Answer
a) $\frac{1}{P}$
b) $=\frac{v-a}{P v}$
Work Step by Step
The volume expansivity and isothermal compressibility are expressed as $$
\beta=\frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_P \text { and } \alpha=-\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T
$$ (a) For an ideal gas $v=R T P$. Thus, $$
\begin{aligned}
& \left(\frac{\partial v}{\partial T}\right)_P=\frac{R}{P} \longrightarrow \beta=\frac{1}{v} \frac{R}{P}=\frac{1}{\mathrm{~T}} \\
& \left(\frac{\partial v}{\partial P}\right)_T=-\frac{R T}{P^2}=-\frac{v}{P} \longrightarrow \alpha=-\frac{1}{v}\left(-\frac{v}{P}\right)=\frac{1}{P}
\end{aligned}
$$ (b) For a gas whose equation of state is $v=R T P+a$, $$
\begin{aligned}
& \left(\frac{\partial v}{\partial T}\right)_P=\frac{R}{P} \longrightarrow \beta=\frac{1}{v} \frac{R}{P}=\frac{R}{R T+a P} \\
& \left(\frac{\partial v}{\partial P}\right)_T=-\frac{R T}{P^2}=-\frac{v-a}{P} \longrightarrow \alpha=-\frac{1}{v}\left(-\frac{v-a}{P}\right)=\frac{v-a}{P v}
\end{aligned}
$$
