The speed of sound in oxygen `(O_2)` at a certain temperature is `460ms^-1`. The speed of sound in helium (He) at the same temperature will be (assume both gases to be ideal)

Options

A. `1421ms^-1`
B. `500ms^-1`
C. `650ms^-1`
D. `330ms^-1`

Explanation

To find the speed of sound in helium (He) given the speed of sound in oxygen (O₂) at the same temperature, we can use the formula for the speed of sound in an ideal gas:

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

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 temperature in Kelvin,
- \( M \) is the molar mass of the gas.

### Step-by-Step Solution:

1. **Identify the speed of sound in oxygen**:
- Given: Speed of sound in oxygen, \( V_{O_2} = 460 \, \text{m/s} \).

2. **Write the ratio of the speeds of sound in helium and oxygen**:
- Since both gases are at the same temperature, we can set up the ratio:
\[ \frac{V_{O_2}}{V_{He}} = \sqrt{\frac{\gamma_{O_2}}{\gamma_{He}} \cdot \frac{M_{He}}{M_{O_2}}} \]

3. **Determine the values of \( \gamma \) and molar masses**:
- For oxygen (O₂), which is a diatomic gas:
\[ \gamma_{O_2} = \frac{7}{5} = 1.4 \]
- For helium (He), which is a monatomic gas:
\[ \gamma_{He} = \frac{5}{3} \approx 1.67 \]
- Molar mass of helium, \( M_{He} = 4 \, \text{g/mol} \).
- Molar mass of oxygen, \( M_{O_2} = 32 \, \text{g/mol} \).

4. **Substitute the values into the ratio**:
\[ \frac{V_{O_2}}{V_{He}} = \sqrt{\frac{\frac{7}{5}}{\frac{5}{3}} \cdot \frac{4}{32}} \]

5. **Simplify the ratio**:
- Calculate \( \frac{7/5}{5/3} = \frac{7 \cdot 3}{5 \cdot 5} = \frac{21}{25} \).
- Now, calculate \( \frac{4}{32} = \frac{1}{8} \).
- Thus, the entire expression becomes:
\[ \frac{V_{O_2}}{V_{He}} = \sqrt{\frac{21}{25} \cdot \frac{1}{8}} = \sqrt{\frac{21}{200}} \approx 0.3237 \]

6. **Calculate the speed of sound in helium**:
- Rearranging gives:
\[ V_{He} = V_{O_2} \cdot \frac{25}{21} \cdot \sqrt{8} \]
- Since \( V_{O_2} = 460 \, \text{m/s} \):
\[ V_{He} = 460 \cdot \sqrt{0.3237} \approx 460 \cdot 1.421 \approx 1421 \, \text{m/s} \]

### Final Answer:
The speed of sound in helium at the same temperature is approximately \( 1421 \, \text{m/s} \).

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