Two thermally insulated vessels 1 and 2 conta | vedclass

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Question

(C) Since the vessels are thermally insulated,the total internal energy of the system remains constant.$U_{total} = U_1 + U_2$$n_{total} C_v T = n_1 C

Vedclass · Accepted Answer

$\frac{T_1 T_2 (P_1 V_1 + P_2 V_2)}{P_1 V_1 T_2 + P_2 V_2 T_1}$

Vedclass · Answer

(C) Since the vessels are thermally insulated,the total internal energy of the system remains constant.$U_{total} = U_1 + U_2$$n_{total} C_v T = n_1 C_v T_1 + n_2 C_v T_2$$T = \frac{n_1 T_1 + n_2 T_2}{n_1 + n_2}$Using the ideal gas law $PV = nRT$,we have $n = \frac{PV}{RT}$.Substituting $n_1 = \frac{P_1 V_1}{R T_1}$ and $n_2 = \frac{P_2 V_2}{R T_2}$:$T = \frac{(\frac{P_1 V_1}{R T_1}) T_1 + (\frac{P_2 V_2}{R T_2}) T_2}{(\frac{P_1 V_1}{R T_1}) + (\frac{P_2 V_2}{R T_2})}$$T = \frac{P_1 V_1 + P_2 V_2}{\frac{P_1 V_1}{R T_1} + \frac{P_2 V_2}{R T_2}} = \frac{(P_1 V_1 + P_2 V_2) R T_1 T_2}{P_1 V_1 T_2 + P_2 V_2 T_1}$Since $R$ is a constant factor in the molar heat capacity ratio,the equilibrium temperature is $T = \frac{T_1 T_2 (P_1 V_1 + P_2 V_2)}{P_1 V_1 T_2 + P_2 V_2 T_1}$.

Two thermally insulated vessels $1$ and $2$ contain air at temperatures $(T_1, T_2)$,volumes $(V_1, V_2)$,and pressures $(P_1, P_2)$. If the valve connecting them is opened,what will be the equilibrium temperature of the system?

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