JEE Main 2023 · 31 January · morning · Single correct · +4 / −1
The correct relation between γ = cₚ/cᵥ and temperature T is :
- (a) γ ∝ T
- (b) γ ∝ 1/√T
- (c) γ ∝ 1/T
- (d) γ ∝ T°
Answer and solution
Answer (d) γ ∝ T°
Solution
Physics sir · 3 steps
Write both heat capacities using degrees of freedom: T never appears, so γ is a constant.
- Energy per mole. By equipartition, each degree of freedom carries 1/2RT per mole. With f degrees of freedom, U = f/2RT.
- Heat capacities. cᵥ is the slope dU/dT = f/2R, so the T disappears when you differentiate. Mayer's relation then gives cₚ = cᵥ + R = (f/2 + 1)R.
γ = cₚ/cᵥ = (f/2+1)/(f/2) = 1 + 2/f
- So. γ depends only on f (for example 5/3 for a monatomic gas and 7/5 for a diatomic gas), not on T. So γ ∝ T⁰, which is option (d).
The idea to keep. Energy grows with T, but heat capacity is the slope of energy against T, so T drops out. For an ideal gas, γ = 1 + 2/f depends only on the degrees of freedom.
Answer from the NTA final answer key. Solution by PrepWiser.