If a gas has n degrees of freedom ratio of specific heats of gas is

Options

A. `(1+n)/(2)`
B. `1+(1)/(n)`
C. `1+(n)/(2)`
D. `1+(2)/(n)`

Explanation

To find the ratio of specific heats (γ) of a gas with n degrees of freedom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Specific Heats**:
- The specific heat at constant volume (Cv) and the specific heat at constant pressure (Cp) are related to the degrees of freedom of the gas.

2. **Expression for Cv**:
- For a gas with n degrees of freedom, the expression for Cv is given by:
\[ Cv = \frac{n}{2} R \]
where R is the universal gas constant.

3. **Relation between Cp and Cv**:
- The relationship between Cp and Cv is given by:
\[ Cp = Cv + R \]

4. **Substituting Cv into the Cp Equation**:
- Substitute the expression for Cv into the equation for Cp:
\[ Cp = \frac{n}{2} R + R \]
- Factor out R:
\[ Cp = R \left(\frac{n}{2} + 1\right) \]

5. **Finding the Ratio (γ)**:
- The ratio of specific heats (γ) is defined as:
\[ \gamma = \frac{Cp}{Cv} \]
- Substitute the expressions for Cp and Cv:
\[ \gamma = \frac{R \left(\frac{n}{2} + 1\right)}{\frac{n}{2} R} \]
- The R cancels out:
\[ \gamma = \frac{\frac{n}{2} + 1}{\frac{n}{2}} \]

6. **Simplifying the Ratio**:
- Rewrite the ratio:
\[ \gamma = \frac{\frac{n}{2} + 1}{\frac{n}{2}} = \frac{n + 2}{n} \]

7. **Final Expression**:
- Therefore, the ratio of specific heats (γ) for a gas with n degrees of freedom is:
\[ \gamma = 1 + \frac{2}{n} \]

### Summary:
The ratio of specific heats (γ) for a gas with n degrees of freedom is given by:
\[ \gamma = 1 + \frac{2}{n} \]