Question

The internal energy per molecule for a diatomic (non-rigid) gas is (where \(k_{B}\) is Boltzmann's constant and \(T\) is absolute temperature).

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

Hint:

Count translational, rotational and vibrational contributions using the law of equipartition of energy.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
By equipartition, each quadratic energy term gives \(\dfrac{1}{2}k_BT\) per molecule.

Step 2: Count the terms
A non-rigid diatomic molecule has 3 translational and 2 rotational degrees of freedom, which give 5 terms.
The vibrational mode stores both kinetic and potential energy, so it adds 2 more terms.

Step 3: Calculation
Total number of quadratic terms \(=3+2+2=7\).
\(U=7\times\dfrac{1}{2}k_BT=\dfrac{7}{2}k_BT\).

Final Answer:
The internal energy per molecule is \(\dfrac{7}{2}k_BT\). \[ \boxed{\text{(C) }\dfrac{7}{2}k_{B}T} \]