The value of `gamma=C_(P)/C_(V)` for a gas is given by `gamma=1+2/f` where f is the number of degrees of freedom of a molecule of a gas
What is the ratio of `(gamma_("monoatomic"))/(gamma_("diatomic"))` ?

Options

A. `25/21`
B. `21/25`
C. `5/7`
D. `3/5`

Explanation

To find the ratio of \(\frac{\gamma_{\text{monoatomic}}}{\gamma_{\text{diatomic}}}\), we will first calculate the values of \(\gamma\) for both monoatomic and diatomic gases using the given formula \(\gamma = 1 + \frac{2}{f}\).

### Step 1: Determine the degrees of freedom for monoatomic and diatomic gases
- For a **monoatomic gas**, the degrees of freedom \(f\) is 3 (translational motion in x, y, and z directions).
- For a **diatomic gas**, the degrees of freedom \(f\) is 5 (3 translational + 2 rotational).

### Step 2: Calculate \(\gamma\) for monoatomic gas
Using the formula:
\[ \gamma_{\text{monoatomic}} = 1 + \frac{2}{f_{\text{monoatomic}}} \]
Substituting \(f_{\text{monoatomic}} = 3\):
\[ \gamma_{\text{monoatomic}} = 1 + \frac{2}{3} = 1 + 0.6667 = \frac{5}{3} \]

### Step 3: Calculate \(\gamma\) for diatomic gas
Using the formula:
\[ \gamma_{\text{diatomic}} = 1 + \frac{2}{f_{\text{diatomic}}} \]
Substituting \(f_{\text{diatomic}} = 5\):
\[ \gamma_{\text{diatomic}} = 1 + \frac{2}{5} = 1 + 0.4 = \frac{7}{5} \]

### Step 4: Calculate the ratio \(\frac{\gamma_{\text{monoatomic}}}{\gamma_{\text{diatomic}}}\)
Now we can find the ratio:
\[ \frac{\gamma_{\text{monoatomic}}}{\gamma_{\text{diatomic}}} = \frac{\frac{5}{3}}{\frac{7}{5}} = \frac{5}{3} \times \frac{5}{7} = \frac{25}{21} \]

### Final Answer
Thus, the ratio of \(\frac{\gamma_{\text{monoatomic}}}{\gamma_{\text{diatomic}}}\) is:
\[ \frac{25}{21} \]