Question

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4V_1$. After this, the gas expands adiabatically and its volume becomes $32V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\text{iso }}$ and $W$ respectively. If the ratio $\frac{W_{\text{iso }}}{W_{\text{adia }}}=f\ln 2$, then $f$ is ?

Answer 1

Answer: 1.78

$W_{\text{iso }}=RT\ln 4$
$T{\left(4V_1\right)}^{2/3}=T^{\prime }{\left(32V_1\right)}^{2/3}$
$\Rightarrow T^{\prime }=\frac{T}{4}$
$W_{\text{adia }}=\frac{2R\left(T-\frac{T}{4}\right)}{3}=\left(\frac{9RT}{8}\right)$
$\frac{W_{\text{iso }}}{W_{\text{adia }}}=\frac{8}{9}\ln 4=\frac{16}{9}\ln 2$
$\Rightarrow f=\frac{16}{9}=1.78$