Derive the expression for the second virial coefficient B₂ in terms of the intermolecular potential  using the approximation  Use the Lennard-Jones potential to outline its calculation.

The virial expansion expresses the pressure of a gas as a power series in the density, capturing the effects of intermolecular interactions beyond the ideal gas approximation. The second virial coefficient $B_2$ accounts for pairwise interactions and is crucial for understanding deviations from ideal behavior.

Starting from the equation of state, the pressure can be written as:

$$
\frac{P}{k_B T} = \rho + B_2 \rho^2 + B_3 \rho^3 + \dots
$$

where $\rho = \frac{N}{V}$ is the number density, $k_B$ is Boltzmann’s constant, and $T$ is the temperature.

In statistical mechanics, the second virial coefficient $B_2$ can be derived from the canonical partition function. For a system of $N$ particles, the configurational integral is:

$$
Q_N = \frac{1}{N! \Lambda^{3N}} \int \dots \int e^{-\beta U(\mathbf{r}^N)} d\mathbf{r}^N
$$

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