Question

1 Kg of an ideal gas at $T_1$ Temperature is heated isochorically till its temperature is tripled. It is then cooled at constant pressure to its original initial temperature. The gas is then allowed to expand to its original volume isothermally . The work done during the cycle is

Answer 1

$W_{net}=W_{12}+W_{23}+W_{31}$

$W_{12}=0$

$\text{Since,}V=0\text{in isochoric process}$

$W_{23}=F_2(V_3-V_2)=R(T_2-3T_1)$

$=-2R{T}_1$

$W_{31}=R{T}_{1}ln\frac{V_1}{V_3}$

In Process 2-3

$\frac{P_2{V}_2}{T_2}=\frac{P_3{V}_3}{T_3}$

$\therefore \frac{V_2}{V_3}=\frac{T_2}{T_3}=3$

$\text{Since,}{P}_2=P_3$

$\therefore W_{31}=R{T}_{1}ln\frac{V_2}{V_3}$ = $R{T}_{1}ln\frac{T_2}{T_3}$

$=R{T}_{1}ln3$

$W_{\text{net}}=0+(-2R{T}_1+R{T}_{1}ln3)$

$=-0.90R{T}_1$

-ve sign means work is done on the gas .