Fick’s first law of diffusion and steady-state flux calculator

Under steady-state diffusion, the solute concentration at any given coordinate does not vary with time (∂c/∂t = 0). Across a flat membrane or barrier of thickness Δx with constant boundary concentrations, the concentration gradient becomes linear (dc/dx = const). Consequently, the diffusion flux J remains uniform across the entire thickness.

Diffusion parameters

Left concentration (c₁) 10.0 mol/m³

Right concentration (c₂) 2.0 mol/m³

Diffusion distance (Δx) 2.0 mm

Diffusion coefficient (D) 1.5 × 10⁻⁹ m²/s

Concentration gradient (dc/dx) -4000.0 mol/m⁴

Steady-state diffusion flux (J) 6.00 × 10⁻⁶ mol/(m²·s)

Flux direction → Left to right

Steady-state concentration profile

Membrane cross-section and particle flux

Mathematical formulation and physical principles

Fick’s first law describes the rate of mass transport of solute species through random thermal motion:

J = -D dcdx

The minus sign indicates that diffusion occurs down the chemical gradient—from regions of higher concentration to regions of lower concentration.

Steady state: According to Fick’s second law, ∂c/∂t = D (∂²c/∂x²). At steady state, ∂c/∂t = 0, which implies that ∂²c/∂x² = 0. Integrating across the barrier results in a constant gradient (dc/dx = Δc/Δx).
Independence of coordinate: Because the slope is identical everywhere across the thickness, the flux J entering the left face precisely equals the flux leaving the right face, preventing any accumulation or depletion of solute within the medium.