Question

If average kinetic energy per molecule is \( 3 \times 10^{-21} \) J, pressure of gas depends on:

• A. Only KE
• B. KE and number density
• C. Only number density
• D. None

Hint:

Remember the "two-factor rule" for pressure: Pressure is about how hard they hit (KE) and how often they hit (number density).

Answer 1

The Correct Option is B

Concept: According to the Kinetic Theory of Gases, pressure is result of the collisions of gas molecules with the walls of the container. The pressure exerted by an ideal gas can be expressed in terms of the average translational kinetic energy of its molecules.
• Pressure Formula: \( P = \frac{1}{3} \frac{Nm}{V} v_{rms}^2 \)
• Relation to Kinetic Energy: Since Average \( KE = \frac{1}{2} m v_{rms}^2 \), we can rewrite the pressure as: \[ P = \frac{2}{3} \left( \frac{N}{V} \right) \left( \frac{1}{2} m v_{rms}^2 \right) = \frac{2}{3} n E \]
• \( n = \frac{N}{V} \): Number density (number of molecules per unit volume).
• \( E \): Average kinetic energy per molecule.

Step 1: Analyzing the relationship between Pressure, KE, and Density.
The derived formula \( P = \frac{2}{3} n E \) shows that the pressure of a gas is directly proportional to both the number of molecules present in a unit volume (number density) and the average kinetic energy of those molecules.

Step 2: Interpreting the dependency.
Even if the average kinetic energy per molecule is a fixed value (as given in the question), the total pressure will still vary if you change the number of molecules in the container (the number density). For example, doubling the number of molecules in the same space will double the pressure, even if their individual energies remain constant.

Step 3: Conclusion.
Therefore, the pressure depends on both the Kinetic Energy (which determines the force of each impact) and the number density (which determines the frequency of impacts).