The correct relation between the degrees of freedom f and the ratio of specific heat γ is :

Solution

Physics sir · 3 steps

Write Cᵥ and Cₚ in terms of f, take their ratio, then turn the formula around to get f.

1. Equipartition. Each degree of freedom carries 1/2RT per mole, so U = f/2RT and Cᵥ = f/2R. Mayer's relation gives Cₚ = Cᵥ + R = (f/2 + 1)R.

   γ = Cₚ/Cᵥ = (f/2+1)/(f/2) = 1 + 2/f

2. Rearrange for f. Move the 1 across: γ - 1 = 2/f. Flip both sides to get f = 2/(γ - 1).

   Option	Put in γ = 5/3 (monatomic)	Should give f = 3
   (a) 2/(γ - 1)	2/(2/3) = 3	matches
   (b) 2/(γ + 1)	2/(8/3) = 3/4	no
   (c) (γ + 1)/2	(8/3)/2 = 4/3	no
   (d) 1/(γ + 1)	1/(8/3) = 3/8	no

3. So. The relation is f = 2/(γ - 1), which is option (a).

The idea to keep. Remember γ = 1 + 2/f. If you are unsure of a rearranged form, put in a monatomic gas, γ = 5/3 and f = 3, and check.

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