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. `(sqrt(3))//5`
D. `(sqrt(6))//5`

Explanation

To find the ratio of the speed of sound in nitrogen gas to that in helium gas at 300K, we can use the formula for the speed of sound in an ideal gas:

\[ V = \sqrt{\frac{\gamma RT}{M}} \]

where:
- \( V \) = speed of sound
- \( \gamma \) = adiabatic constant (ratio of specific heats)
- \( R \) = ideal gas constant
- \( T \) = absolute temperature
- \( M \) = molecular mass of the gas

### Step 1: Calculate the speed of sound in nitrogen gas

For nitrogen gas (N₂):
- Nitrogen is a diatomic molecule, so \( \gamma = \frac{7}{5} \).
- The molecular mass \( M \) of nitrogen is 28 g/mol.
- The temperature \( T \) is 300 K.

Using the formula:

\[ V_{\text{N}_2} = \sqrt{\frac{\frac{7}{5}RT}{28}} \]

### Step 2: Calculate the speed of sound in helium gas

For helium gas (He):
- Helium is a monoatomic gas, so \( \gamma = \frac{5}{3} \).
- The molecular mass \( M \) of helium is 4 g/mol.
- The temperature \( T \) is 300 K.

Using the formula:

\[ V_{\text{He}} = \sqrt{\frac{\frac{5}{3}RT}{4}} \]

### Step 3: Find the ratio of the speeds

Now, we need to find the ratio of the speed of sound in nitrogen to that in helium:

\[ \frac{V_{\text{N}_2}}{V_{\text{He}}} = \frac{\sqrt{\frac{\frac{7}{5}RT}{28}}}{\sqrt{\frac{\frac{5}{3}RT}{4}}} \]

### Step 4: Simplify the ratio

This can be simplified as follows:

\[ \frac{V_{\text{N}_2}}{V_{\text{He}}} = \sqrt{\frac{\frac{7}{5}RT}{28}} \cdot \sqrt{\frac{4}{\frac{5}{3}RT}} = \sqrt{\frac{7}{5} \cdot \frac{4}{28} \cdot \frac{3}{5}} \]

### Step 5: Further simplification

Now simplify the expression inside the square root:

\[ = \sqrt{\frac{7 \cdot 4 \cdot 3}{5 \cdot 5 \cdot 28}} = \sqrt{\frac{84}{700}} = \sqrt{\frac{3}{25}} = \frac{\sqrt{3}}{5} \]

### Conclusion

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

\[ \frac{V_{\text{N}_2}}{V_{\text{He}}} = \frac{\sqrt{3}}{5} \]

### Final Answer

The ratio of the speed of sound in nitrogen gas to that in helium gas at 300K is \( \frac{\sqrt{3}}{5} \).

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