# Intermolecular Potentials

Two experiments that are commonly performed the undergraduate Physical Chemistry (PChem) laboratory
are (i) measuring the second virial coefficient B of gases and (ii) measuring a vapor – liquid equilibrium (VLE)
and observing the phase behavior near the critical point. Often, CO_{2} is used in these experiments because it
exhibits strongly nonideal behavior, yet it is easy to handle and not corrosive or poisonous. The non–ideal
behavior that is quantified by B and the higher virial coefficients, and that manifests in the occurrence of
the phase change and critical behavior, are ultimately caused by the details of the inter–molecular potential
energy surface (PES).

The virial coefficients B, C, . . . are usually defined via an equation of state for a non–ideal gas, using
the number density $\rho = \frac{n}{V}$ (SI units: mol/m^{3}), as :

$p = RT \cdot \rho + RT B(T) \cdot \rho^2 + RT C(T) \cdot \rho^3 + ...$ ..... (Equation 1)

or alternatively, by starting with $\frac{pV_m}{RT} = 1$ for an ideal gas, via the expansion

$\frac{pV_m}{RT} = 1 + \frac{B(T)}{V_m} + \frac{C(T)}{V_m^2} + ...$ ..... (Equation 2)

Here, $V_m = \frac{V}{n} = \rho^{-1}$ is the molar volume (SI units: m^{3}/mol) and B,C, . . . have the units of powers of
the molar volume. If the density and molar volume are given in units of cm^{3}/mol and mol/cm^{3}, respectively, the units of the virial coefficients are given as powers of cm^{3}/mol. For $\rho$ and $V_m$ units of mol/L and L/mol, respectively, are also very common. The virial coefficients B(T), C(T), . . . describe the non–ideal
behavior of the gas at a given temperature T and are therefore related to inter–molecular interactions. In turn, the inter–molecular interactions are ultimately governed by the shape of the inter–molecular PES.