The gas of capacity 20 ml is filled with hydrogen gas. The total average kinetic – energy of translatory motion of its molecules is `1.52 xx10^4`Joules. The pressure of hydrogen in the cylinder is:

Options

A. `2 xx10^5 N//m^2`
B. `3xx10^5 N//m^2`
C. `4 xx10^5 N//m^2`
D. `5 xx10^5 N//m^2`

Explanation

To find the pressure of hydrogen gas in the cylinder, we can use the relationship between the average kinetic energy of gas molecules and pressure. The formula for the average kinetic energy (KE) of translatory motion of gas molecules is given by:

\[ KE = \frac{3}{2} P V \]

where:
- \( KE \) is the total average kinetic energy,
- \( P \) is the pressure of the gas,
- \( V \) is the volume of the gas.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Total average kinetic energy, \( KE = 1.52 \times 10^4 \) Joules
- Volume of the gas, \( V = 20 \) mL = \( 20 \times 10^{-3} \) L = \( 20 \times 10^{-6} \) m³ (since \( 1 \) mL = \( 10^{-6} \) m³)

2. **Substitute the Values into the Formula**:
\[ 1.52 \times 10^4 = \frac{3}{2} P (20 \times 10^{-6}) \]

3. **Rearranging the Equation to Solve for Pressure \( P \)**:
- Multiply both sides by \( \frac{2}{3} \):
\[ P = \frac{2 \times 1.52 \times 10^4}{3 \times (20 \times 10^{-6})} \]

4. **Calculating the Pressure**:
- First, calculate the denominator:
\[ 3 \times (20 \times 10^{-6}) = 60 \times 10^{-6} = 6 \times 10^{-5} \text{ m}^3 \]
- Now substitute this back into the equation for \( P \):
\[ P = \frac{2 \times 1.52 \times 10^4}{6 \times 10^{-5}} \]
- Calculate the numerator:
\[ 2 \times 1.52 \times 10^4 = 3.04 \times 10^4 \]
- Now divide:
\[ P = \frac{3.04 \times 10^4}{6 \times 10^{-5}} = \frac{3.04}{6} \times 10^{4 + 5} = 0.5067 \times 10^9 \text{ Pa} \]
- Converting to a more standard form:
\[ P \approx 5.067 \times 10^5 \text{ Pa} = 5 \times 10^5 \text{ N/m}^2 \]

5. **Final Answer**:
The pressure of hydrogen in the cylinder is approximately \( 5 \times 10^5 \) N/m².