For diffusion in a continuum with concentration‑dependent or constant diffusion coefficient D under isothermal conditions and negligible bulk motion, the time evolution of the local concentration c_i(r,t) obeys the diffusion equation derived from mass conservation and Fick’s first law: ∂c_i/∂t = ∇·(D·∇c_i). For constant D this reduces to ∂c_i/∂t = D·∇^2 c_i. The law describes unsteady (transient) diffusion and provides the relationship between concentration gradients and their temporal relaxation.

Principle

Changes in local concentration arise from the divergence of the diffusive flux: gradients smooth out over time at a rate set by the diffusion coefficient, producing characteristic diffusive time scales that scale like L^2/D for a length scale L. The second law therefore links spatial gradients to temporal evolution under conservation of mass.

Demonstration

Situation: A semi‑infinite quiescent solvent initially has concentration c(x,0) = c_0 for x>0 and a boundary at x=0 suddenly held at concentration c_s for t>0. Recognition: no convection, isothermal conditions, known D. Action: solve the diffusion equation (constant D) to obtain c(x,t) = c_s + (c_0 - c_s)·erf(x/(2√(D t))) (illustrative form). Consequence: The transient concentration profiles and flux at the boundary can be predicted and used to determine uptake as a function of time and to size transient absorbers or predict contaminant spreading.

Misapplication

Using the simple diffusion equation when diffusion coefficient varies strongly with concentration or when reversible/irreversible reactions, adsorption, moving boundaries, or convective transport are important. Another error is applying the constant‑D solution to multicomponent systems with significant cross‑diffusion or to subdiffusive/anomalous regimes where transport does not follow Fickian scaling.

Consequence

Correct use gives time‑dependent concentration fields, transient uptake rates, and characteristic time scales essential for batch reactor charging, pollutant dispersion estimates, and transient mass‑transfer design. Misapplication yields incorrect transient predictions, leading to mis‑sized equipment, underpredicted response times, or faulty environmental dispersion assessments.

Reversal

The diffusion equation form changes when D depends on concentration (nonlinear diffusion), when chemical reactions introduce source/sink terms (reaction‑diffusion equations), when advection adds convective transport (advection‑diffusion), or when anomalous transport (sub‑ or superdiffusion) applies; in those cases the simple linear second law is insufficient.

Boundary

Clearly within: isothermal, diffusion‑dominated transient problems for a single species in a continuum with constant or weakly varying D and negligible bulk flow. Boundary case: concentration‑dependent D requiring nonlinear solutions or effective diffusivities. Clearly outside: systems with strong advection, reactive sinks/sources, moving phase boundaries (stephan problems), nanoscale domains with nonlocal transport or anomalous diffusion.

Semantic Tension

Fick’s second law is in tension with advection‑diffusion and reaction‑diffusion models: while Fick’s second law captures purely diffusive relaxation, real systems often require added convective or reactive terms; choosing the simpler form neglects mechanisms that can dominate transient behavior.

Synthesis

Fick’s second law is the conservation statement that converts Fick’s steady constitutive relation into a time‑dependent partial differential equation; it predicts how concentration gradients relax and sets diffusive time‑scales, but in practice it must be extended (nonlinear D, sources/sinks, advection) to handle many realistic transient transport problems.