N moles of a polyatomic gas (f = 6) must be mixed with two moles of a monoatomic gas so that the mixture behaves as a diatomic gas. The value of N is :

Options

A. 6
B. 3
C. 4
D. 2

Answer

To solve the problem, we need to find the number of moles \( N \) of a polyatomic gas (with degree of freedom \( f = 6 \)) that must be mixed with 2 moles of a monoatomic gas (with degree of freedom \( f = 3 \)) so that the mixture behaves as a diatomic gas (with degree of freedom \( f = 5 \)).

### Step-by-step Solution:

1. **Identify Degrees of Freedom**:
- For the polyatomic gas, \( f_{polyatomic} = 6 \).
- For the monoatomic gas, \( f_{monoatomic} = 3 \).
- For the diatomic gas, \( f_{diatomic} = 5 \).

2. **Set Up the Equation**:
The total degrees of freedom for the mixture can be expressed as:
\[ f_{mixture} = \frac{N \cdot f_{polyatomic} + n_{monoatomic} \cdot f_{monoatomic}}{N + n_{monoatomic}} \]
Here, \( n_{monoatomic} = 2 \) (the number of moles of the monoatomic gas).

3. **Substituting Values**:
Substitute the known values into the equation:
\[ 5 = \frac{N \cdot 6 + 2 \cdot 3}{N + 2} \]

4. **Simplifying the Equation**:
Multiply both sides by \( (N + 2) \):
\[ 5(N + 2) = 6N + 6 \]
Expanding gives:
\[ 5N + 10 = 6N + 6 \]

5. **Rearranging the Equation**:
Rearranging the equation to isolate \( N \):
\[ 10 - 6 = 6N - 5N \]
This simplifies to:
\[ 4 = N \]

6. **Conclusion**:
Thus, the value of \( N \) is \( 4 \).

### Final Answer:
The value of \( N \) is \( 4 \).

---