Answer
The proof is below.
Work Step by Step
When solving for equation 18.4, the book used the following equation:
$W = -nRT \int_{V_1}^{V_2}PdV$
We first solve the van der Waals equation for P:
$P=\frac{nRT}{v-nb}-a(\frac{n^2}{v^2})$
We plug this into the equation for work:
$W = -nRT \int_{V_1}^{V_2}(\frac{nRT}{V-nb}-a(\frac{n^2}{V^2}))dV$
Not surprisingly, when we evaluate this integral, we find:
$W = -nRTln(\frac{V_2}{V_1})$
This is the same as equation 18.4.
