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(C) According to the law of conservation of energy,the total internal energy before thermal contact equals the total internal energy after reaching eq

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$T_f = \frac{3}{2} T_0$

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(C) According to the law of conservation of energy,the total internal energy before thermal contact equals the total internal energy after reaching equilibrium.$U_i = U_f$For nitrogen (diatomic,$f_1 = 5$): $U_{A} = \frac{n_1 f_1 R T_1}{2} = \frac{1 	imes 5 	imes R 	imes T_0}{2} = \frac{5RT_0}{2}$For helium (monatomic,$f_2 = 3$): $U_{B} = \frac{n_2 f_2 R T_2}{2} = \frac{1 	imes 3 	imes R 	imes (7/3)T_0}{2} = \frac{7RT_0}{2}$Total initial energy $U_i = \frac{5RT_0}{2} + \frac{7RT_0}{2} = \frac{12RT_0}{2} = 6RT_0$At equilibrium,the gases share the same temperature $T_f$. The total internal energy is $U_f = \frac{n_1 f_1 R T_f}{2} + \frac{n_2 f_2 R T_f}{2} = \frac{(5+3) R T_f}{2} = 4RT_f$Equating $U_i = U_f$: $6RT_0 = 4RT_f$$T_f = \frac{6}{4} T_0 = \frac{3}{2} T_0$.

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