`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 solve the problem of finding the equivalent degree of freedom for the mixture of gases, we will follow these steps:

### Step 1: Identify the number of moles and degrees of freedom for each gas
- **Monoatomic gas**:
- Moles = \(5n\)
- Degrees of freedom = 3
- **Diatomic gas**:
- Moles = \(n\)
- Degrees of freedom = 5
- **Non-linear polyatomic gas**:
- Moles = \(5n\)
- Degrees of freedom = 6

### Step 2: Calculate the contribution of each gas to the total degrees of freedom
- Contribution from monoatomic gas:
\[ \text{Contribution} = \text{Moles} \times \text{Degrees of freedom} = 5n \times 3 = 15n \]

- Contribution from diatomic gas:
\[ \text{Contribution} = \text{Moles} \times \text{Degrees of freedom} = n \times 5 = 5n \]

- Contribution from non-linear polyatomic gas:
\[ \text{Contribution} = \text{Moles} \times \text{Degrees of freedom} = 5n \times 6 = 30n \]

### Step 3: Sum the contributions from all gases
\[ \text{Total contribution} = 15n + 5n + 30n = 50n \]

### Step 4: Calculate the total number of moles in the mixture
\[ \text{Total moles} = 5n + n + 5n = 11n \]

### Step 5: Calculate the equivalent degree of freedom for the mixture
Using the formula for equivalent degree of freedom:
\[ \text{Equivalent degree of freedom} = \frac{\text{Total contribution}}{\text{Total moles}} = \frac{50n}{11n} = \frac{50}{11} \]

### Final Answer
The equivalent degree of freedom for the mixture is \(\frac{50}{11}\).

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