## Second Virial Coefficient

Approximation techniques are used to study the effect of inter-particle interactions in gases. A common technique is the virial density expansion. The first-order expansion for the ideal gas is given by $$\frac{PV}{NkT} = 1 + \rho B_2(T),$$ where the second virial coefficient is $$B_2(T) = 2\pi \int_0^{\infty} [1 - e^{u(r)/kT}] r^2dr \label{B2}$$ for a spherically symmetric inter-particle potential $u(r)$.

Program `SecondVirialCoefficient` computes $B_2(T)$ for the Lennard-Jones potential.

**Problem: Temperature dependence of $B_2$**

- A potential that captures some of the properties of the Lennard-Jones potential is the square well potential which is defined as \begin{equation} u(r)=\begin{cases} \infty & (r < \sigma), \\ -\epsilon & (\sigma < r < \lambda \sigma),\\ 0 & (r > \lambda\sigma), \end{cases} \label{eq:8/squarewell} \end{equation} where $\sigma$ is the diameter of the hard core, $\lambda \sigma$ is the range of the attractive well, and $\epsilon$ is the well depth. Show that for this potential the Mayer function $f(r)$ is given by \begin{equation} f(r)= \begin{cases} -1 & (r < \sigma), \\ g & (\sigma < r < \lambda \sigma),\\ 0 & (r > \lambda\sigma), \end{cases} \label{eq:8/frsqw} \end{equation} and \begin{equation} \label{eq:8/b2squarewell} B_2(T) = \frac{2\pi\sigma^3}{3}\big[1 - (\lambda^3 - 1)g \big]. \end{equation} where $g=e^{\beta \epsilon} - 1$.
- Plot the temperature dependence of $B_2(T)$ as given by \eqref{eq:8/b2squarewell} and compare it to the approximate temperature dependence given by $B_2 = b - {a \over kT}$.
- Program
`SecondVirialCoefficient`evaluates the integral over $r$ in \eqref{B2} numerically to determine the temperature dependence of $B_2$ for the Lennard-Jones potential. Compare the numerical results for the Lennard-Jones potential with the approximate result $B_2 = b - {a \over kT}$. At what temperature does $B_2$ vanish? How does this temperature compare with that predicted by $B_2 = b - {a \over kT} = 0$?

## Resource

Problem 8.32 in*Statistical and Thermal Physics: With Computer Applications*, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).