\(\therefore\) Internal energy of the system with partition \(=\) Internal energy of the system without partition.

\(n_{1} C_{v} T_{1}+n_{2} C_{v} T_{2}=\left(n_{1}+n_{2}\right) C_{v} T\)

\(\therefore T=\frac{n_{1} T_{1}+n_{2} T_{2}}{n_{1}+n_{2}}\)

But \(n_{1}=\frac{P_{1} V_{1}}{R T_{1}}\) and \(n_{2}=\frac{P_{2} V_{2}}{R T_{2}}\)

\(\therefore T=\frac{\frac{P_{1} V_{1}}{R T_{1}} \times T_{1}+\frac{P_{2} V_{2}}{R T_{2}} \times T_{2}}{\frac{P_{1} V_{1}}{R T_{1}}+\frac{P_{2} V_{2}}{R T_{2}}}=\frac{T_{1} T_{2}\left(P_{1} V_{1}+P_{2} V_{2}\right)}{P_{1} V_{1} T_{2}+P_{2} V_{2} T_{1}}\)