Question

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 :

• A. 6
• B. 3
• C. 4
• D. 2

Answer 1

The Correct Option is C

To solve this problem, we need to determine the number of moles \(N\) of a polyatomic gas with degrees of freedom \(f = 6\) that must be mixed with two moles of a monoatomic gas (which has degrees of freedom \(f = 3\)) so that the mixture behaves like a diatomic gas (which has degrees of freedom \(f = 5\)). 

The degrees of freedom \(f\) for a mixture of gases can be determined using the formula:

\(f_{\text{mix}} = \frac{N_1 \cdot f_1 + N_2 \cdot f_2}{N_1 + N_2}\)

where:

• \(N_1\) and \(f_1\) are the number of moles and degrees of freedom of the first gas (polyatomic).
• \(N_2\) and \(f_2\) are the number of moles and degrees of freedom of the second gas (monoatomic).

We are given:

• Polyatomic gas: \(f_1 = 6\), \(N_1 = N\)
• Monoatomic gas: \(f_2 = 3\), \(N_2 = 2\)
• Desired \(f_{\text{mix}} = 5\)

Substituting these values into the formula:

\(5 = \frac{N \cdot 6 + 2 \cdot 3}{N + 2}\)

We solve for \(N\):

\(5(N + 2) = 6N + 6\)

Expanding both sides:

\(5N + 10 = 6N + 6\)

Rearranging the equation:

\(5N + 10 - 6 = 6N\)

\(5N + 4 = 6N\)

Simplify to solve for \(N\):

\(4 = 6N - 5N\)

\(N = 4\)

Thus, the value of \(N\) that satisfies this condition is 4. Therefore, the correct answer is 4.

Answer 2

The average degrees of freedom for the mixture, \( f_{\text{eq}} \), can be expressed as:  
\(f_{\text{eq}} = \frac{n_1 f_1 + n_2 f_2}{n_1 + n_2}\)
where:  
- \( n_1 = N \) (number of moles of polyatomic gas),  
- \( n_2 = 2 \) (number of moles of monoatomic gas),  
- \( f_1 = 6 \) (degrees of freedom for polyatomic gas),  
- \( f_2 = 3 \) (degrees of freedom for monoatomic gas),  
- \( f_{\text{eq}} = 5 \) (degrees of freedom for diatomic gas).  

Now, we substitute these values into the equation:  
\(5 = \frac{N \times 6 + 2 \times 3}{N + 2}\)

Simplify the equation:  
\(5 = \frac{6N + 6}{N + 2}\)

Multiply both sides by \( (N + 2) \):  
\(5(N + 2) = 6N + 6\)
\(5N + 10 = 6N + 6\)
\(10 - 6 = 6N - 5N\)
\(N = 4\)

Thus, the value of \( N \) is 4.

The Correct Answer is: 4