$N_{1}$ molecules of a gas at temperature $T_{1}$ are mixed with $N_{2}$ molecules at temperature $T_{2}$ . The resulting temperature of the mixture gas is

\frac{(T_{1} - T_{2})}{2}

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\frac{(N_{1} T_{1} - N_{2} T_{2})}{(N_{1} + N_{2})}

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\frac{(N_{1} T_{1} + N_{2} T_{2})}{(N_{1} + N_{2})}

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{\frac{(N_{1} + N_{2})}{2}} {\frac{T_{1} + T_{2}}{2}}

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Solution

The correct option is D $\frac{(N_{1} T_{1} + N_{2} T_{2})}{(N_{1} + N_{2})}$
$(\frac{3}{2} k T_{1}) \times N_{1} + (\frac{3}{2} k T_{2}) \times N_{2}$ (by conservation of energy i.e.,energy will remain constant)
$= (N_{1} + N_{2}) \times \frac{3}{2} k T$
$T_{1} N_{1} + T_{2} N_{2} = (N_{1} + N_{2}) T$
$∴ T = \frac{N_{1} T_{1} + N_{2} T_{2}}{N_{1} + N_{2}}$

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