An ideal gas has molecules with 5 degrees of freedom. The ratio of specific heats at constant pressure `(C_(p))` and at constant volume `(C_(v))` is

Options

A. `(7)/(5)`
B. 6
C. `(7)/(2)`
D. `(5)/(2)`

Answer

To solve the problem of finding the ratio of specific heats \( \frac{C_p}{C_v} \) for an ideal gas with molecules having 5 degrees of freedom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Degrees of Freedom**:
For a gas with \( f \) degrees of freedom, the specific heat at constant volume \( C_v \) can be calculated using the formula:
\[ C_v = \frac{f}{2} R \]
where \( R \) is the universal gas constant.

2. **Calculate \( C_v \)**:
Given that the gas has 5 degrees of freedom (\( f = 5 \)):
\[ C_v = \frac{5}{2} R \]

3. **Using Mayer's Relation**:
Mayer's relation states that:
\[ C_p - C_v = R \]
We can rearrange this to find \( C_p \):
\[ C_p = C_v + R \]

4. **Substituting \( C_v \) into Mayer's Relation**:
Substitute the expression for \( C_v \) into the equation for \( C_p \):
\[ C_p = \frac{5}{2} R + R \]
To combine the terms, express \( R \) as \( \frac{2}{2} R \):
\[ C_p = \frac{5}{2} R + \frac{2}{2} R = \frac{7}{2} R \]

5. **Finding the Ratio \( \frac{C_p}{C_v} \)**:
Now that we have both \( C_p \) and \( C_v \), we can find the ratio:
\[ \frac{C_p}{C_v} = \frac{\frac{7}{2} R}{\frac{5}{2} R} \]
The \( R \) and \( \frac{1}{2} \) cancel out:
\[ \frac{C_p}{C_v} = \frac{7}{5} \]

6. **Final Answer**:
Therefore, the ratio of specific heats at constant pressure and constant volume is:
\[ \frac{C_p}{C_v} = \frac{7}{5} \]