Assuming the Dieterici equation of state

\[P(v-b)=k T \exp (-a / k T v)\]

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 \(T \geq T_{c}\), it yields a unique value of \(v\).

(c) For \(T

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