Question

A vessel of volume 10 liters is filled with \( H_2 \) gas. The total average translational kinetic energy of its molecules is \( 4.5 \times 10^5 \, \text{J} \). The pressure of hydrogen in the vessel is:

• A. \( 3 \times 10^6 \, \text{Nm}^{-2} \)
• B. \( 30 \times 10^6 \, \text{Nm}^{-2} \)
• C. \( 30 \times 10^4 \, \text{Nm}^{-2} \)
• D. \( 3 \times 10^4 \, \text{Nm}^{-2} \)

Hint:

When dealing with gases, use the ideal gas law and the relation between kinetic energy and temperature to calculate the pressure and other properties.

Answer 1

The Correct Option is B

We know that the average translational kinetic energy of molecules in an ideal gas is related to the pressure, volume, and temperature by the ideal gas law: \[ \frac{3}{2} n k_B T = \frac{1}{2} m v^2 \] For a monatomic gas, the average kinetic energy of a molecule is given by: \[ \text{K.E.} = \frac{3}{2} k_B T \] We can also use the equation for the pressure-volume relationship for gases: \[ PV = nRT \] Given the total energy \( E \) is \( 4.5 \times 10^5 \, \text{J} \) and the volume \( V = 10 \, \text{liters} = 10 \times 10^{-3} \, \text{m}^3 \), we can now calculate the pressure by substituting these values into the equation. The pressure \( P \) comes out to be \( 30 \times 10^6 \, \text{Nm}^{-2} \). Thus, the pressure in the vessel is \( \boxed{30 \times 10^6 \, \text{Nm}^{-2}} \).