Question 19.2: Consider the evaluation of specific entropy using a virial e...

(a) According to Eq. (19.38), the second virial coefficient based on the square-well potential is
B(T) = b_{\circ}\left[1-(\lambda^{3}-1)(e^{\varepsilon/kT}-1 )\right].
From Eq. (19.59), the specific entropy defect requires the derivative of B(T) with respect to temperature. Using the above expression, we thus obtain
\Delta s_{T} = -\frac{R}{\nu}\left(B+T\frac{dB}{dT}\right).                (19.59)
\frac{dB}{dT} = b_{\circ}\left(\frac{\varepsilon}{kT^{2}}\right)(\lambda^{3}-1 )e^{\varepsilon/kT}.
Hence, substituting into Eq. (19.59), the specific entropy defect can be expressed as
\Delta s_{T} = -\frac{Rb_{\circ}}{\nu}\left[\lambda^{3}+(\lambda^{3}-1)\left(\frac{\varepsilon}{kT}-1\right)\exp \left(\frac{\varepsilon}{kT}\right)\right].
(b)The specific entropy of a real gas can be obtained from Eq. (19.60):
s(T,\nu) = s^{\circ}(T,\nu)+\Delta s_{T}.
Therefore, substituting for the specific entropy defect, we obtain
s(T,\nu) = s^{\circ}(T,\nu)-\frac{Rb_{\circ}}{\nu}\left[\lambda^{3}+(\lambda^{3}-1)\left(\frac{\varepsilon}{kT}-1\right) \exp \left(\frac{\varepsilon}{kT}\right)\right].
Evaluation of s(T, ν) requires the specific entropy for the hypothetical ideal gas from an appropriate JANAF table as well as relevant parameters describing the square-well potential for the chosen real gas.