Solution
= Solution
With $v_{\rm g}\gg v_{\rm l}$ and $v_{\rm g}=k_BT/p$, the coexistence equation becomes
$$
\frac1p\frac{dp}{dT}=\frac{L}{k_BT^2}.
$$
For constant $L$ this integrates to
$$
p(T)=p_0\exp\left(-\frac{L}{k_BT}\right),
$$
where $p_0$ is constant. For $N$ gas particles, $V_{\rm g}=Nk_BT/p(T)$, and therefore
$$
\boxed{\frac{dV_{\rm g}}{dT}
=V_{\rm g}\left(\frac1T-\frac{L}{k_BT^2}\right).}
$$