A cylinder of capacity 20 litres is filled with H₂ gas. The total average kinetic energy of translatory motion of its molecules is 1.5x10⁵J . The pressure of hydrogen in the cylinder is

The correct option is 5 x 10⁶ N/m².

Step-by-step Derivation and Explanation:

To find the pressure of hydrogen in the cylinder, we can use the kinetic theory of gases. The total average translational kinetic energy of the molecules of an ideal gas is directly related to its pressure and volume.

1. Identify the given values and convert them to SI units:

Volume of the cylinder,
$V=20\text{ litres}$
Since $1\text{ litre}={10}^{-3}{\text{ m}}^3$, we have:
$V=20\times {10}^{-3}{\text{ m}}^3=2\times {10}^{-2}{\text{ m}}^3$

Total average translational kinetic energy of the molecules,
$E_T=1.5\times {10}^5\text{ J}$

2. Formula:

According to the kinetic theory of gases, the relationship between the total translational kinetic energy ($E_T$), pressure ($P$), and volume ($V$) of a gas is given by:
$E_T=\frac{3}{2}PV$

Rearranging the equation to solve for the pressure ($P$):
$P=\frac{2E_T}{3V}$

3. Calculation:

Substitute the given values into the rearranged formula:
$P=\frac{2\times (1.5\times {10}^5)}{3\times (2\times {10}^{-2})}$

Simplify the numerator and the denominator:
$P=\frac{3\times {10}^5}{6\times {10}^{-2}}$

Now perform the final division:
$P=0.5\times {10}^7{\text{ N/m}}^2$

Expressing this in scientific notation:
$P=5\times {10}^6{\text{ N/m}}^2$

Thus, the pressure of hydrogen in the cylinder is 5 x 10⁶ N/m².