Two thermally insulated vessel 1 and 2 are filled with air at temperature `(T_1T_2), volume (V_1V_2)` and pressure `(P_1P_2)` respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be

Options

A. `T_1T_2(P_1V_1+P_2V_2)//(P_1V_1T_2+P_2V_2T_1)`
B. `(T_1+T_2)//2`
C. `T_1+T_2`
D. `T_1T_2(P_1V_1+P_2V_2)//(P_1V_1T_1+P_2V_2T_2)`

Answer

A

Explanation

(a) Here `Q=0 and W=0.` Therefore from first law of thermodynamics `DeltaU=Q+W=0`
`:.` Internal energy of the system with partition
`n_1C_vT_1+n_2C_vT_2=(n_1+n_2)C_vT`
`:. T=(n_1T_1+n_2T_2)/(n_1+n_2)`
But `n_1=(P_1V_1)/(RT_1) and n_2=(P_2V_2)/(RT_2)`
`:. T=((P_1V_1)/(RT_1)xxT_1+(P_2V_2)/(RT_2)xxT_2)/((P_1V_1)/(RT_1)+(P_2V_2)/(RT_2))=(T_1T_2(P_1V_1+P_2V_2))/(P_1V_1T_2+P_2V_2T_1)`