Question 529: Thermodynamics & KTG

Question

An insulated container of gas has two chambers separated  by an insulating partition. One of the chambers has volume $${V_1}$$ and contains ideal gas at pressure $${P_1}$$ and temperature $${T_1}$$. The other chamber has volume $${V_2}$$ and contains ideal gas at pressure $${P_2}$$ and temperature $${T_2}$$. If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be

Accepted Answer

Since, no work is done and system is thermally insulated from surrounding. Therefore, total internal energy is constant, that is, $U=U_1+U_2$ Assuming gas have same degree of freedom, we have
$$
\left(n_1+n_2\right) C_{\mathrm{V}} T=n_1 C_{\mathrm{V}} T_1+n_2 C_{\mathrm{V}} T_2
$$
Therefore,
$$
T=\frac{n_1 R T_1+n_2 R T_2}{R n_1+n_2 R}=\frac{\left(P_1 V_1+P_2 V_2\right)}{\frac{P_1 V_1}{T_1}+\frac{P_2 V_2}{T_2}}=\frac{\left(P_1 V_1+P_2 V_2\right) T_1 T_2}{\left(P_1 V_1 T_2+P_2 V_2 T_1\right)}
$$

Answer

$${{{T_1}{T_2}\left( {{P_1}{V_1} + {P_2}{V_2}} \right)} \over {{P_1}{V_1}{T_2} + {P_2}{V_2}{T_1}}}$$

Answer

$${{{P_1}{V_1}{T_1} + {P_2}{V_2}{T_2}} \over {{P_1}{V_1} + {P_2}{V_2}}}$$

Answer

$${{{P_1}{V_1}{T_2} + {P_2}{V_2}{T_1}} \over {{P_1}{V_1} + {P_2}{V_2}}}$$

Answer

$${{{T_1}{T_2}\left( {{P_1}{V_1} + {P_2}{V_2}} \right)} \over {{P_1}{V_1}{T_1} + {P_2}{V_2}{T_2}}}$$

Thermodynamics & KTG

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