Question

$\gamma_1$ is the ratio of the specific heat capacities of the rigid diatomic gas at constant pressure to that at constant volume and $\gamma_2$ is the corresponding value for a non-rigid diatomic gas molecule with an additional vibrational mode, then the ratio of $\gamma_1$ to $\gamma_2$ is

• A. 49 : 45
• B. 7 : 5
• C. 9 : 7
• D. 7 : 9
• E. 49 : 30

Hint:

Remember that each vibrational mode adds 2 to the degrees of freedom because it involves both potential and kinetic energy terms in the Hamiltonian.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The ratio of specific heats \(\gamma\) is defined as \(\frac{C_p}{C_v}\). It depends on the degrees of freedom (\(f\)) of the gas molecules.
Step 2: Key Formula or Approach:
Specific heat at constant volume: \(C_v = \frac{f}{2}R\).
Specific heat at constant pressure: \(C_p = C_v + R = \left(\frac{f}{2} + 1\right)R\).
Therefore, \(\gamma = 1 + \frac{2}{f}\).
Evaluate \(f\) for both a rigid diatomic gas and a non-rigid diatomic gas with one vibrational mode.
Step 3: Detailed Explanation:
Case 1: Rigid Diatomic Gas
It has 3 translational and 2 rotational degrees of freedom.
Total degrees of freedom, \(f_1 = 5\).
\(\gamma_1 = 1 + \frac{2}{5} = \frac{7}{5}\).
Case 2: Non-rigid Diatomic Gas (1 vibrational mode)
A vibrational mode contributes 2 degrees of freedom (one for kinetic energy and one for potential energy).
Total degrees of freedom, \(f_2 = 5 + 2 = 7\).
\(\gamma_2 = 1 + \frac{2}{7} = \frac{9}{7}\).
Calculate the ratio:
\[ \frac{\gamma_1}{\gamma_2} = \frac{7/5}{9/7} = \frac{7}{5} \times \frac{7}{9} = \frac{49}{45} \] Step 4: Final Answer:
The ratio is 49 : 45.