A particle is performing Brownian motion with a diffusion constant D, and has a
half-life of τ. The root mean square distance it will be displaced from its starting
position before decaying will scale as:
Dτ
D/τ
Correct answer: B) √(Dτ).
Concept of RMS displacement in Brownian motion
For a freely diffusing particle undergoing Brownian motion, the mean squared displacement (MSD) in one dimension is proportional to time:
〈x²〉 = 2Dt. The root mean square displacement is then
xrms = √〈x²〉 = √(2Dt), which shows that the typical displacement grows with the square root of time, not linearly.
Incorporating particle half-life τ
The half-life τ describes the typical time after which the particle has a 50% chance to have decayed, so the characteristic time available for diffusion is on the order of τ.
Substituting t ≈ τ into the Brownian motion formula gives xrms ≈ √(Dτ) (ignoring numerical factors such as 2), which gives the scaling requested in the question.
Evaluation of options
Option A: Dτ
D has units of length
