Question

The value of specific heat at constant volume (\(C_V\)) for diatomic molecules is:

• A. \( \frac{5}{2}R \)
• B. \( \frac{5}{3}R \)
• C. \( \frac{7}{2}R \)
• D. \( \frac{3}{2}R \)

Hint:

Remember the degrees of freedom (f) and corresponding \(C_V\) values for different types of gases:

Monatomic: f=3 (trans only) \(\rightarrow\) \(C_V = \frac{3}{2}R\)
Diatomic: f=5 (3 trans + 2 rot) \(\rightarrow\) \(C_V = \frac{5}{2}R\)
Polyatomic (non-linear): f=6 (3 trans + 3 rot) \(\rightarrow\) \(C_V = 3R\)
(Assuming vibrational modes are frozen).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The molar specific heat at constant volume (\(C_V\)) of a gas is related to its internal energy, which in turn depends on the number of degrees of freedom of its molecules, according to the equipartition theorem. This question asks for the value of \(C_V\) for a diatomic gas.
Step 2: Key Formula or Approach:
According to the equipartition theorem, the average energy per molecule associated with each degree of freedom is \( \frac{1}{2}k_B T \), where \(k_B\) is the Boltzmann constant and T is the temperature.
The molar internal energy U is given by \( U = \frac{f}{2}N_A k_B T = \frac{f}{2}RT \), where 'f' is the number of degrees of freedom.
The molar specific heat at constant volume is defined as \( C_V = \left(\frac{\partial U}{\partial T}\right)_V \).
Step 3: Detailed Explanation:
For a diatomic molecule (like N\(_2\) or O\(_2\)) at moderate temperatures, we consider its translational and rotational degrees of freedom:


Translational degrees of freedom: 3 (motion along x, y, and z axes).
Rotational degrees of freedom: 2 (rotation about two axes perpendicular to the bond axis; rotation about the bond axis itself has a negligible moment of inertia).
Vibrational degrees of freedom are typically not excited at room temperature and are ignored unless stated otherwise.
Total degrees of freedom, \( f = 3 (\text{trans}) + 2 (\text{rot}) = 5 \).
Now, calculate the internal energy per mole:
\[ U = \frac{f}{2}RT = \frac{5}{2}RT \]
Finally, calculate \(C_V\):
\[ C_V = \frac{d}{dT}\left(\frac{5}{2}RT\right) = \frac{5}{2}R \]
Step 4: Final Answer:
The value of specific heat at constant volume (\(C_V\)) for diatomic molecules is \( \frac{5}{2}R \).