16.7: Van der Waals Constants in Terms of Molecular Parameters

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In the last lecture, we saw that for the pair potential

\[u(r) = \begin{cases} \infty & r \leq \sigma \\ -\dfrac{C_6}{r^6} & r > \sigma \end{cases} \label{1} \]

we could write the second virial coefficient as

\[B_2(T) = \dfrac{2}{3} \pi N_0 \sigma^3 \left[ 1 - \dfrac{C_6}{3 k_B T \sigma^6} \right] \label{2} \]

Let us introduce to simplifying variables

\[\begin{align} b &= \dfrac{2}{3} \pi N_0 \sigma^3 \\ a &= \dfrac{2 \pi N_0^2 C_6}{9 \sigma^3} \end{align} \label{3} \]

in terms of which

\[B_2(T) = b - \dfrac{a}{RT} \label{4} \]

With these definitions, the virial equation of state becomes

\[\begin{align} P &= \dfrac{nRT}{V} + \dfrac{n^2}{V^2} RT \left( b - \dfrac{a}{RT} \right) \\ &= \dfrac{nRT}{V} \left(1 + \dfrac{nb}{V} \right) - \dfrac{an^2}{V^2} \end{align} \label{5)} \]

If we assume \(nb/V\) is small, then we can also write

\[1 + \dfrac{nb}{V} \approx \dfrac{1}{1 - \dfrac{nb}{V}} \label{6)} \]

so that

\[P = \dfrac{nRT}{V - nb} - \dfrac{an^2}{V^2} \label{Eq7} \]

which is known as the van der Waals equation of state. Equation 16.7.9 can also be rewritten as

\[\left( P+\dfrac {an^2}{V^2} \right) \left( V-nb \right) = nRT \nonumber \]