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

Step 1: Understanding the Concept:
Gas pressure is a macroscopic phenomenon resulting from countless microscopic collisions between molecules and the container walls.
The total pressure exerted is a function of two distinct factors:
1. The force of each individual collision, which is determined by the kinetic energy (speed and mass) of the molecule.
2. The frequency of collisions, which is determined by the number of molecules present in a given volume (number density).
Even if molecules have high kinetic energy, the pressure will be low if there are very few molecules. Similarly, even if there are many molecules, the pressure will be low if they move very slowly (low KE).
Thus, pressure is fundamentally a joint property of energy and density.
Step 2: Key Formula or Approach:
1. Fundamental Pressure Equation: \(P = \frac{1}{3} n m v_{rms}^2\).
2. Expressed via Kinetic Energy: Since \(E_k = \frac{1}{2} m v_{rms}^2\), we substitute to get \(P = \frac{2}{3} n E_k\).
3. Here \(n\) is number density (\(N/V\)).
Step 3: Detailed Explanation:
The derived equation \(P = \frac{2}{3} n E_k\) highlights that pressure is directly proportional to both the number density \(n\) and the average kinetic energy \(E_k\).
In the given problem, a specific value for kinetic energy is provided (\(3 \times 10^{-21}\) J).
However, this is only one part of the equation. Knowing the energy of one molecule does not tell us the pressure unless we know how many molecules are in the container.
If you keep the energy constant but double the number of molecules, the pressure doubles.
Therefore, pressure is dependent on both kinetic energy and number density.
Step 4: Final Answer:
Pressure depends on both KE and number density.