Let `gamma` denote the ratio of specific heat for an ideal gas. Then choose the correct expression for number of degrees of freedom of a molecule of the same gas.

Options

A. `25/2 (gamma - 1)`
B. `(3gamma - 1)/(2gamma - 1)`
C. `2/(gamma - 1)`
D. `9/2 (gamma - 1)`

Explanation

To find the correct expression for the number of degrees of freedom (f) of a molecule of an ideal gas in terms of the ratio of specific heats (γ), we can follow these steps:

### Step 1: Understand the relationship between specific heats and degrees of freedom
For an ideal gas, the specific heats at constant pressure (Cₚ) and constant volume (Cᵥ) are related to the degrees of freedom (f) of the gas molecules. The relationship is given by:
\[ Cₚ = Cᵥ + R \]
where R is the universal gas constant.

### Step 2: Express Cₚ and Cᵥ in terms of f
The specific heats can also be expressed in terms of the degrees of freedom:
\[ Cᵥ = \frac{f}{2} R \]
\[ Cₚ = \frac{f + 2}{2} R \]
This is because for a monatomic ideal gas, f = 3 (3 translational degrees of freedom), for a diatomic gas, f = 5 (3 translational + 2 rotational), and for polyatomic gases, f can be higher.

### Step 3: Define the ratio γ
The ratio of specific heats is defined as:
\[ \gamma = \frac{Cₚ}{Cᵥ} \]

### Step 4: Substitute the expressions for Cₚ and Cᵥ
Substituting the expressions for Cₚ and Cᵥ into the equation for γ gives:
\[ \gamma = \frac{\frac{f + 2}{2} R}{\frac{f}{2} R} \]

### Step 5: Simplify the expression
The R cancels out:
\[ \gamma = \frac{f + 2}{f} \]
Rearranging this equation allows us to express f in terms of γ:
\[ \gamma f = f + 2 \]
\[ \gamma f - f = 2 \]
\[ f(\gamma - 1) = 2 \]
\[ f = \frac{2}{\gamma - 1} \]

### Conclusion
Thus, the expression for the number of degrees of freedom (f) of a molecule of the ideal gas in terms of the ratio of specific heats (γ) is:
\[ f = \frac{2}{\gamma - 1} \]

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