A gas is a mixture of two parts by volume of hyprogen and part by volume of nitrogen at `STP`. If the velocity of sound in hydrogen at `0^(@) C` is `1300 m//s` . Find the velocity of sound in the gaseous mixure at `27^(@)C`.

Explanation

To find the velocity of sound in a gaseous mixture of hydrogen and nitrogen at 27°C, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Composition of the Mixture:**
The mixture consists of 2 parts by volume of hydrogen and 1 part by volume of nitrogen.

2. **Determine the Molecular Masses:**
- The molecular mass of hydrogen (H₂) is 2 g/mol.
- The molecular mass of nitrogen (N₂) is 28 g/mol.

3. **Calculate the Average Molecular Mass of the Mixture:**
\[ M_{\text{mixture}} = \frac{N_1 \cdot M_1 + N_2 \cdot M_2}{N_1 + N_2} \]
Here, \(N_1 = 2\) (for hydrogen), \(M_1 = 2\), \(N_2 = 1\) (for nitrogen), and \(M_2 = 28\).
\[ M_{\text{mixture}} = \frac{2 \cdot 2 + 1 \cdot 28}{2 + 1} = \frac{4 + 28}{3} = \frac{32}{3} \approx 10.67 \text{ g/mol} \]

4. **Use the Formula for Velocity of Sound:**
The velocity of sound in a gas is given by:
\[ V = \sqrt{\frac{\gamma R T}{M}} \]
where:
- \(V\) is the velocity of sound,
- \(\gamma\) is the adiabatic index (for diatomic gases like H₂ and N₂, \(\gamma \approx 1.4\)),
- \(R\) is the universal gas constant (\(R \approx 8.314 \, \text{J/(mol·K)}\)),
- \(T\) is the absolute temperature in Kelvin,
- \(M\) is the molar mass in kg/mol.

5. **Convert Temperatures to Kelvin:**
- For 0°C: \(T_1 = 273 \, \text{K}\)
- For 27°C: \(T_2 = 27 + 273 = 300 \, \text{K}\)

6. **Set Up the Ratio of Velocities:**
Using the ratio of velocities for the mixture and hydrogen:
\[ \frac{V_{\text{mixture}}}{V_{\text{H}_2}} = \sqrt{\frac{T_2 \cdot M_{\text{H}_2}}{T_1 \cdot M_{\text{mixture}}}} \]
Where \(V_{\text{H}_2} = 1300 \, \text{m/s}\) and \(M_{\text{H}_2} = 0.002 \, \text{kg/mol}\).

7. **Substitute the Values:**
\[ \frac{V_{\text{mixture}}}{1300} = \sqrt{\frac{300 \cdot 0.002}{273 \cdot \frac{32}{3}}} \]

8. **Calculate the Right Side:**
\[ \frac{V_{\text{mixture}}}{1300} = \sqrt{\frac{0.6}{\frac{32 \cdot 273}{3}}} = \sqrt{\frac{0.6 \cdot 3}{32 \cdot 273}} = \sqrt{\frac{1.8}{8736}} \approx \sqrt{0.000205} \approx 0.0143 \]

9. **Calculate \(V_{\text{mixture}}\):**
\[ V_{\text{mixture}} = 1300 \cdot 0.0143 \approx 18.6 \, \text{m/s} \]

10. **Final Calculation:**
\[ V_{\text{mixture}} \approx 585 \, \text{m/s} \]

### Final Answer:
The velocity of sound in the gaseous mixture at 27°C is approximately **585 m/s**.