(a) Show that the isothermal compressibility \(\kappa_{T}\) and the adiabatic compressibility \(\kappa_{S}\) of an ideal Bose gas are given by
\[
\kappa_{T}=\frac{1}{n k T} \frac{g_{1 / 2}(z)}{g_{3 / 2}(z)}, \quad \kappa_{S}=\frac{3}{5 n k T} \frac{g_{3 / 2}(z)}{g_{5 / 2}(z)}
\]
where \(n(=N / V)\) is the particle density in the gas. Note that, as \(z ightarrow 0, \kappa_{T}\) and \(\kappa_{S}\) approach their respective classical values, namely \(1 / P\) and \(1 / \gamma P\). How do they behave as \(z ightarrow 1\) ?
(b) Making use of the thermodynamic relations
\[
C_{P}-C_{V}=T\left(\frac{\partial P}{\partial T}ight)_{V}\left(\frac{\partial V}{\partial T}ight)_{P}=T V \kappa_{T}\left(\frac{\partial P}{\partial T}ight)_{V}^{2}
\]
and
\[
C_{P} / C_{V}=\kappa_{T} / \kappa_{S}
\]
derive equations (7.1.48a) and (7.1.48b).

Data From Equations (7.1.48a) and (7.1.48b).

Transcribed Image Text:

y = Cp Cv 1+ 4 Cv 81/2(2) 9 Nk g3/2(2) 585/2(2)81/2(2). 3 (83/2(2)}2 (48a) (48b)