= Solution
Write $n=N/V$ and $B=B_2(T)$. Differentiating the <virial expansion> $p=k_BT(n+Bn^2)$ at fixed pressure gives
$$
0=(n+Bn^2+TB'n^2)\,dT+T(1+2Bn)\,dn.
$$
Since $dV/V=-dn/n$,
$$
\frac TV\left(\frac{\partial V}{\partial T}\right)_p-1=\frac{n(TB'-B)}{1+2Bn}=n(TB'-B)+O(n^2).
$$
Thus the <virial criterion for Joule-Thomson cooling> gives, \b[for the displayed truncated equation of state],
$$
\boxed{\mu=\frac N{C_p}\frac{TB_2'-B_2}{1+2B_2n}=\frac N{C_p}(TB_2'-B_2)[1-2B_2n+O(n^2)].}
$$
In particular the first density correction to the ideal-gas thermal-expansion factor gives the leading result $\mu=N(TB_2'-B_2)/C_p$. The heat capacity here is the actual $C_p$ at the state considered; replacing it by an ideal-gas value is a further leading-order approximation. On the low-density mechanically stable branch with $C_p>0$ and $1+2B_2n>0$, \b[positive cooling coefficient requires]
$$
\boxed{TB_2'-B_2>0\quad\Longleftrightarrow\quad\frac d{dT}\left(\frac{B_2(T)}T\right)>0.}
$$
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