A thermally insulated vessel contains an ideal gas of molecular mass M and ratio of specific heats 1.4. Vessel is moving with speed v and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases by :
(R = universal gas constant)

Solution

Physics sir · 4 steps

The ordered kinetic energy of the moving gas turns into internal energy, so use Cᵥ per mole.

1. Where energy goes. When the vessel stops, the gas loses its bulk kinetic energy. The vessel is insulated and rigid, so no heat leaves and no work is done. All of that kinetic energy becomes internal energy of the gas.
2. Find Cv. With γ = 1.4, the molar heat capacity at constant volume is Cᵥ = R/(γ - 1) = R/0.4 = 5/2R. Constant volume applies because the vessel does not change size.
3. Energy balance. Take n moles. The gas has mass nM, so its kinetic energy is 1/2nMv². This equals the rise in internal energy, nCᵥ Δ T.

   1/2nMv² = n(5/2R)Δ T

4. So. n cancels, and Δ T = Mv²/5R, which is option (b).

The idea to keep. A rigid vessel does no work, so any energy fed into the gas raises its temperature through Cᵥ, never Cₚ. Get Cᵥ from γ using Cᵥ = R/(γ - 1).

Answer from the NTA final answer key. Solution by PrepWiser.