Discussion
The value of the compressibility factor at the critical point evaluated using the van der Waals equation of state is
Answer: C
The critical compressibility factor is \(\frac{3}{8}\).
For the van der Waals equation, the conditions \((\partial P/\partial V)_{T} = 0\) and \((\partial^{2}P/\partial V^{2})_{T} = 0\) at the critical point give
\[ \begin{aligned} V_{c} &= 3b, \quad T_{c} = \frac{8a}{27Rb} \\ P_{c} &= \frac{a}{27b^{2}} \end{aligned} \]
\[ \begin{aligned} Z_{c} &= \frac{P_{c}V_{c}}{RT_{c}} \\ &= \frac{(a/27b^{2})(3b)}{8a/27b} = \frac{3}{8} \end{aligned} \]
The value 0.375 is the same for every van der Waals gas, higher than the 0.23 to 0.31 measured for real gases. The fractions 2/7, 5/8 and 1/7 do not come out of these critical constants.
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