For a gas `gamma = 9//7`. What is the number of degrees of freedom of the molecules of this gas ?

Explanation

To find the number of degrees of freedom (F) of the molecules of a gas given that \(\gamma = \frac{9}{7}\), we can use the relationship between \(\gamma\), \(C_p\), and \(C_v\):

1. **Understanding the relationship**:
\[ \gamma = \frac{C_p}{C_v} \]
where \(C_p\) is the specific heat at constant pressure and \(C_v\) is the specific heat at constant volume.

2. **Expressing \(C_p\) and \(C_v\) in terms of degrees of freedom**:
The specific heats can be expressed in terms of degrees of freedom (F):
\[ C_p = \frac{F + 2}{2} R \]
\[ C_v = \frac{F}{2} R \]

3. **Substituting \(C_p\) and \(C_v\) into the equation for \(\gamma\)**:
Substitute the expressions for \(C_p\) and \(C_v\) into the equation for \(\gamma\):
\[ \gamma = \frac{C_p}{C_v} = \frac{\frac{F + 2}{2} R}{\frac{F}{2} R} \]

4. **Simplifying the equation**:
The \(R\) and \(\frac{1}{2}\) cancel out:
\[ \gamma = \frac{F + 2}{F} \]

5. **Setting up the equation**:
Given that \(\gamma = \frac{9}{7}\), we have:
\[ \frac{F + 2}{F} = \frac{9}{7} \]

6. **Cross-multiplying to eliminate the fraction**:
\[ 7(F + 2) = 9F \]

7. **Expanding and rearranging the equation**:
\[ 7F + 14 = 9F \]
\[ 14 = 9F - 7F \]
\[ 14 = 2F \]

8. **Solving for F**:
\[ F = \frac{14}{2} = 7 \]

Thus, the number of degrees of freedom of the molecules of this gas is \(F = 7\).