Question

Two identical thermally insulated vessels interconnected by a tube with a valve contain one mole of the same ideal gas each. The gas temperature in one vessel is equal to T₁ a nd in the other, T₂. The molar heat capacity of the gas of constant volume equals C_v. The valve having been opened, the gas comes to a new equilibrium state. Find the entropy increment ΔS of the gas. Demonstrate that  ΔS > 0.

Answer 1

For an ideal gas the internal energy depends on temperature only. We can consider the process in question to be one of simultaneous free expansion. Then the total energy  U = U₁ + U₂. Since

The entropy change is obtained by considering isochoric processes because in effect, the gas remains confined to its vessel.