During an adiabatic process, if the pressure of a gas is found to be proportional to the cube of its absolute temperature, then the ratio of C_P/C_V for the gas is :

Solution

Physics sir · 3 steps

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

1. Remove the volume. For an adiabatic process PV^γ = constant. From the ideal gas law, V ∝ T/P. Substituting gives P (T/P)^γ = constant, which 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 = C_P/C_V = 3/2, which is option (b).

The idea to keep. In an adiabatic process, P ∝ T^(γ/(γ-1)). Derive it from PV^γ and PV = nRT instead of memorising it, and then equate the powers.

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