Show that for any given fluid

\[
\kappa_{T}=ho^{-2}(\partial ho / \partial \mu)_{T}
\]

where \(ho(=N / V)\) is the particle density and \(\mu\) the chemical potential of the fluid. For the ideal Bose gas at \(T
\[
ho=ho_{0}+ho_{e} \approx-\frac{k_{B} T}{V \mu}+\frac{\zeta(d / 2)}{\lambda^{d}}
\]

Using these results, show that \({ }^{11}\)

\[
\kappa_{T} \approx\left(V / k_{B} T\right)\left(ho_{0} / ho\right)^{2} \quad\left(T\]

note that in the thermodynamic limit the reduced compressibility, \(k_{B} T_{\kappa_{T}} / v\), is infinite at all \(T