Assuming the Dieterici equation of state

$$P(v-b)=kT\exp (-a/kTv)$$

evaluate the critical constants $P_c,v_c$, and $T_c$ of the given system in terms of the parameters $a$ and $b$, and show that the quantity $k{T}_c/P_c{v}_c=e^2/2\simeq 3.695$.

Further show that the following statements hold in regard to the Dieterici equation of state:

(a) It yields the same expression for the second virial coefficient $B_2$ as the van der Waals equation does.

(b) For all values of $P$ and for $T\ge T_c$, it yields a unique value of $v$.

(c) For $T<T_c$, there are three possible values of $v$ for certain values of $P$ and the critical volume $v_c$ is always intermediate between the largest and the smallest of the three volumes.

(d) The Dieterici equation of state yields the same critical exponents as the van der Waals equation does.