Heat Capacity at Constant Volume

As shown in the figure, an insulated container is fitted with a thermally conducting but immovable partition ($$P_1$$) and a freely movable but thermally insulated piston ($$P_2$$). The partition $$P_1$$ with thermal conductivity $$K$$, cross sectional area $$A$$ and width $$x$$ divides the container into two sections, $$S_1$$ and $$S_2$$, each containing one mole of a monoatomic gas. The piston $$P_2$$ moves freely such that the gas in $$S_2$$ is always at the atmospheric pressure. Initially, the difference between the temperatures of $$S_1$$ and $$S_2$$ is $$\Delta T_0$$. The time it takes for the temperature difference to become $$\dfrac{\Delta T_0}{2}$$ is $$nxR/KA$$, where $$R$$ is the universal gas constant. The value of $$n$$ is:

[ Given: $$\ln 2\approx 0.7$$ ]