Question

For a gas molecule with 6 degrees of freedom, which one of the following relations between gas constant '$R$' and molar specific heat '$C_v$' is correct?

• A. $R = \frac{C_v}{3}$
• B. $R = \frac{5C_v}{4}$
• C. $R = \frac{C_v}{2}$
• D. $R = \frac{3C_v}{4}$

Hint:

Remember that each degree of freedom contributes exactly $\frac{1}{2}R$ to the total $C_v$ value of a gas. For 6 degrees of freedom, the total capacity accumulates to $6 \times \frac{1}{2}R = 3R$. Isolating $R$ instantly yields $\frac{C_v}{3}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a gas whose molecules possess exactly 6 degrees of freedom. We need to identify the correct formula relating the universal gas constant $R$ to its molar specific heat capacity at constant volume ($C_v$).

Step 2: Key Formula or Approach:
According to the Law of Equipartition of Energy, the molar specific heat at constant volume ($C_v$) of a gas with $f$ degrees of freedom is given by: $$C_v = \frac{f}{2}R$$

Step 3: Detailed Explanation:
Given that the number of degrees of freedom is $f = 6$, let's substitute this value into our kinetic theory formula: $$C_v = \frac{6}{2}R$$ $$C_v = 3R$$ Now, rearrange the terms to express the gas constant $R$ as the subject of the equation: $$R = \frac{C_v}{3}$$

Step 4: Final Answer:
The correct relation is $R = \frac{C_v}{3}$, which corresponds to option (A).