The diffusion coefficient calculator above applies the Stokes-Einstein relation, which links how quickly a particle spreads through a liquid to the temperature, the viscosity of the solvent and the size of the particle. Enter a hydrodynamic radius and it returns the diffusion coefficient; enter a measured coefficient and it returns the radius the particle must effectively have. It then converts that coefficient into the two questions people actually ask about diffusion: how far in a given time, and how long to cover a given distance.
Arb Digital publishes free calculators aimed at the point where a formula meets a real measurement. With diffusion that point is the square root. Almost every intuition people bring to diffusion assumes distance grows in proportion to time, and it does not, which is why a molecule crosses a bacterium in milliseconds and would take years to cross a swimming pool if nothing stirred it.
What This Diffusion Coefficient Calculator Does
Its main job is the Stokes-Einstein calculation for a spherical particle in a continuous viscous medium. You supply the absolute temperature, the dynamic viscosity of the solvent and the hydrodynamic radius, and it returns the translational diffusion coefficient in both SI units and the cm²/s that most published tables use. Reverse mode inverts the same relation, which is what dynamic light scattering and similar techniques do: they measure a diffusion coefficient and report the equivalent hydrodynamic size.
The mean square displacement section then answers the practical question. For a random walk, the root mean square displacement after time t is the square root of 2nDt, where n is the number of dimensions the particle is free to move in. The page lets you choose one, two or three dimensions, because the right choice depends on the geometry: transport along a capillary is effectively one-dimensional, spreading on a membrane surface is two-dimensional, and free solution is three.
The boundary with its nearest neighbour is worth being explicit about. The effusion rate calculator handles Graham's law, which compares how fast two gases escape through a small hole and depends only on molar mass. That is a gas-phase kinetic-theory result with no viscosity and no particle radius in it. This page is about a solute moving through a liquid, where drag on the particle is the whole story. Different physics, different inputs, different units of answer.
How to Use It
- Set the temperature in Celsius or kelvin. The relation uses absolute temperature, so the page converts internally.
- Choose or enter the solvent viscosity. The presets are for water at four temperatures; if you change temperature you must change viscosity too, because the two are not independent.
- Enter the hydrodynamic radius in nanometres, remembering that this is larger than the bare molecular radius because solvent travels with the particle.
- Set the diffusion time and the dimensionality to see the root mean square displacement over that interval.
- Enter a distance to get the reverse figure: how long unassisted diffusion would take to cover it.
The Formula and How It Is Calculated
The Stokes-Einstein equation for a sphere under stick boundary conditions is D = kBT / (6πηr), where kB is the Boltzmann constant, T is absolute temperature in kelvin, η is the dynamic viscosity in pascal seconds and r is the hydrodynamic radius in metres. It comes from setting the Einstein relation between diffusion and mobility against the Stokes drag on a sphere in creeping flow.
Working the default example: at 25 °C, T is 298.15 K. The Boltzmann constant is exactly 1.380649 × 10⁻²³ J/K under the current SI, as published in the NIST CODATA value for the Boltzmann constant. The numerator is 4.1164 × 10⁻²¹ J. With water at 0.890 mPa·s, which is 8.90 × 10⁻⁴ Pa·s, and a 1 nm radius, the denominator is 6π × 8.90 × 10⁻⁴ × 1 × 10⁻⁹ = 1.6776 × 10⁻¹¹. Dividing gives D = 2.454 × 10⁻¹⁰ m²/s, which is 2.454 × 10⁻⁶ cm²/s.
The displacement follows from xrms = √(2nDt). In one dimension over one second that is √(2 × 2.454 × 10⁻¹⁰) = 2.22 × 10⁻⁵ m, or about 22 micrometres. Solvent viscosities at temperature can be looked up from the NIST thermophysical properties of fluid systems database, which returns viscosity and density for a wide range of fluids and conditions.
Why the Square Root Changes Everything
Diffusion is a random walk, and the displacement of a random walk grows as the square root of the number of steps. That single fact controls where diffusion is a useful transport mechanism and where it is not. Take the default coefficient of roughly 2.5 × 10⁻¹⁰ m²/s. Crossing one micrometre takes about two milliseconds. Crossing one hundred micrometres, only a hundred times further, takes twenty seconds, because the time scales with the square of distance. Crossing one centimetre takes over two days, and one metre would take more than fifty years.
This is why cells are small, why a bacterium can rely entirely on diffusion for internal transport while a neuron cannot and evolved active axonal transport instead, and why every industrial process that needs mass transfer over a distance uses stirring, pumping or flow rather than waiting. It is also why the mixing you observe when you add milk to coffee is convection, not diffusion; diffusion would take days at that scale.
Turning that around gives a useful design rule. If you know the diffusion coefficient and the length scale of your system, the characteristic diffusion time x² ÷ 2D tells you whether diffusion will keep up with whatever else is happening. If that time is much shorter than the reaction or flow time, the system is well mixed and you can ignore gradients. If it is comparable or longer, gradients dominate and any model that assumes uniform concentration is wrong.
Hydrodynamic Radius Is Not Molecular Radius
The r in the equation is the radius of an equivalent hard sphere that would experience the same drag, not the radius you would compute from a molecular structure. Two things make it larger. First, solvent molecules are bound to the surface and travel with the particle, so a hydrated ion drags its hydration shell along. Second, the equation assumes a sphere, and real molecules are rarely spherical; an extended or rod-shaped molecule experiences more drag than a compact one of the same mass, so it reports a larger effective radius.
This is why the same protein can give different hydrodynamic radii under different buffer conditions without changing its mass at all. Unfolding increases drag and lowers the diffusion coefficient. Aggregation does the same much more dramatically, which is precisely what makes light scattering sensitive to it. When a reported hydrodynamic radius disagrees with a structural radius, that disagreement is data, not error.
Where you know a molar mass rather than a size, the molar mass calculator and the protein molecular weight calculator give the mass side, but converting mass to hydrodynamic radius requires an assumption about density and shape and is not a fixed conversion.
Where Stokes-Einstein Stops Working
The relation assumes a rigid sphere much larger than the solvent molecules moving through a continuous fluid. When the diffusing particle is comparable in size to the solvent, that continuum picture breaks down, and small molecules in ordinary solvents typically diffuse somewhat faster than the equation predicts. The stick boundary condition can also be replaced with slip, which changes the factor from 6π to 4π and raises D by half.
The assumption also fails in crowded or structured media. Diffusion inside a cell, through a gel, through a porous solid or in a concentrated polymer solution is hindered, and the apparent coefficient can be an order of magnitude below the free-solution value. Near a glass transition, or in any system where the solvent relaxation slows dramatically, the coupling between viscosity and diffusion decouples entirely and the relation fails outright.
Finally, this is translational diffusion only. Rotational diffusion follows a different relation with an r³ dependence, so it is far more sensitive to size, and a measurement of one does not give you the other. If your interest is a settling particle rather than a diffusing one, the Stokes law calculator covers the terminal velocity problem, which uses the same drag expression for a completely different question.
Temperature and Viscosity Move Together
A tempting mistake is to raise T in the numerator and leave η alone, concluding that diffusion increases only in proportion to absolute temperature. In practice viscosity falls much faster than temperature rises. Water at 10 °C has a viscosity of about 1.307 mPa·s and at 37 °C about 0.691 mPa·s, close to a halving, while the absolute temperature rises by less than ten percent. The net effect is that the diffusion coefficient roughly doubles across that range, dominated by the viscosity term.
This is why the presets on this page pair a temperature with the matching water viscosity, and why changing one without the other produces a physically meaningless answer. If you are working in a solvent other than water, take the viscosity at your working temperature from a source that reports it as a function of temperature rather than from a single room-temperature figure. The viscosity converter will move between poise, centipoise, pascal seconds and kinematic units if your source uses a different one, and the temperature converter handles the scale conversions.
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Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Changing temperature without changing viscosity — the viscosity term dominates, and ignoring it gets the direction roughly right and the magnitude badly wrong.
- Using a molecular radius as the hydrodynamic radius — bound solvent and non-spherical shape both make the effective radius larger.
- Assuming distance grows with time — it grows with the square root of time, so ten times the distance is a hundred times the wait.
- Mixing units of viscosity — the formula needs pascal seconds, and centipoise is a thousand times smaller.
- Applying free-solution values inside a gel or a cell — hindered environments can reduce the apparent coefficient by an order of magnitude.
Related Free Tools From Arb Digital
For the gas-phase counterpart, where molar mass alone sets the rate, use the effusion rate calculator. The Stokes law calculator applies the same drag term to settling velocity, the viscosity converter and temperature converter handle the input units, and the osmotic pressure calculator covers the other classic transport property of a dilute solution. For molecular sizes, see the molar mass calculator and the protein molecular weight calculator. The full free online tools hub lists everything else.
Frequently Asked Questions
It states that the diffusion coefficient of a spherical particle in a liquid equals the Boltzmann constant times absolute temperature divided by six pi times the viscosity times the hydrodynamic radius. It links thermal motion to viscous drag.
Square metres per second in SI, though published tables very often use square centimetres per second. One cm²/s equals 10⁻⁴ m²/s, and this page prints both so the conversion is never in doubt.
The root mean square displacement is the square root of 2nDt, where n is one, two or three depending on the dimensions available. For a coefficient of 2.45×10⁻¹⁰ m²/s, one second of one-dimensional diffusion gives about 22 micrometres.
The radius of an equivalent hard sphere that would feel the same drag as the real particle. It includes solvent that travels with the molecule and is inflated by non-spherical shape, so it exceeds the bare molecular radius.
Yes, but mostly indirectly. Absolute temperature appears in the numerator, yet the bigger effect is that viscosity falls sharply as temperature rises. Between 10 and 37 degrees Celsius in water the coefficient roughly doubles.
When the particle is not much larger than the solvent molecules, in crowded or gel-like media where diffusion is hindered, and near a glass transition where viscosity and molecular mobility decouple. It also applies to translational diffusion only.
Graham's law compares gases escaping through a pinhole and depends only on molar mass. Stokes-Einstein describes a solute moving through a liquid, where viscosity and particle size govern the result. They are unrelated physical situations.
This calculator is provided for education and general reference. It computes a published physical relationship from values you supply and is not laboratory, analytical or safety guidance; follow the methods and risk assessments issued by your own institution.
