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$N_1$ 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
$\frac{{\left( {{T_1} - {T_2}} \right)}}{2}$
$\frac{{\left( {{N_1}{T_1} - {N_2}{T_2}} \right)}}{{\left( {{N_1} + {N_2}} \right)}}$
$\frac{{\left( {{N_1}{T_1} + {N_2}{T_2}} \right)}}{{\left( {{N_1} + {N_2}} \right)}}$
$\left\{ {\frac{{{N_1} + {N_2}}}{2}} \right\}\left\{ {\frac{{{T_1} + {T_2}}}{2}} \right\}$
Solution
$\left(\frac{3}{2} k T_{1}\right) \times N_{1}+\left(\frac{3}{2} k T_{2}\right) \times N_{2}$
$=\left(N_{1}+N_{2}\right) \times \frac{3}{2} k T$
$\mathrm{T}_{1} \mathrm{N}_{1}+\mathrm{T}_{2} \mathrm{N}_{2}=\left(\mathrm{N}_{1}+\mathrm{N}_{2}\right) \mathrm{T}$
$\therefore \quad T=\frac{N_{1} T_{1}+N_{2} T_{2}}{N_{1}+N_{2}}$
