- Statistical Mechanics and Thermodynamics of Fluids
- Statistical Mechanics
- Thermodynamics
- Pressure from Energy Equation
- Chemical Potential from Energy Equation
- Pressure and Chemical Potential from Compressibility Equation
- Consistency between Energy Equation and Compressibility Equation
- Strange Temperature Change of Water Density
- Positive Thermal Expansion
- Negative Thermal Expansion
- Fundamental Clarification of Thermodynamic Phenomena in Water

# Statistical Mechanics and Thermodynamics of Fluids

## Statistical Mechanics

Consider a homogeneous isotropic disordered system consisting of *Af* identical particles. The position vector of each particle is **r _{b} r_{2}, **, 1дг, and the potential energy of the system is

**l/jv(r _{b} r_{2}, **• • • ,

**r,**then the grand canonical partition function S is given by the following formula:

_{v}-),

where *m* is the particle mass, *h* is the Planck constant, *к* is the Boltzmann constant, *T* is the absolute temperature, and д is the chemical potential. The canonical partition function for canonical

*The Physics of Liquid Water *Makoto Yasutomi

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-25-1 (Hardcover), 978-1-003-05616-4 (eBook) www.jennystanford.com ensemble *Z^r* is given by

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

Any other thermodynamic quantity can be obtained from S or *Z^.*

The two-pair distribution function p^(r_{1(} **r _{2}) **for a uniform isotropic fluid (generic term for gas and liquid) is defined by the following equation:

where *p* = A'7 *V, ф[гу)* is the potential for the molecule pair (/', *j), *and r,_{;} **= **1**17 **- **Гу **|. The potential energy **r _{2}, **■ ■ • ,

**г**of the

_{Л}")system is given by the following equation:

If the radial distribution function g(r) is obtained, the thermodynamic quantities of the fluid can be obtained from the following equations:

where,

where, *к* is the isothermal compressibility,

Here, *E* is the energy of the whole system. Equations (1.7), (1.8), and (1.9) are respectively the compressibility equation, the energy equation, and the virial equation. Given *g[r*) for a given two- molecule interaction *ф[г*), all thermodynamic quantities can be derived from these equations. Thus, the fluid physics challenge comes down to finding ^(r). Real molecules are often not spherical, but in this book we treat molecules as spherical. Therefore, the intermolecular interaction *ф[г*) is a function of only *r.*

The potential energy *и* of the system per unit volume is called excess internal energy and is given by the following equation:

In this case, Eq. (1.8) can be written as follows:

## Thermodynamics

### Pressure from Energy Equation

If entropy is *S,* the following equation is obtained from the law of conservation of energy:

In this section, we assume *Af* = constant. In this case, the following equation is obtained from Eq. (1.12):

The second-order partial derivative of any thermodynamic function / with respect to any two independent thermodynamic variables *x* and *у* is independent of the order of differentiation, so (3^{2}//3x3y) = (3^{2}//3y3x) holds. Therefore, the following equation is obtained from Eq. (1.13):

From Eq. (1.11), the following equation is obtained:

From Eqs. (1.14) and (1.15), we obtain the following equation: Rewrite Eq. (1.16) to get

If Eq. (1.16) is integrated with *kT* or Eq. (1.17) is integrated with *p, *the pressure *p* is obtained.

### Chemical Potential from Energy Equation

In this section, we assume *V* = constant. In this case, the following equation is obtained from Eq. (1.12):

Using the same method as Section 1.2.1, the following expression is obtained from Eq. (1.18):

Integrating both sides of Eq. (1.20) with *p* gives the chemical potential *p* (see Section 7.4).

### Pressure and Chemical Potential from Compressibility Equation

From the second expression of Eq. (1.7), the following expression is obtained:

where, x,_{e}d is reduced isothermal compressibility. From Gibbs- Duhem equation

If *T =* constant, *dp* = *pdp* from Eq. (1.22), so the following expression is obtained from Eq. (1.21):

Integrating Eq. (1.23) over *p* gives *p* and *p.*

## Consistency between Energy Equation and Compressibility Equation

If the theoiy of thermodynamics and statistical mechanics is strictly correct, the same thermodynamic quantities should match regardless of their derivation paths. Here, we obtain the conditional expression for a physical quantity obtained from the energy equation coinciding with that from the compressibility equation.

Apply *р(д/др)р* to both sides of Eq. (1.20), change the order of differentiation, and Eq. (1.23) gives

This partial differential equation is a conditional expression to be obtained. This is one of the basic equations of Self-Consistent Ornstein-Zernike Approximation (SCOZA) described in Chapter 5.

# Strange Temperature Change of Water Density

Ordinary substances expand when heated and contract when cooled (positive thermal expansion). However, in the case of water, positive thermal expansion occurs when it is above 4°C, and below 4°C it shrinks when heated and expands when cooled (negative thermal expansion). This mysterious nature of water has been investigated for hundreds of years, and various theories have been proposed by many researchers to solve the mystery. However, none of these theories has solved this. The author has recently fundamentally elucidated the thermodynamic mechanism behind negative thermal expansion. It is discussed in detail in this chapter.

## Positive Thermal Expansion

Attractive forces act between the water molecules, which work to increase the density by condensing the water molecule population. On the other hand, each water molecule has a thermal disorder motion, and the energy of the disorder motion increases as it warms up, so that it expands against the intermolecular attractive forces.

*The Physics of Liquid Water *Makoto Yasutomi

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-25-1 (Hardcover), 978-1-003-05616-4 (eBook) www.jennystanford.com

However, when temperature and density are reached such that both shrinkage and expansion are balanced, a thermal equilibrium state is maintained. This is the thermodynamic mechanism of positive thermal expansion. So what is the thermodynamic mechanism that produces negative thermal expansion? It is one of the main purposes of this book to clarify this thermodynamic mechanism.

## Negative Thermal Expansion

In Fig. 2.1, the experimental data (closed circles) [Abascal and Vega (2005); Lide (1990)] show that the density of water *p* changes with temperature *T* under one atmospheric pressure. Above 4°C, the density decreases as the temperature rises (positive thermal expansion), but below 4°C the density increases with temperature (negative thermal expansion). Analyzing this experimental fact based on the laws of thermodynamics, we can get an important conclusion about the thermodynamic mechanism of negative thermal expansion, as explained below.

**Figure 2.1 **Density-temperature relation of liquid water at 1 bar. Closed circles, asterisks, triangles, and open circles show those for experimentally measured data [Abascal and Vega (2005); Lide (1990)], *ф _{л}, ф_{с2},* and

*ф*respectively, where

_{с3},*ф*= 0* in Ref. [Yasutomi (2016)].

_{с1}**Figure 2.2 **Schematic pressure isothermal curves (black lines): The intersections of the constant pressure line (red line) and the pressure isothermal curves give the relationship between density and temperature under constant pressure. When the temperature is raised above 4 C, the isothermal curve of the pressure moves to the high pressure side (a > 0), and positive thermal expansion occurs. Below 4"C, the reverse behavior (a < 0) occurs, so negative thermal expansion occurs.

A curve representing the relationship between pressure *p* and density *p,* when the temperature is kept constant, is called a pressure isothermal curve. For any substance, if the temperature is kept constant, the pressure increases with density, so the isothermal curve becomes a monotonically increasing curve (black lines in the Fig. 2.2). For normal substances and water above 4 C, if the density is constant, the pressure *p* increases with the temperature *T,* so *a = [др/дкТ)„ >* 0. Therefore, the isothermal curve moves to the high pressure side (from bottom to top in the Fig. 2.2) as the temperature rises. In the Fig. 2.2, the intersections of the constant pressure horizontal line (red line) and the isothermal curves (black lines) give the relationship between density and temperature under constant pressure (isobanc curve). Of course, it can be seen that the density decreases with increasing temperature (positive thermal expansion). In other words, positive thermal expansion is equivalent to the pressure isothermal curve moving to the high pressure side as the temperature rises, and it can be seen that this phenomenon occurs when *a >* 0.

From similar considerations, we can see that the negative thermal expansion occurs when *a* < 0 below 4°C. Therefore, the basic elucidation of the mystery of negative thermal expansion is to clarify the thermodynamic mechanism that switches the sign of *a* from positive to negative at 4°C with cooling in relation to the functional form of the interaction between water molecules.

# Fundamental Clarification of Thermodynamic Phenomena in Water

The thermodynamic properties of a material are determined by the interactions between the particles that make the material up. These can be derived using thermodynamic laws and statistical mechanics techniques. Therefore, if the relationship between the functional form of the interparticle interactions and the thermodynamic phenomenon can be established using thermodynamic quantities, it can be said that the mystery of the phenomenon has been fundamentally solved.

Thermodynamic quantities can be derived from various thermodynamic potentials, but in this chapter we use excess internal energy u(p, defined by Eq. (1.10). The reason is that *и* has a clear relationship with the functional form of the interaction between particles, as can be seen from Eq. (1.10). In this sense, *и* is the most suitable for fundamentally elucidating the thermodynamic mechanism that generates negative thermal expansion compared to other thermodynamic potentials.

As can be seen from Eq. (1.16), the sign of a is determined by the competition among *p, u,* and *p{du/dp*)_{T}-Since *p* is always positive, it

*The Physics of Liquid Water *Makoto Yasutomi

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-25-1 (Hardcover), 978-1-003-05616-4 (eBook) www.jennystanford.com works to make *a* positive. However, *и* and *р(ди/др) _{т}* can be positive or negative depending on the thermodynamic state of the liquid. Therefore, it can be seen that these two terms play an important role in switching the sign of

*a.*We need to elucidate these thermal behaviors in relation to the shape of the interparticle interaction

*ф(г).*