Gijsbertus de With Liquid-State Physical Chemistry Related Titles Wolynes, P., Lubchenko, V (eds.) Structural Glasses and Supercooled Liquids Theory, Experiment, and Applications 2012 Print ISBN: 978-1-118-20241-8; also available in electronic formats Schäfer, R., Schmidt, P.C (eds.) Methods in Physical Chemistry 2012 Print ISBN: 978-3-527-32745-4; also available in electronic formats Lalauze, R Physico-Chemistry of Solid–Gas Interfaces 2008 Print ISBN: 978-1-848-21041-7; also available in electronic formats Reichardt, C., Welton, T Solvents and Solvent Effects in Organic Chemistry Fourth Edition 2011 Print ISBN: 978-3-527-32473-6; also available in electronic formats Gijsbertus de With Liquid-State Physical Chemistry Fundamentals, Modeling, and Applications Author Gijsbertus de With Eindhoven Univ of Technology Dept of Chemical Engineering and Chemistry Den Dolech 5612AZ Eindhoven The Netherlands Cover: Martijn de With: Disordered order : an artist’s impression of liquids, 2013 All books published by Wiley-VCH are carefully produced Nevertheless, authors, editors, and publisher not warrant the information contained in these books, including this book, to be free of errors Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at © 2013 Wiley-VCH Verlag & Co KGaA, Boschstr 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages) No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers Registered names, trademarks, etc used in this book, even when not specifically marked as such, are not to be considered unprotected by law Print ISBN: 978-3-527-33322-6 ePDF ISBN: 978-3-527-67678-1 ePub ISBN: 978-3-527-67677-4 Mobi ISBN: 978-3-527-67676-7 oBook ISBN: 978-3-527-67675-0 Typesetting Toppan Best-set Premedia Limited Printing Strowss bmbH, Mörlenbach Cover Design Formgeber, Eppelheim, Germany Printed in the Federal Republic of Germany Printed on acid-free paper There are two quite different approaches to a theory of the liquid state which in fact complement each other Henry Eyring, page 141 in Liquids: Structure, Properties, Solid Interactions, T.J Hughel ed., Elsevier Publ Comp Amsterdam, 1965 VII Contents Preface XV Acknowledgments XIX List of Important Symbols and Abbreviations XXV 1.1 1.2 1.3 1.4 Introduction The Importance of Liquids Solids, Gases, and Liquids Outline and Approach Notation References Further Reading Basic Macroscopic and Microscopic Concepts: Thermodynamics, Classical, and Quantum Mechanics 11 Thermodynamics 11 The Four Laws 11 Quasi-Conservative and Dissipative Forces 15 Equation of State 16 Equilibrium 17 Auxiliary Functions 18 Some Derivatives and Their Relationships 20 Chemical Content 21 Chemical Equilibrium 24 Classical Mechanics 26 Generalized Coordinates 27 Hamilton’s Principle and Lagrange’s Equations 28 Conservation Laws 30 Hamilton’s Equations 33 Quantum Concepts 35 Basics of Quantum Mechanics 35 The Particle-in-a-Box 41 The Harmonic Oscillator 42 The Rigid Rotator 43 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.1.8 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3 2.3.4 VIII Contents 2.4 2.4.1 2.4.2 2.4.3 Approximate Solutions 44 The Born–Oppenheimer Approximation 44 The Variation Principle 45 Perturbation Theory 48 References 51 Further Reading 51 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.7.1 3.7.2 3.7.3 3.8 Basic Energetics: Intermolecular Interactions 53 Preliminaries 53 Electrostatic Interaction 55 Induction Interaction 59 Dispersion Interaction 60 The Total Interaction 63 Model Potentials 65 Refinements 68 Hydrogen Bonding 68 Three-Body Interactions 70 Accurate Empirical Potentials 70 The Virial Theorem 72 References 72 Further Reading 73 4.1 4.2 4.3 4.3.1 Describing Liquids: Phenomenological Behavior 75 Phase Behavior 75 Equations of State 76 Corresponding States 79 Extended Principle 82 References 86 Further Reading 87 The Transition from Microscopic to Macroscopic: Statistical Thermodynamics 89 Statistical Thermodynamics 89 Some Concepts 89 Entropy and Partition Functions 91 Fluctuations 99 Perfect Gases 101 Single Particle 101 Many Particles 102 Pressure and Energy 103 The Semi-Classical Approximation 104 A Few General Aspects 110 Internal Contributions 112 Vibrations 112 Rotations 115 Electronic Transitions 116 5.1 5.1.1 5.1.2 5.1.3 5.2 5.2.1 5.2.2 5.2.3 5.3 5.4 5.5 5.5.1 5.5.2 5.5.3 Contents 5.6 5.6.1 5.6.2 5.6.3 5.6.4 5.6.5 Real Gases 118 Single Particle 118 Interacting Particles 118 The Virial Expansion: Canonical Method 119 The Virial Expansion: Grand Canonical Method 121 Critique and Some Further Remarks 123 References 126 Further Reading 127 6.1 6.2 6.2.1 6.2.2 6.3 6.4 6.5 6.6 Describing Liquids: Structure and Energetics 129 The Structure of Solids 129 The Meaning of Structure for Liquids 132 Distributions Functions 132 Two Asides 136 The Experimental Determination of g(r) 138 The Structure of Liquids 140 Energetics 146 The Potential of Mean Force 150 References 154 Further Reading 154 7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.2.6 7.2.7 7.3 7.4 7.4.1 7.4.2 7.4.3 7.5 7.6 Modeling the Structure of Liquids: The Integral Equation Approach 155 The Vital Role of the Correlation Function 155 Integral Equations 156 The Yvon–Born–Green Equation 156 The Kirkwood Equation 158 The Ornstein–Zernike Equation 159 The Percus–Yevick Equation 161 The Hyper-Netted Chain Equation 162 The Mean Spherical Approximation 162 Comparison 163 Hard-Sphere Results 165 Perturbation Theory 168 The Gibbs–Bogoliubov Inequality 168 The Barker–Henderson Approach 170 The Weeks–Chandler–Andersen Approach 172 Molecular Fluids 174 Final Remarks 174 References 175 Further Reading 175 8.1 8.2 8.3 Modeling the Structure of Liquids: The Physical Model Approach 177 Preliminaries 177 Cell Models 178 Hole Models 187 IX 512 Appendix E Data References Cuadros, F., Cachadina, I., and Ahamuda, W (1996) Mol Eng., 6, 319 Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B (1954) Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., New York Moelwyn-Hughes, E.A (1961) Physical Chemistry, 2nd edn, Pergamon, Oxford Rowlinson, J.S and Swinton, F.L (1982) Liquids and Liquid Mixtures, 3rd edn, Butterworth, London Reid, R.C., Prausnitz, J.M., and Sherwood, T.K (1977) The Properties of Gases and Liquids, 3rd edn, McGraw-Hill, New York Landolt-Börnstein Tabellen (1951) vol I, part 3, Springer, Berlin Bergmann, L (1954) Der Ultraschall, 6th edn, Hirzel Verlag, Stuttgart Marcus, Y (1977) Introduction to Liquid State Chemistry, John Wiley & Sons, Ltd, London Smith, J.M., Van Ness, H.C and Abbott, M.M (2005) Introduction to Chemical 10 11 12 13 14 15 16 Engineering Thermodynamics, 7th edn, McGraw-Hill, New York Pitzer, K.S (1995) Thermodynamics, 3rd edn, McGraw-Hill, New York Connors, K.A (1990) Chemical Kinetics, Wiley-VCH Verlag GmbH, Weinheim, Germany Weast, R.C (ed.) (1972) Handbook of Chemistry and Physics, 53rd edn, Chemical Rubber Company, Cleveland Poling, B.E., Prausnitz, J.M., and O’Connell, J.P (2001) The Properties of Gases and Liquids, 4th and 5th edn, McGraw-Hill Gray, H.B and Haight, G.P (1967) Basic Principles of Chemistry, Benjamin, New York Pauling, L (1960) The Nature of the Chemical Bond, 3rd edn, Cornell University Press, Ithaca (a) Shannon, R.D and Prewitt, C.T (1969) Acta Crystallogr., B25, 925; (b) Shannon, R.D (1976) Acta Crystallogr., A32, 751 513 Appendix F Numerical Answers to Selected Problems 3.11 3.16 Ar: 129 K, HBr: 467 K Taking α1/α2 = 2, we obtain the following results rendering the Berthelot approximation doubtful σ11/σ22 1 2 ε11/ε22 2 Berthelot 1.00 1.41 1.00 1.41 Eq (3.32) 0.80 1.33 0.33 0.34 Ucoh = 14.3 kJ/mol and Hcoh = Ucoh + RT = 16.8 kJ/mol and Ucoh = 28.9 kJ/mol and Hcoh = Ucoh + RT = 31.4 kJ/mol 3.18 The relative contribution is (11/32)α/σ3 = 0.011 5.3 Z = + exp(−ε/kT), p1 = 0.73, p2 = 0.27, U = ε/[exp(ε/kT) + 1], CV = [exp(ε/kT) + 1]−2(ε2/kT2)exp(ε/kT) 5.11 Λ(He) ≅ 4.3 Å with ρ−1/3 ≅ 3.7 Å → classical approximation not valid; Λ(Ar) ≅ 0.30 Å with ρ−1/3 ≅ 4.0 Å → classical approximation valid 5.21 Assuming deviations less than 5% are allowed, one has + θ/3T < 1.05 or T > 7θ 6.1 ρrel = 0.984·ρglass/ρFCC = 0.984·0.637/0.741 = 0.85 Fig 6.3 provides 0.9 < V* < 1.1, hence 0.77 < ρrel < 0.95 6.3 SC BCC FCC HCP 3.17 Shell V/N 6.4 6.6 7.5 7.6 (r/a)2 a3 Nr 12 24 (r/a)2 1⅓ 2⅔ 3⅔ Nr 12 24 (r/a)2 Nr 12 24 12 24 (r/a)2 2⅔ 3⅔ Nr 12 18 12 (1/ )a (1/ )a (4 /3 )a N(1) = 9.2, N(2) = 44.4 a) Nearest neighbour Li-Cl ≅ 2.0 Å, next nearest neighbour Li-Cl ≅ 5.3 Å b) Nearest neighbour Li-Li ≅ Cl-Cl ≅ 3.7 Å PV/KT = + Σn=1(n2 + 3n)ηn v = 1.70; βPvdW = ρ(1 + v0 ρ + 16v02 ρ2 + …); βPCS = ρ(1 + v0 ρ + 10 v02 ρ2 + …); Liquid-State Physical Chemistry, First Edition Gijsbertus de With © 2013 Wiley-VCH Verlag GmbH & Co KGaA Published 2013 by Wiley-VCH Verlag GmbH & Co KGaA 514 Appendix F Numerical Answers to Selected Problems 8.2 10.3 10.5 10.7 10.10 10.11 11.5 11.6 11.10 11.12 12.2 12.3 12.9 12.11 12.13 12.17 12.18 13.1 13.9 13.10 13.11 15.1 15.8 a) vL/vG = a/(a − d) b) Vf = (4π/3)(a − d)3 = (4πa3/3)(vG/vL)3 = 0.008 (4πa3/3) H2O: εr = 78.5, CH3OH: εr = 31.6, C2H5OH: εr = 24.3 α′ = 10.3 Å3, μ = 1.57 D α′ = 6.4 Å3, μ = 2.8 D α′ = 13.8 × 10−30 m3/mol, μ = 0.34 D using the Debye equation The data point at −110 °C is also at the line for the liquid, indicating that the molecule still rotates in the solid phase However, α′ is large and μ is small as compared to independent data (α′ = 3.23 × 10−30 m3/mol, μ = 1.7 D) illustrating the effect of hydrogen bonding μ*/μ = 1.26 V1 = A + B + C[x1(2 − x1)], V2 = A − Cx12 V1 = bx12 a) ΔH = zwx1(1 − x1) = 800 J/mol at x1 = 0.5, hence zw = 3200 and w = 320 J/mol b) ln γ = (wz /kT )x12 = 0.0801 or γ2 = 1.083 Since the expressions are symmetric in x1 and x2, yes ΔsolH = kJ mol−1 > 0, hence ΔT > TΔS > kJ mol−1 or ΔS > 13.3 J K−1 mol−1 ΔS = −(z 2e 2N A /8 πε 0r )ε r−2 ∂ε /∂T Assuming the same structure as in Fig 12.4, the angle is 40° ΔH = A − DT2, ΔS = C−2DT, ΔCP = −2DT κ−1 ≅ 1.8 nm tH = · 10−7 s a) ΔT = 0.31 °C; b) Λm = 0.0273 Ω−1 mol−1 m2; c) I = 14 mA; d) I = 20 mA using η(H2O) = · 10−3 Pa s, a(Ca2+) = 1.0 Å and a(I−) = 2.15 Å; e) t(Ca2+) = 0.44; f) 2.4 K/hour A: L = 10 nm, X = 1.0 nm B: L = 100 nm, X = 3.16 nm C: L = 100 nm, X = 5.47 nm D: L = 100 nm, X = 13.2 nm δ = 18.2 MPa1/2 δ = 18.3 MPa1/2 ϕ1 = 0.10, ϕ2 = 0.41, ϕ3 = 0.49 u(σ ) = 26.1 mJ m−2 γ = 0.0264 N/m The reasonable good agreement for this calculation of cyclohexane with the experimental value 0.0247 N/m is probably fortuitous 515 Index Note: Page references in italics refer to Figures; those in bold refer to Tables a ab initio simulations 216, 217, 218, 364 absolute activity 25, 96 absolute reaction rates 194 absolute scale 288 absolute zero 15 accessibility assumption 51 accessibility range 51, 92, 98 acentric factor 82 acid–base equilibria 286, 287 action 28 activated complex theory 373–379 activation energy 373 activity 25 – absolute 25, 96 – coefficient 25, 26, 258, 267, 268, 277 additive entropy 14 additivity of entropy 14 adiabatic process 13 adiabatic wall 12 affinity 24 altermator 463 ammonia, simulation methods 214–218, 215, 216–218 amorphous solids 2, amphiphilic solutes 418–423 amphoteric solutes 417 analogous models 144 analytic function 464 analyticity of entropy 14 Andersen thermostat 208 angular momentum vector 32 anionic solutes 417 anisotropic phase 21 antisymmetric matrix 469, 470 Antoine constants 510 Antoine equation 436 apolar molecule 55 argon – dimers 70, 71 – self-correlation function for 177, 178 Arrhenius equation 373 Arrhenius-type behavior 378 atactic polymer 325 athermal solvent 334 atomic form factor 138 atomic scattering factor 139 atomic unit system 46 average value 97 Avogadro’s number 22, 90, 311 Axilrod–Teller three-body interaction 71 b background correlation function 151, 161, 200 Barker–Fisher–Watts (BFW) potential 71 Barker–Henderson approach 170, 171, 172 barostat 206 Benedict–Webb–Rubin (BWR) equations 83, 84 Bernal–Fowler rule 239 Berthelot equation 83 Berthelot rule 67, 270, 272 binomial expansion 58, 62, 485, 486 Bjerrum length 320 blend 327 block copolymer 327 blood body-centered cubic (BCC) structure 129, 130 Bogoliubov inequality 169 Bohr radius 46 Boltzmann distribution 137 Boltzmann factor 179 Liquid-State Physical Chemistry, First Edition Gijsbertus de With © 2013 Wiley-VCH Verlag GmbH & Co KGaA Published 2013 by Wiley-VCH Verlag GmbH & Co KGaA 516 Index Boltzmann function 161 Boltzmann statistics 103 Boltzmann’s law 304, 315 bond moments 229 Born approximation 138 Born diagram 287, 287 Born model 289–292 Born repulsion 64 Born–Haber diagram 287 Born–Oppenheimer approximation 44, 45, 53 Bose–Einstein particles (bosons) 39 bosons (Bose–Einstein particles) 39 Bravais lattices 129 Buckingham (exp-6) potential 66, 71 c cage 178 calculus of variations 480, 481 canonical equations of motion 34 canonical holes 130, 130 – relative frequency of 131 capillary waves 413 Carnahan–Starling expression 166, 362 Cartesian components of vector 475 Cartesian coordinates 27 cationic surfactants 417 Cauchy–Riemann conditions 465 cavity field 500 cavity function 151, 161, 200 cell models 178–187, 352 Celsius scale 15 Centigrade scale 15 centrifugal distortion constant 117 chain interaction parameter 335 chain rule 465 characteristic ratio 329, 330 characteristic temperature – rotation 115 – vibration 113 charge of molecule 54 charge–charge Coulomb interaction 57 charge–dipole Coulomb interaction 57 chemical potential 23 chemical work 13 cis conformation 327, 327 Clapeyron equation 434 classical mechanics 26–35 Clausius–Clapeyron equation 435 Clausius–Mossotti equation 224, 233, 236, 237 closed system 12, 96 closure relations 18 cluster expansion 123 cofactor of matrix 471 coherence length 440, 450, 450, 451, 451 colligative properties 260–262 column matrix 469 common ion effect 286 communal entropy 182, 184 commutation 471 complex conjugate of matrix 469 complex fluids 76 complex matrix 469 components 21 composite functions 465, 466 composite reactions 372 compressibility 20, 506 compressibility equation 147–149, 165 compression factor 76 concentration 22 conductivity 311–315, 312, 314, 317–323 – association 320–323 configuration integral 119 configurational part of partition faction 107 conjugated variable 12, 16 conservation laws 30–33 conservative force 29 conservative system 34 contact energy 353 continuous (second-order) phase transition 14, 429 contour length 330 contraction (tensor) 477 conventional scale 288 conversion factors for non-SI units 460 convexity 431 convolution theorem 483 cooling 13 coordinates 11 coordination number 5, 141, 142, 297, 298 copolymer 327 correlated hindered rotation model 329 correlation function for a molecular liquid 142, 143 correlation length, see coherence length correlation operator 135 Coulomb energy 55, 495, 496 Coulomb forces 321 Coulomb interaction 57 Coulomb’s law 242, 303, 495 coupling parameter 149 Cramers’ rule 471 critical micelle concentration (CMC) 418 critical opalescence 440 critical temperature 489 cross product of vectors 474 Index cryoscopic constants 261 crystal ionic radii 289 crystalline solids 2, 3, 129, 130 Cubic-Plus-Association (CPA) 365 cumulants 168 curl (rot) of vector 477 cut-off value 204 d Dalton’s law of partial pressures 254 Darwin–Fowler method 110 Davies equation 308 De Broglie relation 36 Debye-behavior 233 Debye equation 226, 236 Debye–Hückel approximation 309 Debye–Hückel limiting law 307 Debye–Hückel model 321 Debye–Hückel theory 303–308 Debye–Hückel–Onsager model 320, 321, 323 Debye interaction 60 Debye models 233 Debye screening length 310 degenerate energy levels 42 degree of dissociation 301 degree of polymerization 340 degree of reaction 25 degrees of freedom 27 density 3, 4, 22 density functional theory (DFT) 407 density of states 92, 93 density operator 135 determinant of square matrix 470 determinants 469 deuterium 297 diagonal matrix 454, 470 diathermal wall 12 dielectric behavior of gases 224–230 – estimation of dipole moment μ and α 229, 230 dielectric behavior of liquids 231–237 – nonpolar solvent 231, 232 – pure liquids 232 dielectric constant 497 dielectric displacement 497, 498 dielectric sphere in dielectric matrix 498–500 Dieterici equation 83 diffusion control 381–384 dilution law 302 dimethyl ether 70 dipole 495 – in spherical cavity 500, 501 dipole–dipole (Coulomb) interaction 57 dipole moment 55, 56, 506 Dirac bra-ket notion 37 Dirac delta function 185, 482 direct correlation function 160, 161 direct notation 474 directional field 223, 235–237 discontinuity 14 discontinuous (first-order) phase transformation 429 disordered zone 294 dispersion forces 60 dispersion interaction 60–63 displacement 12 dissipative force 16 dissipative work 16 dissociation energy 54 dissolution of salts divergence (scalar) 477 divergence theorem 478 dividing (geometric) plane 395 dividing surface 375 dot product of vectors 474 double zeta (DZP) 215 doublet distribution function 134 Drude model 60–63, 61 e ebullioscopic constants 261 effective polarizability 223 effective potential energy 45, 53 eigenvalue equation 36, 454 Einstein equation 316 Einstein model 114, 132, 133, 195, 197 electric susceptibility 222 electrical work 13 electrolysis, law of 311 electrolytes, strong and weak 300–302 electronic transitions 116, 117 electrophoretic effect 317 electrostatic interaction 55–58 elementary reactions 372 elements of matrix 469 empirical activation energy 378 empirical temperature 12 encounter 380 end-to-end distance 328 energetics 146–150 energy equation 146 energy representation 15 ensemble 90 enthalpy of hydration 294 entropy 13–15, 14, 91–99 entropy representation 15 517 518 Index Eötvös’ rule 69 Eötvös–Ramsey–Shields rule 403 equal a priori probability assumption 99 equations of motion 27, 34 equations of state 17, 76–79, 165 – theories 352–360 equilibrium, system in 12 equilibrium constant 372 equilibrium energy 54 equivalent chain 329, 330 equivalent conductivity 312 ergodic theorem 91 ethanol 70 Euler condition 480 Euler equation 17 Euler–McLaurin formula 116, 485 Euler’s theorem 17, 22, 23, 31, 72, 252, 399, 466 EXAFS (extended X-ray absorption fine structure) 298 excess functions 259 exchange energy 343 excluded volume 333, 334 expansivity 20, 506 expectation value 37 extended Debye–Hückel model 307 extended X-ray absorption fine structure (EXAFS) 298 extensive variables 12 extent of reaction 371 external forces 32 extreme of function 467 Eyring EoS 180 f face-centered cubic (FCC) structure 129, 130, 179, 179 factorial function 481 Faraday constant 311 Fermi–Dirac particles (fermions) 39 Fermi’s golden rule 50 fermions 39 Fick’s first law 316 first approximation (quasi-chemical solution) 190, 282, 487, 489–494, 489 – critical mixing 493, 494 – Helmholtz energy 492, 493 – pair distributions 491, 492 Fisher relation 440 fixed point 452 flexible wall 12 Flory–Huggins model 339–346 – energy 342, 343 – entropy 339–342 – Helmholtz energy 343 – phase behavior 344–346, 345 Flory–Orwoll–Vrij–Eichinger (FOVE) theory 354–356, 356 Flory temperature 332 Flory theorem 332 flow 382 fluctuations 99, 100 flux 382 force 12, 29 force field 205 Fourier transform 138, 139, 160, 483 Fowler–Guggenheim adsorption isotherm 420, 421 Fowler (–Kirkwood–Buff) approximation 406, 408, 408 free volume 132, 178, 179, 185 freely jointed chain model 328 freely rotating chain model 329 fugacity 25 functional 411, 480 functional differentiation 161 fundamental equation 15 g gamma function 481 gas-phase hydration 293, 294 (gas) phase space (Γ-space) 90 gases 3, 4, gauche conformation 327, 329 Gauss’ law 496 Gauss, theorem of 478 Gaussian chain 332 generalized coordinates 27, 374 generalized momentum 33 generalized velocities 27 generic distribution function 133 Gibbs adsorption equation 400 Gibbs–Bogoliubov inequality 168–170 Gibbs–Duhem equation 258, 400 – general 252, 253, 253 Gibbs–Duhem relation 23 Gibbs energy 19, 23, 24, 287, 290, 399 Gibbs entropy 93–95 Gibbs equation 16, 149 – for closed system 18 Gibbs–Helmholtz equation 20, 149, 346 good solvents 334 gradient (tensor) 477 graft 327 grain boundaries grains grand canonical ensemble 147, 148 grand (canonical) partition function 96 Index grand potential 23 (graphical) cluster expansion 161 Greek alphabet 461 Green’s identities 478 Griffiths’ inequality 441 Grotthuss mechanism 315, 315 Güntelberg procedure 320 Gurney force 321 hydration numbers 297 hydrogen bonding 68–70 hydronium ion 299 hydrophilic solutes 417, 424, 425 hydrophobic solutes 423, 424 Hyper-Netted Chain (HNC) equation 164, 165 162, i h Hamilton function 33–36, 90, 225, 226 Hamilton matrix elements 46 Hamilton operator 36, 38, 39, 40 Hamilton’s equations 33–35, 89, 137 Hamilton’s principle 26, 28–30 hard-sphere fluid 152–154, 153 hard-sphere potential 65, 205 harmonic function 465 harmonic oscillation 106, 107 harmonic oscillator 30, 33, 35, 112, 113 Hartree 46 Hartree–Fock (HF) 215 head-to-head addition 325 head-to-tail addition 325 heat 13 heat capacities 20 Heaviside (step) function 482 Heisenberg uncertainty relation 38, 110 Helium atom 46, 47 Helmholtz energy 19, 23, 55, 84, 149, 168, 171, 264 Henry’s law 256–258 Hermite functions 42 heterogeneous system 21 hexagonal close-packed (HCP) structure 129, 130 Hildebrand solubility parameter 350 Hildebrand’s rule 436 hole models 178, 187–194, 352 – basic 189–191 – extended 191–194 Holey Huggins (HH) theory 359 holonomic constraints 27 holonomic system 27 homogeneity of space 31 homogeneity of time 30 homogeneous function 447, 465, 466 – generalized 447, 448 homogeneous system 21 hydration structure 293–300 – enthalpy of 294 – gas-phase hydration 293, 294 – liquid-phase hydration 294–300 ice – structure 238, 239 – tetrahedral configuration 240 “iceberg” structure 294 ideal gas 3, 76, 195 ideal solutions 256–259 impermeable wall 12 implicit function 465, 466 independent hindered rotation model 329 independent migration of ions, law of 313 index notation 475 induction interaction 59, 60 inertial force 32 inertial frame 29 influence parameter 411 inner hydration shell (ordered zone) 294 inner product of vectors 474 inorganic compounds, critical data 509 integral equation approach 155–174 – hard-sphere results 165, 166 – molecular fluids 174 – perturbation theory 168–174 – vital role of correlation function 155, 156 integrals 484, 485 integrals (constants) of motion 30 intensive variables 12 interaction energy 56 interchange energy 188, 263, 346 interfacial tension 396 intermolecular bond 69 intermolecular interactions – accurate empirical potentials 70 – dispersion interaction 60–63 – electrostatic interaction 55–58 – hydrogen bonding 68–70 – induction interaction 59, 60 – model potentials 65 – preliminaries 53–55 – three-body interactions 70 – total interaction 63 – virial theorem 72 internal energy 13 internal field 223, 235–237 internal forces 32 interphase 395 519 520 Index intramolecular bond 69 invariant property 476 inverse of square matrix 471 ionic atmosphere 303 ionic radii 511 ionic solutions 7, 285–323, 387–392 – association 320–323 – conductivity 311–315, 312, 314, 317–323 – double-sphere model 388, 389, 388 – enthalpy of hydration 294 – hydration structure 293–300 – – gas-phase hydration 293, 294 – – liquid-phase hydration 294–300 – influence of ionic strength 390–392, 391 – influence of permittivity 392 – mobility and diffusion 315, 316 – single-sphere model 388–390, 388 – solubility 286–289 – strong and weak electrolytes 300–302 – structure and thermodynamics 308–311 – – correlation function and screening 308–310 – – thermodynamic potentials 310, 311 ionic strength 306 irreversible process 14 isentropic process 14 isochoric conditions 13 isolated system 12, 13, 97 isomerism 325 isotactic polymer 325 isothermal process 14 isotropic phase 21 isotropic (spherical) tensor 476 isotropy of space 32 j Jacobian determinant 472, 473 Jacobian matrix 472 jump rate symmetry 51, 99 k Keesom interaction 57, 63, 64 kelvins 15 Kihara potential 66, 71 kinetic energy of partition function 29 kinetic part 107 kinetics 371–373 Kirchhoff’s equation 435 Kirkwood correlation factor 234, 234 Kirkwood equation 158, 159, 164 Kirkwood hierarchy 159 Kohlrausch expression 312, 317 Kraft temperature 418 Kronecker delta 463 Kuhn length 331 l Lagrange equations 28–30 Lagrange equations of motion 30 Lagrange function 29, 30 Lagrange (undetermined) multipliers 95, 467 lambda expansion 168 Langevin function 226 Langmuir adsorption isotherm 419 Langmuir–Blodgett trough 421, 422 Langmuir isotherm 418, 421 Laplace equation 402, 465, 496 Laplace methods 165 Laplace operator 43, 44, 304, 477 Laplace transform 482–484, 483 lattice fluid (lattice gas) theory 352, 356, 357 lattice gas model 188, 191, 487–494 lattice model 177 lattice with a basis 129 lattices law of electrolysis 311 law of mass action 374 law of rectilinear diameters 80, 195, 445 Legendre equation 498 Legendre transform 19, 23, 33, 468 Lennard-Jones and Devonshire (LJD) theory 180, 181 Lennard-Jones fluid, correlation function for 164, 164 Lennard-Jones parameters 503, 504 Lennard-Jones potential 65, 66, 71, 142, 180, 205, 408 linear Hermitian operator 36 Liouville’s theorem 90, 137 liquid-phase hydration 294–300 liquids – applications 1, – distributions functions 132–136 – experimental determination of pair correlation function g(r) 138–140, 140 – importance of 1, – meaning of structure for 132–137 – schematic structure – static structure 4, 5, – structural features 3, – structure of 140–145 local field 223 London interaction 63 Lorentz–Berthelot rule 67 Lorentz–Lorenz equation 227, 237 Index Lorentz internal field 233 Lorentz rule 66 lower critical solution temperature (LCST) 344, 346 m macro-canonical partition function 96 macro-state 91 macroscopic (linear) dielectric behavior 221, 222 magnitude of vector 473 majority rule 450 Margules equation 278 Markov processes 212, 213 Massieu functions 20 master equation 99 matrix 475 – antisymmetric 469, 470 – cofactor 471 – column 469 – complex 469 – complex conjugate 469 – diagonal 454, 470 – elements 469 – minor 471 – non-symmetric 454 – notation 475 – orthogonal 454 – product of 469 – real 469 – square 454, 469–471 – sum of 469 – symmetric 469 – transpose of 469 – unit 470 maximum term method 111 Maxwell–Boltzmann distribution 207 Maxwell relations 20 Maxwell’s equal area rule 435 Mayer function 120, 159, 161, 171, 334 McMillan–Mayer picture 303 mean 97 mean field theory – approximation – continuous transitions 441–444 – discontinuous transitions 444 – fluid transitions 444–446 mean spherical approximation (MSA) 162, 163, 291 mechanical work 13 meso-cell 204, 205 metal to oxygen distances in metal–water complexes 299 methanol, dipole moment μ 230 micelles 418 micro-canonical (or NVE) ensemble 97, 98, 206 micro-state 91 Mie (pair) potential 54, 65, 66 milk minimum image cut-off 205 minor of matrix 471 mixture 21, 251 mobility 313 molality 22 – activity coefficient 26 – equilibrium constant 26 – scale 277 molar conductivity 312 molar polarization 224 molar quantities 251–253 molar refractivity 227 molarity 22 – activity coefficient 26 – equilibrium constant 26 – scale 277 mole fraction 22 – activity coefficient 25 molecular dynamics 205–210 – simulation 142 molecular-shaped cell 205 molecular solutions 251–283 – activity coefficients 277 – colligative properties 260–262 – empirical improvements on regular solution model 278–280 – ideal and real solutions 256–259 – ideal behavior in statistical terms 262–264 – model based on volume fractions 272–274 – partial (molar) quantities 251–253 – perfect solutions 253, 254, 225 – regular solution model 265–267 – theoretical improvements on regular solution model 281–283 molecularity 372 (molecule) phase space (μ-space) 89 moment of intertia 225 momentoids 225 momentum vector 31 monomer 325, 331 Monte Carlo (MC) simulation 124, 163, 163, 211–214, 309 – for hard-spheres 183 Morse potential 66, 69 most probable thickness 310 521 522 Index multinomial expansion 485, 486 multipole expansion 56, 57 n natural process 14 natural variables 15 Nernst–Einstein equation 316, 382 neutral molecule 54, 385–387 neutron ray diffraction (NRD) 138, 140, 297 Newton’s equations of motion 205, 206 – First Law 29 – Second Law 29 – Third Law 31, 32, 33 non-ionic surfactants 417 non-random mixing (NRM) HH theory 359 non-solvents 335 non-symmetric matrix 454 normal coordinates 114, 374 normal fluids 76 normal pressure 397 normalized function 35 Nosé–Hoover thermostat 208 notation (nth order) differential 464 nuclear magnetic resonance (NMR) spectroscopy – proton 295, 295 NPT ensemble 111 NVE ensemble 98 NVT ensemble 111 o Ohm’s law 311 one-fluid van der Waals (or vdW1) model of mixtures 273, 276 one-to-many jump rate 51 one-to-one jump rate 51 Onsager equation 233, 235–237 open system 12, 95 operator – linear Hermitian 36 – permutation 39 – substitution 463 – correlation 135 – density 135 orbital 39 order–disorder problem 268 order parameter 438 organic compounds, critical data 507 Ornstein–Zernike equation 159–161, 164, 439 orthonormal basis 475 ortho-hydroxybenzoic acid 69 orthogonal curvilinear coordinates orthogonal matrix 454 osmotic coefficient 259 osmotic pressure 261 osmotic virial expansion 262 outer product of vectors 474 overlap matrix elements 46 oxonium ion 299 478 p packing fraction (reduced density) 123 packing function 163 Padé approximant 124, 235 pair correlation function 4, 134, 140, 161, 308, 309 parachor 405, 406 parameters 463 partial derivatives 463, 464 partial (molar) property 21 partial (molar) quantities 251–253 partial pressure 26, 254 partial reaction order 372 partition functions 91–99 Pauli’s principle 39 Pauling radius 289, 290, 292 Peng–Robinson (PR) equation 83, 84 pentane 70 percolation threshold 455, 456, 456 Percus–Yevick equation 161, 162, 164, 165 perfect gas equation of state 76 perfect gas mixtures 195, 253, 254, 376 perfect gases 101–104, 253 – many particles 102, 103 – pressure and energy 103, 104 – single particle 101, 102 perfect solution 253, 254, 255 periodic boundary conditions 204, 204 (periodic) images 204 permanent of wave function 40 permeable wall 12 permittivity 497 permutation operator 39 perturbation theory 48–51 petrol phase 21 phase behavior of fluids 75, 76, 76 phase function 90 phase space 34, 89, 90 phase transitions 429–457 – continuous 437–446 – – critical points 441 – – limiting behavior 438–441 – – mean field theory 441–446 Index – discontinuous 429, 434–437 – – evaporation 435, 436 – – melting 437 – renormalization 451–457 – scaling 447–451 – – scaled potentials 448, 449 – – scaling lattices 449–451 physical constants 460 Planck’s constant 36 Pockels point 422 Poisson equation 304, 315, 496 Poisson–Boltzmann equation 304, 315, 320 polar liquids 221–248 polar molecular 55 polarizability 59, 497 polarizability volume 59, 223, 506 polarization 222, 312, 497 polarization catastrophe 232 polarization volume 506 polycrystalline materials polydispersity index (PDI) 327 polymer configurations 325–333 polymeric solution 8, 325–368 – bond energy 326 – polymer configurations 325–333 – real chains in solution 333–339 – – temperature effects 336–339 – solubility theory 346–351, 349 – tacticity and chain structure 326 polytetrafluorethylene (PTFE), structure 328 poor solvents 335 potential energy 29 potential of mean force 150–155, 161 power (rate of work) 25 pre-exponential factor 373 prefixes 460–461 pressure ensemble 111 pressure (virial) equation 147 (pressure) equilibrium constant 26 pressure equation 165 primitive lattice 129 primitive model 303 principle of corresponding states (PoCS) 80, 81, 405 probability amplitude 38 product of matrices 469 pseudo-momenta 225 q quadrupole moment 55, 56 quantum mechanics 35–44 – basics of 35–41 – harmonic oscillator 42, 43 – particle-in-a-box 41, 42 – rigid rotator 43, 44 quantum number 41 quantum state 91 quasi-chemical approximation 282 quasi-conservative force 16 quasi-equilibrium assumption 376 quasi-ergodic theorem 91 quasi-static process 16 r radial distribution function (RDF) 136, 142 radius of gyration 332 random copolymer 327 Raoult’s law 256–258, 264 rate constant 372 rate of reaction 24, 371 rate of work 25 Rayleigh–Schrödinger perturbation equation 48 reaction control 381, 384, 385 reaction coordinate 376 reaction field 235–237 reaction order 372 reaction product 25 real gases 118–125 – interacting particles 118, 119 – single particle 118 – virial expansion – – canonical method 119–121 – – critique 123–125 – – grand canonical method 121–123 real matrix 469 real solutions 256–259 Redlich–Kister expansion 278 Redlich–Kwong (RK) equation 83, 84 Redlich–Kwong–Soave (RKS) equation 83, 84 reduced vdW equation of state 79, 80 reference configuration – for gases – for liquids – for solids regular (background) part of a function 439 regular solution model 265–267 – activity coefficient 267, 268 – empirical improvements on 278–280 – nature of w and beyond 270–272 – phase separation and vapor pressure 268, 269 – theoretical improvements on 281–283 relative adsorption 400 523 524 Index relative permittivity (dielectric constant) 497 relaxation effect 318 renormalization group (RG) theory 452 representative volume element 204 resistivity 311 response functions 20 restricted primitive model (RPM) 303, 305 retardation effect 64 reversal symmetry 206 reversible process 14 Riedel’s alpha 83 rigid body motions 33 rigid wall 12 rot of vector 477 rotational constant 115 rotations 115, 116 row-into-column rule 469 rule-of-mixtures 348 Rushbrooke ‘s inequality 441 Rydberg 46 s “salting-out” term 308 salts – dissolution of – solubility products 511 scalars 473–477 scaled-particle theory 200–202, 201 scaling 274, 447–451 – homogeneous functions 447, 448 – scaled potentials 448, 489 – scaling lattices 449–451 scattering length 139 Schrödinger’s equation 40, 41, 43, 44 second approximation 282 second partial derivatives 464 second-order phase transition, see continuous phase transition segment 330, 331 self-avoiding random walk (SAW) model 335 self-similar behavior 451 semi-classical approximation 104–110 semi-classical partition function 107 Shannon–Prewitt radius 289–292 Shuttleworth equation 401 SI units 459 significant liquid structure (SLS) theory 194–199, 199, 358 Simha and Somcynski (SS) theory 358–360 simple cell (SC) model 352–354 simple cubic (SC) structure 129, 130 simple point charge (SPC) three-site model 242 simplified hole theory (SHT) 360 simulation approach 203–218 single-crystalline solids single fluids 75 single ion conductivity 313 singlet distribution function 134 singular part of a function 439 Slater determinant 39 Slater-type orbital (STO-xG) 215 “smearing” approximation 181, 182, 192 solids – schematic structure – static structure 4, – structural features 2–4 – structure of 129–133 solubility parameter 274 solution 21 solvation 379 solvation numbers 296 solvent 1, 21 – effects 379–381 – good 334 – poor 335 – physical data 508 – theta 335 space-time 29 specific distribution function 132 (specific) surface internal energy 401 spherical cut-off 205 spherical harmonics 44 spin angular momentum 38 spin orbitals 39 Spinodal (line) 344 (square) gradient theory 409–413 square matrix 454, 469 – determinant of 470 – inverse of 471 square-well (SW) potential 65 standard values 461 Stanley’s lemma 441 state function 13, 17 state variables 12 Statistical Associating Fluid Theory (SAFT) approach 361–368, 364 – Cubic-Plus-Association (SAFT-CPA) 365 – hard-sphere SAFT (SAFT-HS) 365 – heuristic derivation 366–368 – perturbed chain-SAFT (PC-SAFT) 365, 366 – SAFT-LJ 362, 365 statistical thermodynamics 89–122 Stirling’s approximation 264, 341 Index Stirling’s theorem 264 stoichiometric sum 371 Stokes–Einstein equation 316 Stokes’ law 318 Stokes’ theorem 478 strong electrolytes 288 strong electrolytic solution 312 structure-breaking zone (disordered zone) 294 structure factor 138, 139 substitution operator 463 sum of matrices 469 superposition approximation 157 surface (contact) forces 32 surface energy 401 surface excess 396 surface Helmholtz energy 401 surface pressure 421 surface pressure balance 421 surface stress 401 surface tension 396, 401 surfaces, thermodynamics of 395–402 – characteristic adsorption behavior 417–425 – – amphiphilic solutes 418–423 – – hydrophilic solutes 424, 425 – – hydrophobic solutes 423, 424 – one-component liquid 402–409 – statistics of adsorption 415–417 – two-component liquid 413–415 surfactants 417 – anionic 417 – cationic 417 – non-ionic 417 surroundings (thermodynamics) 11 susceptibility 497 symmetric (Hermitian) matrix 469 symplectic requirement 207 syndiotactic polymer 325 system (thermodynamics) 11 Szyszkowski equation 420 t “tail” correction 204 Tait equation 77, 78 tangential pressure 397 Taylor expansion 168, 207, 464 temperature bath 12 tension blob 335 tensor (dyadic) product 474 tensor analysis 477–479 tensors 473–477 – analysis 477–479 – contraction 477 – (dyadic) product 474 – gradient 477 – isotropic (spherical) 476 – trace 477 thermal blob 337 thermal contact 12 thermal equilibrium 17, 18 (thermal) expansion coefficient 20 (thermal) expansivity 20 thermodynamic consistency 149 thermodynamic limit 110 Thermodynamic Perturbation Theory (TPT) 361 thermodynamic potentials 19, 310, 311 thermodynamic probability 97 thermodynamic state 11, 91 thermodynamic temperature 15 thermodynamic vapor pressure constants 504 thermodynamics 11–26 – auxillary function 18–20 – chemical content 21–23 – chemical equilibrium 24–26 – derivatives 20 – dissipative forces 15, 16 – equation of state 16, 17 – equilibrium 17, 18 – laws of 11–15 – – first law 13 – – second law 13–15 – – third law 15 – quasi-conservative force 15, 16 thermometer 12 thermostat 12, 206 theta conditions 332 theta solvents 335 three-body interactions 70, 71 time-dependent Schrödinger equation 36 Tolman length 409 total correlation function 159, 161 total differential 20 trace (tensor) 477 trajectory 90 trans conformation 327, 327, 329 transition state 376 transition state theory (TST) (activated complex theory) 373–379 – activated complex 376–379 – equilibrium constant 373, 374 – potential energy surfaces 374–376, 375 translational invariance 134 transport number 313 transpose of a product 469 transpose of matrix 469 525 526 Index triple product 474 triple zeta (TZP) 215 Trouton constant 505 Trouton’s rule 69, 436 two-fluid van der Waals (vdW2) model 273, 276 (two particle) configuration integral 119 u unit cells 129 unit matrix 470 universality class 441 upper critical solution temperature (UCST) 344, 346 v van der Waals adsorption isotherm 422 van der Waals constants 509 van der Waals equation of state 78, 79, 79, 180 van der Waals interaction model 270 van der Waals interactions 53, 63, 65 van der Waals one-fluid approximation (vdW1) 363, 364 van Laar equations 279 van Laar expansion 279 van’t Hoff factor 302 van’t Hoff equation 373 van’t Hoff law 262 variables 463 – change of 471–473 variance 326 variation principle 45–48 variation theorem 46 vector product 474 vectors 473–477 Verlet’s algorithm 207 vibrations 112–114 virial equation of state 77, 85, 147 virial expansion – canonical method 119–121 – critique 123–125 – grand canonical method 121–123 virial theorem 72 virial theorem expression 214 Vogel constants 505 volume (body) forces 32 volume fractions, model based on 272–274 – one- and two-fluid model 275–277 – solubility parameter approach 274, 275 w wanderer 178 water 1, 238–248 – bonding energy 238, 239, 239 – disorder in 240 – entropy of 241, 242 – Frank model for the structure of water around an ion 295 – internal vibrations 238, 238 – ionic structure 294, 295 – liquid, structure 6, 242–245 – – distorted network model 243, 245 – – interstitial model 243 – – mixture model 243 – – pair-correlation function 244, 244, 245 – liquid-phase hydration 294–300 – models of 242, 243 – – Bjerrum model 242, 242 – – five-site (TIP5P) 242 – – four-site (TIP4) 242, 242 – – Popkie model 242, 242 – – simple point charge (SPC) three-site model 242 – – TIP5P-E 242 – optimal structure of dimer 239 – P-T diagram for 238, 239 – properties 245–248 – Raman spectrum 243 – as solvent 1, wave function 35 weak electrolytes 288, 312 Weeks–Chandler–Andersen (WCA) approach 172, 173, 173 well-behaved function 35 Widom’s inequality 441, 449 Wigner variational theorem 375 Wilson model 280 work 12, 13 x X-ray diffraction (XRD) 138, 298 y Yvon–Born–Green (YBG) equation 156, 157, 164 Yvon–Born–Green hierarchy 157 152, z zeroth approximation (mean field solution) 189, 282, 487–489, 489 zeroth law 12 ...Gijsbertus de With Liquid- State Physical Chemistry Related Titles Wolynes, P., Lubchenko, V (eds.) Structural Glasses and Supercooled Liquids Theory, Experiment, and Applications 2012 Print... Solvents and Solvent Effects in Organic Chemistry Fourth Edition 2011 Print ISBN: 978-3-527-32473-6; also available in electronic formats Gijsbertus de With Liquid- State Physical Chemistry Fundamentals, ... Preface For many processes and applications in science and technology a basic knowledge of liquids and solutions is a must However, the usual curriculum in chemistry, physics, and materials science