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)`

Answer

To find the ratio of the speed of sound in nitrogen gas (N₂) to that in helium gas (He) at a temperature of 300 K, 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 in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 1: Identify the values for nitrogen and helium

1. For nitrogen (N₂):
- Molar mass \( M_{N_2} = 28 \, \text{g/mol} = 0.028 \, \text{kg/mol} \)
- Adiabatic index \( \gamma_{N_2} = \frac{7}{5} = 1.4 \)

2. For helium (He):
- Molar mass \( M_{He} = 4 \, \text{g/mol} = 0.004 \, \text{kg/mol} \)
- Adiabatic index \( \gamma_{He} = \frac{5}{3} \approx 1.67 \)

### Step 2: Write the expressions for the speed of sound in both gases

\[ V_{N_2} = \sqrt{\frac{\gamma_{N_2} R T}{M_{N_2}}} = \sqrt{\frac{\frac{7}{5} R T}{0.028}} \]

\[ V_{He} = \sqrt{\frac{\gamma_{He} R T}{M_{He}}} = \sqrt{\frac{\frac{5}{3} R T}{0.004}} \]

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

Now, we can find the ratio \( \frac{V_{N_2}}{V_{He}} \):

\[ \frac{V_{N_2}}{V_{He}} = \frac{\sqrt{\frac{\frac{7}{5} R T}{0.028}}}{\sqrt{\frac{\frac{5}{3} R T}{0.004}}} \]

This simplifies to:

\[ \frac{V_{N_2}}{V_{He}} = \sqrt{\frac{\frac{7}{5} R T}{0.028}} \cdot \sqrt{\frac{0.004}{\frac{5}{3} R T}} = \sqrt{\frac{\frac{7}{5} \cdot 0.004}{0.028 \cdot \frac{5}{3}}} \]

### Step 4: Simplify the expression

Now, we can simplify the expression further:

\[ = \sqrt{\frac{7 \cdot 0.004 \cdot 3}{5 \cdot 0.028 \cdot 5}} = \sqrt{\frac{7 \cdot 0.012}{0.7}} = \sqrt{\frac{0.084}{0.7}} = \sqrt{\frac{12}{100}} = \sqrt{0.12} \]

### Step 5: Calculate the final value

Calculating \( \sqrt{0.12} \):

\[ \sqrt{0.12} \approx 0.346 \]

### Conclusion

Thus, the ratio of the speed of sound in nitrogen gas to that in helium gas at 300 K is approximately \( 0.346 \).