# An Introduction to the Liquid State

## 1st Edition

**Authors:**P Egelstaff

**eBook ISBN:**9780323159036

**Imprint:**Academic Press

**Published Date:**1st January 1967

**Page Count:**252

## Description

An Introduction to the Liquid State focuses on the atomic motions and positions of liquids. Particularly given importance in this book are internal motion of molecules as a whole and the motion of atoms in a monatomic liquid. Divided into 16 chapters, the book opens by outlining the general properties of liquids, including a comparison of liquid argon and liquid sodium, discussions on theories and methods of studying the liquid state, and thermodynamic relationships. The book proceeds by defining the molecular distribution functions and equation of state, the potential function for non-conducting liquids and metals, and measurement of pair distribution function. Numerical analyses and representations are provided to simplify the functions of equations. The book discusses equilibrium properties wherein calculations on the state of gases and fluids are presented. The text also underlines space and time dependent correlation functions. Given emphasis in this part are neutron scattering, electromagnetic radiation, and various radiation scattering techniques. Other concerns discussed are diffusion and single particle motion, velocity of correlation function, diffusion and viscosity coefficients, liquid-gas critical point, and a comparison of classical and quantum liquids. The selection is a valuable source of information for readers wanting to study the composition and reactions of liquids.

## Table of Contents

Acknowledgments

Preface

General List of Symbols

Chapter 1 General Properties of Liquids

1.1 Introduction

1.2 Comparison of Liquid Argon and Liquid Sodium

1.3 Some Thermodynamic Relationships

1.4 Theories of the Liquid State

1.5 Methods of Studying the Liquid State

Chapter 2 Molecular Distribution Functions and the Equation of State

2.1 Molecular Distribution Functions

2.2 The Pair Distribution Function

2.3 The Internal Energy (E)of a Liquid

2.4 The Equation State

2.5 Cluster and Virial Expansions of the Equation of State

Chapter 3 The Pair Potential Function for Non-conducting Liquids

3.1 A General Restriction on u(r)

3.2 Dipolar Attraction

3.3 Repulsive Terms

3.4 Some Convenient Potentials

3.5 Classical Atom-Atom Scattering Experiments

3.6 Quantum Mechanical Calculation of Atom-Atom Scattering

3.7 Experimental Test of Small (r) and Large (r) Form for u(r)

3.8 Evaluation of the Constants of the Model Potentials

Chapter 4 The Pair Potential Function for Liquid Metals

4.1 Idealized Model for a Metal

4.2 The Ion-Electron Interaction

4.3 The Electron-Electron Interaction

4.4 The Effective Interatomic Potential

4.5 Pair Potential in a Real Metal

4.6 Evaluation of the Repulsive Parameter, β

Chapter 5 Relations between g(r) and u(r)

5.1 A General Relationship

5.2 The Yvon-Born-Green Equation (YBG)

5.3 The Hypernetted Chain Equation (HNC)

5.4 The Percus Yevick Equation (PY)

5.5 Cluster Expansion for the Direct Correlation Function

5.6 Solution of the PY Equation for Hard Spheres

5.7 Physical Significance of Equation (5.6)

5.8 Pair Distribution Equation from Cell Theory

5.9 Discussion of g(r)-u(r) Equations

Chapter 6 Measurement of the Pair Distribution Function

6.1 Diffraction of Radiation

6.2 Neutron Scattering by a Single Atom

6.3 The Measurement of g(r) by Neutron Scattering

6.4 The Measurement of g(r) by X-ray or Electron Scattering

6.5 The Structure Factor S(Q)

6.6 S(Q) for Molecular Liquids

Chapter 7 Discussion of Equilibrium Properties

7.1 Evaluation of g(r) from 7.5 u(r)

7.2 Evaluation of u(r) from g(r)

7.3 The Equation of State for a Dilute Gas

7.4 Equation of State for PY

7.5 Comparisons of Equations of State for a Dense Gas

7.6 Hard-sphere Fluid Problems in the Calculation of Pressure, Energy and Specific Heat for a Liquid

7.7 Conclusions on Equilibrium Properties

Chapter 8 Space and Time Dependent Correlation Functions

8.1 The van Hove Distribution Function

8.2 The Measurement of G(r, τ) by Neutron Scattering

8.3 The Measurement of G(r, τ) by Scattering of Electromagnetic Radiation

8.4 Comparison of Several Radiation Scattering Techniques

8.5 Some Properties of S(Q, ω)

Chapter 9 The Classical Limit of S(Q, ω) and Its Relation to Macroscopic Properties

9.1 The Classical Limit

9.2 The First-Order Quantum Correction

9.3 The Classical Limit of S(Q, ω)

9.4 The Perfect Gas

9.5 Relation between Macroscopic Properties and S(Q, ω)

9.6 The Continuum Limit

Chapter 10 Diffusion and Singh

10.1 Einstein's Random-walk Theory

10.2 Solution of the Diffusion Equation and Measurement of D

10.3 Jump Diffusion

10.4 Mobility and Free Diffusion

10.5 The Langevin Equation Brownian Motion

10.6 Values of D, τ0 and l

Chapter 11 The Velocity Correlation Function

11.1 Relation of Velocity Correlation Function and Diffusion Constant

11.2 The Measurements of z(ω)

11.3 The Moments of z(ω)

11.4 Velocity Correlation Functions for Brownian Motion

11.5 Velocity Correlation for Brownian Motion of Einstein Oscillators

11.6 Comparison of Theoretical and Experimental Values of z(ω)

Chapter 12 Phenomenological Treatments of Diffusion and Viscosity Coefficients

12.1 Problems in the Calculation of the Diffusion Coefficient

12.2 Qualitative Relation between Diffusion and Viscosity Coefficients

12.3 Rate Theory Connection between Diffusion and Viscosity Coefficient

12.4 The Force Correlation Method

12.5 Discussion of Length and Time of a Diffusive Step

12.6 Connection between Viscosity and Thermal Conductivity

Chapter 13 Co-operative Modes of Motion at Low Frequencies

13.1 The Continuum Model

13.2 Summary of Sound-wave Propagation

13.3 Basic Equations of Motion in Visco-elastic Theory

13.4 Propagation of Transverse Modes

13.5 Dispersion of Longitudinal Modes

13.6 Elastic Moduli

13.7 The Velocity of Sound

Appendix A1 Derivation of the Pressure Tensor in Viscoelastic Theory

Appendix A2 Derivation of Frequency-dependent Thermal Expansion

Appendix A3 Derivation of the Microscopic Definition of the Stress in Liquid

Appendix A4 Derivation of the Microscopic Definition of the Elastic Moduli

Chapter 14 Co-operative Modes of Motion at High Frequencies

14.1 Stress Correlation Functions

14.2 Stress Correlation in the Visco-elastic Theory

14.3 Comments on Viscosity Formula

14.4 Measurement of the (z, z) Stress Correlation Function

14.5 Comments on the Thermal Conductivity

14.6 J{G(r, τ)} for a Classical Liquid

14.7 Frequency Wave-number Relation for High-frequency Modes

14.8 Lifetime of Co-operative Modes

14.9 Combined Single-particle and Co-operative Mode Schemes

Chapter 15 The Liquid-Gas Critical Point

15.1 Definition of Critical Point

15.2 Compressibility and Specific Heat

15.3 The Co-existence Curve and Critical Isotherm

15.4 Calculation of Critical Constants

15.5 van der Waals' Equation of State

15.6 Taylor Expansion of the Equation of State

15.7 Widom's Equation of State

15.8 Relationships Between the Critical Exponents

15.9 Behavior of S(Q) near the Critical Point

15.10 Behavior of 5(Q, ω) near the Critical Point

Chapter 16 Quantum Liquids

16.1 Comparison of Classical and Quantum Liquids

16.2 Wave Function for Quantum Liquids with Shortrange Forces

16.3 Calculation of g(r) for a Bose Liquid

16.4 Co-operative Modes of Motion in 4He near T = 0

16.5 Critical Velocity for Superfluid

16.6 The Two-fluid Model of Hell

16.7 Wave Propagation in a Superfluid

16.8 Comment on Critical, Normal and Quantum Liquids

References

Subject Index

## Details

- No. of pages:
- 252

- Language:
- English

- Copyright:
- © Academic Press 1967

- Published:
- 1st January 1967

- Imprint:
- Academic Press

- eBook ISBN:
- 9780323159036