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 a wave in a monatomic gas given the pressure and density, we can use the formula for the speed of sound in a gas, which is given by:

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

where:
- \( v \) is the velocity of the wave (speed of sound),
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( P \) is the pressure of the gas,
- \( \rho \) is the density of the gas.

### Step 1: Identify the values
- Given pressure \( P = 1 \, \text{kPa} = 1000 \, \text{Pa} \)
- Given density \( \rho = 2.6 \, \text{kg/m}^3 \)
- For a monatomic gas, \( \gamma = \frac{5}{3} \)

### Step 2: Substitute the values into the formula
Substituting the known values into the formula:

\[ v = \sqrt{\frac{\frac{5}{3} \times 1000 \, \text{Pa}}{2.6 \, \text{kg/m}^3}} \]

### Step 3: Calculate the numerator
Calculate the numerator:

\[ \frac{5}{3} \times 1000 = \frac{5000}{3} \approx 1666.67 \, \text{Pa} \]

### Step 4: Calculate the fraction
Now, calculate the fraction:

\[ \frac{1666.67}{2.6} \approx 641.03 \]

### Step 5: Take the square root
Now, take the square root to find the velocity:

\[ v = \sqrt{641.03} \approx 25.31 \, \text{m/s} \]

### Final Answer
The velocity of the wave in the monatomic gas is approximately \( 25.31 \, \text{m/s} \).

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