Diffusion Coefficient Calculator: Formula & Examples
The calculator for this page is switched off while it is checked. The formula is below, and the worked example in the article shows how to use it by hand.
This calculator computes the diffusion coefficient (D) using Fick’s first law, based on the diffusion flux and the concentration gradient you provide. Enter the flux (amount of substance per area per time) and the concentration difference over a given distance, and the tool returns the diffusion coefficient in your chosen units. It also gives the equivalent value in cm²/s, the standard unit in most textbooks and reference tables.
Formula
Fick’s first law relates the diffusion flux to the concentration gradient:
J = -D (dC/dx)
Rearranged for the diffusion coefficient:
D = J × Δx / ΔC
Where:
- J = diffusion flux, in mol/(m²·s) – the amount of substance passing through a unit area per second.
- ΔC = concentration difference across the distance, in mol/m³.
- Δx = the distance over which the concentration changes, in meters.
- D = diffusion coefficient, in m²/s (the calculator also shows cm²/s).
The negative sign in Fick’s law indicates that diffusion moves from high to low concentration; the calculator uses the absolute value.
Worked Example
Using the default values from the calculator:
- Flux (J) = 2.5 × 10�� mol/(m²·s)
- Concentration difference (ΔC) = 100 mol/m³
- Distance (Δx) = 0.01 m
Step 1: Multiply flux by distance:
2.5 × 10�� × 0.01 = 2.5 × 10��
Step 2: Divide by the concentration difference:
D = 2.5 × 10�� / 100 = 2.5 × 10�� m²/s
Step 3: Convert to cm²/s (multiply by 10�):
D = 2.5 × 10�� cm²/s
The calculator displays 2.5e-9 m²/s and 2.5e-5 cm²/s, matching the hand calculation.
When to Use This Calculator
This tool is for steady-state diffusion problems where the concentration gradient is constant (or can be approximated as linear). Typical applications include:
- Estimating diffusivity of a gas through a membrane or polymer film.
- Analyzing dopant diffusion in semiconductors (though those often require Fick’s second law).
- Calculating oxygen or nutrient transport in biological tissues.
- Designing barrier materials where diffusion rate matters.
If the concentration changes with time, you need Fick’s second law, which is a partial differential equation – not this simple formula.
How to Read the Result
The diffusion coefficient has units of area per time (m²/s or cm²/s). Larger values mean faster diffusion. For reference:
- Gases in air at room temperature: ~10�� m²/s
- Liquids: ~10�� m²/s
- Solids (e.g., atoms in metals): ~10�¹� m²/s or lower
If your result is far outside these ranges, double-check your input units. A common mistake is mixing cm and m in the distance and concentration gradient.
Common Mistakes and Pitfalls
- Unit mismatch: Flux in mol/(cm²·s) but distance in meters – always convert to consistent units first.
- Concentration units: Using mol/L instead of mol/m³. 1 mol/L = 1000 mol/m³.
- Sign error: Fick’s law has a negative sign; the coefficient itself is always positive.
- Steady-state assumption: This formula only works when the concentration profile is linear (or nearly so). For transient diffusion, use the error function solutions.
- Rounding too early: Keep extra digits during intermediate steps, as small inputs can lead to large relative errors.
Limits of the Formula
Fick’s first law assumes a constant diffusion coefficient (independent of concentration) and a homogeneous medium. In reality, D often depends on temperature, concentration, and the material’s structure. The Arrhenius equation describes the temperature dependence: D = Dâ‚€ exp(-Ea/RT), where Ea is the activation energy. This calculator gives an effective D for your specific conditions; it is not a universal constant.
Also, the formula assumes one-dimensional diffusion. In porous or anisotropic materials, the effective diffusivity may differ from the bulk value.
Frequently Asked Questions
What is the diffusion coefficient in simple terms?
It measures how quickly a substance spreads through another material. Higher D means faster spreading – for example, a drop of dye spreads faster in water (D ≈ 10�� m²/s) than in glass (D ≈ 10�¹� m²/s).
Why are there two different units (m²/s and cm²/s)?
Scientists often use cm²/s because typical values are small numbers (like 10��) that are easier to read in that unit. Engineers may prefer m²/s for consistency with other SI units. The calculator shows both so you can compare directly with literature values.
Can I use this for diffusion in gases?
Yes, if you have the flux and concentration gradient. For gases, you often know partial pressures instead of concentrations; you can convert using the ideal gas law (C = P/RT).
What if my concentration gradient is not linear?
Then Fick’s first law in this simple form doesn’t apply. You need to use the differential form and integrate, or use numerical methods. For many practical situations, a linear approximation is acceptable if the distance is small.
How does temperature affect the diffusion coefficient?
Generally, D increases with temperature. The Arrhenius equation D = D₀ exp(-Ea/RT) is commonly used. A rule of thumb is that D roughly doubles for every 10°C rise in many liquids, but this varies widely.
Is this calculator useful for drug diffusion in the body?
It can give a rough estimate for simple membranes, but biological tissues are complex, heterogeneous, and often involve active transport. For accurate pharmacokinetic modeling, more sophisticated models are needed.