What is the velocity of wave in monatomic gas having pressure 1 kilo pascal and density 2.6 kg/`m^(3)`

Options

A. `8.9 xx 10^(3) m//s `
B. `3.6 m//s `
C. Zero
D. None of these .

Explanation

To find the velocity of sound in a monatomic gas, we can use the formula:

\[ v = \sqrt{\frac{\gamma P}{\rho}} \]

Where:
- \( v \) is the velocity of sound,
- \( \gamma \) is the ratio of specific heats (for a monatomic gas, \( \gamma = \frac{5}{3} \approx 1.66 \)),
- \( P \) is the pressure of the gas,
- \( \rho \) is the density of the gas.

### Step-by-step Solution:

1. **Identify the given values:**
- Pressure, \( P = 1 \text{ kPa} = 1000 \text{ Pa} \) (since 1 kPa = 1000 Pa)
- Density, \( \rho = 2.6 \text{ kg/m}^3 \)
- Ratio of specific heats for a monatomic gas, \( \gamma = 1.66 \)

2. **Substitute the values into the formula:**
\[ v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{1.66 \times 1000 \text{ Pa}}{2.6 \text{ kg/m}^3}} \]

3. **Calculate the numerator:**
\[ \gamma P = 1.66 \times 1000 = 1660 \text{ Pa} \]

4. **Calculate the fraction:**
\[ \frac{\gamma P}{\rho} = \frac{1660 \text{ Pa}}{2.6 \text{ kg/m}^3} \approx 638.46 \text{ m}^2/\text{s}^2 \]

5. **Take the square root to find the velocity:**
\[ v = \sqrt{638.46} \approx 25.24 \text{ m/s} \]

### Final Answer:
The velocity of the wave in the monatomic gas is approximately \( 25.24 \text{ m/s} \).

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