JEE Main 2020 · 6 September · morning · Single correct · +4 / −1
Molecules of an ideal gas are known to have three translational degrees of freedom and two rotational degrees of freedom.The gas is maintained at a temperature of T. The total internal energy, U of a mole of this gas, and the value of
γ ( = Cₚ/Cᵥ ) are given, respectively by:
- (a) U = 5/2RT and γ = 7/5
- (b) U = 5RT and γ = 6/5
- (c) U = 5RT and γ = 7/5
- (d) U = 5/2RT and γ = 6/5
Answer and solution
Answer (a) U = 5/2RT and γ = 7/5
Solution
Physics sir · 3 steps
Five degrees of freedom, each worth 1/2RT per mole: that one count gives you both U and γ.
- Count freedoms. The molecule has 3 translational and 2 rotational degrees of freedom, so f = 5. This is a rigid diatomic gas like N₂ or O₂.
- Internal energy. By equipartition, each degree of freedom carries 1/2RT per mole. So U = f/2RT = 5/2RT.
Cᵥ = dU/dT = 5/2R, Cₚ = Cᵥ + R = 7/2R, γ = Cₚ/Cᵥ = 7/5
- So. U = 5/2RT and γ = 7/5, which is option (a).
The idea to keep. Find f first. Then U = f/2RT per mole, Cᵥ = f/2R and γ = 1 + 2/f. For f = 5 that gives 5/2RT and 7/5.
Answer from the NTA final answer key. Solution by PrepWiser.