The given equation of state (for one mole) of the gas is

\[\begin{equation*}P=R T /(\mathrm{v}-b)-a / \mathrm{v}^{n} \quad(n>1) . \tag{1}\end{equation*}\]

Equating \((\partial P / \partial \mathrm{v})_{T}\) and \(\left(\partial^{2} P / \partial \mathrm{v}^{2}\right)_{T}\) to zero, we get

\[
\mathrm{v}_{c}=\frac{n+1}{n-1} b \text { and } T_{c}=\frac{4 n(n-1)^{n-1}}{(n+1)^{n+1}} \frac{a}{b^{n-1} R} .
\]

Equation (1) then gives

\[
P_{c}=\left(\frac{n-1}{n+1}\right)^{n+1} \frac{a}{b^{n}}
\]

whence \(R T_{c} / P_{c} \mathrm{v}_{c}=4 n /\left(n^{2}-1\right)\).

To determine the critical behaviour of this gas, we write

\[
P=P_{c}(1+\pi), \mathrm{v}=\mathrm{v}_{c}(1+\psi), T=T_{c}(1+t)
\]

The equation of state then takes the form

\[
1+\pi=\frac{4 n(1+t)}{\left(n^{2}-1\right)(1+\psi)-(n-1)^{2}}-\frac{n+1}{(n-1)(1+\psi)^{n}}
\]

Carrying out the usual expansions and retaining only the most important terms, we get

\[
\pi \approx \frac{2 n}{n-1} t-\frac{n(n+1)^{2}}{12} \psi^{3}-\frac{n(n+1)}{n-1} t \psi .
\]

It follows that

(i) at \(t=0, \pi \approx-\left\{n(n+1)^{2} / 12\right\} \psi^{3}\), while at \(\psi=0, \pi \approx\{2 n /(n-\) 1) \(\} t\)

(ii) for \(t<0\), we obtain three values of \(\psi\); while \(\left|\psi_{2}\right| \ll\left|\psi_{1,3}\right|\), implying once again that \(\pi \approx\{2 n /(n-1)\} t, \psi_{1,3} \approx \pm\left\{12 /\left(n^{2}-1\right)\right\}^{1 / 2}|t|^{1 / 2}\),

(iii) the quantity

\[
\begin{aligned}
-\left(\frac{\partial \psi}{\partial \pi}\right)_{t} & \approx \frac{4(n-1)}{n(n+1)\left\{\left(n^{2}-1\right) \psi^{2}+4 t\right\}} \\
& \approx\left\{\begin{array}{ll}
(n-1) / n(n+1) t & (t>0) \\
(n-1) / 2 n(n+1)|t| & (t<0)
\end{array} .\right.
\end{aligned}
\]

Clearly, the critical exponents of this gas are the same as those of the van der Waals gas - regardless of the value of \(n\). The critical amplitudes (as well as the critical constants \(P_{c}, \mathrm{v}_{c}\) and \(T_{c}\) ), however, do vary with \(n\) and hence are model-dependent.