Question

What is the specific heat capacity at constant volume for non-rigid diatomic gas.

Hint:

Do not forget that one active vibrational mode always contributes a full $R$ (or two degrees of freedom) to the specific heat capacity because it involves both kinetic and potential energies!

Answer 1

Step 1: Understanding the Concept:
The molar specific heat capacity at constant volume, denoted as $C_v$, of an ideal gas depends fundamentally on its active degrees of freedom $f$.
The Law of Equipartition of Energy states that each active degree of freedom contributes exactly $\frac{1}{2}RT$ to the internal molar energy of the gas.
Step 2: Key Formula or Approach:
The internal energy for one mole of a gas is given by $U = \frac{f}{2}RT$.
The molar specific heat capacity at constant volume is then defined as the derivative of internal energy with respect to temperature: $C_v = \frac{dU}{dT} = \frac{f}{2}R$.
Step 3: Detailed Explanation:
A standard rigid diatomic gas possesses exactly 5 degrees of freedom (3 translational modes and 2 rotational modes).
However, for a non-rigid diatomic gas, typically considered at higher temperatures, an additional vibrational mode becomes active.
A single active vibrational mode inherently introduces two extra degrees of freedom (one for kinetic energy and one for potential energy).
Therefore, the total degrees of freedom for a non-rigid diatomic gas is $f = 3 + 2 + 2 = 7$.
Substituting $f = 7$ into the $C_v$ formula gives:
\[ C_v = \frac{7}{2}R \] Step 4: Final Answer:
The specific heat capacity at constant volume is $\frac{7}{2}R$.