Question

A cylindrical vessel is divided in two parts by a fixed partition which is perfectly heat conducting. The wall and piston are thermally insulated from surroundings. The left side contains 0.5 moles of gas with $C_v$ = 2R at temperature of 300 K. The right side contains 4 moles of mixture of gas with $C_v$ = 1.75 R at same temperature of 300 K. The piston compresses slowly the right side from volume of $V_o$ to $\frac{V_o}{4}$. If total change in internal energy of gases is $\Delta U=n\times {10}^{4}J.$, find $n$.
(Take : $R=\frac{25}{3}$ S.I unit)

Answer 1

$a)$
$n_1{C}_{v1}dT+n_2{C}_{v2}dT+PdV=0$
$\frac{1}{2}\times 2RdT+4\times \frac{7R}{4}dT+4RT\frac{dV}{V}=0$
$2\int \frac{dT}{T}+\int \frac{dV}{V}=0$
$2\ln \left(\frac{T}{300}\right)+\ln \left(\frac{1}{4}\right)=0$
$\ln \frac{T}{300}=\ln 2$
$T=600\text{K}$

$b)$ Change in internal energy
$\Delta U=\frac{1}{2}\times 2R\times (600-300)$
$+4\times \frac{7R}{4}(600-300)$
$=8R\times 300$
$=8\times \frac{25}{3}\times 300=2\times {10}^{4}J$