us think of the nuclei as being those of sodium, therefore bearing a charge of + 1 1 . The position of each nucleus will repre- sent a deep potential-energy well for the electrons, owing tp the large electro- static attraction. If these wells were infinitely deep, the electrons would all fall into fixed positions on the sodium nuclei, giving rise to l^Zs^/^S-s 1 configurations, typical of isolated sodium atoms. This is the situation shown in (a), Fig. 13.23. But the wells are not infinitely deep, or in other words the potential-energy barriers separating the electrons on different nuclei are not infinitely high. The actual situation is more like the one shown in (b), Fig. 13.23. Now the possibility of a quantum mechanical leakage of electrons through the barriers must be considered. Otherwise expressed, there will be a resonance of electrons between the large number of identical positions. There is always a possibility of an electron on one nucleus slipping through to occupy a position on a neighboring nucleus. We are thus no longer con- cerned with the energy levels of single sodium atoms but with levels of the Sec. 17] CRYSTALS 397 crystal as a whole. Then the Pauli Principle comes into play, and tells us that no more than two electrons can occupy exactly the same energy level. Once the possibility of electrons moving through the structure is admitted, we can no longer consider the energy levels to be sharply defined. The sharp Is energy level in an individual sodium atom is broadened in crystalline sodium into a band of closely packed energy levels. A similar situation arists for the other energy levels, each becoming a band of levels as shown in (b), Fig. 13.23. Each atomic orbital contributes one level to a band. In the lower bands (Is, 2s, 2p) there are therefore just enough levels to accommodate the number of available electrons, so that the bands are completely filled. If an external electric field is applied, the electrons in the filled bands cannot move under its influence, for to be accelerated by the field they would have to move into somewhat higher energy levels. This is impossible for electrons in the interior of a filled band, since all the levels above them are already occupied, and the Pauli Principle forbids their accepting additional tenants. Nor can the elec- trons at the very top of a filled band acquire extra energy, since there are no higher levels for them to move into. Very occasionally, it is true, an electron may acquire a relatively terrific jolt of energy and be knocked completely out of its band into a higher unoccupied band. So much for the electrons in the lower bands. The situation is very different in the uppermost band, the 3s, which is only half filled. An electron in the interior of the 3s band still cannot be accelerated because the levels directly above are already filled. Electrons toward the top of the band, how- ever, can readily move up into unfilled levels within the band. This is what happens when an electric field is applied and a current flows. It will be noticed from the diagram that the topmost band has actually broadened sufficiently to overlap the tops of the potential-energy barriers, so that these electrons can move quite freely through the crystal structure. According to this idealized model in which the nuclei are always arranged at the points of a perfectly periodic lattice, there would indeed be no resist- ance at all offered to the flow of an electric current. The resistance arises from deviations from perfect periodicity. An important loss of periodicity is caused by the thermal vibrations of the lattice nuclei. These vibrations destroy the perfect resonance between the electronic energy levels and cause a resistance to the free flow of electrons. As would be expected, the resistance therefore increases with the temperature. Another illustration of the same principle is found in the increased resistance that results when an alloying constituent is added to a pure metal, and the regular periodicity of the structure is dimin- ished by the foreign atoms. At this point the reader may well be thinking that this is a pretty picture for a univalent metal such as sodium, but what of magnesium with its two 3s electrons and therefore completely filled 3s bands ? Why isn't it an insulator instead of a metal? The answer is that in this, and similar cases, detailed 398 CRYSTALS [Chap. 13 calculations show that the 3p band is low enough to overlap the top of the 35 band, providing a large number of available empty levels. Thus conductors are characterized either by partially filled bands or by overlapping of the topmost bands. Insulators have completely filled lower bands with a wide energy gap between the topmost filled band and the lowest empty band. These models are represented schematically in Fig. 13.24. The energy bands in solids can be studied experimentally by the methods of X-ray emission spectroscopy. 16 For example, if an electron is driven out of the \s level in sodium metal (Fig. 13.23b) the K a X-ray emission occurs when an electron from the 3^ band falls into the hole in the Is level. Since the 3s FILLED IMPURITY LEVELS EMPTY IMPURITY LEVELS (a) Fig. 13.24. Band models of solid types: (a) insulator; (b) metal; (c) semiconductor. electron can come from anywhere within the band of energy levels, the X rays emitted will have a spread of energies (and hence frequencies) exactly corre- sponding with the spread of allowed energies in the 3s band. The following widths (in ev) were found for the conduction bands in a few of the solids investigated : Li Na Be Mg Al 4.1 3.4 14.8 7.6 13.2 18. Semiconductors. Band models for semiconductors are also included in Fig. 13.24. These models possess, in addition to the normal bands, narrow impurity bands, either unfilled levels closely above a filled band or filled levels closely below an empty band. The extra levels are the result of either foreign atoms dissolved in the structure or a departure from the ideal stoichiometric composition. Thus zinc oxide normally contains an excess of zinc, whereas cuprous oxide normally contains an excess of oxygen. Both these compounds behave as typical semiconductors. Their conductivities increase approxi- mately exponentially with the temperature, because the number of conduc- tion electrons depends on excitation of electrons into or out of the impurity levels, and excitation is governed by an e ~* E/HT Boltzmann factor. 16 A review by N. F. Mott gives further references. Prog. Metal Phys. 3, 76-1 14 (1952), " Recent Advances in the Electron Theory of Metals." Sec. 19] CRYSTALS 399 If the energy gap between the filled valence band and the empty con- duction band is narrow enough, a crystal may be a semiconductor even in the absence of effects due to impurities. Germanium with an energy gap of 0.72 ev and grey tin with O.lOev are examples of such intrinsic semi- conductors. 19. Brillouin zones. The band theory of the crystalline state leads to a system of allowed energy levels separated by regions of forbidden energy. In other words, electron waves having a forbidden energy cannot pass through the crystal. Those familiar with radio circuits would say that the periodic crystal structure acts as a band-pass filter for electron waves. In this simple picture we have not considered the variety of periodic patterns that may be encountered by an electron wave, depending on the direction of its path through the crystal. When this is done, it is found that special geometrical requirements are imposed on the band structure, so that it is not necessarily the same for all directions in space. Now we can see qualitatively an important principle. If an electron wave with an energy in a forbidden region were to strike a crystal, it could not be transmitted, but would instead be strongly scattered or "reflected" in the Bragg sense. The Bragg relation therefore defines the geometric structure of the allowed energy bands. This principle was first enunciated by Leon Brillouin, and the energy bands constructed in this way are called the Brillouin zones of the crystal. The quantitative application of the zone theory is still in its early stages. Qualitatively it is clear that the properties of crystals are determined by the nature of the zones and the extent to which they are filled with electrons. This interpretation is especially useful in elucidating the structures of metal alloys. 20. Alloy systems electron compounds. If two pure metals crystallize in the same structure, have the same valence and atoms of about the same size, they may form a continuous series of solid solutions without undergoing any changes in structure. Examples are the systems Cu-Au and Ag-Au. When these conditions are not fulfilled, a more complicated phase diagram will result. An example is that for the brass system, copper and zinc. Pure copper crystallizes in a face-centered-cubic structure and dissolves up to about 38 per cent zinc in this a phase. Then the body-centered-cubic ft phase super- venes. At about 58 per cent zinc, a complex cubic structure begins to form, called "y brass," which is hard and brittle. At about 67 per cent Zn, the hexa- gonal closest packed e phase arises, and finally there is obtained the r\ phase having the structure of pure zinc, a distorted hep arrangement. It is most interesting that a sequence of structure changes very similar to these is observed in a wide variety of alloy systems. Although the com- positions of the phases may differ greatly, the /?, y, and e structures are quite typical. W. Hume-Rothery was the first to show that this regular behavior was related to a constant ratio of valence electrons to atoms for each phase. 400 CRYSTALS [Chap. 13 Examples of these ratios are shown in Table 13.4. The transition metals Fe, Co, Ni follow the rule if the number of their valence electrons is taken as zero. 17 In all these cases, the zone structure determines the crystal structure, and the composition corresponding to each structure is fixed by the number of electrons required to fill the zone. Such alloys are therefore sometimes called electron compounds. TABLE 13.4 ELECTRON COMPOUNDS ILLUSTRATING HUME-ROTHERY RULE Alloy* Electrons Atoms Ratio ft Phases (Ratio 3/2 CuZn 1 -h 2 2 AgCd 1 + 2 2 CuBe 1 + 2 2 AuZn 1 4-2 2 Cu 3 Al 34-3 4 Cu 6 Sn 54-4 6 CoAl 04-3 2 FeAl 04-3 2 3:2 3:2 3:2 3:2 6:4 9:6 3:3 3:2 y Phases (Ratio 21/13) Cu 6 Zn 8 Fe 5 Zn 21 Cu 9 Ga 4 Cu 9 Al 4 Cu 31 Sn 8 54-2x8 13 21 :13 54-2x8 13 21 :13 + 2 x 21 26 42 : 26 943x4 13 21 : 13 9-1-3x4 13 21 :13 31 4-4 x 8 39 63:39 e Phases (Ratio 7/4) CuZn 3 AgCd 3 Cu 3 Sn Cu 3 Ge Au 6 Al 3 AgsAl, 14-2x3 14-2x3 34-4 344 54-3x3 54-3x3 4 4 4 4 8 8 7:4 7:4 7:4 7:4 14:8 14:8 * The alloy composition is variable within a certain range, but the nominal compositions listed always fall within the range. The body-centered /? brass structure illustrates another interesting property of some alloy systems, the order-disorder transition. At low temperatures, the structure is ordered; the copper atoms occupy only the 17 Pauling has pointed out that it seems to be unreasonable to say that iron, which is famous for its great strength, contributes nothing to the bonding in iron alloys. From magnetic moments and other data he concludes that iron actually contributes between 5 and 6 bonding electrons per atom [/. Am. Chem. Soc., 69, 542 (1947)]. Sec. 21] CRYSTALS 401 body-centered positions. At higher temperatures, the various positions are occupied at random by copper and zinc atoms. 21. Ionic crystals. The binding in most inorganic crystals is predomi- nantly ionic in character. Therefore, since coulombic forces are undirected, the sizes of the ions play a most important role in determining the final structure. Several attempts have been made to calculate a consistent set of ionic radii, from which the internuclear distances in ionic crystals could be estimated. The first table, given by V. M. Goldschmidt in 1926, was modified by Pauling. These radii are listed in Table 13.5. TABLE 13.5 IONIC CRYSTAL RADII (A)* o 0.60 0.31 0.20 0.15 1.40 1.36 Me a+ 0.95 K+ 1.33 Rb+ 1.48 Cs+ .69 g f + 0.65 Ca 4 " 4 " 0.99 Sr++ 1.13 Ba + + .35 1 3 + 0.50 Sc 34 ^ 0.81 Y 34 " 0.93 La 3+ .15 4 + 0.41 Ti 44 " 0.68 Zr 4 + 0.80 Ce 44 ^ .01 1.84 Cr* 4 * 0.57 Mo* 4 ^ 0.62 1.81 Cu+ 0.96 Ag 4 " 1.26 Au+ .37 Zn + + 0.74 Cd++ 0.97 Hg++ .10 Se~~ 1.98 Te 2.21 Br- 1.95 I- 2.16 * From L. Pauling, The Nature of the Chemical Bond, 2nd ed. (Ithaca: Cornell Univ. Press, 1940), p. 346. First, let us consider ionic crystals having the general formula CA. They may be classified according to the coordination number of the ions; i.e., the number of ions of opposite charge surrounding a given ion. The CsCl struc- ture, body centered as shown in Fig. 13.25, has eightfold coordination. The NaCi structure (Fig. 13.12) has sixfold co- ordination. Although zinc blende (Fig. 13.19a) is itself covalent, there are a few ionic crys- tals, e.g., BeO, with this structure which has fourfold coordination. The coordination number of a structure is determined primarily by the number of the larger ions, usually the anions, that can be packed around the smaller ion, usually the cation. It should therefore depend upon the radius ratio, /* C ation/ r anion r c /r A . The critical radius ratio is that obtained when the anions packed around a cation are in contact with both the cation and with one another. Consider, for example, the structure of Fig. 13.25. If the anions are at the cube corners and have each a radius a, when they are exactly touching, the unit cube has a side 2a. The length of the cube diagonal is then \/3 2a, and the diameter of the empty hole in the center of the cube is therefore Fig. 13.25. The cesium chloride structure. 402 CRYSTALS [Chap. 13 V/3 2a 2a --= 2a(\/3 1). The radius of the cation exactly filling this hole is thus a(\/3 1), and the critical radius ratio becomes r c /r A = a(\/3 ])/a ----- 0.732. By this simple theory, whenever the ratio falls below 0.732, the structure can no longer have eightfold coordination, and indeed should go over to the sixfold coordination of NaCl. In the sixfold coordination, a given ion at the center of a regular octa- hedron is surrounded by six neighbors at the corners. The critical radius ratio for this structure may readily be shown to be \/2 - 1 0.414. The next lower coordination would be threefold, at the corners of an equilateral triangle, with a critical ratio of 0.225. The structures and ionic-radius ratios of a number of CA compounds are summarized in Table 13.6. The radius-ratio rule, while not infallible, provides the principal key to the occurrence of the different structure types. TABLE 13.6 STRUCTURES AND RADIUS RATIOS OF CA IONIC CRYSTALS Cesium Chloride Structure - Sodium Chloride Structure Zinc Blende or Wurtzite Structure Theoretical Range CsCl CsBr Csl ).732 0.732-0.414 0.414-0.225 0.93 KF 0.98 Rbl 0.69 NaBr 0.49 ZnS 0.40 0.87 0.78 BaO RbF 0.94 0.92 BaSe KBr 0.68 0.68 MgO CaTe 0.46 0.45 MgTe BeO 0.29 0.22 RbCl 0.82 SrS 0.61 LiF 0.44 BeS 0.17 SrO 0.81 BaTe 0.61 Nal 0.44 BeSe 0.16 CsF RbBr BaS 0.80 0.76 0.73 Kl SrSe CaS 0.60 0.57 0.58 MgS MgSe LiBr 0.36 0.33 0.31 BeTe 0.14 KC1 0.73 NaCl 0.53 LiCl 0.30 CaO 0.71 SrTe 0.51 Lil 0.28 NaF 0.70 CaSe 0.50 The structures of CA 2 ionic crystals are found to be governed by the same coordination principles. Four common structures are shown in Fig. 13.26. In fluorite each Ca++ is surrounded by eight F~ ions at the corners of a cube, and each F~ is surrounded by four Ca +4 ~ at the corners of a tetra- hedron. This is an example of 8 : 4 coordination. The structure of rutile illustrates a 6 : 3 coordination, and that of cristobalite a 4 : 2 type. Once again the coordination is determined primarily by the radius ratio. The cadmium-iodide structure illustrates the result of a departure from typically ionic binding. The iodide ion is easily polarized, and one can distinguish definite CdI 2 groups forming a layerlike arrangement. Sec. 22] CRYSTALS 403 (d) Fig. 13.26. CA 2 structures: (a) fluonte; (b) rutile; (c) ft cnstobalite; (d) cadmium iodide. 22. Coordination polyhedra and Pauling's Rule. Many inorganic crystals contain oxygen ions; their size is often so much larger than that of the cations that the structure is largely determined by the way in which they pack to- gether. The oxygens are arranged in coordination polyhedra around the cations, some common examples being the following: Around B: 3 O\s at corners of equilateral triangle Si, Al, Be, B, Zn : 4 O's at corners of tetrahedron Al, Ti, Li, Cr: 6 O's at corners of octahedron For complex structures, Pauling has given a general rule that determines how these polyhedra can pack together. Divide the valence of the positive ion by the number of surrounding negative ions; this gives the fraction of the valence of a negative ion satisfied by this positive ion. For each negative ion, the sum of the contributions from neighboring positive ions should equal its valence. This rule simply expresses the requirement that electro- static lines of force, starting from a positive ion, must end on a negative ion in the immediate vicinity, and not be forced to wander throughout the structure seeking a distant terminus. As an example of the application of the rule, consider the silicate group, (SiO 4 ). The valence of the positive ion, Si +4 , is +4. Therefore each O ion has one valence satisfied by the Si+ 4 ion, i.e., one-half of its total valence of two. It is therefore possible to join each corner of a silicate tetrahedron to another silicate tetrahedron. It is also possible for the silicates to share edges and faces, although these arrangements are less favorable energetically, since they bring the central Si+ 4 ions too close together. In the (A1O 6 ) octahedron, only a valence of \ for each O is satisfied 404 CRYSTALS [Chap. 13 by the central Al+ 3 ion. It is therefore possible to join two aluminum octa- hedra to each corner of a silicate tetrahedron. The various ways of linking the silicate tetrahedra give rise to a great diversity of mineral structures. The following classification was given by W. L. Bragg: (a) Separate SiO 4 groups (b) Separate Si O complexes (c) Extended Si O chains (d) Sheet structures (e) Three-dimensional structures An example from each class is pictured in Fig. 13.27. In many minerals, other anionic groups and cations also occur, but the general principles that (S.0 4 ) 4 " (S.0 3 )< (TETRAHEDRA SHARING CORNERS) (Si 2 o 7 r (0) (Si 2 5 ) z (Si 4 0,,) 6 " (DOUBLE CHAIN) (C) Fig. 13.27. Silicate structures: (a) isolated groups; (b) hexagonal-type sheets; (c) extended chains; (d) three-dimensional framework. (After W. L. Bragg, The Atomic Structure of Minerals, Cornell University Press, 1937.) govern the binding remain the same. The structural characteristics are naturally reflected in the physical properties of the substances. Thus the Sec. 23] CRYSTALS 405 chainlike architecture is found in the asbestos minerals, the sheet arrange- ment in micas and talcs, and the feldspars and zeolites are typical three- dimensional polymers. 23. Crystal energy the Born-Haber cycle. The binding energy in a purely ionic crystal can be calculated via ordinary electrostatic theory. The potential energy of interaction of two oppositely charged ions may be written -z&e 2 be 2 U=- Y~ + 7n (13.8) where r is the internuclear separation and ze the ionic charge. In calculating the electrostatic energy of a crystal, we must take into account not only the attraction between an ion and the oppositely charged ions coordinated around it, but also the repulsions between ions of like sign at somewhat larger separations, then attractions between the unlike ions once removed, and so on. Therefore, for each ion the electrostatic interaction will be a sum of terms, alternately attractive and repulsive, and diminishing in magnitude owing to the inverse-square law. For any given structure this summation amounts to little more than relating all the different internuclear distances to the smallest distance r. Thus, corresponding with eq. (13.8) for an ionic molecule, there is obtained for the potential energy of an ionic crystal per mole u, -<"?" + (,> The constant A, which depends on the type of crystal structure, is called the Madelung constant. 1 * If e is in esu and if is in kcal per mole, one has the following typical A values: NaCl structure, A ----- 1.74756; CsCl, 1.76267; rutile, 4.816. At the equilibrium internuclear distance r , the energy is a minimum, so that (dU/dr) ft 0. Hence for the case z l ----- z 2 , ANe 2 z 2 nBe 2 ANz*t I AM (13.10) n The value of the exponent n in the repulsive term can be estimated from the compressibility of the crystal, since work is done against the repulsive forces in compressing the crystal. Typical values of n range from 6 to 12, indicative of the rapid rise in repulsion as the internuclear separation is narrowed. 18 J. Sherman, Chem. Rev., 77, 93 (1932). 406 CRYSTALS [Chap. 13 The so-called crystal energy is obtained from eqs. (13.9) and (13.10) as This is the heat of reaction of gaseous ions to yield the solid crystal. For example, for rock salt: Na-(g)-| Cl-(g)- NaCl(c) f c Calculated values of E c can be compared with other thermochemical quantities by means of the Born-Haber cycle. For the typical case of NaCl, this has the form: NaCl (c) ------ E - c - -> Na *(g) + Cl" (g) . Na (c) + C1 2 (g) - -- - -> Na (g) 4 Cl (g) The energetic quantities entering into the cycle are defined as follows, all per mole: EC the crystal energy Q the standard heat of formation of crystalline NaCl S = heat of sublimation of metallic Na / the ionization potential of Na A - the electron affinity of Cl D = the heat of dissociation of C1 2 (g) into atoms For the cyclic process, by the First Law of Thermodynamics: c - S f / f iD -A - Q (13.12) All the quantities on the right side of this equation can be evaluated, at least for alkali-halide crystals, and the value obtained for the crystal energy can be compared with that calculated from eq. (13.11). The ionization potentials / are obtained from atomic spectra, and the dissociation energies D can be accurately determined from molecular spectra. Most difficult to measure are the electron affinities A. 19 A summary of the figures obtained for various crystals is given in Table 13.7. When the calculated crystal energy deviates widely from that obtained through the Born-Haber cycle, one may suspect nonionic contributions to the crystal binding. 24. Statistical thermodynamics of crystals: the Einstein model. If one could obtain an accurate partition function for a crystal, it would then be possible to calculate immediately all its thermodynamic properties by making use of the general formulas of Chapter 12. For one mole of a crystalline substance, containing N atoms, there are 19 See, for example, P. P. Sutton and J. E. Mayer, J. Chem. Phys., 2, 146 (1934); 3, 20 (1935). Sec. 24] CRYSTALS 407 TABLE 13.7 THE BORN-HABER CYCLE (Energy Terms in Kilocalories per Mole) Crystal ~Q / 5 D A E e E c * NaCl 99 117 26 54 88 181 190 NaBr 90 117 26 46 80 176 181 Nal 77 117 26 34 71 166 171 KC1 104 99 21 54 88 163 173 KBr 97 99 21 46 80 160 166 Kl 85 99 21 34 71 151 159 RbCl 105 95 20 54 88 159 166 RbBr 99 95 20 46 80 157 161 Rbl 87 95 20 34 71 148 154 * Calculated, Eq. (13.11). 3W degrees of freedom. Except when there is rotation of the atoms within the solid, we can consider that there are 3jV vibrational degrees of freedom, since 3W 6 is to all intents and purposes still 3N. The precise determina- tion of 3N normal modes of vibration for such a system would be an im- possible task, and it is fortunate that some quite simple approximations give sufficiently good answers. First of all, let us suppose that the 3N vibrations arise from independent oscillators, and then that these are harmonic oscillators, which is a good enough approximation at low temperatures, when the amplitudes are small. The model proposed by Einstein in 1906 assigned the same frequency v to all the oscillators. The crystalline partition function according to the Einstein model is, from eqs. ( 12.35) and (12.23), (13.14) (13.15) (13.16) (13.17) z = * It follows immediately that, E - E - 3Nhv(e hv/kT ~ I)" 1 S=3m[ f *l kT --~\n(l e Cy ^= 3yVA:rin(l /2V hv Particularly interesting is the predicted temperature variation of C v . We recall that Dulong and Petit, in 1819, noted that the molar heat capacities of the solid elements, especially the metals, were usually around 3R = 6 calories per degree. Later measurements showed that this figure was merely 408 CRYSTALS [Chap. 13 a high-temperature limiting value, approached by different elements at different temperatures. If we expand the expression in eq. (13.17) and simplify somewhat, 20 we obtain ______ (13 ig\ ^ ' } When r is large, this expression reduces to C v 37*. For smaller T's, a curve like the dotted line in Fig. 13.28 is obtained, the heat capacity being a universal function of (v/T). The frequency v can be determined from one experimental point at low temperatures and then the entire heat-capacity curve can be drawn for the substance. The agreement with the experimental data is good except at the lowest temperatures. It is clear that the higher the fundamental vibration frequency v, the larger is the quantum of vibra- tional energy, and the higher the temperature at which C v attains the classical value of 3R. For example, the frequency for diamond is 2.78 x 10 13 sec- 1 , but for lead it is only 0.19 x 10 13 sec" 1 , so that C v for diamond is only about 1.3 at room temperature, but C v for lead is 6.0. The elements that follow Dulong and Petit's rule are those with relatively low vibration frequencies. 25. The Debye model. If, instead of a single fundamental frequency, a spectrum of vibration frequencies is taken for the crystal, the statistical problem becomes somewhat more complicated. One possibility is to assume that the frequencies are distributed according to the same law as that given on page 261 for the distribution of frequencies in black-body radiation. This problem was solved by P. Debye. Instead of using eq. (13.14), the energy must be obtained by averaging over all the possible vibration frequencies v t of the solid, from to V M the maximum frequency. This gives M hv ~ ~~~' 3N o ***- 1 = i?o 7^-\ (13- 19) Since the frequencies form a virtual continuum the summation is replaced by an integration, by using the distribution function for the frequencies found in eq. (10.14) (multiplied by $ since we have one longitudinal and two transverse vibrations, instead of the two transverse of radiation). Thus dn ^f(v)dv = 1277- ^ r 2 dv (13.20) c^ where c is now the velocity of elastic waves in the crystal. Then eq. (13.19) becomes E ~ E "* "-^ dv (13 - 21) Recalling that cosech x = 2y(e" - -), and e* = 1 + x + (x/2!) + (*/3!) + . . . . Sec. 25] CRYSTALS 409 Before substituting eq. (13.20) in (13.21) we eliminate c by using eq. (10.14), since when n = 3N 9 v = v M9 for each direction of vibration, 4n 3 a _ 47T 3 9N 2 Then eq. (13.21) becomes Jo By differentiation with respect to T y C v = TV vV' Jo (e*"" lkT dv ~r kT*v~* Let us set x = Hv/kT, whereupon eq. (13.23) becomes (13.22) (13.23) krv r v M 7 Jo v - o* (13 ' 24) The Debye theory predicts that the heat capacity of a solid as a function of temperature should depend only on the characteristic frequency V M . If Fig. 13.28. The molar heat capacity of solids. (After F. Seitz, The Modern Theory of Solids, McGraw-Hill, 1940.) the heat capacities of different solids are plotted against kTjhv M , they should fall on a single curve. Such a plot is shown in Fig. 13.28, and the confirma- tion of the theory appears to be very good. Debye has defined a characteristic temperature, & D hv M /k, and some of these characteristic temperatures are listed in Table 13.8 for various solids. The theory of Debye is really adequate for isotropic solids only, and further theoretical work will be necessary 410 CRYSTALS [Chap. 13 before we have a comprehensive theory applicable to crystals with more complicated structures. TABLE 13.8 DEBYE CHARACTERISTIC TEMPERATURES Substance OD Substance BD Substance 0D Na 159 Be 1000 Al 398 K Cu 100 315 Mg Ca 290 230 Ti Pb 350 88 Ag Au 215 180 Zn Hg 235 96 Pt Fe 225 420 KC1 NaCl 227 281 AgCl AgBr 183 144 CaF 2 FeS 2 474 645 The application of eq. (13.24) to the limiting cases of high and very low temperatures is of considerable interest. When the temperature becomes large, e hvlkT becomes small, and the equation may readily be shown to reduce to simply C v ~- 3/?, the Dulong and Petit expression. When the temperature becomes low, the integral may be expanded in a power series to show that C v - aT* (13.25) This r 3 law holds below about 30 K and is of great use in extrapolating heat-capacity data to absolute zero in connection with studies based on the Third Law of Thermodynamics (cf. page 90). PROBLEMS 1. Show that a face-centered-cubic lattice can also be represented as a rhombohedral lattice. Calculate the rhombohedral angle a. 2. To the points in a simple orthorhombic lattice add points at \ \ 0, \ \\ I.e., at the centers of a pair of opposite faces in each unit cell. Prove that the resulting arrangement of points in space is not a lattice. 3. Prove that the spacing between successive planes (hkl) in a cubic lattice is a/Vh* f k 2 -f~ 7 2 where a is the side of the unit cell. 4. The structure of fluorite, CaF 2 , is cubic with Z --- 4, a Q 5.45 A. The Ca++ ions are at the corners and face centers of the cube. The F~ ions are at (Hi, Hi, HI, Hi, *ft, if*. Hi. Hi)- Calculate the nearest distance of approach of Ca Ca, F F, Ca F. Sketch the arrangement of ions in the planes 100, 110, 111. 5. MgO has the NaCl structure and a density of 3.65 g per cc. Calculate the values of (sin 0)/A at which scattering occurs from the planes 100, 110, 111,210. Chap. 13] CRYSTALS 411 6. Nickel crystallizes in the fee structure with a Q 3.52 A. Calculate the distance apart of nickel atoms lying in the 100, 1 10, and 1 1 1 planes. 7. A Debye-Scherrer powder picture of a cubic crystal with radiation of X -~ 1.539 A displayed lines at the following scattering angles: I j i No. of 'me 1 2 | 3 4 5 1 6 7 8 9 0,deg 13.70 15.89 22.75 26.91 28.25 33.15 37.00 37.60 41.95 Intensity w vs s vw m w w m m Note: w weak; s strong; m medium; v very. Index these lines. Calculate a (} for the crystal. Identify the crystal. Explain the intensity relation between lines 5 and 4 in terms of the structure factor. 8. Calculate the atomic volume for spheres of radius 1 A in ccp and hep structures. Give the unit cell dimensions, a Q for cubic, a Q and r for hexagonal. 9. Show that the void volume for spheres in both ccp and hep is 25.9 per cent. What would be the per cent void in a bcc structure with corner atoms in contact with the central atom? 10. White tin is tetragonal with a (} b Q 5.819 A, and c () 3.175 A. Tin atoms are at 000, i J , i j, i |. Calculate the density of the crystal. Grey tin has the diamond structure with a {} -- 6.46 A. Describe how the tin atoms must move in the transformation from grey to white tin. 11. In a powder picture of lead with Cu K a radiation (X 1.539 A) the line from the 531 planes appeared at sin 0.9210. Calculate a and the density of lead. 12. The Debye characteristic temperature of copper is ( H ) =-- 315 U K. Cal- culate the entropy of copper at 0C and 1 atm assuming that a 4.95 x 10~ 5 deg" 1 , Po 7.5 x 10 7 atm l , independent of the temperature. 13. Calculate the proton affinity of NH 3 from the following data (i.e., the A for reaction NH 3 + H f NH 4 f ). NH 4 F crystallizes in the ZnO type structure whose Madelung constant is 1.64. The Born repulsion exponent for NH 4 F is 8, the interionic distance is 2.63 A. The electron affinity of fluorine is 95.0 kcal. The ionization potential of hydrogen is 31 1.9 kcal. The heats of formation from the atoms are: NH 3 279.6; N 2 --= 225; H 2 - 104.1 ; F 2 - 63.5 kcal. The heat of reaction \ N 2 (g) | 2 H 2 (g) -f J F 2 (g) - NH 4 +F~(c)is 11 1.9 kcal. REFERENCES BOOKS 1. Barrett, C. S., Structure of Metals (New York: McGraw-Hill, 1952). 2. Bragg, W. H., and W. L. Bragg, The Crystalline State, vol. I (London: Bell, 1934). 3. Buerger, M. J., X-Ray Crystallography (New York: Wiley, 1942). 412 CRYSTALS [Chap. 13 4. Bunn, C. W., Chemical Crystallography (New York: Oxford, 1945). 5. Evans, R. C., Crystal Chemistry (London: Cambridge, 1939). 6. Hume-Rothery, W., Atomic Theory for Students of Metallurgy (London: Institute of Metals, 1947). 7. Kittel, C., Introduction to Solid-State Physics (New York: Wiley, 1953). 8. Lonsdale, K., Crystals and X-Rays (New York: Van Nostrand, 1949). 9. Phillips, F. C., An Introduction to Crystallography (New York: Long- mans, 1946). 10. Wells, A. F., Structural Inorganic Chemistry (New York: Oxford, 1950). 11. Wilson, A. H., Semiconductors and Metals (London: Cambridge, 1939). 12. Wooster, W. A., Crystal Physics (London: Cambridge, 1938). ARTICLES 1. Bernal, J. D., /. Chem. Soc., 643-66 (1946), "The Past and Future of X-Ray Crystallography." 2. DuBridge, L. A., Am. J. Phys., 16, 191-98 (1948), "Electron Emission from Metal Surfaces." 3. Frank, F. C., Adv. Phys., /, 91-109 (1952), "Crystal Growth and Disloca- tions." 4. Fuoss, R. M., J. Chem. Ed., 19, 190-93, 231-35 (1942), "Electrical Pro- perties of Solids." 5. Lonsdale, K., Endeavour, 6, 139-46 (1947), "X-Rays and the Carbon Atom." 6. Robertson, J. M., /. Chem. Soc., 249-57 (1945), "Diffraction Methods in Modern Structural Chemistry." 7. Sidhu, S. S., Am. J. Phys. 9 16, 199-205 (1948), "Structure of Cubic Crystals." 8. Smoluchowski, R., and J. S. Koehler, Ann. Rev. Phys. Chem., 2, 187-216 (1951), "Band Theory and Crystal Structure." 9. Weisskopf, V. F., Am. J. Phys., 11, 111-12 (1943), "Theory of the Elec- trical Resistance of Metals." CHAPTER 14 Liquids 1. The liquid state. The crystalline and the gaseous states of matter have already been surveyed in some detail. The liquid state remains to be con- sidered. Not that every substance falls neatly into one of these three classifi- cations there is a variety of intermediate forms well calculated to perplex the morphologist : rubbers and resins, glasses and liquid crystals, fibers and protoplasm. Gases, at least in the ideal approximation approached at high tempera- tures and low densities, are characterized by complete randomness on the molecular scale. The ideal crystal, on the other hand, is one of nature's most orderly arrangements. Because the extremes of perfect chaos and perfect harmony are both relatively simple to treat mathematically, the theory of gases and crystals is at a respectably advanced stage. Liquids, however, representing a peculiar compromise between order and disorder, have so far defied a comprehensive theoretical treatment. Thus in an ideal gas, the molecules move independently of one another and interactions between them are neglected. The energy of the perfect gas is simply the sum of the energies of the individual molecules, their internal energies plus their translational kinetic energies; there is no intermolecular potential energy. It is therefore possible to write down a partition function such as that in eq. (12.23), from which all the equilibrium properties of the gas are readily derived. In a crystalline solid, translational kinetic energy is usually negligible. The molecules, atoms, or ions vibrate about equilibrium positions to which they are held by strong intermolecular, interatomic, or interionic forces. In this case too, an adequate partition function, such as that in eq. (13.13), can be obtained. In a liquid, on the other hand, the situation is much harder to define. The cohesive forces are sufficiently strong to lead to a condensed state, but not strong enough to prevent a considerable translational energy of the in- dividual molecules. The thermal motions introduce a disorder into the liquid without completely destroying the regularity of its structure. It has therefore not yet been possible to devise an acceptable partition function for liquids. It should be mentioned that in certain circles it is now considered in- delicate to speak of individual molecules in condensed systems, such as liquids or solids. As James Kendall once put it, we may choose to imagine that "the whole ocean consists of one loose molecule and the removal of a fish from it is a dissociation process." 413 p 414 LIQUIDS [Chap. 14 In studying liquids, it is often helpful to recall the relation between entropy and degree of disorder. Consider a crystal at its melting point. The crystal is energetically a more favorable structure than the liquid to which it melts. It is necessary to add energy, the latent heat of fusion, to effect the melting. The equilibrium situation, however, is determined by the free-energy difference, AF = A// JAS. It is the greater randomness of the liquid, and hence its greater entropy, that finally makes the T&S term large enough to overcome the A// term, so that the crystal melts when the following condition is reached : The sharpness of the melting point is noteworthy. There does not in general appear to be a continuous gradation of properties between liquid CRYSTAL LIQUID GAS Fig. 14.1. Two-dimensional models. and crystal. The sharp transition is due to the extremely rigorous geometrical requirements that must be fulfilled by a crystal structure. It is not possible to introduce small regions of disorder into the crystal without at the same time seriously disturbing the structure over such a long range that the crystalline arrangment is destroyed. Two-dimensional models of the gaseous, liquid, and crystalline states are illustrated in Fig. 14.1. The picture of the liquid was constructed by J. D. Bernal by introducing around "atom" A only five other atoms instead of its normal close-packed coordination of six. Every effort was then made to draw the rest of the circles in the most ordered arrangement possible, with the results shown. The one point of abnormal coordination among some hundred atoms sufficed to produce the long-range disorder believed to be typical of the liquid state. We see that if there is to be any abnormal coordination at all, there has to be quite a lot of it. Herein probably lies an explanation of the sharpness of melting. When the thermal motions in one region of a crystal suffice to destroy the regular structure, the irregularity rapidly spreads throughout the entire specimen; thus disorder in a crystal may be contagious. These remarks should not be taken to imply that all crystals are ideally perfect, and admit of no disorder at all. It is only that the amount of disorder allowed is usually very limited. When the limit is exceeded, complete melting of the crystal occurs. There are two types of defect that occur in crystal structures. There may be vacant lattice positions or "holes," and there may be interstitial positions occupied by atoms or ions. Sec. 2] LIQUIDS 415 It is sometimes convenient to classify liquids, like crystals, from a rather chemical standpoint, according to the kind of cohesive forces that hold them together. Thus there are the ionic liquids such as molten salts, the liquid metals consisting of metal ions and fairly mobile electrons, liquids such as water held together mainly by hydrogen bonds, and finally molecular liquids in which the cohesion is due to the van der Waals forces between essentially saturated molecules. Many liquids fall into this last group, and even when other forces are present, the van der Waals contribution may be large. The nature of these forces will be considered later in this chapter. 2. Approaches to a theory for liquids. From these introductory remarks it may be evident that there are three possible ways of essaying a theory of the liquid state, two cautious ways and one direct way. The cautious approaches are by way of the theory of gases and the theory of solids. The liquid may be studied as an extremely imperfect gas. This is a reasonable viewpoint, since above the critical point there is no distinction at all between liquid and gas, and the so-called "fluid state" of matter exists. On the other hand, the liquid may be considered as similar to a crystal, except that the well-ordered arrangement of units extends over a short range only, five or six molecular diameters, instead of over the whole specimen. This is sometimes called "short-range order and long-range dis- order." This is a reasonable viewpoint, since close to the melting point the density of crystal and liquid are very similar; the solid usually expands about 10 per cent in volume, or only about 3 per cent in intermodular spacing, when it melts. It should be realized too that whatever order exists in a liquid structure is continuously changing because of thermal motions of the in- dividual molecules; it is the time average of a large number of different arrangements that is reflected in the liquid properties. The imperfect-gas theory of liquids would be suitable close to the critical point; the disordered-crystal theory would be best near the melting point. At points between, they might both fail badly. A more direct approach to liquids would abandon these flanking attacks and try to develop the theory directly from the fundamentals of intermolecular forces and statistical mechanics. This is a very difficult undertaking, but a beginning has been made by Max Born, J. G. Kirkwood, and others. We shall consider first some of the resemblances between liquid and crystal structures, as revealed by the methods of X-ray diffraction. 3. X-ray diffraction of liquids. The study of the X-ray diffraction of liquids followed the development of the method of Debye and Scherrer for powdered crystals. As the particle size of the powder decreases, the width of the lines in the X-ray pattern gradually increases. From particles around 100 A in diameter, the lines have become diffuse halos, and with still further decrease in particle size the diffraction maxima become blurred out altogether. If a liquid were completely amorphous, i.e., without any regularity of structure, it should also give a continuous scattering of X rays without 416 LIQUIDS [Chap. 14 maxima or minima. This was actually not found to be the case. A typical pattern, that obtained from liquid mercury, is shown in (a), Fig. 14.2, as a microphotometer tracing of the photograph. This reveals the maxima and minima better than the unaided eye. One or two or sometimes more intensity maxima appear, whose positions often correspond closely to some of the larger interplanar spacings that occur in the crystalline structures. In the case of the metals, these are the close-packed structures. It is interesting that a crystal like bismuth, which has a peculiar and rather loose solid structure, 3i t 2 V 5 10 15 20 25 3.0 (d) DISTANCE FROM CENTRAL SPOT ON FILM-Cm 5 10 II T2 Fig. 14.2. 4 5 6 7 8 9 (b) f- ANGSTROM UNITS (a) Photometric tracing of liquid-mercury picture; (b) radial distribution function for liquid mercury. is transformed on melting into a close-packed structure. We recall that bismuth is one of the few substances that contract in volume when melted. The fact that only a few maxima are observed in the diffraction patterns from liquids is in accord with the picture of short-range order and increasing disorder at longer range. In order to obtain the maxima corresponding to smaller interplanar spacings or higher orders of diffraction, the long-range order of the crystal must be present. The diffraction maxima observed with crystals or liquids should be dis- tinguished from those obtained by the X-ray or electron diffraction of gases. The latter arise from the fixed positions of the atoms within the molecules. The individual molecules are far apart and distributed at random. In deriving on page 327 the diffraction formula for gases, we considered only a single molecule and averaged over all possible orientations in space. With both solids and liquids the diffraction maxima arise from the ordered arrangement Sec. 4] LIQUIDS 417 of the units (molecules or atoms) in the condensed three-dimensional struc- ture. Thus gaseous argon, a monatomic gas, would yield no maxima, but liquid argon displays a pattern similar to that of liquid mercury. It is possible to analyze the X-ray diffraction data from liquids by using the Bragg relation to calculate spacings. A more instructive approach, how- ever, is to consider a liquid specimen as a single giant molecule, and then to use the formulas, such as eq. (11.19), derived for diffraction by single molecules. A simple theory is obtained only in the case of monatomic liquids, such as the metals and group O elements. The arrangement of atoms in such a liquid is described by introducing the radial distribution function g(r). Taking the center of one atom as origin, this g(r) gives the probability of finding the center of another atom at the end of a vector of length r drawn from the origin. The chance of finding another atom between a distance r and r \ dr, irrespective of angular orientation, is therefore 47rr 2 g(r)dr (cf. page 187). It is now possible to obtain, for the intensity of scattered X radiation, an expression similar to that in eq. (11.19), except that instead of a summation over individual scattering centers, there is an integration over a continuous distribution of scattering matter, specified by g(r). Thus 1(0) f 4rrr Wr) ^ dr (14.1) Jo fir sin I - I \^/ As before, /^ = * A By an application of Fourier's integral theorem, this integral can be inverted, 1 yielding By use of this relationship it is possible to calculate a radial-distribution curve, such as that plotted in (b), Fig. 14.2, from an experimental scattering curve, such as that in (a), Fig. 14.2. The regular coordination in the close- packed liquid-mercury structure is clearly evident, but the fact that maxima in the curve are rapidly damped out at larger interatomic distances indicates that the departure from the ordered arrangement becomes greater and greater as one travels outward from any centrally chosen atom. 4. Results of liquid-structure investigations. X-ray diffraction data from liquids are not sufficiently detailed to permit complete structure analyses like those of crystals. This situation is probably inevitable because the diffraction experiments reveal only an average or statistical structure, owing to the continual destruction and reformation of ordered arrangements by the thermal motions of the atoms or molecules in the liquid. One view, however, proposed by G. W. Stewart (around 1930), is that 1 See, for example, H. Bateman, Partial Differential Equations of Mathematical Physics (New York: Dover Publications, 1944), p. 207. 418 LIQUIDS [Chap. 14 there are actually large regions in a liquid that are extremely well ordered. These are called cybotactic groups, and are supposed to contain up to several hundred molecules. These islands of order are dispersed in a sea of almost completely disordered molecules, whose behavior is essentially that of a very dense gas. There is a dynamic equilibrium between the cybotactic groups and the unattached molecules. This picturesque model is probably unsuitable for the majority of liquids and it is usually preferable to think of the disorder as being fairly well averaged throughout the whole structure. The results with liquid metals have already been mentioned. They appear to have approximately close-packed structures quite similar to those of the solids, with the interatomic spacings expanded by about 5 per cent. The number of nearest neighbors in a close-packed structure is twelve. In liquid sodium, each atom is found to have on the average ten nearest neighbors. One of the most interesting liquid structures is that of water. J. Morgan and B. E. Warren 2 have extended and clarified an earlier discussion by Bernal and Fowler. They studied the X-ray diffraction of water over a range of temperatures, and obtained the radial distribution curves. The maximum of the large first peak occurs at a distance varying from about 2.88 A at 1.5C to slightly over 3.00 A at 83C. The closest spacing in ice is at 2.76 A. It might at first be thought that this result is in disagree- ment with the fact that there is a contraction in volume of about 9 per cent when ice melts. Further analysis shows, however, that the coordination in liquid water is not exactly the same as the tetrahedral coordination of four nearest neighbors in ice. The number of nearest neighbors can be estimated from the area under the peaks in the radial-distribution curve, with the following results: Temperature, C: Number nearest neighbors: 1.5 4.4 13 4.4 30 4.6 62 4.9 83 4.9 Thus the tetrahedral arrangement in ice is partially broken down in water, to an extent that increases with temperature. This breakdown permits closer packing, although water is of course far from being a closest-packed struc- ture. The combination of this effect with the usual increase of intermolecular separation with temperature explains the occurrence of the maximum in the density of water at 4C. Among other structures that have been investigated, those of the long- chain hydrocarbons may be mentioned. These molecules tend to pack with parallel orientations of the chains, sometimes suggesting an approach to Stewart's cybotactic models. 5. Liquid crystals. In some substances the tendency toward an ordered arrangement is so great that the crystalline form does not melt directly to a 2 J. Chem. Phys., 6, 666 (1938). This paper is recommended as a clear and excellent example of the X-ray method as applied to liquids. Sec. 5] LIQUIDS 419 liquid phase at all, but first passes through an intermediate stage (the meso- morphic or paracrystalline state), which at a higher temperature undergoes a transition to the liquid state. These intermediate states have been called liquid crystals, since they display some of the properties of each of the adjacent states. Thus some paracrystalline substances flow quite freely but (a) I I Hill 1 1111 II II III! (b) (c) (d) Fig. 14.3. Degrees of order: (a) crystalline orientation and periodicity; (b) smectic orientation and arrangement in equispaced planes, but no periodicity within planes; (c) nematic orientation without periodicity; (d) isotropic fluid neither orientation nor periodicity. are not isotropic, exhibiting interference figures when examined with polar- ized light ; other varieties flow in a gliding stepwise fashion and form "graded droplets" having terracelike surfaces. A compound frequently studied in its paracrystalline state is />-azoxy- anisole, O OCH a CH The solid form melts at 84 to the liquid crystal, which is stable to 150 at which point it undergoes a transition to an isotropic liquid. The compound ethyl /7-anisalaminocinnamate, :H=N/ \ CH=C P/ \_C H=N / \_ CH=CH COOC 2 H 5 420 LIQUIDS [Chap. 14 passes through three distinct paracrystalline phases between 83 and 139. Cholesteryl bromide behaves rather differently. 3 The solid melts at 94 to an isotropic liquid, but this liquid can be supercooled to 67 where it passes over into a metastable liquid-crystalline form. Liquid crystals tend to occur in compounds whose molecules are markedly unsymmetrical in shape. For example, in the crystalline state long-chain molecules may be lined up as shown in (a), Fig. 14.3. On raising the temperature, the kinetic energy may become sufficient to disrupt the binding between the ends of the molecules but insufficient to overcome the strong lateral attractions between the long chains. Two types of anisotropic melt might then be obtained, shown in (b) and (c), Fig. 14.3. In the smectic (ojurjypQL, "soap") state the molecules are oriented in well-defined planes. When a stress is applied, one plane glides over another. In the nematic (*>/7//a, "thread") state the planar structure is lost, but the orientation is pre- served. With some substances, notably the soaps, several different phases, differentiated by optical and flow properties, can be distinguished between typical crystal and typical liquid. It has been suggested that many of the secrets of living substances may be elucidated when we know more about the liquid-crystalline state. Joseph Needham 4 has written : Liquid crystals, it is to be noted, are not important for biology and embryology because they manifest certain properties which can be regarded as analogous to those which living systems manifest (models), but because living systems actually are liquid crystals, or, it would be more correct to say, the paracrystalline state undoubtedly exists in living cells. The doubly refracting portions of the striated muscle fibre are, of course, the classical instance of this arrangement, but there are many other equally striking instances, such as cephalopod spermatozoa, or the axons of nerve cells, or cilia, or birefringent phases in molluscan eggs, or in nucleus and cytoplasm of echinoderm eggs. . . . The paracrystalline state seems the most suited to biological functions, as it combines the fluidity and diffusibility of liquids while preserving the possibilities of internal structure characteristic of crystalline solids. 6. Rubbers. Natural rubber is a polymerized isoprene,' with long hydro- carbon chains of the following structure: ( CH 2 CH CH 2 CH 2 ) n I CH 3 The various synthetic rubbers are al$o long, linear polymers, with similar structures. The elasticity of rubber is a consequence of the different degrees of ordering of these chains in the stretched and unstretched states. An idealized model of the rubber chains when stretched and when contracted 3 J. Fischer, Zeit. physik. Chern., 160A, 110 (1932). 4 Joseph Needham, Biochemistry and Morphogenesis (London: Cambridge, 1942), p. 661. Sec. 6] LIQUIDS 421 is shown in Fig. 14.4. Stretching forces the randomly oriented chains into a much more ordered alignment. The unstretched, disordered configuration is a state of greater entropy, and if the tension is released, the stretched rubber spontaneously reverts to the unstretched condition. Robert Boyle and his contemporaries talked about the "elasticity of a gas," and although we hear this term infrequently today, it is interesting to noie that the thermodynamic interpretations of the elasticity of a gas and of the elasticity of a rubber band are in fact the same. If the pressure is released on a piston that holds gas in a cylinder, the piston springs back as the gas expands. The expanded gas is in a state of higher entropy than the com- (o) (b) Fig. 14.4. Idealized models of chains in rubber: (a) stretched; (b) contracted. pressed gas: it is in a more disordered state since each molecule has a larger volume in which to move. Hence the compressed gas spontaneously expands for the same reason that the stretched rubber band spontaneously contracts. From eq. (6) on page 65, the pressure is (14.3) For a gas, the (dE/3V) T term is small, so that effectively P = T(dS/dV) T , and the pressure varies directly with r, and is determined by the change in entropy with the volume. The analog of eq. (14.3) for a rubber band of length L in which the tension is K is (14.4) It was found experimentally that K varies directly with T, so that, just as in the case of an ideal gas, the term involving the energy must be negligible. It was this observation that first led to the interpretation of rubber elasticity as an entropy effect. 422 LIQUIDS [Chap. 14 7. Glasses. The glassy or vitreous state of matter is another example of a compromise between crystalline and liquid properties. The structure of a glass is essentially similar to that of an associated liquid such as water, so that there is a good deal of truth in the old description of glasses as super- cooled liquids. The two-dimensional models in Fig. 14.5, given by W. H. Zachariasen, illustrate the differences between a glass and a crystal. The bonds are the same in both cases, e.g., in silica the strong electro- static Si O bonds. Thus both quartz crystals and vitreous silica are hard and mechanically strong. The bonds in the glass differ considerably in length and therefore in strength. Thus a glass on heating softens gradually rather (o) (b) Fig. 14.5. Two-dimensional models for (a) crystal and (b) glass. than melts sharply, since there is no one temperature at which all the bonds become loosened simultaneously. The extremely low coefficient of thermal expansion of some glasses, notably vitreous silica, is explicable in terms of a structure such as that in Fig. 14.5. The structure is a very loose one, and just as in the previously discussed case of liquid water, increasing the temperature may allow a closer coordination. To a certain extent, therefore, the structure may "expand into itself." This effect counteracts the normal expansion in interatomic distance with temperature. 8. Melting. In Table 14.1 are collected some data on the melting point, latent heat of fusion, latent heat of vaporization, and entropies of fusion and vaporization of a number of substances. It will be noted that the heats of fusion are much less than the heats of vaporization. It requires much less energy to convert a crystal to liquid than to vaporize a liquid. The entropies of fusion are also considerably lower than the entropies of vaporization. The latter are quite constant, around 21.6 eu (Trouton's rule). The constancy of the former is not so marked. For some classes of sub- stances, however, notably the close-packed metals, the entropies of fusion are seen to be remarkably constant. 9. Cohesion of liquids the internal pressure. We have so far been dis- cussing the properties of liquids principally from the disordered-crystai point of view. Whatever the model chosen for the liquid state, the cohesive Sec. 9] LIQUIDS 423 TABLE 14.1 DATA ON MELTING AND VAPORIZATION ince Heat of Fusion (kcal/mole) Heat of Vaporization (kcal/mole) Melting Point (K) Entropy of Fusion (cal/) Entropy of Vaporiza- tion (cain Metals Na Al K Fe Ag Pt Hg 0.63 24.6 371 1.70 21.1 2.55 67.6 932 2.73 29.0 0.58 21.9 336 1.72 21.0 3.56 96.5 1802 1.97 29.5 2.70 69.4 1234 2.19 27.8 5.33 125 2028 2.63 26.7 0.58 15.5 234 2.48 24.5 Ionic Crystals NaCl 7.22 183 1073 KC1 6.41 165 1043 AgCl 3.15 728 KN0 3 2.57 581 BaCl 2 5.75 1232 K 2 Cr 2 O 7 .77 671 6.72 6.15 4.33 4.42 4.65 13.07 109 93 Molecular Crystals H 2 H 2 A NH 3 C 2 H 5 OH 0.028 1.43 0.280 1.84 1.10 2.35 0.22 11.3 1.88 7.14 10.4 8.3 14 2.0 15.8 273 5.25 30.1 83 3.38 21.6 198 9.30 29.7 156 7.10 29.6 278 8.45 23.5 forces are of primary importance. Ignoring, for the time being, the origin of these forces, we can obtain an estimate of their magnitude from thermo- dynamic considerations. This estimate is provided by the so-called internal pressure. We recall from eq. (3.43) that v- = T - dT p (14.5) In the case of an ideal gas, the internal pressure term />, = (5Ej^V) T is zero since intermolecular forces are absent. In the case of an imperfect gas, the (dE/dV) T term becomes appreciable, and in the case of a liquid it may become much greater than the external pressure. The internal pressure is the resultant of the forces of attraction and the forces of repulsion between the molecules in a liquid. It therefore depends 424 LIQUIDS [Chap. 14 markedly on the volume K, and thus on the external pressure P. This effect is shown in the following data for diethyl ether at 25C. P(atm): 200 P<(atm): 2790 800 2840 2000 2530 5300 2020 7260 40 9200 -1590 11,100 -4380 For moderate increases in P, the P t decreases only slightly, but as P exceeds 5000 atm, the P t begins to decrease rapidly, and goes to large negative values as the liquid is further compressed. This behavior is a reflection, on a larger scale, of the law of force between individual molecules that was illustrated in Fig. 7.8. on page 181. Internal pressures at 1 atm and 25C are summarized in Table 14.2, taken from a compilation by J. H. Hildebrand. With normal aliphatic hydro- carbons there appears to be a gradual increase in P t with the length of the chain. Dipolar liquids tend to have somewhat larger values than nonpolar liquids. The effect of dipole interaction is nevertheless not predominant. As might be expected, water with its strong hydrogen bonds has an exceptionally high internal pressure. TABLE 14.2 INTERNAL PRESSURES OF LIQUIDS (25C and 1 atm) Compound Diethyl ether /^-Heptane /i-Octane Tin tetrachloride Carbon tetrachlork e Benzene Chloroform . Carbon bisulfide Mercury Water . Pi atm 2,370 2,510 2,970 3,240 3,310 3,640 3,660 3,670 13,200 20,000 Hildebrand was the first to point out the significance of the internal pressures of liquids in determining solubility relationships. If two liquids have about the same P t , their solution has little tendency toward positive deviations from Raoult's Law. The solution of two liquids differing con- siderably in P t will usually exhibit considerable positive deviation from ideality, i.e., a tendency toward lowered mutual solubility. Negative devia- tions from ideality are still ascribed to incipient compound or complex formation, 10. Intel-molecular forces. It should be clearly understood from earlier discussions (cf. Chapter 11) that all the forces between atoms and molecules are electrostatic in origin. They are ultimately based on Coulomb's Law of the attraction between unlike, and the repulsion between like charges. One often speaks of long-range forces and short-range forces. Thus a force that Sec. 10] LIQUIDS 425 depends on 1/r 2 will be effective over a longer range than one dependent on 1/r 7 . All these forces may be represented as the gradient of a potential func- tion,/ = 3//3r, and it is often convenient to describe the potential energies rather than the forces themselves (See Fig. 7.8, page 181.) The following varieties of intermolecular and interionic potential energies may then be distinguished: (1) The coulombic energy of interaction between ions with net charges, leading to a long-range attraction, with U ~ r^ 1 . (2) The energy of interaction between permanent dipoles, with U ~ r~ 6 . (3) The energy of interaction between an ion and a dipole induced by it in another molecule, with U ~ r~ 4 . (4) The energy of interaction between a permanent dipole and a dipole induced by it in another molecule, with U ~ r~ 6 . (5) The forces between neutral atoms or molecules, such as the inert gases, with U ~ r~ B . (6) The overlap energy arising from the interaction of the positive nuclei and electron cloud of one molecule with those of another. The overlap leads to repulsion at very close intermolecular separations, with an r~ 9 to r~ 12 potential. The van der Waais attractions between molecules must arise from inter- actions belonging to classes (2), (4), and (5). The first attempt to explain them theoretically was that of W. H. Keesom (1912), based on the interaction between permanent dipoles. Two dipoles in rapid thermal motion may sometimes be oriented so as to attract each other, sometimes so as to repel each other. On the average they are somewhat closer together in attractive configurations, and there is a net attractive energy. This energy was calculated 5 to be ,-- where ^ is the dipole moment. The observed r~ 6 dependence of the interaction energy, or r~ 7 dependence of the forces, is in agreement with deductions from experiment. This theory is of course not an adequate general explanation of van der Waals' forces, since there are considerable attractive forces between molecules, such as the inert gases, with no vestige of a permanent dipole moment. Debye, in 1920, extended the dipole theory to take into account the induction effect. A permanent dipole induces a dipole in another molecule and a mutual attraction results. This interaction depends on the polariza- bility a of the molecules, and leads to a formula, U U = - (14.7) 5 J. E. Lennard-Jones, Proc. Phys. Soc. (London), 43, 461 (1931). 426 LIQUIDS [Chap. 14 This effect is quite small and does not help us to explain the case of the inert gases. In 1930, F. London solved this problem by a brilliant application of quantum mechanics. Let us consider a neutral molecule, such as argon. The positive nucleus is surrounded by a cloud 6 of negative charge. Although the time average of this charge distribution is spherically symmetrical, at any instant the distribution will be somewhat distorted. (This may be visualized very clearly in the case of the neutral hydrogen atom, in which the electron is sometimes on one side of the proton, sometimes on the other.) Thus a "snapshot" taken of an argon atom would reveal a little dipole with a certain orientation. An instant later the orientation would be different, and so on, so that over any macroscopic period of time the instantaneous dipole moments would average to zero. Now it should not be thought that these little snapshot dipoles interact with those of other molecules to produce an attractive potential. This cannot happen since there will be repulsion just as often as attraction; there is no time for the instantaneous dipoles to line up with one another. There is, however, a snapshot-dipole polarization interaction. Each instantaneous argon dipole induces an appropriately oriented dipole moment in neighboring atoms, and these moments interact with the original to produce an in- stantaneous attraction. The polarizing field, traveling with the speed of light, does not take long to traverse the short distances between molecules. Cal- culations show that this dispersion interaction leads to a potential, U m -- -IK^ (14.8) where V Q is the characteristic frequency of oscillation of the charge dis- tribution. 7 The magnitudes of the contributions from the orientation, induction, and dispersion effects are shown in Table 14.3 for a number of simple molecules. It is noteworthy that all the contributions to the potential energy of inter- molecular attraction display an r~ 6 dependence. The complete expression for the inter molecular energy must include also a repulsive term, the overlap energy, which becomes appreciable at very close distances. Thus we may write U=- ~Ar - + Br~ n (14.9) The value of the exponent n is from 9 to 12. 11. Equation of state and intermolecular forces. The calculation of the equation of state of a substance from a knowledge of the intermolecular 6 At least a "probability cloud" see p. 276. 7 The r~* dependence of the potential can be readily derived in this case from electro- static theory. The field due to a dipole varies as 1/r 3 , and the potential energy of an induced dipole in a field F is fcaF 2 . See Harnwell, Electricity and Magnetism, p. 59. For a simple quantum-mechanical derivation of eq. (14.8), see R. H. Fowler, Statistical Mechanics (London: Cambridge, 1936), p. 296. Sec. ii] LIQUIDS TABLE 14.3 RELATIVE MAGNITUDES OF INTERMOLECULAR INTERACTIONS* 427 Molecule Dipole Moment / x 10 18 Polariz- ability a x 10 24 Energy hv (ev) Orienta- tionf Induo tionf Disper- sionf (esu cm) (cc) / ;x |/IVo CO 0.12 1.99 14.3 0.0034 0.057 67.5 HI 0.38 5.4 12 0.35 1.68 382 HBr 0.78 3.58 13.3 6.2 4.05 176 HC1 1.03 2.63 13.7 18.6 5.4 105 NH 3 HoO He 1.5 1.84 2.21 1.48 0.20 16 18 24.5 84 190 10 10 93 47 1.2 A 1.63 15.4 52 Xe 4.00 11.5 217 * J. A. V. Butler, Ann. Rep. Chem. Soc. (London), 34, 75 (1937). | Units of erg cm x 10 60 . forces is in general a problem of great complexity. The method of attack may be outlined in principle, but so far the mathematical difficulties have proved so formidable that in practice a solution has been obtained only for a few drastically simplified cases. We recall that the calculation of the equation of state reduces to cal- culating the partition function Z for the system. From Z the Helmholtz free energy A is immediately derivable, and hence the pressure, P --- ~(dA/dV) T . To determine the partition function, Z = S e ~E i /kT^ ^ Q energy levels of the system must be known. In the cases of ideal gases and crystals it is possible to use energy levels for individual constituents of the system, such as molecules or oscillators, ignoring interactions between them. In the case of liquids, this is not possible since it is precisely the interaction between different molecules that is responsible for the characteristic properties of a liquid. It would therefore be necessary to know the energy levels of the system as a whole, for example, one mole of liquid. So far this problem has not been solved. An indication of the difficulties of a more general theory may be obtained by a consideration of the theory of imperfect gases. In this case we consider that the total energy of the system H can be divided into two terms, the kinetic energy E K , and the intermolecular potential energy U: H = E K + U. For a mole of gas, U is a function of the positions of all the molecules. For the N molecules in a mole there are 37V positional coordinates, q, ft, ft . . . ft^ Therefore, U = U(q l9 ft, ft ft,v)- The partition function may now be written Z = S e - (E *+ U)lkT = 2 e- s * lkT S e~ ulkT (14.10) It is not necessary to consider quantized energy levels, and Z may be 428 LIQUIDS [Chap. 14 written in terms of an integration, rather than a summation over discrete levels. Z=Xe-** l * T S. . .Se- u >" '**** dq l9 dqt. . . dq^ (14.11) The theoretical treatment of the imperfect gas reduces to the evaluation of the so-called configuration integral, f)(T) = /.../ <?-"<' ' '" dq lt dq z . . . dq w (14.12) Since this is a repeated integral over 3 N coordinates, </ it will be easily appreciated that its general evaluation is a matter of unconscionable diffi- culty, so that the general theory has ended in a mathematical cul-de-sac. Physically, however, it is evident that the potential energy of interaction, even in a moderately dense gas, does not extend much beyond the nearest neighbors of any given molecule. This simplification still leaves a problem of great difficulty, which is at present the subject of active research. The only simple approach is to consider interactions between pairs of molecules only. This would be a suitable approximation for a slightly im- perfect gas. One lets <f>(r l9 ) be the potential energy of interaction between two molecules / and j separated by a distance /, and assumes that the total potential energy is the sum of such terms. over pairs When this potential is substituted in eq. (14.12), the configuration integral can be evaluated. The details of this very interesting, but rather long, calculation will not be given here. 8 Efforts have been made to solve the configuration integral for more exact assumptions than the interaction between pairs. These important advances toward a comprehensive theory for dense fluids are to be found in the works of J. E. Lennard- Jones, J. E. Mayer, J. G. Kirkwood, and Max Born. 12. The free volume and holes in liquids. There have been many attempts to devise a workable theory for liquids that would avoid entanglement with the terrible intricacies of the configuration integral. One of the most success- ful efforts has been that of Henry Eyring, based on the concept of a free volume. The liquid is supposed to be in many respects similar to a gas. In a gas, the molecules are free to move throughout virtually the whole container, the excluded volume (four times van der Waals' b) being almost negligible at low densities. In a liquid, however, most of the volume is excluded volume, and only a relatively small proportion is a void space or free volume in which the centers of the molecules can manoeuvre. Eyring then assumes that the partition function for a liquid differs from that for a gas in two respects: (I) the free volume V 1 is substituted for the 8 See J. C. Slater, Introduction to Chemical Physics (New York: McGraw-Hill, 1 939), p. I9l. Sec. 12] LIQUIDS 429 total volume F; (2) the zero point of energy is changed by the subtraction of the latent heat of vaporization from the energy levels of the gas. Thus, instead of eq. (12.28), one obtains ( ' __ / 11(1 ~ Nl[ If \ The idea of a free volume in liquids is supported experimentally by Bridgman's studies of liquid compressibilities. These are high at low pressures, 07 08 T Ac REDUCED TEMPERATURE 0.9 1.0 Fig. 14.6. Law of rectilinear diameters. but after a compression in volume of about 3 per cent, the compressibility coefficient decreases markedly. The initial high compressibility corresponds to "taking up the slack" in the liquid structure or using up the free volume. A useful model is sometimes provided by considering that the free volume is distributed throughout the liquid in the form of definite holes in a more closely packed structure. We should not think of these holes as being of molecular size, since there is probably a distribution of smaller holes of various sizes. The vapor is mostly void space with a few molecules moving at random. The liquid is a sort of inverse of this picture, being mostly material substance with a few holes moving at random. As the temperature of a liquid is raised, the concentration of molecules in its vapor increases and the concentration of holes in the liquid alsp in- creases. Thus as the vapor density increases the liquid density decreases, until they become equal at the critical point. We might therefore expect the 430 LIQUIDS [Chap. 14 average density of liquid and vapor to be constant. Actually, there is a slight linear decrease with temperature. This behavior was discovered by L. Cailletet and E. Mathias (1886), and has been called the law of rectilinear diameters. It may be expressed as p av = p aT, where p av is the arith- metical mean of the densities of the liquid and the vapor in equilibrium with it, and p and a are characteristic constants for each substance. The relation- ship is illustrated in Fig. 14.6 where the data for helium, argon, and ether are plotted in terms of reduced variables to bring them onto the same scale. 13. The flow of liquids. Perhaps most typical of all the properties of fluids is the fact that they begin to flow appreciably as soon as a shearing stress is applied. A solid, on the other hand, apparently supports a very con- siderable shear stress, opposing to it an elastic restoring force proportional to the strain, and given by Hooke's Law,/ KX. Even a solid flows somewhat, but usually the stress must be maintained for a long time before the flow is noticeable. This slow flow of solids is called creep, and it can become a serious concern to designers of metal structural parts. Under high stresses, creep passes over into the plastic deformation of solids, for example, in the rolling, drawing, or forging of metals. These operations proceed by a mechanism involving the gliding of slip planes (page 391). Although creep is usually small, it must be admitted that the flow properties of liquids and solids differ in degree and not in kind. The fact that liquids flow immediately under even a very small shear force does not necessarily mean that there are no elastic restoring forces within the liquid structure. These forces may exist without having a chance to be effective, owing to the rapidity of the flow process. The skipping of a thin stone on the surface of a pond demonstrates the elasticity of a liquid very well. An interesting substance, allied to the silicone rubbers, has been widely exhibited under the name of "bouncing putty." This curious material is truly a hybrid of solid and liquid in regard to its flow properties. Rolled into a sphere and thrown at a wall, it bounces back as well as any rubber ball. Set the ball on a table and it gradually collapses into a puddle of viscous putty. Thus under long-continued stress it flows slowly like a liquid, but under a sudden sharp blow it reacts like a rubber. Some of the hydrodynamic theory of fluid flow was discussed in Chapter 7 (page 173) in connection with the viscosity of gases. It was shown how the viscosity coefficient could be measured from the rate of flow through cylin- drical tubes. This is one of the most convenient methods for use with liquids as well as gases, the viscosity being calculated from the Poiseuille equation, Note that the equation for an incompressible fluid is suitable for liquids, whereas that for a compressible fluid is used for gases. In the Ostwald type of viscometer, one measures the time required for Sec. 14] LIQUIDS 431 a bulb of liquid to discharge through a capillary under the force of its own weight. It is usual to make relative rather than absolute measurements with these instruments, so that the dimensions of the capillary tube and volume of the bulb need not be known. The time 7 required for a liquid of known viscosity ?? , usually water, to flow out of the bulb is noted. The time t 9 for the unknown liquid is similarly measured. The viscosity of the unknown is where p and p x are the densities of water and unknown. Another useful viscometer is the Happier type, based on Stokes' formula [eq.(S.ll)]: \^ JL. = ( m By measuring the rate of fall in the liquid (terminal velocity v) of metal spheres of known radius r and mass w, the viscosity may be calculated, since the force /is equal to (m m Q )g, where m is the mass of liquid dis- placed by the ball. 14. Theory of viscosity. The hydrodynamic theories for the flow of liquids and gases are very similar. The kinetic-molecular mechanisms differ widely, as might be immediately suspected from the difference in the dependence of gas and liquid viscosities on temperature and pressure. In a gas, the viscosity increases with the temperature and is practically independent of the pressure. In a liquid, the viscosity increases with the pressure and decreases exponen- tially with increasing temperature. The exponential dependence of liquid viscosity on temperature was first pointed out by J. deGuzman Carrancio in 1913. Thus the viscosity coefficient may be written 77=-- Ae* K ^ IRT (14.14) The quantity A vl8 is a measure of the energy barrier that must be overcome before the elementary flow process can occur. It is expressed per mole of liquid. The term e ~^ E ^ RT can then be explained as a Boltzmann factor giving the fraction of the molecules having the requisite energy to surmount the barrier. In Table 14.4 are collected the values A vig for a number of liquids, together with values of AZ^p for purposes of comparison. 9 The energy required to create a hole of molecular size in a liquid is A vap . The fact that the ratio of AE vl8 to A is vap about \ to \ for many liquids suggests that the viscous-flow process requires a free space about one-third to one- fourth the volume of a molecule. A noteworthy exception to the constancy of the A vig : A vftp ratio is provided by the liquid metals, for which the 9 R. H. Ewell and Henry Eyring, J. Chem. Phys., 5, 726 (1937). 432 LIQUIDS [Chap. 14 TABLE 14.4 VALUES OF AVis Liquid A V 18 (cal/mole) AEyap (cal/mole) AfVap/AEvis CC14 2500 6600 2.66 QH. 2540 6660 2.62 CH 4 719 1820 2.53 A 516 1420 2.75 N 2 449 1210 2.70 2 398 1470 3.69 CHC1 3 1760 6630 3.76 C 2 H 5 Br 1585 6080 3.84 CS 2 1280 5920 4.63 Na 1450 23,400 16.1 K 1130 19,000 16.7 Ag 4820 60,700 12.5 Hg 650 13,600 20.8 values range from & to 7 V- This low ratio has been interpreted as indicating that the units that flow in liquid metals are ions, whereas the units that vaporize are the much larger atoms. The pressure dependence of the viscosity follows an equation similar in form to eq. (14.14), The AF* is the volume of the hole that must be created for the flow process to occur. As we have seen, AF* for most liquids is about one-quarter of the molar volume, but for liquid metahs AF* is exceptionally small. Thus the viscosity of liquid metals increases only slowly with the pressure. Some properties of the earth provide an interesting confirmation of these ideas. To a depth of about 3500 km the earth consists of silicates. Except for a relatively thin solid c