**1. Introduction**

716 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

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ragazas/stocks-and-planck-s-law/ql47o1qdr604/2# Ragazas, C. (2011) *What is The Matter With de Broglie Waves?* knol This chapter deals with the statistical thermodynamics (statistical mechanics) a modern alternative of the classical (phenomenological) thermodynamics. Its aim is to determine thermodynamic properties of matter from forces acting among molecules. Roots of the discipline are in kinetic theory of gases and are connected with the names Maxwelland Boltzmann. Father of the statistical thermodynamics is Gibbs who introduced its concepts such as the statistical ensemble and others, that have been used up to present.

Nothing can express an importance of the statistical thermodynamics better than the words of Richard Feynman Feynman et al. (2006), the Nobel Prize winner in physics: *If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that* **All things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.**

In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.

The chapter is organized as follows. Next section contains axioms of the phenomenological thermodynamics. Basic concepts and axioms of the statistical thermodynamics and relations between the partition function and thermodynamic quantities are in Section 3. Section 4 deals with the ideal gas and Section 5 with the ideal crystal. Intermolecular forces are discussed in Section 6. Section 7 is devoted to the virial expansion and Section 8 to the theories of dense gases and liquids. The final section comments axioms of phenomenological thermodynamics in the light of the statistical thermodynamics.

## **2. Principles of phenomenological thermodynamics**

The phenomenological thermodynamics or simply thermodynamics is a discipline that deals with the thermodynamic system, a macroscopic part of the world. The thermodynamic state of system is given by a limited number of thermodynamic variables. In the simplest case of one-component, one-phase system it is for example volume of the system, amount of substance (*e.g.* in moles) and temperature. Thermodynamics studies changes of thermodynamic quantities such as pressure, internal energy, entropy, *e.t.c.* with thermodynamic variables.

**3. Principles of statistical thermodynamics**

but in the different microscopic states.

number of particles *N*, volume *V* and energy *E*.

(**NPT**) that will not be considered in this work.

• **Ensemble average of thermodynamic quantity**

**3.2 Axioms of the statistical thermodynamics**

**Axiom on equivalence of average values**

ensemble average

**Axiom on probability**

The statistical thermodynamics is bases on two axioms:

The time average *X<sup>τ</sup>* of a thermodynamic quantity *X* is given by

*<sup>X</sup><sup>τ</sup>* <sup>=</sup> <sup>1</sup> *τ τ* 0

where *X*(*t*) is a value of *X* at time *t* and, *τ* is a time interval of a measurement.

*Xs* = ∑ *i*

where *Xi* is a value in the quantum state *i*, and *Pi* is the probability of the quantum state.

It is postulated that the time average of thermodynamic quantity *X* is equivalent to its

Probability *Pi* of a quantum state *i* is only a function of energy of the quantum state, *Ei*,

The ensemble average *Xs* of a thermodynamic quantity *X* is given by

particles *N*, volume *V* and temperature *T*.

• **Time average of thermodynamic quantity**

The statistical thermodynamics considers thermodynamic system as an assembly of a very large number (of the order of 1023) of mutually interacting particles (usually molecules). It

Statistical Thermodynamics 719

The microscopic state of thermodynamic system is given by positions and velocities of all particles in the language of the Newton mechanics, or by the quantum states of the system in the language of quantum mechanics. There is a huge number of microscopic states that

Statistical ensemble is a collection of all systems that are in the same thermodynamic state

• **Microcanonical ensemble** or **NVE** ensemble is a collection of all systems at a given

• **Canonical ensemble** or **NVT** ensemble is a collection of all systems at a given number of

There is a number of ensembles, *e.g.* the grandcanonical (*μ***VT**) or isothermal isobaric

*X*(*t*) d*t* , (5)

*PiXi* , (6)

*X<sup>τ</sup>* = *Xs* . (7)

*Pi* = *f*(*Ei*). (8)

correspond to a given thermodynamic (macroscopic) state of the system.

**3.1 Basic concepts**

uses the following concepts: • **Microscopic state of system**

• **Statistical ensemble**

The phenomenological thermodynamics is based on six axioms (or postulates if you wish to call them), four of them are called the laws of thermodynamics:

## • **Axiom of existence of the thermodynamic equilibrium**

For thermodynamic system at unchained external conditions there exists a state of the thermodynamic equilibrium in which its macroscopic parameters remain constant in time. The thermodynamic system at unchained external conditions always reaches the state of the thermodynamic equilibrium.

## • **Axiom of additivity**

Energy of the thermodynamic system is a sum of energies of its macroscopic parts. This axiom allows to define extensive and intensive thermodynamic quantities.

### • **The zeroth law of thermodynamics**

When two systems are in the thermal equilibrium, *i.e.* no heat flows from one system to the other during their thermal contact, then both systems have the same temperature as an intensive thermodynamic parameter. If system A has the same temperature as system B and system B has the same temperature as system C, then system A also has the same temperature as system C (temperature is transitive).

#### • **The first law of thermodynamics**

There is a function of state called internal energy *U*. For its total differential d*U* we write

$$\mathbf{d}\mathbf{d}I = \mathbf{d}\mathcal{W} + \mathbf{d}\mathbf{Q} \, \tag{1}$$

where the symbols ¯d*Q* and ¯d*W* are not total differentials but represent infinitesimal values of heat *Q* and work *W* supplied to the system.

#### • **The second law of thermodynamics**

There is a function of state called entropy *S*. For its total differential d*S* we write

$$\text{d}\text{S}=\frac{\text{d}\text{Q}}{T}\_{\text{..}} \qquad \text{[reversible process]}\text{.}\tag{2}$$

$$\text{d}S > \frac{\text{d}Q}{T}, \qquad \text{[irreversible process]}.\tag{3}$$

#### • **The third law of thermodynamics**

At temperature of 0 K, entropy of a pure substance in its most stable crystalline form is zero

$$\lim\_{T \to 0} S = 0.\tag{4}$$

This postulate supplements the second law of thermodynamics by defining a natural referential value of entropy. The third law of thermodynamics implies that temperature of 0 K cannot be attained by any process with a finite number of steps.

Phenomenological thermodynamics using its axioms radically reduces an amount of experimental effort necessary for a determination of the values of thermodynamic quantities. For example enthalpy or entropy of a pure fluid need not be measured at each temperature and pressure but they can be calculated from an equation of state and a temperature dependence of the isobaric heat capacity of ideal gas. However, empirical constants in an equation of state and in the heat capacity must be obtained experimentally.
