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:
Determine the gas constant R in terms of the molar heat capacity C_v for a gas with a known number of degrees of freedom.

Step 2: Key Formula or Approach:
Each degree of freedom contributes (1/2)R to C_v. For f degrees of freedom, C_v = f × (R/2). Thus R = 2C_v/f.

Step 3: Detailed Explanation:
With 6 degrees of freedom, the total molar heat capacity builds up to C_v = 6 × (R/2) = 3R. Rearranging isolates R = C_v/3. This equipartition-based accumulation provides a straightforward pathway from degrees of freedom to the gas constant without referencing specific heat ratios.

Step 4: Final Answer:
The gas constant R equals C_v/3.