Question

If the ratio of specific heats of a gas at constant pressure and at constant volume is $\gamma$, then the number of degrees of freedom of the rigid molecules of the gas is

• A. $\dfrac{3\gamma - 1}{2\gamma - 1}$
• B. $\dfrac{2}{\gamma - 1}$
• C. $\dfrac{9}{2}(\gamma - 1)$
• D. $\dfrac{25}{2}(\gamma - 1)$

Hint:

Remember the formula \( \gamma = \frac{f + 2}{f} \) to relate degrees of freedom with the ratio of specific heats for an ideal gas.

Answer 1

The Correct Option is B

Step 1: Use the relation between degrees of freedom and $\gamma$ For an ideal gas, the ratio of specific heats is given by: \[ \gamma = \frac{C_P}{C_V} = \frac{f + 2}{f} \] where \( f \) is the number of degrees of freedom. 
Step 2: Solve for \( f \) Rearranging the equation: \[ \gamma = \frac{f + 2}{f} \Rightarrow \gamma f = f + 2 \Rightarrow \gamma f - f = 2 \Rightarrow f(\gamma - 1) = 2 \Rightarrow f = \frac{2}{\gamma - 1} \]