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QPages 77-80: Q29: HC Verma: Concepts of Physics - Volume 2 Specific Heat Capacities of Gases Pages 77-80: Q29 Solution: Two vessels A and B of equal volume are connected by a narrow…

Two vessels A and B of equal volume V0V_0 are connected by a narrow tube that can be closed by a valve. The vessels are fitted with pistons that can be moved to change the volumes. Initially, the valve is open and the vessels contain an ideal gas Cp/Cv=γC_p/C_v = \gamma) at atmospheric pressure p0p_0 and atmospheric temperature T0T_0. The walls of vessel A are diathermic and those of B are adiabatic. The valve is now closed and the pistons are slowly pulled out to increase the volumes of the vessels to double the original value. a) Find the temperatures and pressures in the two vessels. b) The valve is now opened for sufficient time so that the gases acquire a common temperature and pressure. Find the new values of the temperature and pressure.

Step-by-Step Verified Solution

**Answer:** (a) TA=T0T_A = T_0, TB=T0/2γT_B = T_0/2^\gamma (b) Final T=T0(1+2−γ)/2T = T_0(1 + 2^{-\gamma})/2. **Step-by-Step Explanation:** Vessel A (diathermic): isothermal expansion to 2V02V_0, so TA=T0T_A = T_0, PA=P0/2P_A = P_0/2. Vessel B (adiabatic): adiabatic expansion to 2V02V_0, so TB=T0/2γ−1T_B = T_0/2^{\gamma-1}, PB=P0/2γP_B = P_0/2^\gamma. After valve opened, gases mix. Final temperature and pressure from energy conservation.

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