Phases of Matter and their Transitions E-Book

Gijsbertus de With

• Herausgeber: Wiley-VCH GmbH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1765

Veröffentlichungsjahr: 2023

Beschreibung

Eine umfassende Einführung in die chemische Physik von Festkörpern, Flüssigkeiten und Gasen mit Schwerpunkt auf den thermodynamischen und strukturellen Aspekten von Phasen und Phasenübergängen, wobei auch Flüssigkristalle, Ferroelektronik und Oberflächenphänomene betrachtet werden.

Leseprobe

Table of Contents

Cover

Table of Contents

Title Page

Copyright

Dedication

Preface

References

Notes

List of Frequently Used Symbols and Abbreviations

SI Units, Physical Constants, and Conversion Factors

Summary of Notation

1 Introduction

1.1 Constituents of Matter

1.2 Matter and Energy: Interaction and Change

1.3 Mass and Charge

1.4 Macroscopic and Microscopic Approaches

1.5 Gases, Liquids, and Solids

1.6 What to Expect?

1.7 Units and Notation

References

Further Reading

Notes

2 Classical Mechanics

2.1 Frames, Particles, and Coordinates

2.2 From Newton to Hamilton

2.3 Hamilton's Principle and Lagrange's Equations

2.4 Conservation Laws

2.5 Hamilton's Equations

2.6 Hamilton's Principle for Continuous Systems

2.7 The Virial Theorem

2.8 Final Remarks

References

Further Reading

Notes

3 Quantum Mechanics

3.1 Quantum Concepts

3.2 Interpretation and Some Exact Solutions

3.3 Approximate Quantum Mechanics Solutions

3.4 Final Remarks

References

Further Reading

Notes

4 Intermolecular Interactions

4.1 The Semi‐classical Approach

4.2 The Quantum Approach

4.3 Model Interactions

4.4 Refinements

4.5 Final Remarks

References

Further Reading

Notes

5 Continuum Mechanics

5.1 The Nature of the Continuum

5.2 Kinematics

5.3 Balance Equations

5.4 Kinetics

5.5 The Stress Tensor

5.6 Mechanical Energy

5.7 Final Remarks

References

Further Reading

Note

6 Macroscopic Thermodynamics

6.1 Classical Thermodynamics

6.2 The Local State and Internal Variables

6.3 Field Formulation

6.4 The Linear Approximation in Non-equilibrium Thermodynamics

6.5 Final Remarks

References

Further Reading

Notes

7 Microscopic Thermodynamics

7.1 Basics of Statistical Thermodynamics

7.2 Noninteracting Particles

7.3 The Semi‐classical Approximation

7.4 Interacting Particles

7.5 Internal Contributions

7.6 Some General Aspects

References

Further Reading

Notes

8 Gases

8.1 Basic Kinetic Theory of Gases

8.2 The Virial Expansion

8.3 Equations of State

8.4 The Principle of Corresponding States

8.5 Transition State Theory

8.6 Dielectric Behavior

References

Further Reading

Notes

9 Liquids

9.1 Approaches to Liquids

9.2 Distribution Functions, Structure, and Energetics

9.3 The Integral Equation Approach

9.4 Comparison: Hard‐Sphere and Lennard‐Jones Results

9.5 Scaled‐Particle Theory

9.6 Structural Models

9.7 The Generalized van der Waals Model

9.8 Phonon Theory of Liquids

9.9 The Quantum Cluster Equilibrium Model

9.10 Some Continuum Aspects

9.11 Dielectric Behavior

References

Further Reading

Notes

10 Solids

10.1 Inorganics and Metals

10.2 Polymers

10.3 Lattice Concepts

10.4 Crystalline Structures

10.5 Bonding: The Quantum‐mechanical Approach

10.6 Bonding: The Empirical Approach

10.7 Lattice Dynamics

10.8 Two Simple Models

10.9 Properties

10.10 Defects

10.11 Thermo‐elasticity

10.12 Final Remarks

References

Further Reading

Notes

11 Interfaces

11.1 Thermodynamics of Interfaces

11.2 One‐Component Surfaces: Semiempirical Considerations

11.3 One‐Component Surfaces: Theoretical Considerations

11.4 Solid Surface Structure

11.5 Adsorption at Interfaces

11.6 Final Remarks

References

Further Reading

Notes

12 Phase Transitions: General Aspects

12.1 Some General Considerations

12.2 The Clapeyron and Clapeyron–Clausius Equation

12.3 The Mosselman Solution for the Clapeyron Equation

12.4 The Ehrenfest–Prigogine–Defay Equations

12.5 Landau and Landau‐like Theory

References

Further Reading

Notes

13 Discontinuous Phase Transitions: Liquids ↔ Gases

13.1 Thermodynamics of Evaporation

13.2 Kinetics of Evaporation

13.3 The Reverse Transition: Condensation

Reference

s

Further Reading

Notes

14 Discontinuous Phase Transitions: Solids ↔ Liquids

14.1 Melting or Fusion

14.2 Mechanical or Bulk Melting

14.3 Thermodynamic or Surface‐Mediated Melting

14.4 Polymer Melting

14.5 The Influence of Pressure

14.6 Other Aspects

14.7 Melting in Perspective

14.8 The Reverse Transition: Freezing or Solidification

References

Further Reading

Notes

15 Continuous Phase Transitions: Liquids ↔ Gases

15.1 Limiting Behavior

15.2 Mean‐Field Theory: Landau Theory

15.3 Scaling

15.4 Renormalization

15.5 Final Remarks

References

Further Reading

Notes

16 The Liquid Crystal Transformation

16.1 Nature and Types

16.2 The Nematic–Isotropic Transformation

16.3 Alternative Approaches

16.4 Some Extensions

16.5 Elastic Energy and Defects

16.6 The Fréedericksz Transformation

References

Further Reading

17 Dielectric Behavior and the Ferroelectric Transformation

17.1 Preliminaries and Dielectric Materials

17.2 Electronic Polarization

17.3 Vibrational Polarization

17.4 Orientational Polarization

17.5 Space–Charge Polarization

17.6 Ferroelectric Materials

17.7 Ferroelectric Behavior

References

Further Reading

Notes

18 The Glass Transition

18.1 What Is a Glass?

18.2 The Thermodynamic Approach

18.3 The Structural Approach

18.4 The Lattice Gas Approach

18.5 Phonon Theory for Glasses

18.6 Mode‐Coupling Theory

18.7 Final Remarks

References

Further Reading

Notes

19 Irreversibility and the Return to Equilibrium

19.1 Some Considerations

19.2 The Boltzmann Approach

19.3 The Gibbs Approach

19.4 The Formal Approach

19.5 The Physical Approach

19.6 The Information Theory Approach

19.7 Closure

References

Further Reading

Notes

Appendix A: Guide to Mathematics Used

A.1 Symbols and Conventions

A.2 Derivatives, Differentials, and Variations

A.3 Composite, Implicit, Homogeneous, Complex, and Analytic Functions

A.4 Extremes and Lagrange Multipliers

A.5 Legendre Transforms

A.6 Coordinate Axes Rotations

A.7 Change of Variables

A.8 Calculus of Variations

A.9 Matrices and Determinants

A.10 The Eigenvalue Problem

A.11 Matrix Decompositions

A.12 Scalars, Vectors, and Tensors

A.13 Tensor Analysis

A.14 Gamma, Dirac, and Heaviside Functions

A.15 Laplace and Fourier Transforms

A.16 Some Useful Expressions

Further Reading

Notes

Appendix B: Elements of Special Relativity Theory

B.1 Lorentz Transformations

B.2 Velocities, Contraction, Dilatation, and Proper Quantities

B.3 Relativistic Lagrange and Hamilton Functions

References

Further Reading

Appendix C: The Lattice Gas Model

C.1 The Lattice Gas Model

C.2 The Zeroth or Mean‐Field Approximation

C.3 The First or Quasi‐Chemical Approximation

C.4 Athermal Entropy for Chain‐Like Molecules

References

Further Reading

Notes

Appendix D: Elements of Electrostatics

D.1 Coulomb, Gauss, Poisson, and Laplace

D.2 A Dielectric Sphere in a Dielectric Matrix

D.3 A Dipole in a Spherical Cavity

Further Reading

Appendix E: Elements of Probability and Statistics

E.1 Probability

E.2 Single Variable

E.3 Multiple Variables

E.4 The Normal Distribution and the Central‐Limit Theorem

References

Further Reading

Notes

Appendix F: Selected Data

References

Appendix G: Answers to Selected Problems

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Some properties of normal water (H

2

O) and heavy water (D

2

O).

Table 1.2 Argon thermodynamic essentials.

Chapter 3

Table 3.1 Summary of the three quantum‐mechanical pictures.

Table 3.2 Bonding data for the H

2

+

and H

2

molecules.

Chapter 4

Table 4.1 Data and vdW energy coefficients (10

−79

 J m

6

) for several m...

Chapter 7

Table 7.1 Vibration, rotation, and dissociation data for several diatomic m...

Chapter 8

Table 8.1 Diffusivity

D

, viscosity

η

, and thermal conductivity

κ

...

Table 8.2 Molecular diameter

σ

...

Table 8.3 Virial coefficients of the hard‐sphere fluid in terms of

B

2

 = 

b

 =...

Table 8.4 Values for

α

,

β

, and

γ

in

B

2

 = 

α

 − 

β

exp(

Table 8.5 The second virial coefficient

B

...

Table 8.6 Critical compression factor and surface tension for several simpl...

Table 8.7 The acentric factor

ω

for the first 20 alip...

Table 8.8 Dipole moment

μ

and polarizability volume

α

/(4π

ε

0

)...

Table 8.9 Permittivity

ε

1

and refractive index

n

1

of several molecules...

Table 8.10 Dipole moments for several chemical bonds using C

−

H

+

 ≡...

Table 8.11 Bond contributions to the molar refraction

R

...

Table 8.12 Experimental and calculated values of

R

m

(cm

3

 mol

−1

) for s...

Chapter 9

Table 9.1 Data for Ar at the triple point

T

tri

.

Table 9.2 Critical data comparison for several theories.

Table 9.3 Vapor pressure

P

, molar volume

V

, and entropy of vaporization Δ

va

...

Table 9.4 Compressibility parameters for a few compounds.

Table 9.5 Parameters for the Tait–Tammann isotherm for H

2

O and CCl

4

.

Table 9.6 Parameters

ε

r

° and

L

for various liquids.

Table 9.7 Onsager dipole moment

μ

0

and Kirkwood factor

g

for several p...

Table 9.8 Values of the Kirkwood factor

g

for some aliphatic ketones.

Chapter 10

Table 10.1 Data for some metals in the Girifalco approach.

Table 10.2 Lattice sums

S

n

for various lattices.

Table 10.3 Madelung constants

M

for nearest‐neighbor distance and unit char...

Table 10.4 Born exponents

n

for various electron configurations.

Table 10.5 The universal bonding (UB) potential parameters

l

(Å) and

r

0

(Å)...

Table 10.6 Debye temperature

θ

D

(K) for various compounds from low‐tem...

Table 10.7 Defect formation enthalpy

h

and entropy

s

for some crystals.

Table 10.8 Intrinsic stacking fault energy

γ

sf

, twin‐boundary energy

γ

...

Table 10.9 Relations between the various elastic constants for isotropic ma...

Table 10.10 Density, lattice constant, and elastic constants for some cubic...

Chapter 11

Table 11.1 Surface tension data for water (W), ben...

Table 11.2 Surface tension

γ

 = 

a

−

bt

as a function of temperatu...

Table 11.3 Eötvös constant

K

and Stefan's ratio

j

for various liquids.

Table 11.4 Surface energy (mJ m

−

2

) for Ag, Au, and Cu.

a

Table 11.5 Surface energy (J m

−

2

) of spinel...

Table 11.6 Parameters

H

E*

 = 

H

...

Chapter 14

Table 14.1 Volumes

V

and enthalpies Δ

H

at the melting point

T

mel

and boilin...

Table 14.2 Simon–Glatzel parameters for various liquids.

Chapter 15

Table 15.1 Critical points and their order parameters.

Table 15.2 Experimental values of critical exponents for various fluids....

Table 15.3 Critical exponents according to various models.

Chapter 17

Table 17.1 Group contributions

P

j

to the molar polarization

P

m

 = Σ

j

P

j

[10];...

Table 17.2 Polarizability

α

of some atoms and alkali halide ions.

Table 17.3 The permittivity

ε

0

and

ε

∞

for various crystals....

Table 17.4 Bulk modulus

K

(GPa) and effective valency

Z

*

(−) for alkali hal...

Table 17.5 Characteristics of some typical ferroelectrics.

Chapter 18

Table 18.1 Frequency of canonical holes.

Table 18.2 Frequency of the number of faces and edges per face of RCP coord...

Appendix A

Table A.1 Laplace transform pairs.

Appendix F

Table F.1 Lennard‐Jones parameters

ε

and

σ

for some molecules from...

Table F.2 Lennard‐Jones parameters

ε

and

σ

for some molecules from...

Table F.3 Thermodynamic vapor pressure constants

a

′,

b

′, and

c

′ for some com...

Table F.4 Trouton constant

C

for some molecules.

Table F.5 Dipole moment

μ

and polarization volume

α

′ = 

α

/4π

ε

...

Table F.6 Density

ρ

, compressibility

κ

, and expansivity

α

for...

Table F.7 Kihara parameters for some molecules with nondimensional core size...

Table F.8 Critical data, acentric factor

ω

, refractive index

n

D

, and no...

Table F.9 Physical data for some liquids at 20 °C (25 °C).

Table F.10 van der Waals constants

a

and

b

for some compounds.

Table F.11 Critical data, acentric factor

ω

, and normal boiling point

T

Table F.12 Antoine constants

A

,

B

, and

C

for the vapor pressure

P

sat

of some...

Table F.13 Structural contributions for the parachor .

Table F.14 Ionic radii according to Pauling

r

P

and Shannon–Prewitt

r

SP

for I...

Table F.15 Pauling EN, IP, EA

a)

and Sanderson data

b)

for some elements.

Table F.16 Unweakened (

E

‴), half weakened (

E

″), and fully weakened bond ener...

Table F.17

B

‐values for some elements according to Sanderson.

a)

Table F.18 Data for the orbital electronegativity model as given by Hinze.

a)

Table F.19 Covalent radii

r

cov

and bond energy

D

AA

according to the orbital ...

List of Illustrations

Chapter 1

Figure 1.1 Relativity. (a) Constant relative frequency difference Δ between ...

Figure 1.2 Schematic of structure and coordination of (a) a (crystalline) so...

Figure 1.3 The pair correlation function

g

(

r

). (a) A solid with a regular ar...

Figure 1.4 Phase relationships for a simple fluid between the solid (S), liq...

Chapter 4

Figure 4.1 The Mie pair potential for a diatomic molecule in units of

ε

Figure 4.2 The interaction

ρ

1

φ

2

of two charge distributions

ρ

Figure 4.3 (a) The interaction at a distance

r

of a point charge

q

1

with a d...

Figure 4.4 Schematic of the momentary interaction between the electrons (o) ...

Figure 4.5 (a) Comparison of the Lennard–Jones (LJ), Kihara (K), and Bucking...

Chapter 5

Figure 5.1 Schematic picture of a continuous body. (a) Deformation from

A

to...

Figure 5.2 (a) Cauchy's tetrahedron; (b) stress vector

t

, normal stress vect...

Chapter 6

Figure 6.1 (a) If in energy space

U

(

a

1

,

a

2

) points 1 and 2 are isentropic and...

Figure 6.2 (a) The calculation of the chemical potentials

μ

1

and

μ

Figure 6.3 (a) The Maxwell element with a spring and dashpot in series. (b) ...

Figure 6.4 (a) The dynamical pressure as illustrated by a gas in a cylinder....

Chapter 7

Figure 7.1 Phase space for a 1D oscillator indicating phase space between

n

 ...

Figure 7.2 Normal modes for water: (a) mode 1 or symmetric stretching (3652 ...

Figure 7.3 Terms in the Gibbs–Bogoliubov inequality for

〈

1

〉

0

 < ...

Chapter 8

Figure 8.1 (a) Approaching the perfect gas EoS for gases as exemplified by H

Figure 8.2 Compression factor

Z

as a function of the reduced pressure for va...

Figure 8.3 (a) The curve of

B

/

V

cri

versus

T

/

T

cri

for Ar, Kr, Xe, and CH

4

. Th...

Figure 8.4 (a) The reduced vapor pressure

P

sat

/

P

cri

versus

T

cri

/

T

for variou...

Figure 8.5 (a) Contour map of the potential energy surface using structural ...

Figure 8.6 Molar polarization

P

of (a) HCl, HBr, and HI and (b) CH

4

, CH

3

Cl, ...

Figure 8.7 (a) The relative permittivity

ε

1

(—) and

ε

2

(‐ ‐) as a ...

Chapter 9

Figure 9.1 (a) The location of volume element d

r

1

located at

r

1

and the volu...

Figure 9.2 Correlation functions. (a) For Ar as determined using neutron‐ray...

Figure 9.3 The hard‐sphere potential

φ

HS

, the overlap function

A

HS

, the...

Figure 9.4 (a) The pair correlation function

g

(

r

) and EoS for a hard‐sphere ...

Figure 9.5 (a) The correlation function

g

(

r

) for a Lennard‐Jones fluid at

η

...

Figure 9.6 The self‐correlation function for Ar at 84.5 K showing the probab...

Figure 9.7 (a) A two‐dimensional (2D) representation of the FCC lattice for ...

Figure 9.8 (a) The equation of state for three‐dimensional (3D) hard spheres...

Figure 9.9 Hole theory results. (a) The logarithm of the reduced vapor press...

Figure 9.10 Typical properties of several liquids. (a) The fraction of holes...

Figure 9.11 (a) The fraction

p

of non H‐bonded OH groups in liquids H

2

O, CH

3

Figure 9.12 Generalized vdW model using the SW potential. (a) Coordination n...

Figure 9.13 The heat capacity

C

V

according to phonon theory. (a) For Ar in t...

Figure 9.14 QCE results for water in the 18‐cluster 3‐21G model. (a) Dominat...

Figure 9.15 (a) The molar polarization

P

m

of the polar molecule C

6

H

5

NO

2

in t...

Chapter 10

Figure 10.1 (a) Schematic of a crystalline solid. (b) Schematic of an amorph...

Figure 10.2 Schematics of bonding. (a) An ionically bonded material; (b) a c...

Figure 10.3 Microstructural features of polymers. (a) Network characteristic...

Figure 10.4 (a) The 14 Bravais lattices in 7 crystal systems. (b) The Wigner...

Figure 10.5 Three basic crystal structures: (a) FCC, (b) BCC, and (c) HCP.

Figure 10.6 Two ionic crystal structures: (a) CsC and (b) NaCl.

Figure 10.7 Covalent crystal structures.(a) diamond, (b) graphite.

Figure 10.8 The crystallographic structure of (a) pyrazine [13] and (b, c) o...

Figure 10.9 (a) Bragg reflection at the zone boundary. (b) Energy bands in t...

Figure 10.10 Brillouin zones with symmetry points indicated. (a) BCC lattice...

Figure 10.11 (a) The symmetry of the four basic types of interactions betwee...

Figure 10.12 The band structure (left) and density of states

d

(

E

) (right) fo...

Figure 10.13 (a) The volume–pressure relation for MgAl

2

O

4

and the HP form of...

Figure 10.14 (a) The nodes in 1s, 2s, and 3s wavefunctions. (b) The shape of...

Figure 10.15 A bond diagram [61] showing the character of an A—B bond by its...

Figure 10.16 The polar covalence approach for gaseous and solid alkali halid...

Figure 10.17 The orbital electronegativity approach.(a) Calculated bond ...

Figure 10.18 (a) The universal bonding (UB) potential

φ

(

r

) = 

U

(

a

)/|

U

0

| ...

Figure 10.19 (a) Density of states for Si (

T

mel

 = 1687 K) with the correspon...

Figure 10.20 (a) Schematic of a vacancy (I), interstitial (II), substitution...

Figure 10.21 (a) Binary cubic lattice containing a mixed dislocation. At poi...

Figure 10.22 Image of a bubble raft, providing a 2D analogue of crystalline ...

Chapter 11

Figure 11.1 (a) Schematic representation of the density profile over a liqui...

Figure 11.2 (a) Schematic representation of a sliding‐wire experiment to ext...

Figure 11.3 Surface energy

U

m

(σ)

as a function of

T

/

T

cri

for several li...

Figure 11.4 (a) The TLK model. (b) A 2D Wulff plot showing several orientati...

Figure 11.5 Kosterlitz–Thouless trajectories

y

as a function of

x

.(a) Al...

Figure 11.6 (a) Surface roughness

R

for {100} SC surfaces as a function of

1

...

Figure 11.7 The regular solution model. (a) The enthalpy of mixing

H = ΔmixH

...

Figure 11.8 (a) Schematic representation of a Langmuir–Blodgett trough. (b) ...

Figure 11.9 SDS adsorption.(a) Surface tension

γ

as a function of S...

Figure 11.10 (a) The Langmuir isotherm

θ(c) = K(c/c0)/[1 + K(c/c0)]

...

Figure 11.11 Results for the Feinerman model.(a) The ratio

(γ − γ2)/(γ1 

...

Chapter 12

Figure 12.1 Schematic of the behavior of the Gibbs energy

G

for two phases a...

Figure 12.2 (a) Schematic of the phase equilibrium between the solid (S), li...

Figure 12.3 The disappearing of the meniscus of benzene along the coexistenc...

Figure 12.4 (a) Concave and convex intervals of a function; (b) The relation...

Figure 12.5 The effect of pressure on the melting point. (a)

V

m

(S) < 

V

m

(L) l...

Figure 12.6 (a) The vaporization enthalpy Δ

vap

H

(kJ mol

−1

) of ethanol ...

Figure 12.7 The Helmholtz energy curve

f

for a continuous transition along t...

Figure 12.8 The discontinuous transition: (a) the nondimensional internal va...

Chapter 13

Figure 13.1 (a) Surface temperature

T

sur

of water evaporating at an initial ...

Figure 13.2 (a) Molar energy of surface formation

U

(σ)

for water, calcu...

Figure 13.3 Schematic of nucleation.(a) Volume and surface contribution ...

Figure 13.4 Relative nucleation rate

J

rel

 = 

J

exp

/

J

the

for water. The circles...

Figure 13.5 Relationship between

z

R

and

t

S

of molecules in −3.0 < 

z

R

 < 3.0. ...

Figure 13.6 (a) Staying time distribution

t

S

for reflection molecules (dark ...

Figure 13.7 Molecular distribution functions. (a) Molecular velocity distrib...

Figure 13.8 Condensation coefficient

ζ

V

as a function of incoming veloc...

Chapter 14

Figure 14.1 Melting characteristics for elements. (a) Melting temperature

T

m

...

Figure 14.2 The LJD fusion model for Ar at

P

 = 0. (a) The pressure as a func...

Figure 14.3 DTM illustrated. (a) Helmholtz energy

F

as a function of disloca...

Figure 14.4 Interstitialcy theory illustrated. (a) Interstitialcy configurat...

Figure 14.5 MD Argon simulations. (a) The energy per atom Ar for a simulatio...

Figure 14.6 The size dependence of the melting point. (a)

T

mel

for Au as a f...

Figure 14.7 (a) Schematic of a lattice with about 10% vacancies, showing the...

Figure 14.8 DTM according to [71]. (a) The lhs (L) and rhs (R) for

T

 < 

T

sin

...

Figure 14.9 Polymer melting. (a) The Hoffman–Lauritzen model. (b) Melting te...

Figure 14.10 Melting curves for various compounds with ∘ experimental data p...

Figure 14.11 The odd–even effect. (a) Melting point and boiling of the first...

Figure 14.12 Schematics of nucleation. (a) According to classical nucleation...

Figure 14.13 Heterogeneous nucleation. (a) The heterogeneity factor

f

(

m

,

x

) a...

Figure 14.14 Polymer crystallization. (a) The left part represents crystalli...

Chapter 15

Figure 15.1 The reduced density

D

 = (

ρ

L

 − 

ρ

V

)/2

ρ

cri

of CO

2

ve...

Figure 15.2 The difference in volume fraction

φ

(1)

in the upper phase a...

Figure 15.3 The coherence length

ξ

for the lattice gas model for

T

 ≪ 

T

c

...

Figure 15.4 Lattice scaling. (a) Terminology used, illustrated for a 2D squa...

Figure 15.5 Images (cathode ray tube (CRT) screen shots) of lattice gas mode...

Figure 15.6 Percolation and renormalization by taking three spheres together...

Chapter 16

Figure 16.1 (a) Structure of

N‐(4‐methoxybenzylidene)‐4‐butylaniline

...

Figure 16.2 While solids have preferred orientation and periodicity and liqu...

Figure 16.3 The function

P

2

(

θ

) = ½(3cos

2

θ

 − 1) as a function of po...

Figure 16.4 (a) The entropy

S

(

ξ

) as a function of the internal variable...

Figure 16.5 The helix–coil transformation. (a) Schematic of the configuratio...

Figure 16.6 (a) The average 〈|sin

γ

|〉 as a function of

ξ

. (b) The H...

Figure 16.7 (a) The Helmholtz energy

F

as a function of

ξ

for two value...

Figure 16.8 Isotropic–nematic phase coexistence for hard sphero‐cylinders as...

Figure 16.9 Deformations for a nematic liquid crystal showing equilibrium, s...

Figure 16.10 Volterra's distortions. (a) Reference cylinder with defect line...

Figure 16.11 The orientation pattern of molecules for the various type of di...

Chapter 17

Figure 17.1 The various polarization mechanisms.

Figure 17.2 (a) Real and imaginary part of the electronic susceptibility

χ

...

Figure 17.3 (a) Schematic of the behavior of the permittivity

ε

, refrac...

Figure 17.4 (a) Polarization

P

of a ferroelectric upon reversal of the elect...

Figure 17.5 (a) [001] projection of the crystal structure of the unit cell o...

Figure 17.6 (a) The unit cell of perovskite ABO

3

. (b) Schematic of 180° and ...

Figure 17.7 Dielectric behavior in linear and nonlinear approximations. (a) ...

Chapter 18

Figure 18.1 (a) Two‐dimensional (2D) schematic of an amorphous structure wit...

Figure 18.2 (a) The five canonical holes of Bernal.(b) The volume per sp...

Figure 18.3 (a) The viscosity

η

(dPa s) of SiO

2

and glycerol as functio...

Figure 18.4 Change in properties for glycerol glass as a function of tempera...

Figure 18.5 (a) Temperature dependence of the enthalpy Δ

H

and Gibbs energy Δ

Figure 18.6 Example of time–temperature equivalence demonstrated on data for...

Figure 18.7 Results from the Gutzow–Milchev–Petroff analysis for glasses [22...

Figure 18.8 (a) Typical MCT prediction for

F

(

q

,

t

) of a supercooled liquid as...

Chapter 19

Figure 19.1 Behavior of a system of particles with the positions and momenta...

Figure 19.2 The same system as in Figure 19.1, showing (a)

v

and (b)

S

as a ...

Figure 19.3 (a) Matrix representation of a slowly varying operator in the en...

Figure 19.4 Schematic of the behavior of entropy

S

given a description inclu...

Appendix A

Figure A.1 (a) Coordinate rotation; (b) cylindrical coordinates; (c) spheric...

Figure A.2 Scheme of vector properties showing a vector

a

, its negative −

a

, ...

Appendix B

Figure B.1 (a) The relationship between inertial frames I and I′ having a re...

Appendix C

Figure C.1 The curves

y

 = 

s

and

y

 = tanh(

as

) for (a)

a

 = 0.5, (b)

a

 = 1.0, a...

Figure C.2 The composition |

s

| = |2

x

2

 − 1| where phase separation occurs (a)...

Appendix D

Figure D.1 Schematic for the potential of a dipole

μ

 = 

qd

at distance

r

Figure D.2 The pillbox construction to derive the continuity of

D

at the bou...

Guide

Cover

Table of Contents

Title Page

Copyright

Dedication

Preface

List of Frequently Used Symbols and Abbreviations

SI Units, Physical Constants, and Conversion Factors

Summary of Notation

Begin Reading

Appendix A Guide to Mathematics Used

Appendix B Elements of Special Relativity Theory

Appendix C The Lattice Gas Model

Appendix D Elements of Electrostatics

Appendix E Elements of Probability and Statistics

Appendix F Selected Data

Appendix G Answers to Selected Problems

Index

End User License Agreement

Pages

iii

iv

v

xvi

xvii

xviii

xix

xx

xxi

xxii

xxiii

xxiv

xxv

xxvi

xxvii

xxviii

xxix

xxx

xxxi

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

659

660

661

662

663

664

665

666

667

668

669

670

671

672

673

Phases of Matter and their Transitions

Concepts and Principles for Chemists, Physicists, Engineers, and Materials Scientists

Gijsbertus de With

Author

Gijsbertus de With

Professor Emeritus, Department of Chemical Engineering and Chemistry

Eindhoven University of Technology

Helix Building

PO Box 513

5600MB Eindhoven

The Netherlands

Cover image designed by Martijn de With (2023)

All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do 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 http://dnb.d-nb.de.

© 2024 WILEY‐VCH GmbH, Boschstraße 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: 978‐3‐527‐35031‐5

ePDF ISBN: 978‐3‐527‐83693‐2

ePub ISBN: 978‐3‐527‐83692‐5

Nothing has such power to broaden the mind as the ability to investigate systematically and truly all that comes under thy observation in life.

Marcus Aurelius (121–180)

Progress is made by trial and failure; the failures are generally a hundred times more numerous than the successes, yet they are usually left unchronicled.

William Ramsay (1852–1916)

If you thought that science was certain? Well, that is just an error on your part.

Richard Feynman (1918–1988)

Measure what is measurable and make measurable what is not so.

Galileo Galilei (1564–1642)

Preface

At some point in time, I got intrigued by the fact that attention for transitions between phases is more limited than for transformations within one phase. For example, in many books on the chemistry, physics, or thermodynamics of solids, melting is mentioned in passing, often merely stating the rather large change in properties between solids and liquids accompanying this discontinuous transition. This interest led to the present book. Discussing phase transitions requires knowledge of the phases themselves as well as various other disciplines. The physical chemistry, or chemical physics, of phases of matter and their transitions is wide, and therefore choices had to be made. This is always a difficult process, considering the interests of both readers and the author, coherency of the material covered, and size of the book.

As I always felt that a book, as far as is reasonably possible, should be more or less self‐contained, the text is divided into three parts. Extending the metaphor of “stones” and “house” by Poincaré1 (1854–1912), the first part (Chapters 2–7) deals with the various disciplines involved, the “tools” or Basics, supported by some further background (Appendices A–E). In the second part (Chapters 8–11), we discuss the “stones,” mostly “bricks” nowadays, or the Phases themselves. These parts are necessarily compact and selective. In the third part (Chapters 12–19), the “houses” or the Transitions are dealt with. In the same vein, the houses are part of the “city” of materials science, many cities form the “country” of natural science, what used to be called natural philosophy, and many countries constitute the “world” of science(s). Clearly, some of the same types of bricks are used for different houses, meaning that some topics could be considered to belong to either gases, liquids, or solids. The choice is evidently somewhat arbitrary. The topic at hand is also inevitably connected with some mathematics, and, contrary to present popular approaches, the mathematics required is presented concisely in Appendix A. This choice attempts to avoid entangling physical and mathematical issues and to make the more general use of these mathematical topics clear, quite apart from where they are really used in this book. Evidently, these “bricks” are all imported from the “city” of mathematics.

Given the “three‐fold way,” it is still clear that not all aspects of each of the three ways can be discussed, not even superficially. Making these choices is a matter of personal choice as well as one of what is of broad interest. The contents section makes the choices made clear: the focus is on thermodynamics and properties of dielectric behavior. Although the book focuses on phase transitions (i.e. changes from one phase to another), reviewers of the content outline pointed out that the inclusion of some aspects of phase transformations (i.e. structural changes within one phase) would be desirable. To remain in line with the focus on dielectric behavior, the phase transformation part deals with ferroelectrics and liquid crystals, as these are the most closely related to dielectric behavior (and, for ferroelectrics, within my primary interest). What to choose to discuss for liquids is particularly precarious, and I opted to include, what I now call, structural models. No apology is offered for that, as I am convinced that these models are there to stay for quite some time.

Another difficult choice is how to present the matter. There are essentially two ways to go. The first option, which may be more Anglo‐Saxon in style, is to demonstrate concepts through examples of applications and thereafter generalize, that is, use an inductive approach. The second option, which may be more Franco‐Roman in style, is to postulate concepts and thereafter specialize in specific examples of application, that is, use a deductive style. In view of the “Zeitgeist,” the spirit of times, most books nowadays use the former approach. Nevertheless, I have chosen the second approach, as that is closer to my heart. To alleviate possible concerns, I also opted to include examples at many stages of the formal development to keep readers connected to practical issues. Nevertheless, it is important to keep in mind, as I learned already during my scientific education from Inorganic Chemistry by Phillips and Williams, that it is necessary “to distinguish quite clearly between the usefulness of a model in predicting or correlating certain types of experimental observation and the general validity of the picture that this model seems to convey. Usefulness and validity often go hand in hand, but they do not always do so”. For several topics, different approaches, not necessarily compatible but all describing the facts reasonably well, exist, indicating that neither all is clear nor that all authors agree. Wherever appropriate, I pointed to that. As illustration, I refer for thermodynamics to Maugin, who jokingly (but I think not 100%) quoted, what he called a faked true common citation: Some many thermodynamicists, some many thermodynamics. For glasses, a field recognized to be loaded with uncertainty, Weitz has even joked that: There are more theories of the glass transition than there are theorists who propose them. This clearly suffices to underline that not all is certain.

In many cases, the relevant derivations are given in some detail, so that readers need not retrieve the original source right away if they are trying to follow the road towards the result instead of looking only for the results. This is another, at least partial, explanation for the inclusion of what I called Basics (Chapters 2–7, Appendices A–E).

Each chapter contains at the end a section on Further Reading with a list of books recommended. For each of them, I made a small comment, hopefully helping readers assess what they can expect. The fact that a book is on the list implies that I like that book, and most of the time I refrain from further recommendations.

To some, the Basics as described in Chapters 2–7 may seem, to quote Pippard (1920–2008) in the introduction of his Classical Thermodynamics, to be “concentrated too much on the dry bones and too little on the flesh.” He continued with, “but I would ask such critics to concede at least that the bones have an austere beauty of themselves.” I cannot agree more. On a more personal note, I remember that as a youngster of about 22 years, I was highly impressed by The Principles of Quantum Mechanics by Dirac (1902–1984), not only because of the clarity but also because of the conciseness and beauty of presentation.

Not all of the text is completely new, as I used some, generally heavily rewritten, parts of my earlier books. Still, the content is essentially new, with a focus on what the title promises. The concept of justification vignettes for dealing with more detailed or advanced arguments, as used before, is retained, as is the concept of examples, these being essentially solved problems. As I know from experience (in lecturing and when studying new topics), full answers to problems invite the reader, after a short time, to accept the implicit invitation to have a look at these solutions. Therefore, as a compromise, I provided in an appendix the final answers to selected problems and sometimes their full solutions.

On a somewhat different tone, Kuhn (1921–1996) has advocated that scientific progress is made by large leaps. Sometimes it is even advocated that a single person is mainly responsible for a certain topic, in one sense or another. Sometimes that seems true; for example, for the theory of special relativity, it is often stated that it was largely single‐handedly created by Einstein (1879–1955). Actually, many of the relevant ingredients were already there, but Darrignol made clear that Einstein was most likely the first to take the principle of relativity and the constancy of the speed of light at face value. To quote another example, classical mechanics is often said to be based on the three laws formulated by Newton (1642–1727), implying that they were the result of his effort only, but in reality, many predecessors and successors contributed: Galilei (1564–1642), Huygens (1629–1695), Borelli (1608–1679), and so on. In fact, what is now known as Newton's first law, the principle of inertia, was formulated first by Galilei and reformulated by Huygens. The second law started with Galilei, was reformulated by Huygens and Newton, while the form f = ma as we know it now was first given by Euler (1750), as made clear by Maltese and Maronne and Panza. Another example is d'Alembert's principle (1743), first formulated in the form we know it by Lagrange (1788). For the history of mechanics, I refer to Dugas (1897–1957) and Dijksterhuis (1892–1965), while a critical assessment of the basis of mechanics was given by Mach (1838–1916). Highly readable in this connection is also the chapter on classical mechanics in the book by Lindsay and Margenau, which contains quite relevant parts on other parts of physics. For a brief review of energy and inertia, admittedly from a physicist's point of view, see a paper by von Laue and the highly relevant papers by Hecht referred to in the Introduction.

For those who object that this is a rather physical picture, I plead guilty. As an antidote, I like to refer to the concept of an atom, which is a good example of a converging chemical idea over time (see the books by Leicester or Partington). Accepted by chemists long before it was accepted by physicists, it is now a concept generally accepted by almost all scientists. Well known is probably that the physical chemist Ostwald (1853–1932), after being an ardent opponent for quite some time, in a later stage of his long career nevertheless accepted the idea.2 On the contrary, Mach, the influential physicist, thinker, and contemporary of Ostwald, never accepted the concept.

In general, history is not always kind to authors, but many labels referring to scientists, although historically incorrect, have been so frequently used that it would be foolish to change them. Nevertheless, one should realize that labels indicating a responsible scientist more often than not are incorrect, or at least largely incomplete.

Finally, errors are hopefully minor, but, as Callen (1919–1993) stated in the introduction to his book Thermodynamics and an Introduction to Thermostatistics, “no error, numerical or textual, is truly minor to the student reader.” I am sure that despite all efforts, the text will not be free of errors. The responsibility is entirely mine, and any comments, corrections, or indications of omissions will be appreciated.

Writing is done for a large part outside regular office hours, and I am, again, indebted to my wife Ada for her patience and forbearance. Although writing is done alone, improving written text is not. I would like to express my gratitude to my former PhD students, Dr. Gökhan Kaçar (Trakya University, Edirne, Turkey) and Dr. Bernette Oosterlaken (Leyden‐Jar Technologies, Eindhoven) for helpful comments and suggestions on several sections. Further, to Prof. E. Hecht (formerly at Adelphi University, New York), Dr. K. Trachenko (Queen Mary, London), Prof. Erik Nies (KU Leuven), Prof. Helmut Cölfen, (University of Konstanz), Dr. Liesbeth Janssen, C. Laudicina, MSc, ir. Peter Koets, Dr. Mark Vis, and Prof. Dick Broer (all TU/e, Eindhoven). Finally, I would like to thank the staff of Wiley-VCH, in particular Dr. Frank Weinreich (initial Commissioning Editor), Dr. Sarah Higginbotham (final Commissioning Editor), Satvinder Kaur Sandhu (Managing Editor), and Judit Anbu Hena Daniel (Content Refinement Specialist) for their efforts during the production of this book.

Having said all this, I hope that the aim, purpose, and approach of the book have been made clear.

March 2023

Gijsbertus de With

TU Eindhoven

References

Callen, H.B. (1985).

Thermodynamics and an Introduction to Thermostatistics

. New York: John Wiley & Sons.

Darrigol, O. (2006).

The Genesis of the Theory of Relativity

, Ch. 1 in

Einstein, 1905–2005: Poincaré Seminar 2005

,

Progress

in

Mathematical Physics

, Vol. 47 (ed. T. Damour, O. Darrigol, B. Duplantier, and V. Rivasseau), p. 1. Basel: Birkhäuser Verlag.

Dijksterhuis, E.J. (1950, 1977, 3e).

De Mechanisering van het Wereldbeeld

(in Dutch), 3e. Amsterdam: Meulenhoff; (1961).

The Mechanization of the World Picture

. Oxford: Clarendon Press.

Dirac, P.A.M. (1958).

The Principles of Quantum Mechanics

, 4e. Oxford: Oxford University Press.

Dugas, R. (1955).

A History of Mechanics

. Neuchâtel, Switzerland: Éditions du Griffon (also Dover, 1988).

Jack, R.S. and Scholz, F. (2017).

Wilhelm Ostwald, The Autobiography

. Cham, Switzerland: Springer.

Kuhn, T.S. (1996).

The Structure of Scientific Revolutions

, 3e. Chicago: University of Chicago Press.

von Laue, M. (1949).

Inertia and Energy

, Ch. II.19 in

Albert Einstein Philosopher‐Scientist

(ed. P.A. Schilpp), p. 501. Evanston: Ill., Library of Living Philosophers; (1969). 3e. New York: MJF Books.

Leicester, H.M. (1956).

The Historical Background of Chemistry

. New York: John Wiley & Sons (also Dover, 1971).

Lindsay, R.B. and Margenau, H. (1936).

Foundations of Physics

. New York: John Wiley & Sons (also Dover, 1957).

Mach, E. (1912).

Die Mechanik in Ihrer Entwicklung Historisch‐Kritisch Dargestellt

, 7e, F.A. Brockhaus, Leipzig (1921, 8e and 1933, 9e); (1960).

The Science of Mechanics

. 6e. LaSalle, IL: Open Court Publishing Company (after the 7th German ed. of 1912, with revisions through the 9th German ed.).

Maltese, G. (2000). On the relativity of motion in Leonhard Euler's science.

Arch. Hist. Exact Sci.

54

: 319.

Maronne, A. and Panza, M. (2014). Euler, reader of Newton: mechanics and algebraic analysis.

Adv. Hist. Stud.

3

: 12.

Maugin, G.A. (1999).

The Thermomechanics of Nonlinear Irreversible Behaviors

. Singapore: World Scientific Publishing Company.

Partington, J.R. (1957).

A short History of Chemistry

. New York: MacMillan & Co. (also Dover, 1989).

Phillips, C.S.G. and Williams, R.J.P. (1965).

Inorganic Chemistry

,

Principles and Non‐metals

, Vol. 1. New York: Oxford University Press.

Pippard, A. (1966).

Classical Thermodynamics

. Cambridge: Cambridge University Press.

Poincaré, H. (1902).

La Science et l'Hypothèse

. Paris: Flammarion; (1952).

Science and Hypothesis

. New York: Dover.

Weitz, D.A. (2008). In an interview by K. Chang in the

New York Times

(29 July).

www.nytimes.com/2008/07/29/science/29glass.html

.

Notes

1

“Le savant doit ordonner ; on fait la science avec des faits comme une maison avec des pierres ; mais une accumulation de faits n'est pas plus une science qu'un tas de pierres n'est une maison.” or “The man of science must work with method. Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.”

2

The story is told by Ostwald himself in his reissued autobiography, translated by Jack and Scholz.

List of Frequently Used Symbols and Abbreviations

(X)

dimension dependent on property, (–) dimensionless property

ϑ

kinetic energy (J)

Λ

thermal wavelength (m)

Ξ

grand partition function (–)

Φ

potential energy (J)

Ψ

wave function (–)

voltage (V)

Ω

external potential energy (J)

grand potential (J)

atomic volume (m

3

)

α

constant (X)

activity coefficient (–)

thermal expansivity (K

−1

)

polarizability (C

2

 m

2

 J

−1

)

β

constant (X)

1/

kT

(J

−1

)

activity coefficient (–)

γ

constant (X)

activity coefficient (–)

δ

solubility parameter (MPa

1/2

)

Dirac delta function (–)

ε

small scalar (X)

energy (J)

ε

j

strain component

j

(–)

η

(bulk) viscosity (Pa s)

packing density (–)

internal variable (X)

κ

compressibility (m

2

 N

−1

)

θ

characteristic temperature

λ

parameter (X)

wavelength (m)

μ

dipole moment (C m)

ν

frequency (s

−1

)

ξ

internal variable (X)

ρ

number density (m

−3

)

mass density (g m

−3

)

radius of curvature (m)

ρ

′

mass density (kg m

−3

)

σ

(hard‐sphere) diameter (m)

σ

j

stress component

j

(X)

τ

characteristic time (s)

φ

(pair) potential energy (J)

electrical potential (V m

−1

)

volume fraction (–)

χ

susceptibility (X)

ω

circular frequency (s

−1

)

D

dielectric displacement (A s m

−2

)

E

electric field (V m

−1

)

F

force (N )

P

polarization (A s m

−2

)

Q

quadrupole moment (C m

2

)

e

unit vector (m)

f

force (N)

p

generalized momentum (X)

q

generalized coordinate (X)

r

coordinate (m)

u

displacement (m)

v

velocity (m s

−1

)

x

coordinate (m)

affinity

Hamilton function

Lagrange function

molar mass

Hamilton density

Lagrange density

A

area (m

2

)

generalized force (X)

constant (X)

B

constant (X)

generalized force (X)

B

i

virial coefficients

C

constant (X)

heat capacity

D

diffusivity (m

2

 s

−1

)

E

energy (J)

electric field (V m

−1

)

F

Helmholtz energy (J)

force (N)

Faraday constant (C mol

−1

)

G

Gibbs energy (J)

H

enthalpy (J)

I

moment of inertia (kg m

2

)

J

flux (x s

−1

 m

−2

)

K

equilibrium constant (X)

bulk modulus (N m

−2

)

L

length (m)

M

molecular weight

N

number of molecules (particles) (–)

N

i

number of molecules of type

i

(–)

N

A

Avogadro's number (mol

−1

)

P

pressure (MPa)

probability (–)

polarization (m

3

)

Q

configurational partition function (–)

R

gas constant (J K

−1

 mol

−1

)

radius (m)

distance (m)

S

entropy (J K

−1

)

T

temperature (K)

U

(internal) energy (J)

V

potential energy (J)

volume (m

3

)

W

work (J)

Z

canonical partition function (–)

a

generalized displacement (X)

constant (X)

spring constant (N m

−1

)

b

second virial coefficient (m

3

mol

−1

)

constant (X)

generalized displacement (X)

c

constant (X)

inverse spring constant (compliance) (m N

−1

)

e

unit charge (C)

f

(volume) fraction (–)

specific Helmholtz energy (J kg

−1

)

force (N)

spring constant (N m

−1

)

rate constant (X)

g

specific Gibbs energy (J kg

−1

)

density of states (J

−1

)

h

Planck's constant

j

flux (X m

−2

 s

−1

)

k

Boltzmann's constant (J K

−1

)

spring constant (N

−1

)

rate constant (X)

l

length (m)

m

Mie constant (–)

mass (kg)

n

Mie constant (–)

material constant (X)

number of moles (–)

number density (m

−3

)

n

α

number of moles of component α (–)

p

j

probability of state

j

(–)

q

charge (C)

r

distance (m)

rate (X s

−1

)

s

specific entropy (J K

−1

 kg

−1

)

t

time (s)

u

(pair) potential energy (J)

v

volume (m

3

)

w

regular solution parameter (J)

x

α

mole fraction of component α (X)

z

single‐particle partition function (–)

coordination number (–)

z

j

valency of particle

j

(–)

Main Abbreviations

AAD

average absolute difference

AB

acid–base

AO

atomic orbital

BCC

body‐centered cubic

BZ

Brillouin zone

CM

continuum mechanics

CMC

critical micelle concentration

CN

coordination number

CNT

classical nucleation theory

CWT

capillary wave theory

DFT

density functional theory

DoF

degrees of freedom

DoS

density of states

EN

electronegativity

EoS

equation of state

FCC

face‐centered cubic

FVT

free volume theory

GAE

Gibbs adsorption equation

HCP

hexagonal closed packed

HNC

hyper‐netted chain equation

HS

hard sphere

IR

infrared

MC

Monte Carlo

MCT

mode coupling theory

MD

molecular dynamics

MO

molecular orbital

MT

macroscopic thermodynamics

LCAO

linear combination of AOs

LJ(D)

Lennard‐Jones(‐Devonshire)

NFE

nearly free electron

NMR

nuclear magnetic resonance

NRD

neutron‐ray diffraction

OZ

Ornstein–Zernike

PM

particle mechanics

PME

principle of maximum entropy

PoCS

principle of corresponding states

PY

Percus–Yevick equation

(Q)HA

(quasi‐) harmonic approximation

QM

quantum mechanics

RCP

random close packed

RDF

radial distribution function

RS

regular solution

RVE

representative volume element

SC

simple cubic

SCF

self‐consistent field

SLS

significant liquid structure

ST

statistical thermodynamics

STP

standard temperature and pressure

SW

square well

T

tetragonal

TB

tight binding

TEM

transmission electron microscopy

TST

transition state theory

UV

ultraviolet

VFT

free volume theory

XRD

X‐ray diffraction

YBG

Yvon–Born–Green equation

lhs

left‐hand side

rhs

right‐hand side

vdW

van der Waals

≅

approximately equal

≈

very approximately equal

≡

identical

∼

proportional to

⇔

corresponds with

order of magnitude

x

Superscripts

E

excess

∞

infinite dilution

°

standard

*

pure substance

‡

activated complex

id

ideal

Subscripts

f

formation reaction

fus

fusion

g

glass

mix

mixing

m

molar

mel

melting

r

reaction in general

sol

solution

sub

sublimation

trs

transition

vap

vaporization

SI Units, Physical Constants, and Conversion Factors

In 2019, the Système International d'Unités or International System of Units (SI) was revised to comprise a coherent system of units of measurement in terms of defined physical constants.

Basic SI Units

Quantity

Unit

Symbol

Length

Metre

m

Mass

Kilogram

kg

Time

Second

s

Electric current

Ampere

A

Temperature

Kelvin

K

°C

t

/°C = 

T

/K − 273.15

Amount of substance

Mole

mol

Luminosity

Candela

cd

Derived SI Units

Force

Newton

N = kg m s

−2

Work, energy, heat

Joule

J = N m

Power

Watt

W = J s

−1

Pressure

Pascal

Pa = N m

−2

Frequency

Hertz

Hz = s

−1

Electrical charge

Coulomb

C = A s

Electrical potential

Volt

V = J C

−1

Electrical resistance

Ohm

Ω = V A

−1

SI Defining Physical Constants

(… indicating that more digits are in the definition)

Constant

Symbol

Value

Avogadro's number

N

A

6.022 141… × 10

23

 mol

−1

Elementary charge

e

1.602 177… × 10

−19

 C

Boltzmann's constant

k

1.380 649 × 10

−23

 J K

−1

8.617 337 eV K

−1

Planck's constant

h

6.626 070… × 10

−34

 J s

ℏ

 = 

h

/2π

1.054 572… × 10

−34

 J s

Speed of light

c

2.997 925… × 10

10

 m s

−1

Hyperfine transition frequency of Cs

–

9.192 632… × 10

9

 Hz

Luminous efficacy (of 540 THz radiation)

K

cd

683 lm W

−1

Other Constants

Electron rest mass

m

e

9.109 384 × 10

−31

 kg

Proton rest mass

m

p

1.672 622 × 10

−27

 kg

Neutron rest mass

m

n

1.674 927 × 10

−27

 kg

Gas constant

R

 = 

kN

A

8.314 463 J mol

−1

 K

−1

Standard acceleration of gravity

g

9.806 65 m s

−2

Faraday's constant

F

 = 

eN

A

9.648 533 × 10

4

 C mol

−1

Permeability of vacuum

μ

0

1.000 000 × 4π × 10

−7

 N A

−2

Permittivity of vacuum

ε

0

 = 1/

μ

0

c

2

8.854 187 × 10

−12

 C

2

 N

−1

 m

−2

Conversion Factors for Non‐SI Units

1 dyne

=

10

−5

 N

1 bar

=

10

5

 Pa

1 atm

=

1.013 25 bar

1 mm Hg (torr)

=

1/760 atm

1 D

=

3.336 641 × 10

−30

 C m

1 L

=

1 dm

3

 = 10

−3

 m

3

1 eV

=

1.602 176… × 10

−19

 J

1 erg

=

10

−7

 J

1 eV/particle

=

96.485 332 kJ mol

−1

1 Å

=

10

−10

 m

1 cm

−1

=

1.986 302 × 10

−23

 J

hc

/

k

=

1.438 777 cm K

1 cal

th

(refers to the

thermochemical calorie

)

≡

4.184 J

1 u (Da) (unified atomic mass unit, dalton)

=

1.660 539 × 10

−27

 kg

Prefixes

femto, f

=

10

−15

pico, p

=

10

−12

nano, n

=

10

−9

micro, μ

=

10

−6

milli, m

=

10

−3

kilo, k

=

10

3

mega, M

=

10

6

giga, G

=

10

9

tera, T

=

10

12

Greek Alphabet

A, α

Alpha

B, β

Beta

Γ, γ

Gamma

Δ, δ

Delta

E, ε

Epsilon

Z, ς

Zeta

H, η

Eta

Θ, θ, ϑ

Theta

I, ι

Iota

K, κ

Kappa

Λ, λ

Lambda

M, μ

Mu

N, ν

Nu

Ξ, ξ

Xi

O, o

Omicron

∏, π

Pi

P, ρ

Rho

Σ, σ

Sigma

T, τ

Tau

Y, υ

Upsilon

Φ, ϕ, φ

Phi

X, χ

Chi

Ψ, ψ

Psi

Ω, ω

Omega

Standard Values

Until 1982, standard temperature and pressure