The Physics of Liquid Water

Makoto Yasutomi

• Publisher: Jenny Stanford Publishing
• Year: 2021
• Print ISBN: 9789814877251
• eBook ISBN: 9781000091663
• Topic: Physical Sciences
• Subtopic: Condensed Matter

Unraveling the mystery of the negative thermal expansion of liquid water has been a challenge for scientists for centuries. Various theories have been proposed so far, but none has been able to solve this mystery. Since the thermodynamic properties of matter are determined by the interaction between particles, the mystery can be solved fundamentally if the thermodynamic physical quantities using the laws of thermodynamics and statistical mechanics are determined, the experimental results are reproduced, and the phenomena in relation to the shape of the interaction between particles are elucidated. In this sense, this book has fundamentally unraveled this mystery. In addition, it discusses the mysteries of isothermal compressibility, structural diversity, as well as liquefaction and boiling points of water in relation to the shape of the interaction between particles. It carefully explains the analysis and calculation methods so that they can be easily understood by the readers.

Chapter 1
Statistical Mechanics and Thermodynamics of Fluids

1.1 Statistical Mechanics

Consider a homogeneous isotropic disordered system consisting of N identical particles. The position vector of each particle is r₁, r₂, …, r_N, and the potential energy of the system is U_N (r₁, r₂, …, r_N), then the grand canonical partition function $\text{Ξ}$ is given by the following formula:

$$\Xi ={\displaystyle \sum _{N=0}^{\infty }{z}^N}{Q}_{\mathcal{N}},$$ (1.1)

$$Q_{\mathcal{N}}=\frac{1}{\mathcal{N}!}{\displaystyle \int d}{r}_1{\displaystyle \int d}{r}_2\cdots {\displaystyle \int d}{r}_{\mathcal{N}}\exp \left[-\beta U_{\mathcal{N}}\left(r_1,r_2,\cdots ,r_{\mathcal{N}}\right)\right],$$

$$z={\left(\frac{2\pi m}{\beta h^2}\right)}^{3/2}\exp (\beta \mu ),\beta =\frac{1}{kT},$$

where m is the particle mass, h is the Planck constant, k is the Boltzmann constant, T is the absolute temperature, and μ is the chemical potential. The canonical partition function for canonical ensemble ZN is given by

$$Z_{\mathcal{N}}={\left(\frac{2\pi m}{\beta h^2}\right)}^{(3/2)\mathcal{N}}{Q}_{\mathcal{N}}.$$ (1.2)

The following equation holds between the pressure p, volume V, and free energy F:

$$pV=kT\log \Xi ,F=-kT\log Z_{\mathcal{N}}.$$ (1.3)

Any other thermodynamic quantity can...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Preface
• 1 Statistical Mechanics and Thermodynamics of Fluids
• 2 Strange Temperature Change of Water Density
• 3 Fundamental Clarification of Thermodynamic Phenomena in Water
• 4 Variety of Shapes of Water Molecule Interactions
• 5 Ornstein–Zernike Equation
• 6 Calculation Procedure of SCOZA
• 7 Pressure and Chemical Potential
• 8 Thermodynamic Properties of Subcritical Fluids
• Appendix A Integration Region
• Appendix B Q^0(n),Q1(n,k),Q2(n,k)
• Appendix C F0(k), F1(k), F2(k, n), F3(k, n), F4(k, n), F5(k, n)
• Appendix D A,B
• Appendix E ∂xj/∂u
• Appendix F Stability of Solutions of Diffusion Equations
• Appendix G Solutions of Dn and G n (1 ≥ n ≥ 6) for Water below Tc for φc1
• Bibliography
• Index