Two vessels of volume `(V_1) and (V_2)` contain the same ideal gas. The pressures in the vessels are `(P_1) and (P_2) ` and the temperatures are `(T_1) and (T_2)` respectively . The two vessels are now connected to each other through a narrow tube. Assuming that no heat is exchanged between the surrounding and the vessels, find the common pressure and temperature attained after the connection.

Explanation

On opening the valve, the air will flow from the vessel at heigher pressure to the vessel at lower pressure till both vessels have the same air pressure. If this air pressure is `p`, the total volume of the air in the two vessels will br `(V_1 + V_2)`. Also if `v_1` and `v_2` be the number of moles of air initially in the two vessel, we have
`p_1 V_1 = v_1 RT_1` and `p_2 V_2 = v_2 RT_2` ....(1)
After the air is mixed up, the total number of moles are `(v_1 + v_2)` and the mixture is at temperature `T`
Hence `p(V_1 + V_2) = (v_1 + v_2)RT` ....(2)
Let us look at the two portions of air as one single system. Since this system is contained in a thermally insulated vessel, no heat excharge is involved in the process. That is, total heat transfer for the combined system `Q = 0`
Moreover, this combined system does not perform mechanical work either. The walls of the containers are rigid and there are no pistons etc to be pushed, looking at the total system, we know `A = 0`
Hence, internal energy of the combined system does not change in the process. Initially energy of the combined system if equal to the sum of internal energies of the two portions of air :
`U_l = U_1 + U_2 = (v_1 RT)/(gamma - 1)+(v_2 R T_2)/(gamma - 1)` ...(3)
Final internal energy of `(n_1 + n_2)` moles of air at temperature `T` is given by
`U_f = ((v_1 + v_2)RT)/(gamma - 1)` ....(4)
Therefore, `U_l = U_f` implies :
`T = (v_1 T_1 + v_2 T_2)/(v_1 + v_2)=(p_1 V_1 + p_2 V_2)/((p_1 V_1//T_1)+(p_2 V_2//T"_2)) = T_1 T_2 (p_1 V_1 + p_2 V_2)/(p_1 V_1 T_2 + p_2 V_2 T_2)`
From (2). therefore, final pressure is given by :
`p = (v_1+ v_2)/(V_1 + V_2) RT = (R)/(V_1 + V_2) (v_1 T_1 + v_2 T_2) = (p_1 V_1 + p_2 V_2)/(V_1 + V_2)`
This process in an example of free adiabatic expansion of ideal gas.