`5`n , n and `5`n moles of a monoatomic , diatomic and non-linear polyatomic gases (which do not react chemically with each other ) are mixed at room temperature . The equivalent degree of freedom for the mixture is :-

Options

A. `(25)/(7)`
B. `(48)/(11)`
C. `(52)/(11)`
D. `(50)/(11)`

Explanation

To find the equivalent degree of freedom for the mixture of gases, we can follow these steps:

### Step 1: Identify the degree of freedom for each type of gas
- **Monoatomic gas**: The degree of freedom (F1) is 3.
- **Diatomic gas**: The degree of freedom (F2) is 5.
- **Non-linear polyatomic gas**: The degree of freedom (F3) is 6.

### Step 2: Identify the number of moles for each type of gas
- **Monoatomic gas**: n1 = 5n moles
- **Diatomic gas**: n2 = n moles
- **Non-linear polyatomic gas**: n3 = 5n moles

### Step 3: Use the formula for equivalent degree of freedom
The formula for the equivalent degree of freedom (F) for the mixture is given by:

\[ F = \frac{F1 \cdot n1 + F2 \cdot n2 + F3 \cdot n3}{n1 + n2 + n3} \]

### Step 4: Substitute the values into the formula
Substituting the values we have:

\[ F = \frac{(3 \cdot 5n) + (5 \cdot n) + (6 \cdot 5n)}{5n + n + 5n} \]

### Step 5: Simplify the equation
Calculating the numerator:

\[ = \frac{(15n) + (5n) + (30n)}{11n} \]

Combine the terms in the numerator:

\[ = \frac{50n}{11n} \]

### Step 6: Final calculation
Now, simplifying further gives:

\[ F = \frac{50}{11} \]

### Conclusion
Thus, the equivalent degree of freedom for the mixture is:

\[ \frac{50}{11} \]