• **Statistical ensemble**

Statistical ensemble is a collection of all systems that are in the same thermodynamic state but in the different microscopic states.


There is a number of ensembles, *e.g.* the grandcanonical (*μ***VT**) or isothermal isobaric (**NPT**) that will not be considered in this work.

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

$$\overline{X}\_{\tau} = \frac{1}{\tau} \int\_{0}^{\tau} X(t) \,\mathrm{d}t \,\mathrm{d}t \,\tag{5}$$

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

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

$$\overline{X}\_s = \sum\_{i} P\_i X\_i \,\tag{6}$$

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

### **3.2 Axioms of the statistical thermodynamics**

The statistical thermodynamics is bases on two axioms:

### **Axiom on equivalence of average values**

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

$$
\overline{X}\_{\mathsf{T}} = \overline{X}\_{\mathsf{s}}.\tag{7}
$$

### **Axiom on probability**

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

$$P\_{\bar{l}} = f(E\_{\bar{l}}) \,. \tag{8}$$

Unfortunately, the partition function is known only for the simplest cases such as the ideal gas (Section 4) or the ideal crystal (Section 5). In all the other cases, real gases and liquids

Statistical Thermodynamics 721

A relation between entropy *S* and probabilities *Pi* of quantum states of a system can be proved

*i*

where *W* is a number of accessible states. This equation (with log instead of ln) is written in

The ideal gas is in statistical thermodynamics modelled by a assembly of particles that do not mutually interact. Then the energy of i-th quantum state of system, *Ei*, is a sum of energies of

> *N* ∑ *i*=1

In this way a problem of a determination of the partition function of system is dramatically

*<sup>Q</sup>* <sup>=</sup> *<sup>q</sup><sup>N</sup> N*!

The partition function of molecule may be further simplified. The energy of molecule can be approximated by a sum of the translational *�*trans, the rotational *�*rot, the vibrational *�*vib, and

where *�*<sup>0</sup> is the zero point energy. The partition function of system then becomes a product

Consequently all thermodynamic quantities of the ideal gas become sums of the

= *kBT* ln *N*! + *U*<sup>0</sup> − *NkBT* ln *q*tr − *NkBT* ln *q*rot − *NkBT* ln *q*vib − *NkBT* ln *q*el

where *U*<sup>0</sup> = *N�*<sup>0</sup> and *A*tr, *A*rot, *A*vib, *A*el are the translational, rotational, vibrational,

= *kBT* ln *N*! + *U*<sup>0</sup> + *A*tr + *A*rot + *A*vib + *A*el , (27)

*Ei* =

*q* = ∑ *j*

the electronic *�*el contributions (subscript *j* in *�<sup>j</sup>* is omitted for simplicity of notation)

*<sup>Q</sup>* <sup>=</sup> exp(−*Nβ�*0)

corresponding contributions. For example the Helmholtz free energy is

electronic contributions to the Helmholtz free energy, respectively.

*Pi* ln *Pi* . (20)

*�i*,*<sup>j</sup>* . (22)

, (23)

exp(−*β�j*) (24)

*�* = *�*<sup>0</sup> + *�*trans + *�*rot + *�*vib + *�el* , (25)

*<sup>N</sup>*! *<sup>q</sup>*trans*q*rot*q*vib*q*el . (26)

*S* = *kB* ln *W* , (21)

*<sup>S</sup>* = −*kB* ∑

the grave of Ludwig Boltzmann in Central Cemetery in Vienna, Austria.

simplified. For one-component system of *N* molecules it holds

considered here, it can be determined only approximatively.

For the microcanonical ensemble a similar relation holds

**3.5 Probability and entropy**

in the canonical ensemble

**4. Ideal gas**

where

individual particles

is the partition function of molecule.

*A* = −*kBT* ln *Q*

#### **3.3 Probability in the microcanonical and canonical ensemble**

From Eq.(8) relations between the probability and energy can be derived:

#### **Probability in the microcanonical ensemble**

All the microscopic states in the microcanonical ensemble have the same energy. Therefore,

$$P\_{\bar{l}} = \frac{1}{W} \quad \text{for} \quad \bar{i} = 1, 2, \dots, W \,\tag{9}$$

where *W* is a number of microscopical states (the statistical weight) of the microcanonical ensemble.

#### **Probability in the canonical ensemble**

In the canonical ensemble it holds

$$P\_i = \frac{\exp(-\beta E\_i)}{Q} \,\, \,\, \,\tag{10}$$

where *<sup>β</sup>* <sup>=</sup> <sup>1</sup> *kBT* , *kB* is the Boltzmann constant, *<sup>T</sup>* temperature and *<sup>Q</sup>* is the *partition function*

$$Q = \sum\_{i} \exp(-\beta E\_i) \, , \tag{11}$$

where the sum is over the microscopic states of the canonical ensemble.

#### **3.4 The partition function and thermodynamic quantities**

If the partition function is known thermodynamic quantities may be determined. The following relations between the partition function in the canonical ensemble and thermodynamic quantities can be derived

$$A = -k\_B T \ln Q \tag{12}$$

$$
\Delta U = k\_B T^2 \left(\frac{\partial \ln Q}{\partial T}\right)\_V \tag{13}
$$

$$S = k\_B \ln Q + k\_B T \left(\frac{\partial \ln Q}{\partial T}\right)\_V \,. \tag{14}$$

$$\mathcal{C}\_{V} = \left(\frac{\partial \mathcal{U}}{\partial T}\right)\_{V} = k\_{B} \, T^{2} \frac{\partial^{2} \ln \mathcal{Q}}{\partial T^{2}} + 2k\_{B} \, T \left(\frac{\partial \ln \mathcal{Q}}{\partial T}\right)\_{V} \, \tag{15}$$

$$p = -\left(\frac{\partial A}{\partial V}\right)\_T = k\_B T \left(\frac{\partial \ln Q}{\partial V}\right)\_T \tag{16}$$

$$H = \mathcal{U} + pV = k\_B T^2 \left(\frac{\partial \ln Q}{\partial T}\right)\_V + Vk\_B T \left(\frac{\partial \ln Q}{\partial V}\right)\_T \tag{17}$$

$$G = A + pV = -k\_B T \ln Q + V k\_B T \left(\frac{\partial \ln Q}{\partial V}\right)\_T \tag{18}$$

$$\mathbb{C}\_p = \left(\frac{\partial H}{\partial T}\right)\_V = \mathbb{C}\_V + Vk\_B \frac{\partial^2 \ln Q}{\partial V \partial T} \,. \tag{19}$$

*A* is Helmholtz free energy, *U* internal energy, *S* entropy, *CV* isochoric heat capacity, *p* pressure, *H* enthalpy, *G* Gibbs free energy and *Cp* isobaric heat capacity.

4 Will-be-set-by-IN-TECH

All the microscopic states in the microcanonical ensemble have the same energy. Therefore,

where *W* is a number of microscopical states (the statistical weight) of the microcanonical

*Pi* <sup>=</sup> exp(−*βEi*)

If the partition function is known thermodynamic quantities may be determined. The following relations between the partition function in the canonical ensemble and

*Q* = ∑ *i*

where the sum is over the microscopic states of the canonical ensemble.

 *∂* ln *Q ∂T*

 *V*

> *∂* ln *Q ∂T*

<sup>=</sup> *kB <sup>T</sup>*<sup>2</sup> *<sup>∂</sup>*<sup>2</sup> ln *<sup>Q</sup>*

= *kB T*

*G* = *A* + *pV* = −*kB T* ln *Q* + *V kB T*

pressure, *H* enthalpy, *G* Gibbs free energy and *Cp* isobaric heat capacity.

= *CV* + *VkB*

 *V*

*<sup>∂</sup>T*<sup>2</sup> <sup>+</sup> <sup>2</sup>*kB <sup>T</sup>*

 *V*

*∂*<sup>2</sup> ln *Q*

*A* is Helmholtz free energy, *U* internal energy, *S* entropy, *CV* isochoric heat capacity, *p*

 *T*

 *∂* ln *Q ∂V*

 *∂* ln *Q ∂T*

**3.4 The partition function and thermodynamic quantities**

*S* = *kB* ln *Q* + *kB T*

 *∂A ∂V T*

*H* = *U* + *pV* = *kB T*<sup>2</sup>

 *∂H ∂T V*

 *∂U ∂T V*

thermodynamic quantities can be derived

*CV* =

*Cp* =

*p* = −

*U* = *kBT*<sup>2</sup>

*kBT* , *kB* is the Boltzmann constant, *<sup>T</sup>* temperature and *<sup>Q</sup>* is the *partition function*

*A* = −*kB T* ln *Q* (12)

 *∂* ln *Q ∂T*

+ *V kB T*

 *∂* ln *Q ∂V*

 *V*

 *∂* ln *Q ∂V*

> *T*

*<sup>W</sup>* for *<sup>i</sup>* <sup>=</sup> 1, 2, . . . , *<sup>W</sup>* , (9)

*<sup>Q</sup>* , (10)

exp(−*βEi*), (11)

. (14)

, (16)

 *T*

*<sup>∂</sup>V∂<sup>T</sup>* . (19)

, (15)

, (17)

, (18)

(13)

**3.3 Probability in the microcanonical and canonical ensemble**

**Probability in the microcanonical ensemble**

**Probability in the canonical ensemble** In the canonical ensemble it holds

ensemble.

where *<sup>β</sup>* <sup>=</sup> <sup>1</sup>

From Eq.(8) relations between the probability and energy can be derived:

*Pi* <sup>=</sup> <sup>1</sup>

Unfortunately, the partition function is known only for the simplest cases such as the ideal gas (Section 4) or the ideal crystal (Section 5). In all the other cases, real gases and liquids considered here, it can be determined only approximatively.

#### **3.5 Probability and entropy**

A relation between entropy *S* and probabilities *Pi* of quantum states of a system can be proved in the canonical ensemble

$$S = -k\_B \sum\_{i} P\_i \ln P\_i \,. \tag{20}$$

For the microcanonical ensemble a similar relation holds

$$S = k\_B \ln \mathcal{W} \,\,\,\,\tag{21}$$

where *W* is a number of accessible states. This equation (with log instead of ln) is written in the grave of Ludwig Boltzmann in Central Cemetery in Vienna, Austria.

### **4. Ideal gas**

The ideal gas is in statistical thermodynamics modelled by a assembly of particles that do not mutually interact. Then the energy of i-th quantum state of system, *Ei*, is a sum of energies of individual particles

$$E\_{\vec{l}} = \sum\_{i=1}^{N} \varepsilon\_{\vec{l}, \vec{j}} \,. \tag{22}$$

In this way a problem of a determination of the partition function of system is dramatically simplified. For one-component system of *N* molecules it holds

$$Q = \frac{q^N}{N!} \, ' \tag{23}$$

where

$$q = \sum\_{j} \exp(-\beta \epsilon\_j) \tag{24}$$

is the partition function of molecule.

The partition function of molecule may be further simplified. The energy of molecule can be approximated by a sum of the translational *�*trans, the rotational *�*rot, the vibrational *�*vib, and the electronic *�*el contributions (subscript *j* in *�<sup>j</sup>* is omitted for simplicity of notation)

$$
\epsilon = \epsilon\_0 + \epsilon\_{\text{trans}} + \epsilon\_{\text{rot}} + \epsilon\_{\text{vib}} + \epsilon\_{\text{el}} \,\prime \tag{25}
$$

where *�*<sup>0</sup> is the zero point energy. The partition function of system then becomes a product

$$Q = \frac{\exp(-N\beta\varepsilon\_0)}{N!} q\_{\text{trans}} q\_{\text{rot}} q\_{\text{vib}} q\_{\text{el}} \,. \tag{26}$$

Consequently all thermodynamic quantities of the ideal gas become sums of the corresponding contributions. For example the Helmholtz free energy is

$$\begin{split} A &= -k\_B T \ln Q \\ &= k\_B T \ln N! + \mathcal{U}\_0 - N k\_B T \ln q\_{\text{fr}} - N k\_B T \ln q\_{\text{rot}} - N k\_B T \ln q\_{\text{vib}} - N k\_B T \ln q\_{\text{el}} \\ &= k\_B T \ln N! + \mathcal{U}\_0 + A\_{\text{fr}} + A\_{\text{rot}} + A\_{\text{vib}} + A\_{\text{el}} \end{split} \tag{27}$$

where *U*<sup>0</sup> = *N�*<sup>0</sup> and *A*tr, *A*rot, *A*vib, *A*el are the translational, rotational, vibrational, electronic contributions to the Helmholtz free energy, respectively.

with *n* a number of atoms in molecule, *mi* their atomic masses and *ri* their distances from the

Statistical Thermodynamics 723

*<sup>A</sup>*rot <sup>=</sup> <sup>−</sup>*RT* ln *<sup>q</sup>*rot <sup>=</sup> <sup>−</sup>*RT* ln <sup>8</sup>*π*<sup>2</sup> *IkBT*

*σh*<sup>2</sup>

8*π*<sup>2</sup>*kBT h*2

3/2

where *IA*, *IB* and *IC* the principal moments of inertia. Contributions to the thermodynamic

1 *σ*

3/2

8*π*<sup>2</sup>*kBT h*2

(*πIA IB IC*)1/2

*p*rot = 0 , (50)

*H*rot = *U*rot , (52) *G*rot = *A*rot , (53)

*Cp*,rot = *CV*,rot . (55)

Vibrations of atoms in molecule around their equilibrium states may be at not very high

In a diatomic molecule there is only one vibrational motion. Its partition function is

*q*vib = [1 − exp(*hν*0/*kBT*)]

(*πIA IB IC*)

3/2

+ 3 2

*RT* , (51)

*R* , (54)

*σh*<sup>2</sup>

*p*rot = 0 , (41) *U*rot = *RT* , (42) *H*rot = *U*rot , (43) *G*rot = *F*rot , (44) *CV*,rot = *R* , (45) *Cp*,rot = *CV*,rot . (46)

+ *R* , (40)

, (39)

1/2 , (47)

*R* , (49)

<sup>−</sup><sup>1</sup> , (56)

, (48)

(*πIA IB IC*)1/2

center of mass. Contributions to the thermodynamic quantities are

The partition function of rotation of a non-linear molecule is

*<sup>q</sup>*rot <sup>=</sup> <sup>1</sup> *σ*

*<sup>A</sup>*rot <sup>=</sup> <sup>−</sup>*RT* ln *<sup>q</sup>*rot <sup>=</sup> <sup>−</sup>*RT* ln

1 *σ* 8*π*<sup>2</sup>*kBT h*2

*<sup>S</sup>*rot <sup>=</sup> *<sup>R</sup>* ln

*<sup>U</sup>*rot <sup>=</sup> <sup>3</sup> 2

*CV*,rot <sup>=</sup> <sup>3</sup>

**4.3 Vibrational contributions**

**4.3.1 Diatomic molecules**

2

temperatures approximated by harmonic oscillators.

**4.2.2 Non-linear molecules**

quantities are

*<sup>S</sup>*rot <sup>=</sup> *<sup>R</sup>* ln <sup>8</sup>*π*<sup>2</sup> *IkBT*

#### **4.1 Translational contributions**

Translational motions of a molecule are modelled by a particle in a box. For its energy a solution of the Schrödinger equation gives

$$\mathfrak{e}\_{\rm tr} = \frac{h^2}{8m} \left( \frac{n\_x^2}{a^2} + \frac{n\_y^2}{b^2} + \frac{n\_z^2}{c^2} \right) \, , \tag{28}$$

where *h* is the Planck constant, *m* mass of molecule, and *abc* = *V* where *V* is volume of system. Quantities *nx*, *ny*, *nz* are the quantum numbers of translation. The partition function of translation is

$$q\_{\rm tr} = \left(\frac{2\pi mk\_B T}{h^2}\right)^{3/2} V.\tag{29}$$

Translational contribution to the Helmholtz energy is

$$A\_{\rm tr} = -RT\ln q\_{\rm tr} = -RT\ln\left(\lambda^{-3}V\right) \, , \tag{30}$$

where *R* = *NkB* is the gas constant and *λ* = *h*/ <sup>√</sup>2*πmkBT* is the Broglie wavelength. The remaining thermodynamic functions are as follows

$$S\_{\rm tr} = -\left(\frac{\partial A\_{\rm tr}}{\partial T}\right)\_V = R\ln\left(\lambda^{-3}V\right) + \frac{3}{2}R\_{\prime} \tag{31}$$

$$p\_{\rm tr} = -\left(\frac{\partial A\_{\rm tr}}{\partial V}\right)\_{T} = \frac{RT}{V} \,\text{,}\tag{32}$$

$$\mathcal{U}\_{\rm tr} = A\_{\rm tr} + T\mathcal{S}\_{\rm tr} = \frac{3}{2}RT\,\tag{33}$$

$$H\_{\rm tr} = \mathcal{U}\_{\rm tr} + p\_{\rm tr}V = \frac{5}{2}RT\,\,\,\,\tag{34}$$

$$G\_{\rm tr} = A\_{\rm tr} + p\_{\rm tr}V = -RT\ln\left(\lambda^{-3}V\right) + RT\,\tag{35}$$

$$\mathbf{C}\_{V, \text{tr}} = \left(\frac{\partial \mathcal{U}\_{\text{tr}}}{\partial T}\right)\_V = \frac{3}{2} \mathcal{R}\_{\text{\textdegree}} \tag{36}$$

$$\mathbb{C}\_{p, \text{tr}} = \left(\frac{\partial H\_{\text{tr}}}{\partial T}\right)\_p = \frac{5}{2}R.\tag{37}$$

#### **4.2 Rotational contributions**

Rotations of molecule are modelled by the rigid rotator. For linear molecules there are two independent axes of rotation, for non-linear molecules there are three.

#### **4.2.1 Linear molecules**

For the partiton function of rotation it holds

$$q\_{\rm rot} = \frac{8\pi \, I \, k\_B \, T}{\sigma \, h^2} \,, \tag{38}$$

where *σ* is the symmetry number of molecule and *I* its moment of inertia

$$I = \sum\_{1}^{n} m\_i r\_i^2 \,\mu$$

6 Will-be-set-by-IN-TECH

Translational motions of a molecule are modelled by a particle in a box. For its energy a

where *h* is the Planck constant, *m* mass of molecule, and *abc* = *V* where *V* is volume of system. Quantities *nx*, *ny*, *nz* are the quantum numbers of translation. The partition function

> 2*πmkBT h*2

> > = *R* ln *λ*−3*V* + 3 2

<sup>=</sup> *RT*

2

2

<sup>=</sup> <sup>3</sup> 2

Rotations of molecule are modelled by the rigid rotator. For linear molecules there are two

*<sup>q</sup>*rot <sup>=</sup> <sup>8</sup>*<sup>π</sup> I kB <sup>T</sup>*

*I* = *n* ∑ 1 *mir* 2 *i* ,  *λ*−3*V* 

*A*tr = −*RT* ln *q*tr = −*RT* ln

 *V*

 *T* *n*2 *y <sup>b</sup>*<sup>2</sup> <sup>+</sup>

3/2

 *λ*−3*V* 

<sup>√</sup>2*πmkBT* is the Broglie wavelength.

*<sup>V</sup>* , (32)

*RT* , (33)

*RT* , (34)

*R* , (36)

*R* . (37)

*<sup>σ</sup> <sup>h</sup>*<sup>2</sup> , (38)

*n*2 *z c*2

, (28)

*V* . (29)

, (30)

*R* , (31)

+ *RT* , (35)

 *n*2 *x <sup>a</sup>*<sup>2</sup> <sup>+</sup>

*�*tr <sup>=</sup> *<sup>h</sup>*<sup>2</sup> 8*m*

*q*tr =

 *∂A*tr *∂T*

 *∂A*tr *∂V*

*<sup>U</sup>*tr <sup>=</sup> *<sup>A</sup>*tr <sup>+</sup> *TS*tr <sup>=</sup> <sup>3</sup>

*<sup>H</sup>*tr <sup>=</sup> *<sup>U</sup>*tr <sup>+</sup> *<sup>p</sup>*tr*<sup>V</sup>* <sup>=</sup> <sup>5</sup>

 *∂U*tr *∂T*

 *∂H*tr *∂T*

independent axes of rotation, for non-linear molecules there are three.

where *σ* is the symmetry number of molecule and *I* its moment of inertia

*G*tr = *A*tr + *p*tr*V* = −*RT* ln

 *V*

> *p* <sup>=</sup> <sup>5</sup> 2

**4.1 Translational contributions**

of translation is

solution of the Schrödinger equation gives

Translational contribution to the Helmholtz energy is

The remaining thermodynamic functions are as follows

*S*tr = −

*p*tr = −

*CV*,tr =

*Cp*,tr =

**4.2 Rotational contributions**

For the partiton function of rotation it holds

**4.2.1 Linear molecules**

where *R* = *NkB* is the gas constant and *λ* = *h*/

with *n* a number of atoms in molecule, *mi* their atomic masses and *ri* their distances from the center of mass. Contributions to the thermodynamic quantities are

$$A\_{\rm rot} = -RT \ln q\_{\rm rot} = -RT \ln \left(\frac{8\pi^2 I k\_B T}{\sigma h^2}\right) \tag{39}$$

$$S\_{\rm rot} = R \ln \left( \frac{8 \pi^2 I k\_B T}{\sigma h^2} \right) + R\_\prime \tag{40}$$

$$p\_{\rm rot} = 0\_{\prime} \tag{41}$$

$$
\hat{\mathbf{u}}\_{\text{rot}} = \mathbf{R} \mathbf{T} \,\tag{42}
$$

$$H\_{\rm rot} = \mathcal{U}\_{\rm rot},\tag{43}$$

$$G\_{\rm rot} = F\_{\rm rot} \tag{44}$$

$$
\mathbb{C}\_{V, \text{rot}} = \mathbb{R},
\tag{45}
$$

$$
\mathbb{C}\_{p, \text{rot}} = \mathbb{C}\_{V, \text{rot}} \,. \tag{46}
$$

#### **4.2.2 Non-linear molecules**

The partition function of rotation of a non-linear molecule is

$$q\_{\rm rot} = \frac{1}{\sigma} \left(\frac{8\pi^2 k\_B T}{h^2}\right)^{3/2} \left(\pi I\_A I\_B I\_C\right)^{1/2} \text{ .}\tag{47}$$

where *IA*, *IB* and *IC* the principal moments of inertia. Contributions to the thermodynamic quantities are

$$A\_{\rm rot} = -RT\ln q\_{\rm rot} = -RT\ln\left[\frac{1}{\sigma}\left(\frac{8\pi^2 k\_B T}{h^2}\right)^{3/2} \left(\pi I\_A I\_B I\_C\right)^{1/2}\right],\tag{48}$$

$$S\_{\rm rot} = R \ln \left[ \frac{1}{\sigma} \left( \frac{8\pi^2 k\_B T}{h^2} \right)^{3/2} \left( \pi I\_A I\_B I\_C \right)^{1/2} \right] + \frac{3}{2} R \,\tag{49}$$

$$p\_{\text{rot}} = 0\_{\text{A}} \tag{50}$$

*<sup>U</sup>*rot <sup>=</sup> <sup>3</sup> 2 *RT* , (51)

$$\begin{aligned} H\_{\text{rot}} &= U\_{\text{rot}} \\ G\_{\text{rot}} &= A\_{\text{rot}} \end{aligned} \tag{52}$$

$$\mathbf{C}\_{V, \text{rot}} = \frac{3}{2} \mathbf{R}\_{\prime} \tag{54}$$

$$
\begin{array}{ccc}
\circ \text{"} & & & \\
\circ \text{"} & & & \\
\circ \text{"} & & & \\
\end{array}
\tag{52}
$$

$$
\mathbb{C}\_{p, \text{rot}} = \mathbb{C}\_{V, \text{rot}} \,. \tag{55}
$$

#### **4.3 Vibrational contributions**

Vibrations of atoms in molecule around their equilibrium states may be at not very high temperatures approximated by harmonic oscillators.

#### **4.3.1 Diatomic molecules**

In a diatomic molecule there is only one vibrational motion. Its partition function is

$$q\_{\rm vib} = \left[1 - \exp\left(h\nu\_0/k\_B T\right)\right]^{-1} \,\tag{56}$$

where *xi* <sup>=</sup> *<sup>h</sup>ν<sup>i</sup>*

*kBT* .

**4.4 Electronic contributions**

The electronic partition function reads

Therefore they are not written here.

**4.5 Ideal gas mixture**

fraction. Then

**5. Ideal crystal**

partition function of crystal is

where *U*<sup>0</sup> is the lattice energy.

Debye approximation.

*q*el =

mixture is straightforward). The partition function of mixture is

∞ ∑ =0

*g*el,*e*

Statistical Thermodynamics 725

where *ε*el, the energy level , and *g*el, is its degeneracy. In most cases the electronic contributions to the thermodynamic functions are negligible at not very hight temperatures.

Let us consider two-component mixture of *N*<sup>1</sup> non-interacting molecules of component 1 and *N*<sup>2</sup> non-interacting molecules of component 2 (extension to the case of a multi-component

where *q*<sup>1</sup> and *q*<sup>2</sup> are the partition functions of molecules 1 and 2, respectively. Let us denote

We will call the ideal crystal an assembly of molecules displayed in a regular lattice without any impurities or lattice deformations. Distances among lattice centers will not depend on temperature and pressure. For simplicity we will consider one-atomic molecules. The

We will discuss here two models of the ideal crystal: the Einstein approximation and the

*Q* = *e*

*qN*<sup>2</sup> 2

*RT Vm*

+ *x*<sup>2</sup> *RT Vm*

*U* = *x*1*Um*,1 + *x*2*Um*,2 , (80) *H* = *x*1*Hm*,1 + *x*2*Hm*,2 , (81) *CV* = *x*1*CVm*,1 + *x*2*CVm*,2 , (82) *Cp* = *x*1*Cpm*,1 + *x*2*Cpm*,2 . (83)

<sup>−</sup>*U*0/*kB TQ*vib , (84)

*A* = *RT* (*x*<sup>1</sup> ln *x*<sup>1</sup> + *x*<sup>2</sup> ln *x*2) + *x*1*Am*,1 + *x*2*Am*,2 (76) *S* = −*R* (*x*<sup>1</sup> ln *x*<sup>1</sup> + *x*<sup>2</sup> ln *x*2) + *x*1*Sm*,1 + *x*2*Sm*,2 , (77) *G* = *RT* (*x*<sup>1</sup> ln *x*<sup>1</sup> + *x*<sup>2</sup> ln *x*2) + *x*1*Gm*,1 + *x*2*Gm*,2 , (78)

*<sup>Q</sup>* <sup>=</sup> *<sup>q</sup>N*<sup>1</sup> 1 *N*1!

*Xm*,*<sup>i</sup>* the molar thermodynamic quantity of pure component *<sup>i</sup>*, *<sup>i</sup>* = 1, 2 and *xi* = *Ni*

*p* = *x*<sup>1</sup> *pm*,1 + *x*<sup>2</sup> *pm*,2 = *x*<sup>1</sup>

<sup>−</sup>*ε*el,/*kBT* , (74)

*<sup>N</sup>*2! (75)

, (79)

*<sup>N</sup>*1+*N*<sup>2</sup> its mole

where *ν*<sup>0</sup> is the fundamental harmonic frequency. Vibrational contributions to thermodynamic quantities are

$$A\_{\rm vib} = -RT \ln q\_{\rm vib} = RT \ln \left( 1 - e^{-x} \right) \, , \tag{57}$$

$$S\_{\rm vib} = R \frac{\varkappa e^{-\chi}}{1 - e^{-\chi}} - R \ln\left(1 - e^{-\chi}\right) \tag{58}$$

*p*vib = 0 , (59)

$$\mathcal{U}\_{\rm vib} = RT \frac{\varkappa e^{-\varkappa}}{1 - e^{-\varkappa}} \,\prime\,\tag{60}$$

$$H\_{\rm vib} = \mathcal{U}\_{\rm vib} \,\,\,\,\,\,\tag{61}$$

*G*vib = *A*vib , (62)

$$\mathcal{C}\_{V, \text{vib}} = R \frac{\mathbf{x}^2 e^{-\mathbf{x}}}{(1 - e^{-\mathbf{x}})^2} \,' \,. \tag{63}$$

*Cp*,vib = *CV*,vib , (64)

where *<sup>x</sup>* <sup>=</sup> *<sup>h</sup>ν*<sup>0</sup> *kBT* .

#### **4.3.2 Polyatomic molecules**

In *n*-atomic molecule there is *f* fundamental harmonic frequencies *ν<sup>i</sup>* where

$$f = \begin{cases} 3n - 5 & \text{linear molecule} \\ 3n - 6 & \text{non-linear molecule} \end{cases}$$

The partition function of vibration is

$$q\_{\rm vib} = \prod\_{i=1}^{f} \frac{1}{1 - \exp(-h\nu\_i/k\_B T)}\,. \tag{65}$$

For the thermodynamic functions of vibration we get

$$A\_{\rm vib} = -RT \ln q\_{\rm vib} = RT \sum\_{i=1}^{f} \ln \left( 1 - e^{-\mathbf{x}\_{i}} \right) \tag{66}$$

$$S\_{\rm vib} = R \sum\_{i=1}^{f} \frac{\mathbf{x}\_{i} e^{-\mathbf{x}\_{i}}}{1 - e^{-\mathbf{x}\_{i}}} - R \sum\_{i=1}^{f} \ln\left(1 - e^{-\mathbf{x}\_{i}}\right) \tag{67}$$

$$p\_{\rm vib} = 0\_{\prime} \tag{68}$$

$$\mathcal{U}L\_{\text{vib}} = RT \sum\_{i=1}^{f} \frac{\mathbf{x}\_{i}e^{-\mathbf{x}\_{i}}}{1 - e^{-\mathbf{x}\_{i}}} \, \prime \tag{69}$$

*H*vib = *U*vib , (70)

*G*vib = *A*vib , (71)

$$\mathcal{C}\_{V, \text{vib}} = R \sum\_{i=1}^{f} \frac{\mathbf{x}\_i^2 e^{-\mathbf{x}\_i}}{(1 - e^{-\mathbf{x}\_i})^2} \,\prime \tag{72}$$

$$\mathsf{C}\_{p,\text{vib}} = \mathsf{C}\_{V,\text{vib}}.\tag{73}$$

where *xi* <sup>=</sup> *<sup>h</sup>ν<sup>i</sup> kBT* .

8 Will-be-set-by-IN-TECH

where *ν*<sup>0</sup> is the fundamental harmonic frequency. Vibrational contributions to thermodynamic

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*<sup>x</sup>* <sup>−</sup> *<sup>R</sup>* ln �

1 − *e* −*x*�

*p*vib = 0 , (59)

*H*vib = *U*vib , (61) *G*vib = *A*vib , (62)

*Cp*,vib = *CV*,vib , (64)

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*<sup>x</sup>* , (60)

(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*x*)<sup>2</sup> , (63)

<sup>1</sup> <sup>−</sup> exp(−*hνi*/*kBT*) . (65)

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*xi* , (69)

(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*xi*)<sup>2</sup> , (72)

, (66)

, (67)

1 − *e* −*x*� , (57)

, (58)

*<sup>A</sup>*vib <sup>=</sup> <sup>−</sup>*RT* ln *<sup>q</sup>*vib <sup>=</sup> *RT* ln �

*<sup>S</sup>*vib <sup>=</sup> *<sup>R</sup> xe*−*<sup>x</sup>*

*<sup>U</sup>*vib <sup>=</sup> *RT xe*−*<sup>x</sup>*

*CV*,vib <sup>=</sup> *<sup>R</sup> <sup>x</sup>*2*e*−*<sup>x</sup>*

In *n*-atomic molecule there is *f* fundamental harmonic frequencies *ν<sup>i</sup>* where

*f* ∏ *i*=1

*A*vib = −*RT* ln *q*vib = *RT*

*f* ∑ *i*=1

*f* ∑ *i*=1

*xie*−*xi* <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*xi* <sup>−</sup> *<sup>R</sup>*

*xie*−*xi*

*x*2 *<sup>i</sup> <sup>e</sup>*−*xi*

*f* ∑ *i*=1

3*n* − 5 linear molecule

3*n* − 6 non-linear molecule

1

*f* ∑ *i*=1 ln � 1 − *e* −*xi* �

*f* ∑ *i*=1 ln � 1 − *e* −*xi* �

*p*vib = 0 , (68)

*H*vib = *U*vib , (70) *G*vib = *A*vib , (71)

*Cp*,vib = *CV*,vib , (73)

*f* =

For the thermodynamic functions of vibration we get

*S*vib = *R*

*U*vib = *RT*

*CV*,vib = *R*

⎧ ⎨ ⎩

*q*vib =

quantities are

where *<sup>x</sup>* <sup>=</sup> *<sup>h</sup>ν*<sup>0</sup>

*kBT* .

**4.3.2 Polyatomic molecules**

The partition function of vibration is

### **4.4 Electronic contributions**

The electronic partition function reads

$$q\_{\rm el} = \sum\_{\ell=0}^{\infty} g\_{\rm el,\ell} e^{-\varepsilon\_{\rm el,\ell}/k\_B T} \, \, \, \, \, \tag{74}$$

where *ε*el, the energy level , and *g*el, is its degeneracy. In most cases the electronic contributions to the thermodynamic functions are negligible at not very hight temperatures. Therefore they are not written here.

#### **4.5 Ideal gas mixture**

Let us consider two-component mixture of *N*<sup>1</sup> non-interacting molecules of component 1 and *N*<sup>2</sup> non-interacting molecules of component 2 (extension to the case of a multi-component mixture is straightforward). The partition function of mixture is

$$Q = \frac{q\_1^{N\_1}}{N\_1!} \frac{q\_2^{N\_2}}{N\_2!} \tag{75}$$

where *q*<sup>1</sup> and *q*<sup>2</sup> are the partition functions of molecules 1 and 2, respectively. Let us denote *Xm*,*<sup>i</sup>* the molar thermodynamic quantity of pure component *<sup>i</sup>*, *<sup>i</sup>* = 1, 2 and *xi* = *Ni <sup>N</sup>*1+*N*<sup>2</sup> its mole fraction. Then

$$A = RT\left(\mathbf{x}\_1 \ln \mathbf{x}\_1 + \mathbf{x}\_2 \ln \mathbf{x}\_2\right) + \mathbf{x}\_1 A\_{\mathfrak{m},1} + \mathbf{x}\_2 A\_{\mathfrak{m},2} \tag{76}$$

$$S = -R\left(\mathbf{x}\_1 \ln \mathbf{x}\_1 + \mathbf{x}\_2 \ln \mathbf{x}\_2\right) + \mathbf{x}\_1 S\_{m,1} + \mathbf{x}\_2 S\_{m,2} \tag{77}$$

$$G = RT\left(\mathbf{x}\_1 \ln \mathbf{x}\_1 + \mathbf{x}\_2 \ln \mathbf{x}\_2\right) + \mathbf{x}\_1 G\_{m,1} + \mathbf{x}\_2 G\_{m,2} \,\prime \,\tag{78}$$

$$p = x\_1 p\_{m,1} + x\_2 p\_{m,2} = x\_1 \frac{RT}{V\_m} + x\_2 \frac{RT}{V\_m} \, , \tag{79}$$

$$\mathbf{U} = \mathbf{x}\_1 \mathbf{U}\_{m,1} + \mathbf{x}\_2 \mathbf{U}\_{m,2} \tag{80}$$

$$H = \mathbf{x}\_1 H\_{m,1} + \mathbf{x}\_2 H\_{m,2} \, \tag{81}$$

$$\mathbf{C}\_{V} = \mathbf{x}\_{1}\mathbf{C}\_{Vm,1} + \mathbf{x}\_{2}\mathbf{C}\_{Vm,2} \,. \tag{82}$$

$$\mathbb{C}\_p = \mathbf{x}\_1 \mathbb{C}\_{pm,1} + \mathbf{x}\_2 \mathbb{C}\_{pm,2} \,. \tag{83}$$

### **5. Ideal crystal**

We will call the ideal crystal an assembly of molecules displayed in a regular lattice without any impurities or lattice deformations. Distances among lattice centers will not depend on temperature and pressure. For simplicity we will consider one-atomic molecules. The partition function of crystal is

$$\mathbf{Q} = e^{-\mathbf{L}\_0/k\_B T} \mathbf{Q}\_{\text{vib}}.\tag{84}$$

where *U*<sup>0</sup> is the lattice energy.

We will discuss here two models of the ideal crystal: the Einstein approximation and the Debye approximation.

#### **5.1 Einstein model**

An older and simpler Einstein model is based on the following postulates

1. Vibrations of molecules are independent:

$$Q\_{\rm vib} = q\_{\rm vib}^N \,\,\,\,\,\tag{85}$$

where *u* = Θ*D*/*T* and

temperature

*<sup>D</sup>*(*u*) = <sup>3</sup>

 *T* Θ*<sup>D</sup>*

*CV* = 3*R*

Both models give a correct high-temperature limit (the Dulong-Petit law)

*CV* = 36*R*

while the Einstein model incorrectly gives

The same is true for the zero temperature limit

intermolecular force must be included explicitly.

*<sup>Q</sup>* <sup>=</sup> <sup>1</sup> *N*!

> ··· (*V*)

**5.3 Beyond the Debye model**

but very good theories for solids.

*Z* is the *c*onfigurational integral

*Z* = (*V*) (*V*)

**6. Intermolecular forces**

*u*3

 *u* 0

Statistical Thermodynamics 727

It can be proved that at low temperatures the heat capacity becomes a cubic function of

<sup>3</sup> <sup>∞</sup> 0

 Θ*<sup>E</sup> T*

*CV* = 3*R* .

Both the Einstein and the Debye models assume harmonicity of lattice vibrations. This is not true at high temperatures near the melting point. The harmonic vibrations are not assumed in the lattice theories (the cell theory, the hole theory, ...) that used to be popular in forties and fifties of the last century for liquids. It was shown later that they are poor theories of liquids

Up to now forces acting among molecules have been ignored. In the ideal gas (Section 4) molecules are assumed to exert no forces upon each other. In the ideal crystal (Section 5) molecules are imprisoned in the lattice, and the intermolecular forces are counted indirectly in the lattice energy and in the Einstein or Debye temperature. For real gases and liquids the

where *q*int = *q*rot*q*vib*q*el is the partition function of the internal motions in molecule. Quantity

int <sup>2</sup>*πmkBT h*2

3 <sup>2</sup> *N*

exp[−*βuN*(*r*1,*r*2,...,*rN*)]d*r*<sup>1</sup> d*r*<sup>2</sup> ... d*rN* , (93)

*Z* . (92)

*CV* = 0 .

lim *T*→0

Thermodynamic functions cannot be obtained analytically in the lattice theories.

**6.1 The configurational integral and the molecular interaction energy** The partition function of the real gas or liquid may be written in a form

exp(−*Nβ�*0)*q<sup>N</sup>*

<sup>2</sup> *e* <sup>−</sup>Θ*E*/*<sup>T</sup>* .

*x*3 *<sup>e</sup><sup>x</sup>* <sup>−</sup> <sup>1</sup> <sup>d</sup>*<sup>x</sup>* .

*x*3

*<sup>e</sup><sup>x</sup>* <sup>−</sup> <sup>1</sup> <sup>d</sup>*<sup>x</sup>* <sup>=</sup> *a T*<sup>3</sup> ,

where *q*vib is the vibrational partition function of molecule.

2. Vibrations are isotropic:

$$
\eta\_{\rm vib} = \eta\_{\rm x} \eta\_{\rm y} \eta\_{\rm z} = \eta\_{\rm x}^3. \tag{86}
$$

3. Vibrations are harmonical

$$q\_X = \sum\_{\upsilon=0}^{\infty} e^{-\epsilon\_{\upsilon}/k\_B T} \, \, \, \, \, \tag{87}$$

where

$$
\mathfrak{e}\_{\upsilon} = h\nu \left( \upsilon + \frac{1}{2} \right),
$$

is the energy in quantum state *v* and *ν* is the fundamental vibrational frequency. Combining these equations one obtains

$$Q = e^{-l\mathcal{U}\_0/k\_B T} \left(\frac{e^{-\Theta\_\mathbb{E}/(2T)}}{1 - e^{-\Theta\_\mathbb{E}/T}}\right)^{3N} \text{ .} \tag{88}$$

where

$$
\Theta\_E = \frac{h\nu}{k\_B}
$$

is the *Einstein characteristic temperature*. For the isochoric heat capacity it follows

$$\mathcal{C}\_{V} = 3Nk\_{B} \left(\frac{\Theta\_{E}}{T}\right)^{2} \frac{e^{-\Theta\_{E}/T}}{\left(1 - e^{-\Theta\_{E}/T}\right)^{2}}.\tag{89}$$

#### **5.2 Debye model**

Debye considers crystal as a huge molecule (*i.e* he replaces the postulates of independence and isotropy in the Einstein model) of an ideal gas; the postulate of harmonicity of vibrations remains. From these assumptions it can be derived for the partition function

$$\ln Q = -\frac{\mathcal{U}\_0}{k\_B T} - \frac{9}{8} N \frac{\Theta\_D}{T} - 9N \left(\frac{T}{\Theta\_D}\right)^3 \int\_0^{\Theta\_D/T} \mathbf{x}^2 \ln(1 - e^{-\mathbf{x}}) \,\mathrm{d}x,\tag{90}$$

where

$$
\Theta\_D = \frac{h\nu\_{\text{max}}}{k\_B}
$$

is the *Debye characteristic temperature* with *ν*max being the highest frequency of crystal. For the isochoric heat capacity it follows

$$\mathcal{C}\_V = 3\mathcal{R}\left(4D(u) - \frac{3u}{e^u - 1}\right) \,. \tag{91}$$

where *u* = Θ*D*/*T* and

10 Will-be-set-by-IN-TECH

*Q*vib = *q<sup>N</sup>*

*q*vib = *qxqyqz* = *q*<sup>3</sup>

 *v* + 1 2 

*e*−Θ*E*/(2*T*) <sup>1</sup> − *<sup>e</sup>*−Θ*E*/*<sup>T</sup>*

<sup>2</sup> *e*−Θ*E*/*<sup>T</sup>* 

> <sup>3</sup> <sup>Θ</sup>*D*/*<sup>T</sup>* 0

<sup>3</sup>*<sup>N</sup>*

∞ ∑ *v*=0 *e*

*qx* =

*�<sup>v</sup>* = *hν*

is the energy in quantum state *v* and *ν* is the fundamental vibrational frequency.

<sup>Θ</sup>*<sup>E</sup>* <sup>=</sup> *<sup>h</sup><sup>ν</sup> kB*

Debye considers crystal as a huge molecule (*i.e* he replaces the postulates of independence and isotropy in the Einstein model) of an ideal gas; the postulate of harmonicity of vibrations

> *T* Θ*<sup>D</sup>*

<sup>Θ</sup>*<sup>D</sup>* <sup>=</sup> *<sup>h</sup>ν*max *kB*

<sup>4</sup>*D*(*u*) <sup>−</sup> <sup>3</sup>*<sup>u</sup>*

*<sup>e</sup><sup>u</sup>* − <sup>1</sup>

 Θ*<sup>E</sup> T*

−*U*0/*kB T*

vib , (85)

<sup>−</sup>*�v*/*kB <sup>T</sup>* , (87)

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−Θ*E*/*T*<sup>2</sup> . (89)

*<sup>x</sup>*<sup>2</sup> ln(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*

*<sup>x</sup>* . (86)

, (88)

<sup>−</sup>*x*) d*x* , (90)

. (91)

An older and simpler Einstein model is based on the following postulates

where *q*vib is the vibrational partition function of molecule.

*Q* = *e*

*CV* = 3*NkB*

remains. From these assumptions it can be derived for the partition function

*<sup>T</sup>* <sup>−</sup> <sup>9</sup>*<sup>N</sup>*

is the *Debye characteristic temperature* with *ν*max being the highest frequency of crystal.

*CV* = 3*R*

**5.1 Einstein model**

2. Vibrations are isotropic:

3. Vibrations are harmonical

Combining these equations one obtains

is the *Einstein characteristic temperature*. For the isochoric heat capacity it follows

ln *<sup>Q</sup>* <sup>=</sup> <sup>−</sup> *<sup>U</sup>*<sup>0</sup>

For the isochoric heat capacity it follows

*kB <sup>T</sup>* <sup>−</sup> <sup>9</sup> 8 *<sup>N</sup>* <sup>Θ</sup>*<sup>D</sup>*

where

where

**5.2 Debye model**

where

1. Vibrations of molecules are independent:

$$D(\mu) = \frac{3}{\mu^3} \int\_0^\mu \frac{x^3}{e^x - 1} \,\mathrm{d}x \dots$$

It can be proved that at low temperatures the heat capacity becomes a cubic function of temperature

$$\mathbf{C}\_{V} = \mathfrak{H}\mathbf{R} \left(\frac{T}{\Theta\_{D}}\right)^{3} \int\_{0}^{\infty} \frac{\mathbf{x}^{3}}{e^{\mathbf{x}} - 1} \,\mathrm{d}\mathbf{x} = a \,\mathrm{T}^{3}\,\mathrm{s}$$

while the Einstein model incorrectly gives

$$\mathbf{C}\_{V} = \Im \mathcal{R} \left(\frac{\Theta\_{\mathcal{E}}}{T}\right)^{2} e^{-\Theta\_{\mathcal{E}}/T} \, .$$

Both models give a correct high-temperature limit (the Dulong-Petit law)

$$\mathsf{C}\_{V} = \mathsf{3R}\text{ .}$$

The same is true for the zero temperature limit

$$\lim\_{T \to 0} C\_V = 0$$

### **5.3 Beyond the Debye model**

Both the Einstein and the Debye models assume harmonicity of lattice vibrations. This is not true at high temperatures near the melting point. The harmonic vibrations are not assumed in the lattice theories (the cell theory, the hole theory, ...) that used to be popular in forties and fifties of the last century for liquids. It was shown later that they are poor theories of liquids but very good theories for solids.

Thermodynamic functions cannot be obtained analytically in the lattice theories.

#### **6. Intermolecular forces**

Up to now forces acting among molecules have been ignored. In the ideal gas (Section 4) molecules are assumed to exert no forces upon each other. In the ideal crystal (Section 5) molecules are imprisoned in the lattice, and the intermolecular forces are counted indirectly in the lattice energy and in the Einstein or Debye temperature. For real gases and liquids the intermolecular force must be included explicitly.

#### **6.1 The configurational integral and the molecular interaction energy**

The partition function of the real gas or liquid may be written in a form

$$Q = \frac{1}{N!} \exp(-N\beta\epsilon\_0) q\_{\rm int}^N \left(\frac{2\pi mk\_B T}{h^2}\right)^{\frac{3}{2}N} Z. \tag{92}$$

where *q*int = *q*rot*q*vib*q*el is the partition function of the internal motions in molecule. Quantity *Z* is the *c*onfigurational integral

$$Z = \int\_{(V)} \int\_{(V)} \cdots \int\_{(V)} \exp[-\beta u\_N(\vec{r}\_1, \vec{r}\_2, \dots, \vec{r}\_N)] \, \text{d}\vec{r}\_1 \, \text{d}\vec{r}\_2 \dots \, \text{d}\vec{r}\_N \,. \tag{93}$$

**6.2.3 Lennard-Jones potential**

potential energy on distance

hard ellipsoids, and so on.

*u*(*r*, *θ*1, *θ*2, *φ*) = 4*�*

where *μ* is the dipole moment.

(2003) and references therein.

**6.3 The three-body potential**

calculations Malijevský et al. (2007).

**6.2.5 Pair potentials of real molecules**

dipole moment

therein.

well.

This well known pair intermolecular potential realistically describes a dependence of pair

Statistical Thermodynamics 729

. (98)

. (99)

*<sup>r</sup>*<sup>3</sup> [2 cos *<sup>θ</sup>*<sup>1</sup> cos *<sup>θ</sup>*<sup>2</sup> <sup>−</sup> sin *<sup>θ</sup>*<sup>1</sup> sin *<sup>θ</sup>*<sup>2</sup> cos *<sup>φ</sup>*] , (100)

*rst* (3 cos *<sup>θ</sup>*<sup>1</sup> cos *<sup>θ</sup>*<sup>2</sup> cos *<sup>θ</sup>*<sup>3</sup> <sup>+</sup> <sup>1</sup>) , (101)

 *σ r* <sup>12</sup> − *σ r* 6 

 *σ r n* − *σ r <sup>m</sup>*

There are analogues of hard spheres for non-spherical particles: hard diatomics or dumbbells made of two fused hard spheres, hard triatomics, hard multiatomics, hard spherocylinders,

Examples of soft pair potentials are Lennard-Jones multiatomics, molecules whose atoms

Another example is the Stockmayer potential, the Lennard-Jones potential with an indebted

The above model pair potentials, especially the Lennard-Jones potential and its extensions, may be used to calculate properties of the real substances. In this case their parameters, for example *�* and *σ*, are fitted to the experimental data such as the second virial coefficients,

More sophisticated approach involving a realistic dependence on the interparticle separation with a number of adjustable parameters was used by Aziz, see Aziz (1984) and references

For simple molecules, there is a fully theoretical approach without any adjustable parameters utilizing the first principle quantum mechanics calculations, see for example Slaviˇcek et al.

The three-body intermolecular interactions are caused by polarizablilities of molecules. The

where *ν* is a strength parameter. It is a first term (DDD, dipole-dipole-dipole) in the multipole expansion. Analytical formulae and corresponding strength parameters are known for higher order terms (DDQ, dipole-dipole-quadrupole, DQQ, dipole-quadrupole-quadrupole,. . . ) as

More accurate three-body potentials can be obtained using quantum chemical *ab initio*

*u*(*r*) = 4*�*

*u*(*r*) = 4*�*

*�* is a depth of potential at minimum, and 21/6*σ* is its position.

More generally, the Lennard-Jones n-m potential is

**6.2.4 Pair potentials of non-spherical molecules**

interact according to the Lennard-Jones potential (98).

 *σ r* <sup>12</sup> − *σ r* 6 − *μ*2

rare-gas transport properties and molecular properties.

simplest and the most often used is the Axilrod-Teller-Muto term *<sup>u</sup>*(*r*,*s*, *<sup>t</sup>*) = *<sup>ν</sup>*

where symbol

$$\int\_{(V)} \cdot \cdot \cdot \mathbf{d}\vec{r}\_i = \int\_0^L \int\_0^L \int\_0^L \cdot \cdot \mathbf{d}x\_i \mathbf{d}y\_i \mathbf{d}z\_i \quad \text{and} \quad L^3 = V \,. \, L$$

The quantity *uN*(�*r*1,�*r*2,...,�*rN*) is the potential energy of an assembly of *N* molecules. Here and in Eq.(93) one-atomic molecules are assumed for simplicity. More generally, the potential energy is a function not only positions of centers of molecules�*ri* but also of their orientations *ω*� *<sup>i</sup>*. However, we will use the above simplified notation.

The interaction potential energy *uN* of system may be written as an expansion in two-body, three-body, *e.t.c* contributions

$$\mu\_N(\vec{r}\_1, \vec{r}\_2, \dots, \vec{r}\_N) = \sum\_{i$$

Most often only the first term is considered. This approximation is called *the rule of pairwise additivity*

$$\mu\_N = \sum\_{i$$

where *u*<sup>2</sup> is the pair intermolecular potential. The three-body potential *u*<sup>3</sup> is used rarely at very accurate calculations, and *u*<sup>4</sup> and higher order contributions are omitted as a rule.

#### **6.2 The pair intermolecular potential**

The pair potential depends of a distance between centers of two molecules *r* and on their mutual orientation *ω*� . For simplicity we will omit the angular dependence of the pair potential (it is true for the spherically symmetric molecules) in further text, and write

$$
\mu\_2(\vec{r}\_{i\prime}\vec{r}\_j) = \mu\_2(r\_{i\dot{j}\prime}\vec{\omega\_{i\dot{j}}}) = \mu(r).
$$

where subscripts 2 and *ij* are omitted, too.

The following model pair potentials are most often used.

#### **6.2.1 Hard spheres**

It is after the ideal gas the simplest model. It ignores attractive interaction between molecules, and approximates strong repulsive interactions at low intermolecular distances by an infinite barrier

$$u(r) = \begin{cases} \infty & r < \sigma \\ 0 & r > \sigma \end{cases} \tag{96}$$

where *σ* is a diameter of molecule.

#### **6.2.2 Square well potential**

Molecules behave like hard spheres surrounded by an area of attraction

$$u(r) = \begin{cases} \infty & r < \sigma \\ -\varepsilon & \sigma < r < \lambda \sigma \\ 0 & r > \lambda \sigma \end{cases} \tag{97}$$

Here *σ* is a hard-sphere diameter, *�* a depth of the attractive well, and the attraction region ranges from *σ* to *λσ*.

12 Will-be-set-by-IN-TECH

The quantity *uN*(�*r*1,�*r*2,...,�*rN*) is the potential energy of an assembly of *N* molecules. Here and in Eq.(93) one-atomic molecules are assumed for simplicity. More generally, the potential energy is a function not only positions of centers of molecules�*ri* but also of their orientations

The interaction potential energy *uN* of system may be written as an expansion in two-body,

Most often only the first term is considered. This approximation is called *the rule of pairwise*

where *u*<sup>2</sup> is the pair intermolecular potential. The three-body potential *u*<sup>3</sup> is used rarely at very accurate calculations, and *u*<sup>4</sup> and higher order contributions are omitted as a rule.

The pair potential depends of a distance between centers of two molecules *r* and on their mutual orientation *ω*� . For simplicity we will omit the angular dependence of the pair potential

*u*2(�*ri*,�*rj*) = *u*2(*rij*, *ω*�*ij*) = *u*(*r*)

It is after the ideal gas the simplest model. It ignores attractive interaction between molecules, and approximates strong repulsive interactions at low intermolecular distances by an infinite

> � ∞ *r* < *σ* 0 *r* > *σ*

∞ *r* < *σ* −*� σ* < *r* < *λσ* 0 *r* > *λσ*

Here *σ* is a hard-sphere diameter, *�* a depth of the attractive well, and the attraction region

*u*2(�*ri*,�*rj*) + ∑

*i*<*j*<*k*

*u*3(�*ri*,�*rj*,�*rk*)) + ··· (94)

(96)

(97)

*u*2(�*ri*,�*rj*), (95)

··· <sup>d</sup>*xi*d*yi*d*zi* and *<sup>L</sup>*<sup>3</sup> <sup>=</sup> *<sup>V</sup>* .

where symbol

*additivity*

� (*V*)

three-body, *e.t.c* contributions

**6.2 The pair intermolecular potential**

where subscripts 2 and *ij* are omitted, too.

where *σ* is a diameter of molecule.

**6.2.2 Square well potential**

ranges from *σ* to *λσ*.

**6.2.1 Hard spheres**

barrier

The following model pair potentials are most often used.

··· d�*ri* =

*ω*� *<sup>i</sup>*. However, we will use the above simplified notation.

*uN*(�*r*1,�*r*2,...,�*rN*) = ∑

� *L* 0

� *L* 0

*i*<*j*

(it is true for the spherically symmetric molecules) in further text, and write

*u*(*r*) =

Molecules behave like hard spheres surrounded by an area of attraction

⎧ ⎪⎪⎨

⎪⎪⎩

*u*(*r*) =

*uN* = ∑ *i*<*j*

� *L* 0

#### **6.2.3 Lennard-Jones potential**

This well known pair intermolecular potential realistically describes a dependence of pair potential energy on distance

$$
\mu(r) = 4\epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right]. \tag{98}
$$

*�* is a depth of potential at minimum, and 21/6*σ* is its position. More generally, the Lennard-Jones n-m potential is

$$
\mu(r) = 4\epsilon \left[ \left( \frac{\sigma}{r} \right)^n - \left( \frac{\sigma}{r} \right)^m \right]. \tag{99}
$$

#### **6.2.4 Pair potentials of non-spherical molecules**

There are analogues of hard spheres for non-spherical particles: hard diatomics or dumbbells made of two fused hard spheres, hard triatomics, hard multiatomics, hard spherocylinders, hard ellipsoids, and so on.

Examples of soft pair potentials are Lennard-Jones multiatomics, molecules whose atoms interact according to the Lennard-Jones potential (98).

Another example is the Stockmayer potential, the Lennard-Jones potential with an indebted dipole moment

$$u(r, \theta\_1, \theta\_2, \phi) = 4\epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right] - \frac{\mu^2}{r^3} \left[ 2\cos\theta\_1 \cos\theta\_2 - \sin\theta\_1 \sin\theta\_2 \cos\phi \right] \tag{100}$$

where *μ* is the dipole moment.

#### **6.2.5 Pair potentials of real molecules**

The above model pair potentials, especially the Lennard-Jones potential and its extensions, may be used to calculate properties of the real substances. In this case their parameters, for example *�* and *σ*, are fitted to the experimental data such as the second virial coefficients, rare-gas transport properties and molecular properties.

More sophisticated approach involving a realistic dependence on the interparticle separation with a number of adjustable parameters was used by Aziz, see Aziz (1984) and references therein.

For simple molecules, there is a fully theoretical approach without any adjustable parameters utilizing the first principle quantum mechanics calculations, see for example Slaviˇcek et al. (2003) and references therein.

#### **6.3 The three-body potential**

The three-body intermolecular interactions are caused by polarizablilities of molecules. The simplest and the most often used is the Axilrod-Teller-Muto term

$$u(r,s,t) = \frac{\nu}{rst} \left( 3\cos\theta\_1 \cos\theta\_2 \cos\theta\_3 + 1 \right) \tag{101}$$

where *ν* is a strength parameter. It is a first term (DDD, dipole-dipole-dipole) in the multipole expansion. Analytical formulae and corresponding strength parameters are known for higher order terms (DDQ, dipole-dipole-quadrupole, DQQ, dipole-quadrupole-quadrupole,. . . ) as well.

More accurate three-body potentials can be obtained using quantum chemical *ab initio* calculations Malijevský et al. (2007).

**7.3 Higher virial coefficients**

**7.4 Virial coefficients of mixtures**

molecule 1 and molecule 2. The third virial coefficient reads

**8. Dense gas and liquid**

straightforward.

crystal.

function *g*(*r*)

*B*<sup>3</sup> = *x*<sup>3</sup>

ideal-gas values using the virial expansion.

**8.1 Internal structure of fluid**

Expressions for higher virial coefficients become more and more complicated due to an increasing dimensionality of the corresponding integrals and their number. For example, the ninth virial coefficient consists of 194 066 integrals with the Mayer integrands, and their dimensionalities are up to 21 Malijevský & Kolafa (2008) in a simplest case of spherically symmetric molecules. For hard spheres the virial coefficients are known up to ten, which is at

Statistical Thermodynamics 731

<sup>1</sup>*B*2(11) + <sup>2</sup>*x*1*x*2*B*2(12) + *<sup>x</sup>*<sup>2</sup>

where *xi* are the mole fractions, *B*2(*ii*) the second virial coefficients of pure components and *B*2(12) the crossed virial coefficient representing an influence of the interaction between

<sup>1</sup>*x*2*B*3(112) + <sup>3</sup>*x*1*x*<sup>2</sup>

Extensions of these equations on multicomponent mixtures and higher virial coefficients is

Determination of thermodynamic properties from intermolecular interactions is much more difficult for dense fluids (for gases at high densities and for liquids) than for rare gases and

Free energy has a minimum in equilibrium at constant temperature and volume. At high temperatures and low densities the term *TS* dominates because not only temperature but also entropy is high. A minimum in *A* corresponds to a maximum in *S* and system, thus, is in the gas phase. Ideal gas properties may be calculated from a behavior of individual molecules only. At somewhat higher densities thermodynamic quantities can be expanded from their

At low temperatures the energy term in equation (111) dominates because not only temperature but also entropy is small. For solids we may start from a concept of the ideal

No such simple molecular model as the ideal gas or the ideal crystal is known for liquid and dense gas. Theoretical studies of liquid properties are difficult and uncompleted up to now.

There is no internal structure of molecules in the ideal gas. There is a long-range order in the crystal. The fluid is between of the two extremal cases: it has a local order at short

The fundamental quantity describing the internal structure of fluid is the pair distribution

intermolecular distances (as crystal) and a long-range disorder (as gas).

solids. This fact can be explained using a definition of the Helmholtz free energy

<sup>2</sup>*B*3(122) + *<sup>x</sup>*<sup>3</sup>

*A* = *U* − *TS* . (111)

<sup>2</sup>*B*2(22), (109)

<sup>2</sup>*B*3(222). (110)

the edge of a present computer technology Labík et al. (2005).

*B*<sup>2</sup> = *x*<sup>2</sup>

<sup>1</sup>*B*3(111) + <sup>3</sup>*x*<sup>2</sup>

For binary mixture of components 1 and 2 the second virial coefficient reads

### **7. The virial equation of state**

The virial equation of state in the statistical thermodynamics is an expansion of the compressibility factor *z* = *pV RT* in powers of density *<sup>ρ</sup>* <sup>=</sup> *<sup>N</sup> V*

$$z = 1 + B\_2 \rho + B\_3 \rho^2 + \cdots,\tag{102}$$

where *B*<sup>2</sup> is the second virial coefficient, *B*<sup>3</sup> the third, *e.t.c.* The virial coefficients of pure gases are functions of temperature only. For mixtures they are functions of temperature and composition.

The first term in equation (102) gives the equation of state of ideal gas, the first two terms or three give corrections to non-ideality. Higher virial coefficients are not available experimentally. However, they can be determined from knowledge of intermolecular forces. The relations among the intermolecular forces and the virial coefficients are exact, the pair and the three-body of potentials are subjects of uncertainties, however.

#### **7.1 Second virial coefficient**

For the second virial coefficient of spherically symmetric molecules we find

$$B = -2\pi \int\_0^\infty f(r) \, r^2 \mathrm{d}r = -2\pi \int\_0^\infty \left( e^{-\beta u(r)} - 1 \right) r^2 \mathrm{d}r \, , \tag{103}$$

where

$$f(r) = \exp[-\beta \mu(r)] - 1$$

is the *Mayer function*. For linear molecules we have

$$B = -\frac{1}{4} \int\_0^\infty \int\_0^\pi \int\_0^\pi \int\_0^{2\pi} \left[ e^{-\not\mu (r\_1 \theta\_1 \theta\_2 \phi)} - 1 \right] r^2 \sin \theta\_1 \sin \theta\_2 \text{d}r \, \text{d}\theta\_1 \text{d}\theta\_2 \text{d}\phi \,. \tag{104}$$

For general non-spherical molecules we obtain

$$B = -\frac{2\pi}{\int\_{\vec{\omega}1} \int\_{\vec{\omega}2} \mathbf{d}\vec{\omega}\_1 \mathbf{d}\vec{\omega}\_2} \int\_0^\infty \int\_{\omega\_1} \int\_{\omega\_2} \left[e^{-\beta u(r\vec{\omega}\_1, \vec{\omega}\_2)} - 1\right] r^2 \mathbf{d}r \mathbf{d}\vec{\omega}\_1 \mathbf{d}\vec{\omega}\_2 \,. \tag{105}$$

#### **7.2 Third virial coefficient**

The third virial coefficient may be written for spherically symmetric molecules as

*C* = *C*add + *C*nadd , (106)

where

$$\mathcal{C}\_{\text{add}} = -\frac{8}{3}\pi^2 \int\_0^\infty \int\_0^r \int\_{|r-s|}^{r+s} \left(e^{-\beta u(r)} - 1\right) \left(e^{-\beta u(s)} - 1\right) \left(e^{-\beta u(t)} - 1\right) r \, s \, t \, d\mathbf{r} \, \text{ds } \text{dt} \,\tag{107}$$

and

$$\mathcal{C}\_{\text{nadd}} = \frac{8}{3}\pi^2 \int\_0^\infty \int\_0^r \int\_{|r-s|}^{r+s} e^{-\beta u(r)} e^{-\beta u(s)} e^{-\beta u(t)} \left\{ \exp[-\beta u\_3(r,s,t)] - 1 \right\} r \, s \, t \, d\mathbf{r} \, \text{ds} \, \text{d}t,\tag{108}$$

where *u*3(*r*,*s*, *t*) is the three-body potential. Analogous equations hold for non-spherical molecules.

14 Will-be-set-by-IN-TECH

The virial equation of state in the statistical thermodynamics is an expansion of the

where *B*<sup>2</sup> is the second virial coefficient, *B*<sup>3</sup> the third, *e.t.c.* The virial coefficients of pure gases are functions of temperature only. For mixtures they are functions of temperature and

The first term in equation (102) gives the equation of state of ideal gas, the first two terms or three give corrections to non-ideality. Higher virial coefficients are not available experimentally. However, they can be determined from knowledge of intermolecular forces. The relations among the intermolecular forces and the virial coefficients are exact, the pair and

2d*<sup>r</sup>* <sup>=</sup> <sup>−</sup>2*<sup>π</sup>*

*f*(*r*) = exp[−*βu*(*r*)] − 1

<sup>−</sup>*βu*(*r*,*θ*1,*θ*2,*φ*) <sup>−</sup> <sup>1</sup>

 ∞ 0 *e*

> *r*

<sup>−</sup>*βu*(*r*,*<sup>ω</sup>* 1,*<sup>ω</sup>* <sup>2</sup>) <sup>−</sup> <sup>1</sup>

 *e*

*V*

*<sup>z</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>B</sup>*2*<sup>ρ</sup>* <sup>+</sup> *<sup>B</sup>*3*ρ*<sup>2</sup> <sup>+</sup> ··· , (102)

<sup>−</sup>*βu*(*r*) <sup>−</sup> <sup>1</sup>

 *r*

 *r*

*C* = *C*add + *C*nadd , (106)

<sup>−</sup>*βu*(*t*) <sup>−</sup> <sup>1</sup>

<sup>−</sup>*βu*(*t*) {exp[−*βu*3(*r*,*s*, *<sup>t</sup>*)] <sup>−</sup> <sup>1</sup>} *rst* <sup>d</sup>*<sup>r</sup>* <sup>d</sup>*<sup>s</sup>* <sup>d</sup>*<sup>t</sup>* , (108)

2d*r*, (103)

2d*r*d*ω* 1d*ω* <sup>2</sup> . (105)

*rst* d*r* d*s* d*t* , (107)

<sup>2</sup> sin *θ*<sup>1</sup> sin *θ*2d*r* d*θ*1d*θ*2d*φ* . (104)

*RT* in powers of density *<sup>ρ</sup>* <sup>=</sup> *<sup>N</sup>*

the three-body of potentials are subjects of uncertainties, however.

 ∞ 0

For the second virial coefficient of spherically symmetric molecules we find

*f*(*r*)*r*

**7. The virial equation of state**

compressibility factor *z* = *pV*

**7.1 Second virial coefficient**

*<sup>B</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> 4 ∞ 0

**7.2 Third virial coefficient**

3 *π*2 ∞ 0

3 *π*2 ∞ 0

*B* = −2*π*

is the *Mayer function*. For linear molecules we have

 *π* 0

For general non-spherical molecules we obtain

 *r* 0

 *r* 0

 *r*+*s* |*r*−*s*|

 *r*+*s* |*r*−*s*| *e* −*βu*(*r*) *e* −*βu*(*s*) *e*

 *e*

*<sup>B</sup>* <sup>=</sup> <sup>−</sup> <sup>2</sup>*<sup>π</sup> ω* <sup>1</sup> 

 *π* 0

*<sup>ω</sup>* <sup>2</sup> d*ω* 1d*ω* <sup>2</sup>

 2*π* 0

 *e*

 ∞ 0

 *ω*<sup>1</sup> *ω*<sup>2</sup> *e*

The third virial coefficient may be written for spherically symmetric molecules as

<sup>−</sup>*βu*(*r*) <sup>−</sup> <sup>1</sup>

 *e*

where *u*3(*r*,*s*, *t*) is the three-body potential. Analogous equations hold for non-spherical

<sup>−</sup>*βu*(*s*) <sup>−</sup> <sup>1</sup>

composition.

where

where

and

*<sup>C</sup>*add <sup>=</sup> <sup>−</sup><sup>8</sup>

*<sup>C</sup>*nadd <sup>=</sup> <sup>8</sup>

molecules.

## **7.3 Higher virial coefficients**

Expressions for higher virial coefficients become more and more complicated due to an increasing dimensionality of the corresponding integrals and their number. For example, the ninth virial coefficient consists of 194 066 integrals with the Mayer integrands, and their dimensionalities are up to 21 Malijevský & Kolafa (2008) in a simplest case of spherically symmetric molecules. For hard spheres the virial coefficients are known up to ten, which is at the edge of a present computer technology Labík et al. (2005).

### **7.4 Virial coefficients of mixtures**

For binary mixture of components 1 and 2 the second virial coefficient reads

$$B\_2 = \mathbf{x}\_1^2 B\_2(11) + 2\mathbf{x}\_1 \mathbf{x}\_2 B\_2(12) + \mathbf{x}\_2^2 B\_2(22) \, , \tag{109}$$

where *xi* are the mole fractions, *B*2(*ii*) the second virial coefficients of pure components and *B*2(12) the crossed virial coefficient representing an influence of the interaction between molecule 1 and molecule 2.

The third virial coefficient reads

$$B\_3 = \mathbf{x}\_1^3 B\_3(111) + 3\mathbf{x}\_1^2 \mathbf{x}\_2 B\_3(112) + 3\mathbf{x}\_1 \mathbf{x}\_2^2 B\_3(122) + \mathbf{x}\_2^3 B\_3(222). \tag{110}$$

Extensions of these equations on multicomponent mixtures and higher virial coefficients is straightforward.

## **8. Dense gas and liquid**

Determination of thermodynamic properties from intermolecular interactions is much more difficult for dense fluids (for gases at high densities and for liquids) than for rare gases and solids. This fact can be explained using a definition of the Helmholtz free energy

$$A = \mathcal{U} - \mathcal{TS}.\tag{111}$$

Free energy has a minimum in equilibrium at constant temperature and volume. At high temperatures and low densities the term *TS* dominates because not only temperature but also entropy is high. A minimum in *A* corresponds to a maximum in *S* and system, thus, is in the gas phase. Ideal gas properties may be calculated from a behavior of individual molecules only. At somewhat higher densities thermodynamic quantities can be expanded from their ideal-gas values using the virial expansion.

At low temperatures the energy term in equation (111) dominates because not only temperature but also entropy is small. For solids we may start from a concept of the ideal crystal.

No such simple molecular model as the ideal gas or the ideal crystal is known for liquid and dense gas. Theoretical studies of liquid properties are difficult and uncompleted up to now.

### **8.1 Internal structure of fluid**

There is no internal structure of molecules in the ideal gas. There is a long-range order in the crystal. The fluid is between of the two extremal cases: it has a local order at short intermolecular distances (as crystal) and a long-range disorder (as gas).

The fundamental quantity describing the internal structure of fluid is the pair distribution function *g*(*r*)

$$\log(r) = \frac{\rho(r)}{\rho} \, \text{,} \tag{112}$$

**8.3 Integral equation theories**

and the Percus-Yevick approximation

**8.4 Computer simulations**

phenomenological axioms.

Ornstein-Zernike equation

it follows

where

Among the integral equation theories the most popular are those based on the

Statistical Thermodynamics 733

 (*V*)

where *h*(*r*) = *g*(*r*) − 1 is the total correlation function and *c*(*r*) the direct correlation function. This equation must be closed using a relation between the total and the direct correlation functions called *the closure* to the Ornstein-Zernike equation. From the diagrammatic analysis

*γ*(*r*) = *h*(*r*) − *c*(*r*) is the indirect (chain) correlation function and *B*(*r*) is the bridge function, a sum of elementary diagrams. Equation (120) does not yet provide a closure. It must be completed by an approximation for the bridge function. The mostly used closures are in listed in Malijevský &

Let us compare the perturbation and the integral equation theories. The first ones are simpler but they need an extra input - the structural and thermodynamic properties of a reference system. The accuracy of the second ones depends on a chosen closure. Their examples shown

Besides the above theoretical approaches there is another route to the thermodynamic quantities called the computer experiments or pseudoexperiments or simply simulations. For a given pair intermolecular potential they provide values of thermodynamic functions in the dependence on the state variables. In this sense they have characteristics of real experiments. Similarly to them they do not give an explanation of the bulk behavior of matter but they serve as tests of approximative theories. The thermodynamic values are free of approximations, or more precisely, their approximations such as a finite number of molecules in the basic box or a finite number of generated configurations can be systematically improved Kolafa et al. (2002). The computer simulations are divided into two groups: the Monte Carlo simulations and the molecular dynamic simulations. The Monte Carlo simulations generate the ensemble averages of structural and thermodynamic functions while the molecular dynamics simulations generate their time averages. The methods are described in detail in the

In Section 2 the axioms of the classical or phenomenological thermodynamics have been listed. The statistical thermodynamics not only determines the thermodynamic quantities from knowledge of the intermolecular forces but also allows an interpretation of the

*h*(*r*13)*c*(*r*32) d*r*<sup>3</sup> . (119)

*B*(*r*) = 0 (121)

*B*(*r*) = *γ*(*r*) − ln[*γ*(*r*) + 1] . (122)

*h*(*r*) = exp[−*βu*(*r*) + *γ*(*r*) + *B*(*r*)] − 1 , (120)

*h*(*r*12) = *c*(*r*12) + *ρ*

Kolafa (2008). The simplest of them are the hypernetted chain approximation

here, the hypernetted chain and the Percus-Yevick, are too simple to be accurate.

monograph of Allen and Tildesley Allen & Tildesley (1987).

**9. Interpretation of thermodynamic laws**

where *ρ*(*r*) is local density at distance *r* from the center of a given molecule, and *ρ* is the average or macroscopic density of system. Here and in the next pages of this section we assume spherically symmetric interactions and the rule of the pair additivity of the intermolecular potential energy.

The pair distribution function may be written in terms of the intermolecular interaction energy *uN*

$$\log(r) = V^2 \frac{\int\_{(V)} \cdots \int\_{(V)} e^{-\beta u\_N(\vec{r}\_1, \vec{r}\_2, \dots, \vec{r}\_N)} \, d\vec{r}\_3 \ldots \, d\vec{r}\_N}{\int\_{(V)} \cdots \int\_{(V)} e^{-\beta u\_N(\vec{r}\_1, \vec{r}\_2, \dots, \vec{r}\_N)} \, d\vec{r}\_1 \ldots \, d\vec{r}\_N} \,. \tag{113}$$

It is related to the thermodynamic quantities using the pressure equation

$$z \equiv \frac{pV}{RT} = 1 - \frac{2}{3}\pi\rho\beta \int\_0^\infty \frac{du(r)}{\mathbf{dr}} g(r) r^3 \mathbf{dr} \,\tag{114}$$

the energy equation

$$\frac{\mathcal{U}}{RT} = \frac{\mathcal{U}^0}{RT} + 2\pi\rho\mathcal{g} \int\_0^\infty u(r)g(r)r^2 \mathrm{d}r \,\,\,\,\,\tag{115}$$

where *U*<sup>0</sup> internal energy if the ideal gas, and the compressibility equation

$$\beta \left( \frac{\partial p}{\partial \rho} \right)\_{\beta} = \left\{ 1 + 4 \pi \rho \int\_{0}^{\infty} \left[ g(r) - 1 \right] r^{2} dr \right\}^{-1} \,. \tag{116}$$

Present mainstream theories of liquids can be divided into two large groups: perturbation theories and integral equation theories Hansen & McDonald (2006), Martynov (1992).

#### **8.2 Perturbation theories**

A starting point of the perturbation theories is a separation of the intermolecular potential into two parts: a harsh, short-range repulsion and a smoothly varying long-range attraction

$$
\mu(r) = \mu^0(r) + \mu^p(r) \,. \tag{117}
$$

The term *u*0(*r*) is called *the reference potential* and the term *up*(*r*) *the perturbation potential*. In the simplest case of the first order expansion of the Helmholtz free energy in the perturbation potential it holds

$$\frac{A}{RT} = \frac{A^0}{RT} + 2\pi\rho\beta \int\_0^\infty u^p(r)g^0(r)r^2 \,\mathrm{d}r \,\,\tag{118}$$

where *A*<sup>0</sup> is the Helmholtz free energy of a reference system.

In the perturbation theories knowledge of the pair distribution function and the Helmholtz free energy of the reference system is supposed. On one hand the reference system should be simple (the ideal gas is too simple and brings nothing new; a typical reference system is a fluid of hard spheres), and the perturbation potential should be small on the other hand. As a result of a battle between a simplicity of the reference potential (one must know its structural and thermodynamic properties) and an accuracy of a truncated expansion, a number of methods have been developed.

#### **8.3 Integral equation theories**

Among the integral equation theories the most popular are those based on the Ornstein-Zernike equation

$$h(r\_{12}) = \mathcal{c}(r\_{12}) + \rho \int\_{(V)} h(r\_{13}) \mathcal{c}(r\_{32}) \, \text{d}\vec{r\_3} \,. \tag{119}$$

where *h*(*r*) = *g*(*r*) − 1 is the total correlation function and *c*(*r*) the direct correlation function. This equation must be closed using a relation between the total and the direct correlation functions called *the closure* to the Ornstein-Zernike equation. From the diagrammatic analysis it follows

$$h(r) = \exp[-\beta u(r) + \gamma(r) + B(r)] - 1,\tag{120}$$

where

16 Will-be-set-by-IN-TECH

*<sup>g</sup>*(*r*) = *<sup>ρ</sup>*(*r*)

where *ρ*(*r*) is local density at distance *r* from the center of a given molecule, and *ρ* is the average or macroscopic density of system. Here and in the next pages of this section we assume spherically symmetric interactions and the rule of the pair additivity of the

The pair distribution function may be written in terms of the intermolecular interaction energy

(*V*) *<sup>e</sup>*−*βuN* (*<sup>r</sup>*1,*<sup>r</sup>*2, ...,*rN* )<sup>d</sup>*<sup>r</sup>*<sup>3</sup> ...d*rN*

(*V*) *<sup>e</sup>*−*βuN* (*<sup>r</sup>*1,*<sup>r</sup>*2, ...,*rN* )<sup>d</sup>*<sup>r</sup>*<sup>1</sup> ...d*rN*

*du*(*r*) <sup>d</sup>*<sup>r</sup> <sup>g</sup>*(*r*)*<sup>r</sup>*

*u*(*r*)*g*(*r*)*r*

[*g*(*r*) − 1]*r*

2d*r* −<sup>1</sup>

*u*(*r*) = *u*0(*r*) + *up*(*r*). (117)

 ∞ 0

 ∞ 0

Present mainstream theories of liquids can be divided into two large groups: perturbation

A starting point of the perturbation theories is a separation of the intermolecular potential into two parts: a harsh, short-range repulsion and a smoothly varying long-range attraction

The term *u*0(*r*) is called *the reference potential* and the term *up*(*r*) *the perturbation potential*. In the simplest case of the first order expansion of the Helmholtz free energy in the perturbation

> ∞ 0

In the perturbation theories knowledge of the pair distribution function and the Helmholtz free energy of the reference system is supposed. On one hand the reference system should be simple (the ideal gas is too simple and brings nothing new; a typical reference system is a fluid of hard spheres), and the perturbation potential should be small on the other hand. As a result of a battle between a simplicity of the reference potential (one must know its structural and thermodynamic properties) and an accuracy of a truncated expansion, a number of methods

*up*(*r*)*g*0(*r*)*r*

*RT* <sup>+</sup> <sup>2</sup>*πρβ*

intermolecular potential energy.

the energy equation

**8.2 Perturbation theories**

potential it holds

have been developed.

*g*(*r*) = *V*<sup>2</sup>

 (*V*) ···

 (*V*) ···

*<sup>z</sup>* <sup>≡</sup> *pV*

*U RT* <sup>=</sup> *<sup>U</sup>*<sup>0</sup>

*A RT* <sup>=</sup> *<sup>A</sup>*<sup>0</sup>

where *A*<sup>0</sup> is the Helmholtz free energy of a reference system.

*β ∂p ∂ρ β* = 

It is related to the thermodynamic quantities using the pressure equation

*RT* <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>

where *U*<sup>0</sup> internal energy if the ideal gas, and the compressibility equation

3 *πρβ* ∞ 0

*RT* <sup>+</sup> <sup>2</sup>*πρβ*

1 + 4*πρ*

theories and integral equation theories Hansen & McDonald (2006), Martynov (1992).

*uN*

*<sup>ρ</sup>* , (112)

. (113)

3d*r*, (114)

2d*r*, (115)

<sup>2</sup> d*r*, (118)

. (116)

$$\gamma(r) = h(r) - c(r)$$

is the indirect (chain) correlation function and *B*(*r*) is the bridge function, a sum of elementary diagrams. Equation (120) does not yet provide a closure. It must be completed by an approximation for the bridge function. The mostly used closures are in listed in Malijevský & Kolafa (2008). The simplest of them are the hypernetted chain approximation

$$B(r) = 0\tag{121}$$

and the Percus-Yevick approximation

$$B(r) = \gamma(r) - \ln[\gamma(r) + 1].\tag{122}$$

Let us compare the perturbation and the integral equation theories. The first ones are simpler but they need an extra input - the structural and thermodynamic properties of a reference system. The accuracy of the second ones depends on a chosen closure. Their examples shown here, the hypernetted chain and the Percus-Yevick, are too simple to be accurate.

#### **8.4 Computer simulations**

Besides the above theoretical approaches there is another route to the thermodynamic quantities called the computer experiments or pseudoexperiments or simply simulations. For a given pair intermolecular potential they provide values of thermodynamic functions in the dependence on the state variables. In this sense they have characteristics of real experiments. Similarly to them they do not give an explanation of the bulk behavior of matter but they serve as tests of approximative theories. The thermodynamic values are free of approximations, or more precisely, their approximations such as a finite number of molecules in the basic box or a finite number of generated configurations can be systematically improved Kolafa et al. (2002). The computer simulations are divided into two groups: the Monte Carlo simulations and the molecular dynamic simulations. The Monte Carlo simulations generate the ensemble averages of structural and thermodynamic functions while the molecular dynamics simulations generate their time averages. The methods are described in detail in the monograph of Allen and Tildesley Allen & Tildesley (1987).

#### **9. Interpretation of thermodynamic laws**

In Section 2 the axioms of the classical or phenomenological thermodynamics have been listed. The statistical thermodynamics not only determines the thermodynamic quantities from knowledge of the intermolecular forces but also allows an interpretation of the phenomenological axioms.

**9.5 Statistical thermodynamics and the arrow of time**

• **Cosmological time**

what will be tomorrow. • **Thermodynamic time**

**10. Acknowledgement**

**9.6 The third law of thermodynamics**

Republic under the project No. 604 613 7316.

to future. • **Psychological time**

Direction of time from past to future is supported by three arguments

of time "**from coffin to the cradle**" but again with a very, very low probability.

probability *P*<sup>0</sup> = 1. By substituting to the equation we get *S* = 0.

List of symbols

A Helmholtz free energy B second virial coefficient B*<sup>i</sup>* i-th virial coefficient *B*(*r*) bridge function C*<sup>V</sup>* isochoric heat capacity C*<sup>p</sup>* isobaric capacity

*c*(*r*) direct correlation function

*g*(*r*) pair distribution function

*h*(*r*) total correlation function k*<sup>B</sup>* Boltzmann constant

q partition function of molecule

N number of molecules, Avogadro number

R (universal) gas constant (8.314 in SI units)

 energy of molecule G Gibbs free energy

E energy

H enthalpy h Planck constant

P probability p pressure Q heat

S entropy T temperature

Q partition function

The cosmological time goes according the standard model of Universe from the Big Bang

Statistical Thermodynamics 735

We as human beings remember (as a rule) what was yesterday but we do not "*remember*"

Time goes in the direction of the growth of entropy in the direction given by equation (125). The statistical thermodynamics allows due to its probabilistic nature a change of a direction

Within the statistical thermodynamics the third law may be easily derived from equation (20) relating entropy and the probabilities. The state of the ideal crystal at *T* = 0 K is one. Its

This work was supported by the Ministry of Education, Youth and Sports of the Czech

## **9.1 Axiom on existence of the thermodynamic equilibrium**

This axiom can be explained as follows. There is a very, very large number of microscopic states that correspond to a given macroscopic state. At unchained macroscopic parameters such as volume and temperature of a closed system there is much more equilibrium states then the states out of equilibrium. Consequently, a spontaneous transfer from non-equilibrium to equilibrium has a very, very high probability. However, a spontaneous transfer from an equilibrium state to a non-equilibrium state is not excluded.

Imagine a glass of whisky on rocks. This two-phase system at a room temperature transfers spontaneously to the one-phase system - a solution of water, ethanol and other components. It is not excluded but it is highly improbable that a glass with a dissolved ice will return to the initial state.

## **9.2 Axiom of additivity**

This axiom postulates that the internal energy of the macroscopic system is a sum of its two macroscopic parts

$$\mathcal{U} = \mathcal{U}\_1 + \mathcal{U}\_2. \tag{123}$$

However, if we consider a macroscopic system as an assembly of molecules, equation (123) does not take into account intermolecular interactions among molecules of subsystem 1 and molecules of subsystem 2. Correctly the equation should be

$$
\mathcal{U}\mathcal{U} = \mathcal{U}\_1 + \mathcal{U}\_2 + \mathcal{U}\_{12} \,. \tag{124}
$$

Due to the fact that intermolecular interactions vanish at distances of the order of a few molecule diameters, the term *U*<sup>12</sup> is negligible in comparison with *U*.

### **9.3 The zeroth law of thermodynamics and the negative absolute temperatures**

The statistical thermodynamics introduces temperature formally as parameter *<sup>β</sup>* <sup>=</sup> <sup>1</sup> *kBT* in the expression (11) for the partition function

$$Q = \sum\_{i} \exp(-\beta E\_i) \ .$$

As energies of molecular systems are positive and unbounded, temperature must be positive otherwise the equation diverges. For systems with bounded energies

$$E\_{\rm min} \le E\_{\rm i} \le E\_{\rm max}$$

both negative and positive temperatures are allowed. Such systems are in lasers, for example.

### **9.4 The second law of thermodynamics**

From equation (3) it follows that entropy of the adiabatically isolated system either grows for spontaneous processes or remains constant in equilibrium

$$\text{d}\mathcal{S} \ge 0.\tag{125}$$

Entropy in the statistical thermodynamics is connected with probability via equation (20)

$$S = -k\_B \sum\_{i} P\_i \ln P\_i \dots$$

Thus, entropy may spontaneously decrease but with a low probability.

18 Will-be-set-by-IN-TECH

This axiom can be explained as follows. There is a very, very large number of microscopic states that correspond to a given macroscopic state. At unchained macroscopic parameters such as volume and temperature of a closed system there is much more equilibrium states then the states out of equilibrium. Consequently, a spontaneous transfer from non-equilibrium to equilibrium has a very, very high probability. However, a spontaneous transfer from an

Imagine a glass of whisky on rocks. This two-phase system at a room temperature transfers spontaneously to the one-phase system - a solution of water, ethanol and other components. It is not excluded but it is highly improbable that a glass with a dissolved ice will return to the

This axiom postulates that the internal energy of the macroscopic system is a sum of its two

However, if we consider a macroscopic system as an assembly of molecules, equation (123) does not take into account intermolecular interactions among molecules of subsystem 1 and

Due to the fact that intermolecular interactions vanish at distances of the order of a few

exp(−*βEi*).

As energies of molecular systems are positive and unbounded, temperature must be positive

*E*min ≤ *Ei* ≤ *E*max both negative and positive temperatures are allowed. Such systems are in lasers, for example.

From equation (3) it follows that entropy of the adiabatically isolated system either grows for

Entropy in the statistical thermodynamics is connected with probability via equation (20)

*i*

*Pi* ln *Pi* .

*<sup>S</sup>* = −*kB* ∑

Thus, entropy may spontaneously decrease but with a low probability.

*U* = *U*<sup>1</sup> + *U*<sup>2</sup> . (123)

*U* = *U*<sup>1</sup> + *U*<sup>2</sup> + *U*<sup>12</sup> . (124)

d*S* ≥ 0 . (125)

*kBT* in the

**9.1 Axiom on existence of the thermodynamic equilibrium**

equilibrium state to a non-equilibrium state is not excluded.

molecules of subsystem 2. Correctly the equation should be

expression (11) for the partition function

**9.4 The second law of thermodynamics**

molecule diameters, the term *U*<sup>12</sup> is negligible in comparison with *U*.

otherwise the equation diverges. For systems with bounded energies

spontaneous processes or remains constant in equilibrium

**9.3 The zeroth law of thermodynamics and the negative absolute temperatures** The statistical thermodynamics introduces temperature formally as parameter *<sup>β</sup>* <sup>=</sup> <sup>1</sup>

> *Q* = ∑ *i*

initial state.

**9.2 Axiom of additivity**

macroscopic parts
