Question

An ideal gas in a thermally insulated container, at internal pressure $P_1,$ volume $V_1$ and absolute temperature $T_1$ expands irreversibly against zero external pressure so that the pressure of gas becomes $P_2,$ volume $V_2$ and temperature $T_2$. Which of the following statement(s) is / are incorrect?

• A. $\Delta S_{\text{surrounding}}=0$
• B. $\Delta S_{\text{System}}=nR\ln \frac{V_2}{V_1}$
• C. $\Delta H=0$
• D. $P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$

Answer 1

The correct option is D $P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$

$P_{\text{ext}}=0W=-P_{\text{ext}}\Delta V=0\text{Also}q=0,\Delta U=0,\Delta H=0\Delta S_{\text{Sorrounding}}=0\Delta S_{\text{System}}=n{C}_v\ln \frac{T_2}{T_1}+nR\ln \frac{V_2}{V_1}\Rightarrow \Delta S_{\text{System}}=nR\ln \frac{V_2}{V_1}(\because T_1=T_2)$

Since the gas is expanding against vacuum, work done is zero, temperature change is zero and hence $PV$ is constant for the process.