Two moles of helium are mixed with n moles of hydrogen. The root mean spure (rms) speed of the gas molecules in the mexture is `sqrt2` times the speed of sound in the mixture. Then value of n is

Options

A. 1
B. 3
C. 2
D. `3//2`

Answer

To solve the problem, we need to find the value of \( n \) when two moles of helium are mixed with \( n \) moles of hydrogen, given that the root mean square (rms) speed of the gas molecules in the mixture is \( \sqrt{2} \) times the speed of sound in the mixture.

### Step-by-Step Solution:

1. **Understand the relationship between rms speed and speed of sound:**
\[ V_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
\[ V_{\text{sound}} = \sqrt{\frac{\gamma RT}{M}} \]
Given that \( V_{\text{rms}} = \sqrt{2} \times V_{\text{sound}} \), we can write:
\[ \sqrt{\frac{3RT}{M}} = \sqrt{2} \times \sqrt{\frac{\gamma RT}{M}} \]

2. **Square both sides to eliminate the square roots:**
\[ \frac{3RT}{M} = 2 \times \frac{\gamma RT}{M} \]
Since \( RT \) and \( M \) are common on both sides, we can cancel them out (assuming \( RT \neq 0 \)):
\[ 3 = 2\gamma \]
Therefore, we have:
\[ \gamma = \frac{3}{2} \]

3. **Calculate \( \gamma \) for the mixture:**
The specific heat ratio \( \gamma \) is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
For a mixture of gases, it can be expressed as:
\[ \gamma = \frac{n_1 C_{p1} + n_2 C_{p2}}{n_1 C_{v1} + n_2 C_{v2}} \]
Where:
- For helium (\( n_1 = 2 \)):
- \( C_{p1} = \frac{5}{2}R \)
- \( C_{v1} = \frac{3}{2}R \)
- For hydrogen (\( n_2 = n \)):
- \( C_{p2} = \frac{7}{2}R \)
- \( C_{v2} = \frac{5}{2}R \)

4. **Substituting the values into the equation for \( \gamma \):**
\[ \gamma = \frac{2 \cdot \frac{5}{2}R + n \cdot \frac{7}{2}R}{2 \cdot \frac{3}{2}R + n \cdot \frac{5}{2}R} \]
Simplifying this gives:
\[ \gamma = \frac{5 + \frac{7}{2}n}{3 + \frac{5}{2}n} \]

5. **Set the equation equal to \( \frac{3}{2} \):**
\[ \frac{5 + \frac{7}{2}n}{3 + \frac{5}{2}n} = \frac{3}{2} \]

6. **Cross-multiply to solve for \( n \):**
\[ 2(5 + \frac{7}{2}n) = 3(3 + \frac{5}{2}n) \]
Expanding both sides:
\[ 10 + 7n = 9 + \frac{15}{2}n \]
Rearranging gives:
\[ 10 - 9 = \frac{15}{2}n - 7n \]
\[ 1 = \frac{15}{2}n - \frac{14}{2}n \]
\[ 1 = \frac{1}{2}n \]
Therefore:
\[ n = 2 \]

### Conclusion:
The value of \( n \) is \( 2 \).