= Solution
Use the <additive-volume two-level system> implicit in the model: each excited particle increases the volume by the same increment. The difference between the two endpoint volumes is $V_0$, so that increment is $v=V_0/N$.
If $N_1$ particles are excited, $V=V_0+(V_0/N)N_1$. Linearity of the expectation gives $\langle N_1\rangle=NP_1$, hence
$$
\boxed{V(T)=V_0\left(1+\frac{3e^{-x}}{1+3e^{-x}}\right)
=V_0\frac{1+6e^{-x}}{1+3e^{-x}}}.
$$
As $T\to0$, $V\to V_0$. As $T\to\infty$, only three quarters of the particles are excited on average, giving $V\to7V_0/4$, rather than the all-excited endpoint $2V_0$. These limits also check the inclusion of <state degeneracy>.
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