# 7: Statistical Physics

- Page ID
- 25624

## Degrees of freedom

A molecule consisting of \(n\) atoms has \(s=3n\) degrees of freedom. There are 3 translational degrees of freedom, a linear molecule has \(s=3n-5\) vibrational degrees of freedom and a non-linear molecule \(s=3n-6\). A linear molecule has 2 rotational degrees of freedom and a non-linear molecule 3.

Because vibrational degrees of freedom account for both kinetic and potential energy they count double. So, for linear molecules this results in a total of \(s=6n-5\). For non-linear molecules this gives \(s=6n-6\). The average energy of a molecule in thermodynamic equilibrium is \(\left\langle E_{\rm tot} \right\rangle=\frac{1}{2}skT\). Each degree of freedom of a molecule has in principle the same energy: the *principle of equipartition*.

The rotational and vibrational energy of a molecule modeled as a rigid harmonic oscillator are:

\[W_{\rm rot}=\frac{\hbar^2}{2I}l(l+1)=Bl(l+1)~,~~W_{\rm vib}=(v+\frac{1}{2})\hbar\omega_0\]

The vibrational levels are excited if \(kT\approx\hbar\omega\), the rotational levels of a hetronuclear molecule are excited if \(kT\approx2B\). For homonuclear molecules additional selection rules apply so the rotational levels are well coupled if \(kT\approx6B\).

## The energy distribution function

The general form of the equilibrium velocity distribution function is \(P(v_x,v_y,v_z)dv_xdv_ydv_z=P(v_x)dv_x\cdot P(v_y)dv_y\cdot P(v_z)dv_z\) with

\[P(v_i)dv_i=\frac{1}{\alpha\sqrt{\pi}}\exp\left(-\frac{v_i^2}{\alpha^2}\right)dv_i\]

where \(\alpha=\sqrt{2kT/m}\) is the *most probable velocity* of a particle. The average velocity is given by \(\left\langle v \right\rangle=2\alpha/\sqrt{\pi}\), and \(\left\langle v^2 \right\rangle=\frac{3}{2}\alpha^2\). The distribution as a function of the absolute value of the velocity is given by:

\[\frac{dN}{dv}=\frac{4N}{\alpha^3\sqrt{\pi}}~v^2\exp\left(-\frac{mv^2}{2kT}\right)\]

The general form of the energy distribution function then becomes:

\[P(E)dE=\frac{c(s)}{kT}\left(\frac{E}{kT}\right)^{\frac{1}{2}s-1}\exp\left(-\frac{E}{kT}\right)dE\]

where \(c(s)\) is a normalization constant, given by:

- Even \(s\): \(s=2l\): \(\displaystyle c(s)=\frac{1}{(l-1)!}\)
- Odd \(s\): \(s=2l+1\): \(\displaystyle c(s)=\frac{2^l}{\sqrt{\pi}(2l-1)!!}\)

## Pressure on a wall

The number of molecules that collides with a wall with surface \(A\) within a time \(\tau\) is given by:

\[\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int d^3N=\int\limits_0^\infty \int\limits_0^\pi \int\limits_0^{2\pi} nAv\tau\cos(\theta)P(v,\theta,\varphi)dvd\theta d\varphi\]

From this follows for the particle flux on the wall: \(\Phi= \frac{1}{4} n\left\langle v \right\rangle\). For the pressure on the wall it then follows that:

\[d^3p=\frac{2mv\cos(\theta)d^3N}{A\tau}~,~~\mbox{so}~~p=\frac{2}{3}n\left\langle E \right\rangle\]

## The equation of state

If intermolecular forces and the volume of the molecules can be neglected then for gases from \(p=\frac{2}{3}n\left\langle E \right\rangle\) and \(\left\langle E \right\rangle=\frac{3}{2}kT\) it can be derived that:

\[pV=n_sRT=\frac{1}{3}Nm\left\langle v^2 \right\rangle\]

Here, \(n_s\) is the number of *moles* of molecules and \(N\) is the total number of molecules within volume \(V\). If the molecular volume and the intermolecular forces cannot be neglected the *Van der Waals* equation can be derived:

\[\left(p+\frac{an_s^2}{V^2}\right)(V-bn_s)=n_sRT\]

There is an isotherm with a horizontal point of inflection. In the *Van der Waals* equation this corresponds with the *critical temperature, pressure* and *volume* of the gas. This is the upper limit of the area of coexistence between liquid and vapor. From \(dp/dV=0\) and \(d^2p/dV^2=0\) follows:

\[T_{\rm cr}=\frac{8a}{27bR}~,~~p_{\rm cr}=\frac{a}{27b^2}~,~~V_{\rm cr}=3bn_s\]

At the critical point: \(p_{\rm cr}V_{m,\rm cr}/RT_{\rm cr}=\frac{3}{8}\), which differs from the value of 1 which follows from the ideal gas law.

Scaled using the critical quantities, with \(p^*:=p/p_{\rm cr}\), \(T^*=T/T_{\rm cr}\) and \(V_m^*=V_m/V_{m,\rm cr}\) with \(V_m:=V/n_s\) one obtains:

\[\left(p^*+\frac{3}{(V_m^*)^2}\right)\left(V_m^*-\mbox{$\frac{1}{3}$}\right)= \mbox{$\frac{8}{3}$}T^*\]

Gases behave the same for equal values of the reduced quantities: the *law of the corresponding states*. A *virial expansion *can be used for an even more accurate picture:

\[p(T,V_m)=RT\left(\frac{1}{V_m}+\frac{B(T)}{V_m^2}+\frac{C(T)}{V_m^3}+\cdots\right)\]

The *Boyle temperature* \(T_{\rm B}\) is the temperature for which the 2nd virial coefficient is 0. In a Van der Waals gas, this happens at \(T_{\rm B}=a/Rb\). The *inversion temperature* \(T_{\rm i}=2T_{\rm B}\).

An equation of state for solids and liquids is given by:

\[\frac{V}{V_0}=1+\gamma_p\Delta T-\kappa_T\Delta p= 1+\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}\Delta T+\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T}\Delta p\]

## Collisions between molecules

The collision probability of a particle in a gas that is moving over a distance \(dx\) is given by \(n\sigma dx\), where \(\sigma\) is the *cross section*. The mean free path is given by \(\displaystyle\ell=\frac{v_1}{nu\sigma}\) with \(u=\sqrt{v_1^2+v_2^2}\) the relative velocity between the particles. If \(m_1\ll m_2\) then: \(\displaystyle\frac{u}{v_1}=\sqrt{1+\frac{m_1}{m_2}}\), so \(\displaystyle\ell=\frac{1}{n\sigma}\). If \(m_1=m_2\) then: \(\displaystyle\ell=\frac{1}{n\sigma\sqrt{2}}\). This means that the average time between two collisions is given by \(\displaystyle\tau=\frac{1}{n\sigma v}\). If the molecules are approximated by hard spheres the cross section is: \(\sigma= \frac{1}{4} \pi(D_1^2+D_2^2)\). The average distance between two molecules is \(0.55n^{-1/3}\). Collisions between molecules and small particles in a solution result in *Brownian motion*. For the average motion of a particle with radius \(R\) it can be derived that: \(\left\langle x_i^2 \right\rangle=\frac{1}{3}\left\langle r^2 \right\rangle=kTt/3\pi\eta R\).

A gas is called a *Knudsen gas* if \(\ell\gg\) the volume of the gas, something that can easily occur at low pressures. The equilibrium condition for a vessel which has a hole with surface area \(A\) in it if \(\ell\gg\sqrt{A/\pi}\) is: \(n_1\sqrt{T_1}=n_2\sqrt{T_2}\). Together with the general gas law follows: \(p_1/\sqrt{T_1}=p_2/\sqrt{T_2}\).

If two plates slide along each other at a distance \(d\) with velocity \(w_x\) the *viscosity* \(\eta\) is given by: \(\displaystyle F_x=\eta\frac{Aw_x}{d}\). The velocity profile between the plates is in that case given by \(w(z)=zw_x/d\). It can be derived that \(\eta=\frac{1}{3}\varrho\ell\left\langle v \right\rangle\) where \(v\) is the *thermal velocity*.

The heat conductance in a stationary gas is described by: \(\displaystyle\frac{dQ}{dt}=\kappa A\left(\frac{T_2-T_1}{d}\right)\), which results in a temperature profile \(T(z)=T_1+z(T_2-T_1)/d\). It can be derived that \(\kappa=\frac{1}{3}C_{mV}n\ell\left\langle v \right\rangle/N_{\rm A}\). Also: \(\kappa=C_V\eta\). A better expression for \(\kappa\) can be obtained with the *Eucken correction*: \(\kappa=(1+9R/4c_{mV})C_V\cdot\eta\) with an error \(<\)5%.

## Interaction between molecules

For dipole interaction between molecules it can be derived that \(U\sim-1/r^6\). If the distance between two molecules approaches the molecular diameter \(D\) a repulsive force between the electron clouds appears. This force can be described by \(U_{\rm rep}\sim\exp(-\gamma r)\) or \(V_{\rm rep}=+C_s/r^s\) with \(12\leq s\leq20\). This results in the *Lennard-Jones* potential for intermolecular forces:

\[U_{\rm LJ}=4\epsilon\left[\left(\frac{D}{r}\right)^{12}-\left(\frac{D}{r}\right)^6\right]\]

with a minimum \(\epsilon\) at \(r=r_{\rm m}\) the following holds: \(D\approx0.89r_{\rm m}\). For the Van der Waals coefficients \(a\) and \(b\) and using critical quantities: \(a=5.275 N_{\rm A}^2D^3\epsilon\), \(b=1.3N_{\rm A}D^3\), \(kT_{\rm kr}=1.2\epsilon\) and \(V_{\rm m,kr}=3.9N_{\rm A}D^3\).

A simpler model for intermolecular forces assumes a potential \(U(r)=\infty\) for \(r<D\), \(U(r)=U_{\rm LJ}\) for \(D\leq r\leq3D\) and \(U(r)=0\) for \(r\geq3D\). This gives for the potential energy of one molecule: \(\displaystyle E_{\rm pot}=\int_D^{3D}U(r)F(r)dr\) with \(F(r)\) the spatial distribution function in spherical coordinates, which for a homogeneous distribution is given by: \(F(r)dr=4n\pi r^2dr\).

Some useful mathematical relations are:

\[\int\limits_0^\infty x^n{\rm e}^{-x}dx=n!~~,~~ \int\limits_0^\infty x^{2n}{\rm e}^{-x^2}dx=\frac{(2n)!\sqrt{\pi}}{n!2^{2n+1}}~~,~~ \int\limits_0^\infty x^{2n+1}{\rm e}^{-x^2}dx=\mbox{$\frac{1}{2}$}n!\]