Questions: Diffusion and Fick's Laws

Question 1

Fick's first law (J = −D ∂c/∂x) tells you that diffusion flux is proportional to which quantity?

• A. The absolute concentration c at a given point
• B. The concentration gradient ∂c/∂x
• C. The rate of change of flux over time
• D. The square root of the diffusion coefficient D

Answer: (B)

Flux (J) is driven by the concentration gradient — the spatial rate of change of concentration — not by absolute concentration. A region of uniformly high concentration has no net flux; a steep gradient drives strong flux. The negative sign indicates flow runs from high to low concentration.

Question 2

Fick's first law describes how a concentration profile changes over time in a non-steady-state diffusion system.

• T. True
• F. False

Answer: False

This describes Fick's second law: ∂c/∂t = D(∂²c/∂x²). Fick's first law applies to steady-state conditions where the flux is constant in time and space (∂c/∂t = 0). The two laws address different physical situations: first law for steady-state flux, second law for transient spreading.

Question 3

What physical meaning does the relation ⟨x²⟩ = 2Dt convey about diffusion?

Model answer: The mean squared displacement of a diffusing particle grows linearly with time t, so the characteristic distance a particle spreads scales as √(2Dt) — not linearly with time. This reflects the random-walk nature of diffusion: many small random steps accumulate, but the net displacement grows more slowly than if particles moved in a straight line.

This result, derived from solving the diffusion equation for a point source, is one of the most important in all of transport theory. It distinguishes diffusion (⟨x²⟩ ∝ t) from directed drift (⟨x⟩ ∝ t). In practice, measuring ⟨x²⟩ vs. t in dynamic light scattering gives D directly, from which the Stokes-Einstein equation yields particle size.