The root mean square velocity of gas molecules of `0^(@)C` will be if at N.T.P. its density is 1.43 kg/`m^(3)`

Options

A. 461m/s
B. 164m/s
C. 461cm/s
D. 164cm/s

Explanation

To find the root mean square (RMS) velocity of gas molecules at 0°C and given the density at NTP (Normal Temperature and Pressure), we can use the formula for RMS velocity:

\[ v_{rms} = \sqrt{\frac{3P}{\rho}} \]

Where:
- \( P \) is the pressure in Pascals (Pa)
- \( \rho \) is the density in kg/m³

### Step-by-Step Solution:

1. **Identify the given values:**
- Density (\( \rho \)) = 1.43 kg/m³
- At NTP, the pressure (\( P \)) = 1.01 × 10⁵ Pa

2. **Substitute the values into the RMS velocity formula:**
\[ v_{rms} = \sqrt{\frac{3 \times P}{\rho}} = \sqrt{\frac{3 \times (1.01 \times 10^5)}{1.43}} \]

3. **Calculate the numerator:**
- First, calculate \( 3 \times 1.01 \times 10^5 \):
\[ 3 \times 1.01 \times 10^5 = 3.03 \times 10^5 \]

4. **Divide by the density:**
\[ \frac{3.03 \times 10^5}{1.43} \approx 2.12 \times 10^5 \]

5. **Take the square root:**
\[ v_{rms} = \sqrt{2.12 \times 10^5} \approx 460 \text{ m/s} \]

### Final Answer:
The root mean square velocity of the gas molecules at 0°C and a density of 1.43 kg/m³ at NTP is approximately **460 m/s**.