Question

Which statements are correct about degrees of freedom?
(A) A molecule with n degrees of freedom has n² different ways of storing energy.
(B) Each degree of freedom is associated with (1/2)RT average energy per mole.
(C) A monatomic gas molecule has 1 rotational degree of freedom whereas diatomic molecule has 2 rotational degrees of freedom.
(D) CH₄ has a total of 6 degrees of freedom.

Choose the correct answer from the option given below:

• (B) and (C) only
• (B) and (D) only
• (A) and (B) only
• (C) and (D) only

Answer 1

The Correct Option is B

To determine which statements about degrees of freedom are correct, let's analyze each statement individually:

1. Statement (A): A molecule with n degrees of freedom has n² different ways of storing energy.
   • A molecule with n degrees of freedom can store energy independently in each of these degrees of freedom. However, the statement about n² ways is incorrect because the number of energy storage mechanisms isn't equal to n², it's simply n.
2. Statement (B): Each degree of freedom is associated with \(\left(\frac{1}{2}\right)RT\) average energy per mole.
   • This is correct based on the equipartition theorem which states that each degree of freedom contributes \(\left(\frac{1}{2}\right) kT\) to the energy of the molecule, or \(\left(\frac{1}{2}\right)RT\) per mole.
3. Statement (C): A monatomic gas molecule has 1 rotational degree of freedom whereas a diatomic molecule has 2 rotational degrees of freedom.
   • Monatomic gases do not have any rotational degrees of freedom because they are considered as point masses. Diatomic molecules have 2 rotational degrees of freedom (along axes perpendicular to the bond), so this statement is incorrect.
4. Statement (D): CH₄ has a total of 6 degrees of freedom.
   • CH₄ (methane) is a polyatomic molecule. For a nonlinear polyatomic molecule, we use 3 translational, 3 rotational, and the remaining as vibrational degrees of freedom. Specifically for small molecules like CH₄, it can efficiently have 6 degrees of freedom in practical terms under simple conditions, comprising 3 translational and 3 rotational.

Based on the above analysis:

• Statement (B) is correct as it accurately reflects the distribution of energy per degree of freedom.
• Statement (D) is also correct when considering smaller molecules with limited vibrational analysis required under basic conditions.

Therefore, the correct answer is:

(B) and (D) only