Question

If \( r_1 = \frac{c_p}{c_v} \) of a rigid diatomic gas and \( r_2 = \frac{c_p}{c_v} \) of a non-rigid diatomic gas, find \( r_1 \) and \( r_2 \).

Hint:

For diatomic gases, the value of \( r \) (the ratio \( \frac{c_p}{c_v} \)) depends on the degrees of freedom available to the molecules. A rigid diatomic gas has 5 degrees of freedom, while a non-rigid diatomic gas has 7 degrees of freedom.

Answer 1

Step 1: Definition of \( r_1 \) and \( r_2 \).
For a gas, the ratio \( r = \frac{c_p}{c_v} \),
where:
- \( c_p \) is the specific heat at constant pressure,
- \( c_v \) is the specific heat at constant volume.

Step 2: Value of \( r_1 \) for a rigid diatomic gas.
For a rigid diatomic gas, \( r_1 = \frac{c_p}{c_v} = \frac{7}{5} \). This is because the degrees of freedom for a rigid diatomic gas are 5 (3 translational and 2 rotational), and using the ideal gas law: \[ r_1 = 1 + \frac{2}{f} = 1 + \frac{2}{5} = \frac{7}{5} \]
Step 3: Value of \( r_2 \) for a non-rigid diatomic gas.
For a non-rigid diatomic gas, the number of degrees of freedom increases because the molecule can undergo vibration as well. The specific value of \( r_2 \) depends on the exact nature of the gas, but for most practical cases, it is approximately: \[ r_2 = \frac{9}{7} \] This is because for non-rigid diatomic gases, the number of degrees of freedom is 7 (3 translational, 2 rotational, and 2 vibrational), and the corresponding ratio \( r \) is: \[ r_2 = 1 + \frac{2}{7} = \frac{9}{7} \]