γ_A is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. γ_B is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If γ_A/γ_B=(1+1/n), then the value of n is ________ .

Solution

Physics sir · 3 steps

One vibrational mode carries both kinetic and potential energy, so it adds 2 to f, not 1.

1. Gas A. A monoatomic gas has only 3 translational degrees of freedom, so f_A = 3. Using γ = 1 + 2/f gives γ_A = 1 + 2/3 = 5/3.
2. Gas B. Each vibrational mode stores kinetic and potential energy, so it adds 2 to f. That gives f_B = 3 + 3 + 2 = 8, and γ_B = 1 + 2/8 = 5/4.

   γ_A/γ_B = (5/3)/(5/4) = 4/3 = 1 + 1/3

3. So. Comparing with 1 + 1/n gives n = 3.

The common slip: 3.375. Counting the vibrational mode as only one degree of freedom gives f_B = 7, γ_B = 9/7 and n = 27/8 = 3.375, not 3.

The idea to keep. Translational and rotational degrees each add 1 to f, but every vibrational mode adds 2. Find f first, then use γ = 1 + 2/f.

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