[NOTES/SM-04009] The Imperfect Gas

Perfect gas behaviour is expected to be a good approximation only when the density is low. When the density increases we want to obtain corrections to the equation of state $pV=NkT$. The total energy can be written as $$ E=\sum_{i=1}^N \epsilon_i+\sum_{i>j}^N \Phi(r_{ij}). $$ where\\ $\sum_i\epsilon_i=$ energy of molecules when no forces are present. \\ $\sum\Phi(\vec{r}_{ij})$= potential energy between gas molecules. The canonical partition function is given by \begin{align*} Z = & \int_V d^3r_1d^3r_2\cdots d^3r_N\int d^3p_1d^3p_2 \cdots e^{-\beta E}\\ =&\int_V d^3r_1d^3r_2\cdots d^3r_N \exp\left(-\beta\sum\Phi(r_{ij})\right) \int d{\bf R} \exp\left(-\beta\sum\epsilon_i\right). \end{align*} Here \(\vec r_1, \vec r_2, ...\) are the position vectors of the centre of mass of the molecules and \({\bf R}\) collectively denote the remaining variables. \begin{align*} p=&kT \frac{\partial}{\partial V} \ln Z\\ \ln Z=&\ln\int_Vd^3r_1\cdots d^3r_n\exp\left(-\beta\sum\Phi(r_{ij})\right)+ \text{terms indep of $V$}.\\ p=&kT\frac{\partial}{\partial V}(\ln Z_\text{conf}), \end{align*} where $$ Z_\text{conf} = \int_V d^3r_1\cdots d^3r_N\exp\left(-\beta\sum\Phi (r_{ij})\right). $$ is the configuration part of the partition function. If the density of the gas is not too large, we may assume only two body interactions. When $\Phi=0$, $Z_\text{conf}=V^N$ and we get perfect gas equation. We can then write \(\Phi= \sum_{i>j}\Phi(r_{ij}\) \begin{align*} \exp(-\beta\sum_{i>j}\Phi(r_{ij}))=&\prod_{i>j}\exp(-\beta\Phi(r_{ij}))\\ =&\prod_{i>j}\left(1+(e^{-\beta\Phi(r_{ij})}-1)\right). \end{align*} Let $\exp(-\beta\Phi(r_{ij})-1)=u_{ij}$, then \begin{align*} \exp\left(-\beta\sum\Phi(r_{ij})\right)=&\prod_{i>j}(1+u_{ij})\\ =&(1+u_{21})(1+u_{31})(1+u_{32})(\cdots)\\ =&(1+u_{21}+u_{31}+\cdots)+\cdots\qquad\text{higher powers of $u$}. \end{align*} Therefore \begin{align*} Z_\text{conf}=&\int_Vd^3r_1\cdots d^3r_N \exp\left(-\beta\sum\Phi(r_{ij})\right)\\ \approx&\int_V d^3r_1\cdots d^3r_N\left(1+\sum_{i>j}^Nu_j\right)+\cdots \end{align*} $u_{ij}$ represent the deviation from the perfect gas.\\ $+\cdots$ denote higher powers of $u_{ij}$ and products $u_{12}u_{23}$ etc. The inter molecular forces are short range forces and $+\cdots$ terms are nonzero only when three or more molecules are close. We shall neglect these terms. \begin{align*} Z_{conf}\simeq& \int_V d^3r_1\cdots d^3r_N\left(1+\sum_{i>j}u_{ij}\right)\\ = & V^N+\int_V d^3r_1\cdots d^3r_N \sum_{i>j}u_{ij}. \end{align*} \begin{equation} \sum_{i>j}\int d^3r_1\cdots d^3r_Nu_{ij}=V^{N-2} \sum_{i>j}^N \int_V d^3r_id^3r_j u_{ij}. \end{equation} \begin{align*} Z_\text{conf}=&V^N+V^{N-2}\frac{1}{2}N(N-1)\int_V d^3r_1d^3r_2u_{12}(r_{12}).\\ {r}_{12}=&|\vec{r}_1-\vec{r}_2|. \end{align*} Next we change variable to $\vec{r}=(\vec{r}_1-\vec{r}_2)$ \begin{align*} Z_\text{conf}\approx & V^N+V^{N-2}\frac{1}{2}N(N-1)\int_V d^3r_1 d^3r_2 u(r)\\ \approx& V^N+V^{N-1}\frac{1}{2}N(N-1)\int d^3r u(r)\\ Z_\text{conf} \cong& V^N-V^{N-1}N^2B(T).\\ B(T) =&\frac{1}{2}\int_Vd^3r[1-\exp(-\beta\Phi(r)]. \end{align*} The function $\Phi(r)$ is nonzero when \(r\) very small, $r\lesssim10^{-8}$cm , and $B(T)$ independent of volume of the gas. \begin{align*} \ln Z_\text{conf}\approx& \ln\left(V^N-N^2V^{N-1}B(T)\right)\\ =&\ln V^N(1-N^2 B(T)/V)\\ =&\ln V^N+\ln\left(1-\frac{N^2B(T)}{V}\right)\\ \approx&\ln V^N-\frac{N^2B(T)}{V}.\\ p=&kT\frac{\partial}{\partial V}\left[N \ln V -V^{-1}N^2B(T)\right]\\ =&kT\left[\frac{N}{V}+\frac{N^2}{V^2}B(T)+\cdots\right].\\ pv=&NkT\left[1+\frac{NB(T)}{V}+\frac{N^2}{V^2}C(T)+\cdots\right]. \end{align*} Thus we will get a series in $(N/V)$. $B(T)$ is known as the second virial coefficient, the next term $C(T)$ is called the third virial coefficient and so on. \subsubsection{Form of intermolecular potential} The inter molecular potential \(\Phi(r)\) tends to infinity as \(r\to 0\). For large distance it tends to zero very fast. It can be modeled by Lenard Jones potential given by \begin{align*} \Phi(r)=&\epsilon_0\left[\left(\frac{r_0}{r}\right)^{12}-2\left(\frac{r_0}{r}\right)^6\right] \end{align*} with parameter values $\epsilon_0\approx 10^{-2}$eV and $r_0\approx2\times10^{-8}$ cm. % \FigHere Another useful potential has the form \begin{align*} \Phi(r) = \begin{cases} \infty & r<2r_s\\ -\epsilon & 2r_s<r<2r_a\\ 0 & r>2r_a\end{cases}. \end{align*} This potential is a square well with a hard core. Therefore \begin{align*} 1-\exp(-\beta\Phi(r))=\begin{cases} 1 & r<2r_s\\ 1-\exp(\beta\epsilon)&2r_s2r_s\end{cases}. \end{align*} This form can be used to compute virial coefficients. We shall skip this calculation. \subsubsection{van der Waals equation} We assume $\Phi(r)$ to be approximated by a square well with hard core. Compute

\begin{align*}
B(T) =&\frac{1}{2} \int d^3r(1-\exp(-\beta\Phi(r))\\
=&\frac{1}{2}\int_0^\infty r^2dr\int \sin\theta d\theta\int d\phi
(1-\exp(-\beta\Phi))\\
=&2\pi\int_0^\infty r^2(1-\exp(-\beta\Phi))dr\\
=&2\pi\int_0^{2r_s}
r^2dr+\int_{2r_s}^{2r_a}r^2dr(1-e^{+\beta\epsilon_0})\\
=&2\pi\frac{r^2}{3}\Big|_{0}^{2r_s}+\left.\frac{r^3}{3}(1-e^{+\beta\epsilon_0})\right|_{2r_s}^{2r_a}\\
=&\frac{16\pi}{3}\left(r^3_s-(r^3_a-r^3_s)(1-e^{\beta\epsilon_0})\right).\\
\cong&\frac{16\pi}{3}\left(r^3_s-r^3_a\beta\epsilon_0\right)\qquad \text{since }\beta\epsilon_0\ll1.
\end{align*}

We write the Avogadro number as \(N_0\) and \(R=N_0k\) and define two parameters \(a,b\) by \begin{equation} a=\frac{16\pi}{3}r^3_a\epsilon_0N_0^2.\qquad b= \frac{16\pi}{3} r^3_sN_0 \end{equation} then \begin{equation*} N_0B(T) \equiv\left(b-\frac{a}{RT}\right)N_0 \end{equation*} \begin{align*} pV =& NkT\left[1+\frac{NB(T)}{V}\right]\\ =&nRT\left[1+\frac{n}{V}\left(b-\frac{a}{RT}\right)\right]\\ =&nRT\left(1+\frac{nb}{V}\right)-nRT\left(\frac{na}{VRT}\right)\\ =&nRT\Big(1+\frac{nb}{V}\Big)-\frac{an^2}{V}. \end{align*} Here \(n=N/N_0\) is the number of moles. \begin{align*} \left(p+\frac{an^2}{V^2}\right)V=&nRT\left(1+\frac{nb}{V}\right)\\ \approx& nRT\left(1-\frac{nb}{V}\right)^{-1}. \end{align*} Ths we get the famous van der Waals equation \begin{align*} \left(p+\frac{an^2}{V^2}\right)(V-nb)=&nRT, \end{align*} where \(b\) is related to the volume occupied by molecules \(nb\). This volume \(nb\), \begin{equation} nb=4\left(\frac{4\pi}{3}r^3_3\right)N. \end{equation} is much smaller than the total volume \(V\). The constant $a$ represents the attractive forces between the molecules and $b$ is a measure of the repulsive interaction between the molecules.