During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of Cp/Cv for the gas is :

Solution

Physics sir · 3 steps

Rewrite the adiabatic law in terms of P and T only, then match the power of T to 3.

1. Remove the volume. For an adiabatic process PV^γ is constant. From the ideal gas law, V = nRT/P, so V ∝ T/P. Substituting gives P(T/P)^γ = constant, that is P^(1-γ)T^γ = constant.

   P ∝ T^(γ/(γ - 1))

2. Match the exponent. The question says P ∝ T³, so γ/(γ - 1) = 3. This gives γ = 3γ - 3, so 2γ = 3.
3. So. γ = Cₚ/Cᵥ = 3/2, which is option (b).

The idea to keep. In an adiabatic process, P ∝ T^(γ/(γ-1)). Derive it from PV^γ = constant and PV = nRT instead of memorising it, and the sign of the exponent will not trip you.

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