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Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$ , while box $B$ contains one mole of helium at temperature $(7/3)\, T_0$ . The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of boxes), then the final temperature of gases $T_f$ , in terms of $T_0$ is
${T_f} = \frac{3}{7}{T_0}$
${T_f} = \frac{7}{3}{T_0}$
${T_f} = \frac{3}{2}{T_0}$
${T_f} = \frac{5}{2}{T_0}$
Solution
$\Delta \mathrm{U}_{\mathrm{A}}=-\Delta \mathrm{U}_{\mathrm{B}}$
$\Rightarrow \mathrm{n} \mathrm{C}_{\mathrm{vA}}(\Delta \mathrm{T})_{\mathrm{A}}=-\mathrm{n} \mathrm{C}_{\mathrm{vB}}(\Delta \mathrm{T})_{\mathrm{B}}$
$\Rightarrow 1 \times \frac{5}{2} \mathrm{R}\left(\mathrm{T}-\mathrm{T}_{0}\right)=-1 \times \frac{3}{2} \mathrm{R}\left(\mathrm{T}-\frac{7}{3} \mathrm{T}_{0}\right)$
$\Rightarrow 5 \mathrm{T}-5 \mathrm{T}_{0}=-3 \mathrm{T}+7 \mathrm{T}_{0}$
$\Rightarrow 8 \mathrm{T}=12 \mathrm{T}_{0} \Rightarrow \mathrm{T}=\frac{3}{2} \mathrm{T}_{0}$
