1. Specific heat ratio formula

For an ideal gas with total degrees of freedom $f$, $C_V = \frac{f}{2}R, \qquad C_P = C_V + R = \frac{f+2}{2}R$ Hence, $\gamma = \frac{C_P}{C_V} = \frac{f+2}{f}$

2. For monoatomic gas $A$

A monoatomic gas has only $3$ translational degrees of freedom. So, $f_A = 3$ Therefore, $\gamma_A = \frac{3+2}{3} = \frac{5}{3}$

3. For polyatomic gas $B$

Given degrees of freedom:

• $3$ translational
• $3$ rotational
• $1$ vibrational mode

Now, one vibrational mode contributes 2 degrees of freedom (one kinetic + one potential). So total degrees of freedom: $f_B = 3+3+2 = 8$ Thus, $\gamma_B = \frac{8+2}{8} = \frac{10}{8} = \frac{5}{4}$

4. Compute the ratio

$\frac{\gamma_A}{\gamma_B} = \frac{\frac{5}{3}}{\frac{5}{4}} = \frac{4}{3}$

Given, $\frac{\gamma_A}{\gamma_B} = 1 + \frac{1}{n}$ So, $1 + \frac{1}{n} = \frac{4}{3}$

Subtract $1$ from both sides: $\frac{1}{n} = \frac{4}{3} - 1 = \frac{1}{3}$ Hence, $n=3$

5. Comparison with stored answer

Derived answer is $3$, which matches the stored correct answer.