Vessels 1 and 2 contain $
u$ moles of gaseous helium each. The ratio of the vessels' volumes is $V_

Question

Vessels 1 and 2 contain $
u$ moles of gaseous helium each. The ratio of the vessels' volumes is $V_2/V_1 = \alpha$, and the ratio of the absolute temperatures of helium in them is $T_1/T_2 = \beta$. Assuming the gas to be ideal, which of the following expressions correctly represent the difference of gas entropies in these vessels, $S_2 - S_1$?

Professor PC Thomas Classes · Answer

$\Delta S = S_2 - S_1 = 
u C_v \ln(T_2/T_1) + 
u R \ln(V_2/V_1)$. For He (monoatomic), $C_v = \frac{3}{2}R$. Thus $\Delta S = 
u \frac{3}{2}R \ln(1/\beta) + 
u R \ln \alpha = 
u R (\ln \alpha - \frac{3}{2} \ln \beta) = 
u R \ln(\alpha \beta^{-1.5})$. Options A, B, and C are mathematically identical representations of this result.