Diffusion Coefficient Calculator
Calculate the diffusion coefficient (D) using four methods: the Stokes-Einstein equation for spherical or non-spherical particles in liquids, Fick's First Law from a concentration gradient, the diffusion time scaling relation, or the mean squared displacement (MSD) from experimental tracking data. Switch between temperature units (K or degC), viscosity units (Pa·s, mPa·s/cP), and output units (m2/s through um2/s). Results update as you type.
What is the diffusion coefficient?
The diffusion coefficient D (also called diffusivity) quantifies how fast a substance spreads through a medium by random Brownian motion. A larger D means faster spreading: in one second, a particle with D = 100 um2/s moves an RMS distance of about 14 um in 2D. D depends on the size and shape of the diffusing particle, the viscosity of the surrounding fluid, and the temperature. In SI units D is expressed in m2/s, but um2/s is more convenient for biological and nanoscale work.
Stokes-Einstein equation and particle shape
For a spherical particle in a viscous fluid, D = kB T / (6 pi eta a), where kB is the Boltzmann constant (1.38e-23 J/K), T is absolute temperature in kelvin, eta is the dynamic viscosity of the solvent, and a is the hydrodynamic radius of the sphere. The denominator 6 pi eta a is the Stokes friction coefficient for a sphere. Non-spherical particles have different friction coefficients: a disk moving face-on has xi = 16 eta a, edge-on xi = (32/3) eta a, and for an elongated ellipsoid moving lengthways xi = 4 pi eta a / (ln(2a/b) + 0.5), where b is the semi-minor axis. In all cases D = kB T / xi, which is the Einstein-Smoluchowski relation.
Fick's First Law and the concentration gradient
Fick's First Law relates the steady-state diffusive flux J (mol per m2 per second) to the diffusion coefficient and the concentration gradient: J = -D (dC/dx). A negative sign appears because diffusion flows from high to low concentration. If you know the flux across a membrane and the concentration on each side, you can solve for D = |J| / |dC/dx|. This approach is used to characterise transport across biological membranes and synthetic films. For a linear gradient across a layer of thickness x, dC/dx is simply (C1 - C2) / x.
Diffusion time scaling and the MSD method
The diffusion time scaling relation t = x2 / (2n D) estimates how long it takes a molecule to travel a distance x in n dimensions, where n = 1, 2, or 3. This is useful for quick estimates in cell biology: glucose (D ~ 670 um2/s) takes about 0.7 ms to cross a 1 um membrane. The mean squared displacement (MSD) method extracts D from single-particle tracking data using MSD = 2n D dt, where dt is the time lag. Plotting MSD against dt gives a straight line; the slope is 2n D. This calculator implements the single-point version; for a robust estimate, fit a series of (dt, MSD) pairs and take the slope.
Typical diffusion coefficients in water at 25 degC
| Species | D (um2/s) | D (m2/s) | Type |
|---|---|---|---|
| Water molecule (self) | 2300 | 2.3e-9 | Small molecule |
| O2 dissolved | 2100 | 2.1e-9 | Small molecule |
| Na+ ion | 1330 | 1.33e-9 | Ion |
| Cl- ion | 2030 | 2.03e-9 | Ion |
| Glucose | 670 | 6.7e-10 | Small molecule |
| Sucrose | 520 | 5.2e-10 | Small molecule |
| Myoglobin (~17 kDa) | 113 | 1.13e-10 | Protein |
| Albumin (~67 kDa) | 59 | 5.9e-11 | Protein |
| IgG antibody (~150 kDa) | 40 | 4.0e-11 | Protein |
| Ribosome (~2.5 MDa) | 4 | 4e-12 | Complex |
| Tobacco mosaic virus (~40 MDa) | 0.46 | 4.6e-13 | Virus particle |
| 100 nm nanoparticle | 4.3 | 4.3e-12 | Nanoparticle |
Reference values from published literature for common species in aqueous solution at 25 degC. Values vary with temperature, ionic strength, and solvent viscosity.
Frequently asked questions
What units does the diffusion coefficient have?
In SI units D is measured in m2/s (square metres per second). Because biological and nanoscale diffusion coefficients are very small in SI (typically 1e-10 to 1e-12 m2/s), researchers often report them in um2/s (square micrometres per second), where the same values fall in the range 0.01 to 2000. This calculator lets you display D in m2/s, mm2/s, cm2/s, or um2/s.
What is a typical diffusion coefficient for a protein?
In water at 25 degC, small proteins (around 10-20 kDa) diffuse at roughly 100-150 um2/s, medium-sized proteins (50-100 kDa) at 40-80 um2/s, and large proteins or antibodies (150-600 kDa) at 10-50 um2/s. Large complexes like ribosomes fall below 10 um2/s. In the cytoplasm of a cell, values are 3-10 times lower due to crowding.
How does temperature affect diffusion?
D is proportional to T and inversely proportional to viscosity (D = kBT / xi). Because the viscosity of water decreases roughly 2-3 % per degree Celsius, and T itself rises, D approximately doubles between 10 degC and 37 degC for typical aqueous systems. This is why enzymatic reactions speed up at body temperature and why diffusion coefficients are always quoted with the temperature.
What is the difference between Fick's First and Second Law?
Fick's First Law describes steady-state diffusion, where the concentration profile does not change with time: J = -D dC/dx. Fick's Second Law describes how concentration evolves over time: dC/dt = D d2C/dx2. The first law is sufficient when you have a stable gradient (like across a membrane held at fixed concentrations); the second law is needed for time-varying profiles such as a pulse of dye spreading in a gel.
When should I use the MSD method?
The MSD method is used when you have single-particle or single-molecule tracking data, typically from fluorescence microscopy. You record the position of a particle at successive time points, compute the squared displacement for each time lag, and average over many trajectories. The linear slope of MSD vs. time lag gives D = slope / (2n). Deviations from linearity reveal anomalous diffusion: sub-linear MSD indicates confined or obstructed motion; super-linear MSD suggests active transport.
Can I use the Stokes-Einstein equation for gases?
No. The Stokes-Einstein equation applies to particles suspended in a viscous liquid. For binary gas diffusion at low pressure, use the Chapman-Enskog equation or the Wilke-Lee empirical correlation instead. Diffusion coefficients in gases are orders of magnitude larger than in liquids (typically 1e-5 to 1e-4 m2/s vs. 1e-12 to 1e-9 m2/s in water).