The ratio of the speed of sound in nitrogen gas to that in helium gas, at 300K is

Options

A. `(sqrt(2//7))`
B. `(sqrt(1//7))`
C. `(sqrt3)//5`
D. `(sqrt6)//5`

Explanation

To find the ratio of the speed of sound in nitrogen gas (N₂) to that in helium gas (He) at 300K, we can use the formula for the speed of sound in a gas:

\[ V = \sqrt{\frac{\gamma R T}{M}} \]

Where:
- \( V \) is the speed of sound,
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

Since we are looking for the ratio of the speeds, we can simplify the expression as follows:

1. **Write the ratio of the speeds**:
\[ \frac{V_{N_2}}{V_{He}} = \frac{\sqrt{\frac{\gamma_{N_2} R T}{M_{N_2}}}}{\sqrt{\frac{\gamma_{He} R T}{M_{He}}}} \]

2. **Simplify the ratio**:
\[ \frac{V_{N_2}}{V_{He}} = \sqrt{\frac{\gamma_{N_2}}{M_{N_2}} \cdot \frac{M_{He}}{\gamma_{He}}} \]

3. **Substitute the known values**:
- For nitrogen (\(N_2\)):
- \( \gamma_{N_2} = \frac{7}{5} \)
- \( M_{N_2} = 28 \, \text{g/mol} \)
- For helium (\(He\)):
- \( \gamma_{He} = \frac{5}{3} \)
- \( M_{He} = 4 \, \text{g/mol} \)

Now, substituting these values into the ratio:
\[ \frac{V_{N_2}}{V_{He}} = \sqrt{\frac{\frac{7}{5}}{28} \cdot \frac{4}{\frac{5}{3}}} \]

4. **Calculate the ratio**:
- First, calculate \( \frac{4}{\frac{5}{3}} = 4 \cdot \frac{3}{5} = \frac{12}{5} \).
- Now, substitute this back into the equation:
\[ \frac{V_{N_2}}{V_{He}} = \sqrt{\frac{\frac{7}{5}}{28} \cdot \frac{12}{5}} \]
- Simplifying further:
\[ = \sqrt{\frac{7 \cdot 12}{5 \cdot 5 \cdot 28}} = \sqrt{\frac{84}{700}} = \sqrt{\frac{3}{25}} = \frac{\sqrt{3}}{5} \]

5. **Final result**:
The ratio of the speed of sound in nitrogen gas to that in helium gas at 300K is:
\[ \frac{V_{N_2}}{V_{He}} = \frac{\sqrt{3}}{5} \]