Use this diffusion coefficient calculator
This diffusion coefficient calculator is designed for students and routine checking. Choose a measured mean-square-displacement calculation or a Stokes–Einstein fluid estimate. The workspace keeps the selected method, known values, result, formula, and substituted arithmetic together so you can see what produced the answer instead of receiving an unexplained number. Change one input at a time when checking homework, and confirm that the direction of change agrees with the stated relationship.
The result uses the model and assumptions stated on this page together with the values shown in the active form. It does not infer values, identities, or conditions that the active method does not ask for. If your course, data sheet, or laboratory method specifies a different reference state or sign convention, use that source consistently before comparing answers.
Diffusion coefficient formula and input guide
The calculation uses D = ⟨r²⟩/(2dt) or D = kBT/(6πηr). MSD uses m² and seconds; Stokes–Einstein uses kelvin, Pa·s, metres, and the exact SI Boltzmann constant. Inputs that must be positive, integral, dimensionless, or above absolute zero are checked before a result is shown. A hidden default is never used for a selected unknown, and changing a method or known value invalidates the previous result until you calculate again.
Read each visible field label as part of the active formula. A label distinguishes quantities that may share a symbol or unit but use a different basis. Keep every input on the basis named by the selected method, and use the visible substitution steps as a unit audit before accepting the result.
Work through a diffusion coefficient example
An MSD of 1×10⁻¹² m² over 1 s in three dimensions gives D ≈ 1.67×10⁻¹³ m²/s. The page displays the intermediate scale or correction before the final answer, which makes sign, exponent, and unit mistakes easier to spot. Recalculate the example independently and compare the displayed substitution rather than matching only the last rounded digits.
A useful reverse check substitutes the reported answer back into the original relationship. The reconstructed known quantity should agree at the displayed precision. If it does not, inspect the active fields, unit prefixes, selected method, and stated model limits before recalculating.
Interpret the diffusion coefficient result
The numerical answer describes only the quantity named in the result card under the selected method and its stated assumptions. Do not treat one calculated quantity as proof of a different property, mechanism, or experimental outcome. Preserve the sign and unit when copying the result because removing either can reverse or erase its physical meaning.
Report significant figures that match the least precise input or source constant required by your work. Additional digits are shown when they are needed to keep the visible equation internally consistent, not to claim laboratory accuracy. For consequential research, manufacturing, medical, environmental, or safety decisions, use validated measurements and a method approved for that application.
Model limits, sources, and private use
Anomalous trajectories, boundaries, active motion, slip, nonspherical particles, and molecular-scale solvent structure can violate the selected model. The principal reference basis is Einstein–Smoluchowski and Stokes–Einstein relations with NIST 2022 CODATA kB. When the page uses a named correlation or constant, it keeps that source within the range stated here; unsupported inputs are rejected instead of silently extrapolated or completed with invented values. A classroom approximation can be useful without being a universal property prediction.
The values are calculated in the current browser tab and are not uploaded as part of the calculation. No account is required. Copying is a deliberate action after the answer and steps are visible, while Reset restores the example and Clear removes the current result. Refreshing or leaving the page clears the working state, so save only the information you need for later study.