Two ideal polyatomic gases at temperatures   and are mixed so that there is no loss of energy. If and , and , and be the degrees of freedom, masses, number of molecules of the first and second gas respectively, the temperature of mixture of these two gases is :     Option: 1 Option: 2 Option: 3   Option: 4

Let the final temperature of the mixture be T.

Since, there is no loss in energy.
$\Delta \mathrm{U}=0$

$\begin{array}{l} \Rightarrow \frac{F_{1}}{2} n_{1} R \Delta T+\frac{F_{2}}{2} n_{2} R \Delta T=0 \\ \\ \Rightarrow \frac{F_{1}}{2} n_{1} R\left(T_{1}-T\right)+\frac{F_{2}}{2} n_{2} R\left(T_{2}-T\right)=0 \\ \\ \Rightarrow T=\frac{F_{1} n_{1} R T_{1}+F_{2} n_{2} R T_{2}}{F_{1} n_{1} R+F_{2} n_{2} R} \Rightarrow \frac{F_{1} n_{1} T_{1}+F_{2} n_{2} T_{2}}{F_{1} n_{1}+F_{2} n_{2}} \end{array}$

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