If the speed of sound in helium be `960 ms^(-1)`, when will be the speed of sound in hydrogen at the same temperature ? The value of `gamma` for He and `H_2` are 1.67 and 1.40 respectively and the ratio of the molecular masses is 2:1.

Answer

To find the speed of sound in hydrogen when the speed of sound in helium is given, we can use the relationship between the speed of sound in different gases based on their properties. The formula for the speed of sound \( v \) in a gas is given by:

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

Where:
- \( \gamma \) 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.

### Step-by-step Solution:

1. **Identify Given Values**:
- Speed of sound in helium, \( v_{He} = 960 \, \text{m/s} \)
- \( \gamma_{He} = 1.67 \)
- \( \gamma_{H_2} = 1.40 \)
- Ratio of molar masses \( M_{He} : M_{H_2} = 2 : 1 \)

2. **Express the Speed of Sound for Helium**:
Using the formula for the speed of sound, we can express it for helium:
\[ v_{He} = \sqrt{\frac{\gamma_{He} R T}{M_{He}}} \]

3. **Express the Speed of Sound for Hydrogen**:
Similarly, for hydrogen:
\[ v_{H_2} = \sqrt{\frac{\gamma_{H_2} R T}{M_{H_2}}} \]

4. **Set Up the Ratio of Speeds**:
To find the relationship between the speeds of sound in helium and hydrogen, we can take the ratio:
\[ \frac{v_{He}}{v_{H_2}} = \sqrt{\frac{\gamma_{He} R T / M_{He}}{\gamma_{H_2} R T / M_{H_2}}} \]

This simplifies to:
\[ \frac{v_{He}}{v_{H_2}} = \sqrt{\frac{\gamma_{He} M_{H_2}}{\gamma_{H_2} M_{He}}} \]

5. **Substitute Known Values**:
Given \( M_{He} = 2 \) and \( M_{H_2} = 1 \):
\[ \frac{v_{He}}{v_{H_2}} = \sqrt{\frac{1.67 \times 1}{1.40 \times 2}} \]

6. **Calculate the Ratio**:
\[ \frac{v_{He}}{v_{H_2}} = \sqrt{\frac{1.67}{2.80}} = \sqrt{0.5964} \]

7. **Calculate the Speed of Sound in Hydrogen**:
Rearranging gives:
\[ v_{H_2} = v_{He} \cdot \sqrt{\frac{1.40 \times 2}{1.67}} = 960 \cdot \sqrt{0.5964} \]

Now, calculate \( v_{H_2} \):
\[ v_{H_2} \approx 960 \cdot 0.772 = 741.12 \, \text{m/s} \]

### Final Answer:
The speed of sound in hydrogen at the same temperature is approximately \( 741.12 \, \text{m/s} \).