Fick's Law Diffusion Calculator

Example Data Table

Formula Used

Fick’s First Law (steady state, 1D):

J = -D\,\frac{dC}{dx} where J is flux, D diffusivity, and C concentration.

For a linear gradient across a slab of thickness L:

\frac{dC}{dx} \approx \frac{C_2 - C_1}{L} and \dot{N} = J\,A gives the molar transfer rate through area A.

Fick’s Second Law (transient):

\frac{\partial C}{\partial t} = D\,\frac{\partial^2 C}{\partial x^2}

A common diffusion time estimate uses mean-square displacement:

t \approx \frac{L^2}{2D} and a conservative scale t \approx \frac{L^2}{D}.

For a semi-infinite medium with fixed surface concentration C_s at x=0:

C(x,t) = C_s + (C_0 - C_s)\,\mathrm{erf}\!\left(\frac{x}{2\sqrt{Dt}}\right)

How to Use This Calculator

1. Select a calculation mode for steady or transient diffusion.
2. Enter D and choose units that match your data.
3. For first-law cases, enter C1, C2, and L.
4. For transfer rate, also enter area A.
5. For transient profile, provide x, t, C0, and Cs.
6. Press Calculate to show results above the form.
7. Use CSV or PDF buttons to export the results panel.

Diffusion Notes and Practical Context

1) Why diffusion modeling matters

Diffusion controls how species spread when there is a concentration difference. It appears in polymer curing, battery electrodes, semiconductor doping, corrosion barriers, and gas transport through membranes. Reliable diffusion estimates help you forecast equilibration time, set thickness targets, and interpret measured flux from experiments.

2) Steady transport with the first law

Fick’s first law assumes a stable gradient that does not change with time. In a flat layer, the calculator approximates the gradient as a linear drop from C1 to C2 over thickness L. It then returns flux J and, if requested, the total molar transfer rate Ṅ through area A.

3) Interpreting sign and direction

The negative sign in J = −D(dC/dx) means transport occurs from higher concentration toward lower concentration. If C2 < C1 and x increases from side 1 to side 2, the gradient is negative and flux becomes positive, indicating net transport in the +x direction.

4) Units that keep results consistent

Flux is reported in mol/(m²·s) after converting inputs to SI internally. Concentration conversions are common sources of error; for example, 1 mol/L equals 1000 mol/m³. Keeping D in m²/s and lengths in meters makes scaling clearer and simplifies comparisons across materials.

5) Typical diffusivity ranges

Diffusivity depends strongly on temperature and microstructure. Dense solids often fall near 10⁻¹³ to 10⁻¹⁰ m²/s, while liquids commonly range around 10⁻¹⁰ to 10⁻⁹ m²/s. Gases can be higher, sometimes near 10⁻⁵ m²/s. Use literature values when possible, then refine using measured flux.

6) Time scale estimates from the second law

Transient diffusion is frequently summarized by a characteristic time t ≈ L²/(2D). This estimate is rooted in mean-square displacement and is useful for early-stage planning. For tighter safety margins, t ≈ L²/D provides a more conservative scale, especially when boundary effects matter.

7) Concentration profiles and the error function

For a semi-infinite domain held at a fixed surface concentration, the profile follows an error-function form. The calculator evaluates η = x/(2√(Dt)) and estimates erf(η) to compute C(x,t). As time increases, the penetration depth grows approximately with √(Dt), broadening the transition region.

8) Assumptions and limitations

These models assume diffusion-dominated transport, constant D, and one-dimensional geometry. Reactions, convection, porosity changes, or finite boundaries can alter results. If your system shows strong temperature gradients or concentration-dependent diffusivity, treat outputs as baseline estimates and validate against experimental data.

FAQs

1) What is the difference between the first and second law modes?

The first law computes steady flux from a fixed gradient. The second law handles transient behavior, including time scale estimates and concentration profiles that change with time.

2) Why does the flux equation include a negative sign?

The negative sign enforces transport from higher concentration to lower concentration. It also encodes direction relative to your chosen x-axis, so consistent coordinate definition is important.

3) Can I use mol/L for concentration inputs?

Yes. The calculator converts mol/L to mol/m³ internally using 1 mol/L = 1000 mol/m³. This keeps flux outputs consistent with SI units.

4) What does “transfer rate” mean in the first-law section?

Transfer rate is total molar flow through a surface. It is computed as Ṅ = J·A, where A is the cross-sectional area available for diffusion.

5) How should I choose the characteristic length L for time estimates?

Use the distance over which concentration must significantly change, such as a coating thickness or diffusion path in a particle. For layered systems, choose the limiting thickness.

6) What does the semi-infinite profile assumption imply?

It assumes the domain is large enough that the far boundary does not influence the profile during the time of interest. If diffusion reaches the boundary, a finite-domain model is needed.

7) My outputs look too large or too small. What should I check first?

Check units and orders of magnitude: D units, concentration units, and thickness. A factor-of-1000 error often comes from mixing mol/L and mol/m³ or using cm instead of m.