Question

A diatomic ideal gas is expanded isothermally to $32$ times of its initial volume then it is cooled to restore to initial entropy at constant volume. Calculate ratio of the initial temperature to final temperature.

Answer 1

$Entropy=n{C}_V\ln \left(\frac{T_2}{T_1}\right)+nR\ln \left(\frac{V_2}{V_1}\right)$

considering initial entropy to be zero.

Step $1:T-$ constant

$\Delta S_1=nR\ln 32$

Step $2:V-$ constant

$\Delta S_2=n\times \frac{5}{2}R\ln \left(\frac{T_f}{T_1}\right)$

$⟹\frac{5}{2}R\ln \left(\frac{T_f}{T_1}\right)=-R\ln 32$ $(\Delta S_1+\Delta S_2=0)$

$\ln \left(\frac{T_f}{T_i}\right)=\ln {\left(\frac{1}{2^5}\right)}^{2/5}$

$⟹\frac{T_f}{T_i}=\frac{1}{4}$

$⟹\frac{T_i}{T_f}=4$