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

The amount of the gas in vessel 1 is `n_1 = (P_(1) V_(1))/(RT_(1))`
and that in vessel 2 is `n_2 = (P_(2) V_(2))/(RT_(2)`.
If `p'` and T'` be the common pressure and temperature after the connection is made, the amounts are
`n_1' = (p'V_(1))/(RT') and n_2^'`
or, `(p_1 V_(1))/(RT_(1)) + (p_(2) V(2))/(RT_(2)) = (p' V_(1))/(RT') + (p' V_(2))/(RT')`
or, `p'/T' = 1/(V_ (1) + V_(2) ((p_(1) V_(1)/T_(1) + (p_2 V_(2)/(T_(2))`
or, `T'/p' = (T_(1) T_(2) (V_(1) + V_(2))/(p_1 V_(1) T_(2) + p_(2) V_(2) T_(1)`. ... (i)
As the vessels have fixed volume, no work is done by the gas plus the vessels system. Also, no heat is exchanged with the surrounding. Thus, the internal energy of the total system remains constant. The internal energy of an ideal gas is
`U = nC_(v) T`
`= C_(v) (pV)/(R)`.
The internal energy of the gases before the connection
`(C_(v) p_(1) V_(1))/(R) + (C_(v) p_(2) V_(2))/(R)`
and after the connection
`= (C_v p' (V_(1) + V(2))/(R)`
Neglecting the change in internal energy of the vessels (the heat capacity of the vessels is assumed negligible).
`(C_(v) p_(1) V_(1)/(R) + (C_(v) p_(2) V_(2))/(R) = (C_(v) p' IV_(1) + V_(2))/(R)`
or, `p' = (p_(1) V_(1) + p_(2) V_(2))/(V_(1) + V_(2))`
From (i), `T' = (T_(1) T_(2) (p_(1) V_(1) + p_(2) V_(2))/(p_(1) V_(1) T_(2) + p_(2) V_(2) T_(1))`.