If the density of oxygen is 1.44 `kg//m^(3)` at pressure of `10^(5)N//m^(2)`, then the root-mean-square velocity of oxygen molecules in m/s will be

Options

A. 469
B. 456
C. 120
D. 270

Explanation

To find the root-mean-square (RMS) velocity of oxygen molecules, we can use the formula derived from the kinetic theory of gases. The formula for the root-mean-square velocity (Vrms) is given by:

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

where:
- \( P \) is the pressure of the gas,
- \( \rho \) is the density of the gas.

### Step-by-Step Solution:

**Step 1: Identify the given values.**
- Density of oxygen (\( \rho \)) = 1.44 kg/m³
- Pressure (\( P \)) = \( 10^5 \) N/m²

**Step 2: Substitute the values into the RMS velocity formula.**
Using the formula:

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

Substituting the values we have:

\[ V_{rms} = \sqrt{\frac{3 \times 10^5 \, \text{N/m}^2}{1.44 \, \text{kg/m}^3}} \]

**Step 3: Calculate the numerator.**
Calculate \( 3 \times 10^5 \):

\[ 3 \times 10^5 = 300000 \, \text{N/m}^2 \]

**Step 4: Divide by the density.**
Now, divide by the density:

\[ \frac{300000}{1.44} \]

Calculating this gives:

\[ \frac{300000}{1.44} \approx 208333.33 \]

**Step 5: Take the square root.**
Now, take the square root of the result:

\[ V_{rms} = \sqrt{208333.33} \]

Calculating the square root gives:

\[ V_{rms} \approx 456.25 \, \text{m/s} \]

**Step 6: Round the result.**
Rounding to the nearest whole number, we find:

\[ V_{rms} \approx 456 \, \text{m/s} \]

### Final Answer:
The root-mean-square velocity of oxygen molecules is approximately **456 m/s**.