The given equation of state is

$$\begin{array}{}P=\frac{kT}{v-b}{e}^{-a/kTv}& \text{(1)}\end{array}$$

It follows that

$$\text{Misplaced \&}$$

At the critical point, both these derivatives vanish - with the result that

$$\text{Extra left or missing right}$$

whence $v_c=2b$ and $k{T}_c=a/4b$. Equation (1) then gives: $\text{Extra left or missing right}$ and hence $k{T}_c/P_c{v}_c=e^2/2\simeq 3.695$.

(a) For large v, the given equation of state may be approximated as

$$\text{Extra left or missing right}$$

Comparing this with eqns. (10.3.7-10), we see that the coefficient $B_2$ in the present case is formally the same as the one for the van der Waals gas, viz. $b-\left(a/kT\right)$.

(b) We note that the derivative $(\partial P/\partial v{)}_T$ for the Dietrici gas can be written as

$$\text{Misplaced \&}$$

Clearly, if $T>T_c$, then $(\partial P/\partial v{)}_T$ is definitely negative; the same is true at $T=T_c$ - except for the special case $v=a/2k{T}_c=2b$ when $(\partial P/\partial v{)}_T$ is zero. In any case, for all $T\ge T_c,P$ is a monotonically decreasing function of $v-$ with the result that, for any given $T$ and $P$, we have a unique $v$.

(c) For $T<T_c,P$ is a non-monotonic function of $v-$ generally decreasing with $v$ but increasing between the values

$$\text{Extra left or missing right}$$

For any given $T$, we now have (for a certain range of $P$ ) three possible values of $v$ such that

$$v_1>v_{max}>v_2>v_{min}>v_3;$$

see Figs. 12.2 and 12.3. We further note that

$$\text{Extra left or missing right}$$

Clearly, $v_{min}<v_c<v_{max}$ and, hence, $v_3<v_c<v_1$.