JEE Main 2022 · 25 June · evening · Single correct · +4 / −1
The ratio of specific heats ( C_P/C_V ) in terms of degree of freedom (f) is given by :
- (a) ( 1 + f/3 )
- (b) ( 1 + 2/f )
- (c) ( 1 + f/2 )
- (d) ( 1 + 1/f )
Answer and solution
Answer (b) ( 1 + 2/f )
Solution
Physics sir · 4 steps
Each degree of freedom holds 1/2RT per mole: build C_V from that, add R, then divide.
- Internal energy. By equipartition, each degree of freedom carries 1/2k_BT per molecule, which is 1/2RT per mole. So one mole of an ideal gas has U = f/2RT, and C_V = dU/dT = f/2R.
- Add R for CP. Mayer's relation gives C_P = C_V + R = f/2R + R = (f/2 + 1)R.
γ = C_P/C_V = (f/2+1)R/(f/2R) = 1 + 2/f
- Quick check. A monatomic gas has f = 3, so γ = 1 + 2/3 = 5/3. A diatomic gas has f = 5, so γ = 1 + 2/5 = 7/5. Both are the familiar values.
- So. C_P/C_V = 1 + 2/f, which is option (b).
The idea to keep. Remember γ = 1 + 2/f. More degrees of freedom means γ moves closer to 1. Test any formula with f = 3: it must give 5/3.
Answer from the NTA final answer key. Solution by PrepWiser.