Question

Consider the adjacent figure, a piston divides a cylindrical container into two parts. The left part contains $1$ mole of helium gas and the right part contains two moles of oxygen gas. The initial temperatures and pressures of the gases in the left chamber and $T_0$ and $P_0$ and the right chamber are $\frac{T_0}{2}$ and $P_0$ respectively, as shown in the figure. The piston as well as the walls of the container are adiabatic. After removal of the piston, gas mix homogeneously and the final pressure becomes $P=\frac{4n}{13}{P}_0$. Find the value of $n$.

Answer 1

$\frac{P_0{V}_0}{T_0}+\frac{2P_0{V}_0}{T_0}=\frac{2P{V}_0}{T}$
$\Rightarrow P=\frac{3}{2}\frac{P_{0}T}{T_0}$
$△U_1+△U_2=0$
$\frac{3}{2}R(T_0-T)=\frac{25}{2}R\left(T\frac{T_0}{2}\right)$
$T=\frac{8}{13}{T}_0$
Solving $P=\frac{12P_0}{13}\Rightarrow n=3$.