JEE Main 2020 · 8 January · morning · Single correct · +4 / −1

The plot that depicts the behavior of the mean free time t (time between two successive collisions) for the molecules of an ideal gas, as a function of temperature (T), qualitatively, is:
(Graphs are schematic and not drawn to scale)

  1. (a)
    Option (a)
  2. (b)
    Option (b)
  3. (c)
    Option (c)
  4. (d)
    Option (d)
Answer and solution

Answer (b)

Solution

Physics sir · 4 steps

Mean free time is distance over speed: the free path stays fixed while the speed grows as √T.

  1. Time between collisions. A molecule covers one mean free path λ between collisions at an average speed v̄. So t = λ / v̄.
  2. Free path. λ = 1/(√2 π d² n) depends only on the molecular diameter d and the number density n. For a fixed amount of gas in a fixed volume, n is constant, so λ does not change with T.
  3. Average speed. v̄ = √(8RT/(π M)), so v̄ ∝ √T. Hotter molecules move faster and reach their next collision sooner.

    t = λ/v̄ ∝ 1/√T

  4. So. A plot of t against 1/√T is a rising straight line, which is graph (b).

The idea to keep. For any time-between-events question in kinetic theory, write it as distance over speed. Speed always goes as √(T/M), then check what the distance depends on.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Kinetic Theory of Gases

Asked about 130 times in JEE Main since 2020. This question is from 2020.