γ A \gamma_{\char"0391 } γ A is the specific-heat ratio of a monatomic gas A having 3 translational degrees of freedom. γ B \gamma_{B} γ B is the specific-heat ratio of a polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If γ A \gamma_{A} γ A / γ B \gamma_{B} γ B = 1 + 1/n, find n.
Physics · Thermodyanmics · kinetic therory of gaseous
JEE Mains 2025 2 April Morning Shift

Question

γ A \gamma_{\char"0391 } γ A ​ is the specific-heat ratio of a monatomic gas A having 3 translational degrees of freedom. γ B \gamma_{B} γ B ​ is the specific-heat ratio of a polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If γ A \gamma_{A} γ A ​ / γ B \gamma_{B} γ B ​ = 1 + 1/n, find n.

G1 Physics · Accepted Answer

For an ideal gas with f active quadratic degrees of freedom, γ = (f+2)/f. For monatomic gas A: fA = 3, γA = (3+2)/3 = 5/3. For gas B: 3 translational + 3 rotational + 1 vibrational mode. A vibrational mode contributes two quadratic energy terms (kinetic + potential), so fB = 3+3+2 = 8. Therefore γB = (8+2)/8 = 10/8 = 5/4. Hence γA/γB = (5/3)/(5/4) = 4/3. Given 4/3 = 1 + 1/n, we get 1/n = 1/3 ⇒ n = 3. Answer: 3. (The printed PDF has an intermediate typo for γB, but the key n=3 is consistent with…