Two perfect gases at absolute temperature $T_{1}$ and $T_{2}$ are mixed. There is no loss of energy. Find the temperature of the mixture if masses of molecules are $m_{1}$ and $m_{2}$ and the number of moles of the gases are $n_{1}$ and $n_{2}$ respectively.

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Solution

Energy is conserved. Therefore, initial energy = final energy
Let final temperature is $T$ .
Initial Energy= $\frac{3}{2} n_{1} R T_{1} + \frac{3}{2} n_{2} R T_{2}$
Final Energy = $\frac{3}{2} (n_{1} + n_{2}) R T$
Equating initial energy and final energy we get,
$T = \frac{n_{1} T_{1} + n_{2} T_{2}}{n_{1} + n_{2}}$ .

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