Diffusion Rate Calculator
Diffusion rate answers a simple question: how fast does a substance spread from where there is more of it to where there is less? Chemists and physicists answer it three different ways, and most ranking calculators each pick only one. This calculator does all three: Graham’s law for comparing gas diffusion rates from molar masses, Fick’s first law for the total rate through an area from a concentration gradient, and the observed rate from a measured mass, area, and time.
Pick the tab that matches the data you have. Every mode shows the formula it uses, and the Graham’s law mode can also run backwards to find an unknown molar mass from two measured rates.
R2 = R1 / sqrt(M2 / M1). Reverse: M2 = M1 x (R1 / R2)^2
Quick Answer
Key takeaways
- There is no single diffusion rate formula. Graham's law compares two gases, Fick's law predicts a flow from a gradient, and the observed rate reports what a measurement actually did. Picking the wrong one is the most common source of nonsense answers.
- Light gases move fast: the rate ratio is the inverse square root of the mass ratio, so a gas 8 times lighter diffuses only about 2.83 times faster, not 8 times.
- Fick's law rewards thin barriers and steep gradients: halving the distance doubles the rate, and so does doubling the concentration difference or the area.
- Units must be consistent inside any one calculation, and the Fick unit selector must match the values you entered. In centimeter mode the tool converts every input to SI before calculating, and the rate is always reported in mol/s.
- Quick Answer
- How to use the diffusion rate calculator
- The three formulas, and when each one applies
- Worked examples, verified against the tool
- What actually changes a diffusion rate
- Common mistakes to avoid
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
- References
How to use the diffusion rate calculator
Graham's law tab answers "which gas diffuses faster, and by how much?" Enter the known rate and molar mass of gas 1 plus the molar mass of gas 2. Switch "Solve for" to run it backwards and find an unknown molar mass from two measured rates.
Fick's law tab answers "how much substance crosses this barrier per second?" First pick the unit system that matches your values, centimeter based or SI, then enter the diffusion coefficient D, the area A, the concentration difference ΔC, and the distance Δx in that system. In centimeter mode the tool converts each input to SI before calculating: D in cm2/s and A in cm2 are multiplied by 0.0001, ΔC in mol/cm3 is multiplied by 1,000,000, and Δx in cm is multiplied by 0.01. The result working shows the converted SI values it used, and the rate is reported in mol/s either way.
Observed rate tab answers "what rate did my experiment actually produce?" Enter the mass that crossed, the area it crossed, and the time it took. This is the form membrane and permeability worksheets usually ask for.
If your problem gives you concentrations rather than ready-made differences, work out ΔC first as the high-side concentration minus the low-side concentration. Our concentration calculator helps when the concentrations themselves still need to be computed from masses and volumes, and the chemical dilution calculator covers the common lab step of preparing the lower concentration from a stock solution.
The three formulas, and when each one applies
Fick's first law (total rate): Rate = (D × A × ΔC) ÷ Δx
Observed rate: Rate = mass ÷ (area × time)
Graham's law says the diffusion rate of a gas is inversely proportional to the square root of its molar mass, at the same temperature and pressure. It compares two gases and needs no other information, which is why it dominates textbook problems: given one gas as a reference, the other's rate follows from the masses alone. The same relationship can be written with densities instead of molar masses, since molar mass is proportional to gas density at fixed conditions; if a problem hands you densities, our density calculator covers that side of the relationship.
Fick's first law describes steady flow through a barrier. Textbooks often write it as a flux, J = −D × (ΔC ÷ Δx), in moles per area per second. This calculator multiplies the flux by the area to give the total rate in mol/s, which is the quantity most lab and engineering questions actually ask for. The negative sign in the flux form only records direction, from high concentration toward low; this calculator reports the magnitude, and ΔC is entered as the positive difference between the two sides. Unit handling is part of the calculation, not just a label: the Fick calculation runs internally in SI, so centimeter inputs are converted first (D and A by 0.0001, ΔC by 1,000,000, Δx by 0.01) and the working line in the result shows those converted values.
The observed rate makes no assumptions at all. It is what you get when you weigh or measure what crossed and divide by the area and the elapsed time. It needs no diffusion coefficient and no molar mass, and it is the ground truth the other two formulas are trying to predict.
| Method | You need | You get | Typical use |
|---|---|---|---|
| Graham's law | One known rate and both molar masses, or both rates and one mass | The second gas's rate, or its molar mass | Comparing gases, identifying an unknown gas |
| Fick's first law | D, area, concentration difference, distance | Total rate in mol/s | Membranes, films, tissue, reactor walls |
| Observed rate | Measured mass, area, time | Rate in g/cm2/s | Reporting an experiment, permeability tests |
Worked examples, verified against the tool
Graham's law. Oxygen (M1 = 32 g/mol) diffuses at R1 = 2 units. For helium (M2 = 4 g/mol): R2 = 2 ÷ sqrt(4 ÷ 32) = 2 ÷ 0.3536 = 5.656854. Helium diffuses 5.656854 ÷ 2 = 2.828 times as fast. Run the same tab backwards with R2 = 5.657 and "Solve for" set to molar mass, and the tool returns M2 = 4.0000 g/mol (using the full 5.656854 gives exactly 4; the rounded 5.657 input gives 3.9998 g/mol), which is how an unknown gas gets identified from a measured rate.
Fick's law. A membrane with D = 1 × 10-9 m2/s, area 0.05 m2, concentration difference 50 mol/m3, and thickness 0.002 m: (1e-9 × 0.05 × 50) ÷ 0.002 = 1.2500e-6 mol/s. The same membrane entered in centimeter units (D = 1 × 10-5 cm2/s, area 500 cm2, concentration difference 5 × 10-5 mol/cm3, thickness 0.2 cm, with centimeter based selected) gives the identical 1.2500e-6 mol/s, because the tool converts those values to the SI values above before calculating. Notice what drives the number: the barrier is thin and the gradient is steep (50 mol/m3 across 2 mm), so even a tiny diffusion coefficient produces a measurable flow.
Observed rate. In a permeability test, 0.02 g crosses a 5 cm2 membrane in 200 s: 0.02 ÷ (5 × 200) = 2.0000e-5 g/cm2/s. If a Fick prediction and an observed rate disagree badly on the same setup, suspect the diffusion coefficient or a unit mix-up before suspecting the arithmetic.
Concentration units deserve one extra sentence, because mol/cm3 and mol/m3 differ by a factor of a million and silently wreck Fick calculations when the selected system does not match the entered value. If your concentrations arrive as molarity, the dilution molarity calculator handles the solution arithmetic, and you then express the difference in the unit system you selected in this tool: mol/m3 for SI, or mol/cm3 for centimeter based, which this tool then converts to SI internally.
What actually changes a diffusion rate
Molar mass sets the Graham's law comparison: lighter is faster, with a square root softening the effect. Temperature speeds diffusion up, because warmer molecules move faster; Graham comparisons assume both gases share the same temperature, and a diffusion coefficient quoted at 20°C will understate flow at body temperature. The concentration gradient is the engine in Fick's law: double the difference across the same distance and the rate doubles. Distance works against you: double the membrane thickness and the rate halves. Area scales the total rate without changing the flux, which is why lungs pack an enormous surface into a small chest. The medium matters through D: gases diffuse through air orders of magnitude faster than small molecules diffuse through water, and solids are slower still.
Common mistakes to avoid
Mixing unit systems in Fick's law. A diffusion coefficient in cm2/s with an area in m2 produces an answer off by powers of ten that still looks plausible. Choose the system that matches your values first and enter every input in that one system. The tool converts centimeter inputs to SI for you, but it cannot detect a value typed in the wrong system.
Confusing flux with total rate. Flux is per unit area; total rate is not. If a textbook answer is in mol/cm2/s and this tool reports mol/s, both can be right: divide the total rate by the area to get the flux.
Expecting Graham's law to need temperature or pressure inputs. It does not, as long as both gases are compared under the same conditions, because those factors cancel in the ratio. If the two gases are at different temperatures, the simple ratio no longer applies.
Using molar mass where density belongs, or the reverse. Graham's law works with either, because they are proportional for gases, but the ratio must be built consistently: rate ratio equals the square root of the inverse mass ratio, and flipping the ratio upside down is the single most common algebra slip.
What competing calculators miss
Ranking diffusion calculators each serve a single intent. The best-known Graham's law page does that one law well but offers no Fick mode and no way to report a measured rate. A popular engineering wiki's Graham page shows its headline formula with the mass ratio the wrong way up in the summary line, the kind of slip that flips which gas is faster. Fick's law tools elsewhere tend to compute flux only, in centimeters only, leaving SI users to convert by hand and total-rate users to multiply by area themselves. Here the Fick tab accepts either system and converts centimeter inputs to SI itself. None of them reverse-solve Graham's law for an unknown molar mass. This calculator covers all three intents in one place, with reverse solving, both unit systems for Fick's law with automatic conversion to SI, and copy and share on every result.
Frequently Asked Questions
Use Graham's law when you know molar masses and want to compare how fast two gases diffuse. Use Fick's law when you know the diffusion coefficient, the area, and the concentration gradient driving the flow. Use the observed-rate tab when you measured an actual mass crossing an area over a time.
Yes. On the Graham's law tab, set "Solve for" to molar mass of gas 2, then enter the known gas's rate and molar mass plus the measured rate of the unknown gas. The tool applies M2 = M1 x (R1 / R2)^2. With oxygen as the reference (R1 = 2, M1 = 32 g/mol) and a measured rate of 5.656854, it returns 4.0000 g/mol, identifying the gas as helium.
Any consistent set, with the unit selector set to match. The calculator accepts centimeter-based units (cm2/s, cm2, mol/cm3, cm) or SI units (m2/s, m2, mol/m3, m). The calculation runs internally in SI: centimeter inputs are converted before calculating (D and A are multiplied by 0.0001, the concentration difference by 1,000,000, and the distance by 0.01), and the result working shows the converted values. Both systems give the rate in mol/s for the same physical setup. Mixing systems in one calculation is the most common error.
A property of the substance and the medium together, found in literature or measured experimentally. Typical gas-in-air values are around 0.1-0.2 cm2/s; small molecules in water are around 1 x 10^-5 cm2/s.
Diffusion is spreading through a medium or across a concentration gradient; effusion is gas escaping through a tiny hole. Graham's law describes both, which is why some calculators call it the law of effusion.
The sign indicates direction (down the gradient is usually taken as positive). This calculator reports the magnitude.
No. That is a different topic: pricing for text-to-image models based on tokens or steps. This page covers the physics and chemistry meaning of diffusion rate.
Add This Calculator to Your Website
References
- OpenStax, Rice University, Chemistry 2e, Chapter 9: Gases, on Graham's law of effusion and diffusion and the inverse square root relationship with molar mass. OpenStax: Chemistry 2e (free textbook). Accessed October 2026.
- OpenStax, Rice University, University Physics Volume 2, on the kinetic theory of gases and molecular transport, the background to diffusion speeds, temperature effects, and diffusion coefficients. OpenStax: University Physics Volume 2 (free textbook). Accessed October 2026.
- Encyclopaedia Britannica, "Diffusion" (physics), on the factors that govern diffusion rates, including temperature, medium, and concentration gradient. Britannica: Diffusion, physics overview. Accessed October 2026.