Fick's Law Diffusion Calculator
Work out how fast a solute moves through a layer of fluid, from the diffusion coefficient, the concentrations and the distance.
What it does
Drop a spoonful of dye into a glass of water and nothing seems to happen. Leave it a while and the colour spreads through the whole glass, always moving from the crowded side towards the empty side. That spreading is diffusion, and it is described by a law Adolf Fick wrote in 1855, one of the few laws in science that people still quote almost word for word.
The chemistry text behind this page states the first law plainly: the flux equals minus the diffusion coefficient times the change in concentration over distance. The flux is the amount of material crossing an area in a unit time, the diffusion coefficient has units of area per time, and the concentration gradient is the change in concentration over the distance. The minus sign marks the direction of travel, from high concentration to low.
The second law adds the time: the rate of change of concentration equals the diffusion coefficient times the curvature of the concentration profile. In practice most working problems use the first law with a linear gradient, which is exactly the steady state: solute entering one face at the same rate it leaves the other, so the concentration in the middle stops changing.
This page runs that steady state calculation. Give it a diffusion coefficient, a concentration on each side of a layer, the thickness of the layer and the area you are interested in, and it returns the flux, the rate in moles per second, the amount that moves in a chosen time, and the characteristic time the layer needs to reach that steady profile.
The limits are worth keeping in mind. A linear gradient is an idealisation, so the answer is exact only when the layer is well stirred on both sides and nothing is happening inside it. The diffusion coefficient depends on the solute, the solvent and the temperature, and it changes with all three. And diffusion is slow: in water, molecular diffusion alone moves a salt only millimetres in an hour.
How to use
- Choose metric or imperial for the distance, the area, the concentrations and the answer.
- Enter the diffusion coefficient D for your solute in your solvent. It depends on the temperature, so use the value for the temperature you care about.
- Enter the concentration on the high side and the concentration on the low side. The tool needs the first to be larger than the second.
- Enter the distance across the layer, which is the thickness of the barrier the solute has to cross.
- Enter the area of the layer if you want the rate and the amount as well as the flux.
- Enter the time over which the amount should accumulate, then read the flux, the rate and the amount.
Formula
- J
- Amount of solute crossing one unit of area in one unit of time, in moles per square metre per second.
- D
- Diffusion coefficient of the solute in that solvent, in square metres per second.
- c1
- Concentration on the high side of the layer, in moles per cubic metre.
- c2
- Concentration on the low side of the layer, in moles per cubic metre.
- L
- Distance across the layer, in metres.
The molar rate is the flux times the area, and the amount transferred is that rate times the time. The characteristic diffusion time is L squared over D, which is the time scale on which the concentration profile settles. The book version of the law is written with a negative sign, J = -D dc over dx, because the concentration falls as the distance grows, and the minus sign turns that into a positive flux in the direction of travel.
Worked example
Example: a solute diffusing through 1 cm of water
Defaults: metric units, a diffusion coefficient of 5e-10 m2 per s, concentrations of 10 and 2 mol per m3, a layer 1 cm thick, an area of 0.01 m2 and a time of 3600 s.
- The concentration difference across the layer is 10 - 2 = 8 mol per m3.
- The gradient is that difference over the distance: 8 / 0.01 = 800 mol per m4.
- The flux is the diffusion coefficient times the gradient: 5e-10 x 800 = 4e-7 mol per m2 per s.
- Through an area of 0.01 m2 the rate is 4e-7 x 0.01 = 4e-9 mol per s.
- Over 3600 s the amount moved is 4e-9 x 3600 = 1.44e-5 mol.
- The characteristic time is the square of the distance over the diffusion coefficient: 0.01 squared / 5e-10 = 200000 s, which is 55.6 hours.
- The tool shows 4e-7 mol per m2 per s, 4e-9 mol per s, 1.44e-5 mol, 800 mol per m4 and 200000 s.
Reference table
| Concentration difference | Flux per m2 per s | Amount through 0.01 m2 in one hour |
|---|---|---|
| 2 mol per m3 | 1e-7 | 3.6e-6 mol |
| 4 mol per m3 | 2e-7 | 7.2e-6 mol |
| 6 mol per m3 | 3e-7 | 1.08e-5 mol |
| 8 mol per m3 | 4e-7 | 1.44e-5 mol |
| 10 mol per m3 | 5e-7 | 1.8e-5 mol |
Limits
- The calculation assumes a linear concentration profile, which is the steady state with well stirred conditions on both faces of the layer.
- The diffusion coefficient depends on the solute, the solvent and the temperature, and a value taken from a table may not match your conditions.
- It covers diffusion only. Convection, turbulence and electric or osmotic forces usually dominate the transport in real equipment.
- It says nothing about what happens after the solute arrives, so it cannot predict any concentration at the far side of a receiving chamber.
Common mistakes
- Forgetting that the concentration has to be in the same units as the distance. Mixing mol per litre with metres shifts the answer by a factor of 1000.
- Using the concentration difference instead of the gradient. The distance has to divide the difference before it meets the diffusion coefficient.
- Leaving out the area. The flux is per unit area, so the rate in moles per second needs the area multiplied in.
- Using the value of D at the wrong temperature. The coefficient falls by roughly half for every 10 degrees rise in water.
- Expecting diffusion alone to move things quickly. In water the characteristic time for a 1 cm layer is more than two days.
Try it yourself
Quick checks. Answers sit tucked under each question.
What is the flux when the concentration difference is doubled to 18 mol per m3?
9e-7 mol per m2 per s, which is 2.25 times the 4e-7 of the default case. The gradient is 1800 mol per m4 and the flux follows it directly, so 3.24e-5 mol passes through the 0.01 m2 area in an hour.
What happens when the layer is twice as thick?
The flux falls to 2e-7 mol per m2 per s, and the amount over an hour falls to 7.2e-6 mol. The characteristic time also goes from 200000 s to 800000 s, because that time goes with the square of the distance.
How much solute crosses the 0.01 m2 area in an hour in the default case?
1.44e-5 mol. The rate is 4e-9 mol per s, and there are 3600 s in an hour, so 4e-9 x 3600 = 1.44e-5 mol.
Questions
What does Fick's first law say?
That the flux is proportional to the concentration gradient, with the diffusion coefficient as the constant of proportionality. The chemistry text behind this page writes it as J = -D dc over dx, and the minus sign shows that material moves from high concentration to low.
What is a realistic diffusion coefficient?
About 5e-10 m2 per s for a small molecule in water at room temperature, and about 2e-5 m2 per s for a gas in air. Those two numbers differ by a factor of 40000, which is why gases spread quickly and dissolved salts do not.
Why does the flux fall as the layer gets thicker?
Because the same concentration difference is spread over a longer distance, so the gradient is smaller. Halving the gradient halves the flux, while the time the layer needs to settle goes with the square of the distance.
Is diffusion the same as convection?
No. Diffusion is the random movement of molecules and it always acts from high concentration to low. Convection is bulk movement of fluid, and it can move material in any direction, often much faster.
What is the characteristic time for?
It is the square of the distance over the diffusion coefficient, and it gives the rough time the concentration profile needs to reach its steady shape. If you wait much less than that, the profile inside the layer is still changing and the steady state answer is too high.
Does temperature matter?
Yes. The diffusion coefficient of a solute in water falls by roughly half for every 10 degree rise, and a higher temperature also speeds up the random motion that drives the whole process.
Sources
- Chemistry LibreTexts, chapter 9: diffusion, with Fick's first and second laws
- University of California Irvine chemistry notes: Fick's laws of diffusion and electrochemistry
- California Institute of Technology biology notes: diffusion and the Fick equation