Diffusion Coefficient (Einstein-Stokes) Calculator
Find the translational diffusion coefficient of a spherical particle in a fluid using the Stokes-Einstein relation.
🧬 What is the Diffusion Coefficient (Einstein-Stokes) Calculator?
The diffusion coefficient calculator finds the translational diffusion coefficient D of a spherical particle moving through a viscous fluid due to Brownian motion, using the Stokes-Einstein relation D=kT/(6πηr). This single formula connects a particle's size, the medium's temperature and viscosity, and how quickly the particle spreads out through random thermal motion.
This calculator is used across several real-world contexts. Biophysics and biochemistry students use it to estimate how fast a protein, nucleic acid, or small molecule diffuses through a cell or a test tube. Researchers running dynamic light scattering (DLS) experiments use the Stokes-Einstein relation in reverse, measuring D experimentally and solving for the particle's hydrodynamic radius. Pharmaceutical scientists use diffusion coefficients to model how quickly a drug molecule crosses a membrane or diffuses through tissue.
A common misconception is that diffusion coefficients are roughly the same for all biomolecules. They are not, D is inversely proportional to particle radius, so a large protein complex diffuses far slower than a small ion or metabolite under identical conditions. Another point worth noting is that the raw SI result (m²/s) is an extremely small number for anything biologically relevant, which is why this calculator also reports D in µm²/s, the unit typically used in biophysics literature.
This calculator is useful because it handles the unit conversions (Celsius to Kelvin, nanometers to meters, millipascal-seconds to pascal-seconds) that are the most common source of errors when computing D by hand, and it shows how D changes across a realistic range of particle sizes with an interactive chart.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Small Globular Protein (Lysozyme-Scale)
r=2 nm, T=25°C, water viscosity η=0.89 mPa·s
Example 2 — Virus-Scale Particle
r=50 nm, same conditions: T=25°C, water viscosity η=0.89 mPa·s
Example 3 — Same Small Protein at Body Temperature
r=2 nm, T=37°C, water viscosity at body temperature η=0.692 mPa·s
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Stokes-Einstein equation?
The Stokes-Einstein equation, D=kT/(6πηr), gives the translational diffusion coefficient D of a spherical particle moving through a viscous fluid due to Brownian motion, where k is the Boltzmann constant, T is absolute temperature, η is the fluid's dynamic viscosity, and r is the particle's hydrodynamic (Stokes) radius.
What is the diffusion coefficient of a small protein like lysozyme?
For a small globular protein with a hydrodynamic radius around 2 nm in water at 25°C (viscosity 0.89 mPa·s), the Stokes-Einstein relation gives a diffusion coefficient of about 123 µm²/s, consistent with experimentally measured values for lysozyme-scale proteins.
How does particle size affect the diffusion coefficient?
D is inversely proportional to the hydrodynamic radius r, so a particle 25 times larger diffuses 25 times slower. A 50 nm virus-scale particle diffuses about 25 times slower than a 2 nm protein under the same temperature and viscosity, exactly the D∝1/r scaling predicted by the formula.
What units does this calculator use for viscosity?
Millipascal-seconds (mPa·s), which is numerically identical to centipoise (cP), the traditional viscosity unit still common in reference tables. Water at room temperature has a viscosity close to 1 mPa·s (1 cP), making it an easy mental reference point.
Why does the diffusion coefficient increase at body temperature (37°C) compared to room temperature?
Two effects both push D higher at 37°C: the temperature T in the numerator directly increases, and water's viscosity η drops from about 0.89 mPa·s at 25°C to about 0.692 mPa·s at 37°C, and since η is in the denominator, a lower viscosity further increases D.
What is the hydrodynamic (Stokes) radius?
The hydrodynamic radius is the radius of an idealized hard sphere that would diffuse at the same rate as the actual particle, including any bound solvent or hydration shell. It is often slightly larger than a particle's dry, crystallographic radius because of this effective solvent shell.
Why is the SI diffusion coefficient such a small number?
In SI units (m²/s), typical biomolecular diffusion coefficients fall between roughly 1e-10 and 1e-11, because diffusion over the meter-scale unit happens extremely slowly at the molecular level. Biophysicists conventionally use µm²/s instead, where the same values become easy-to-read numbers in the range of tens to hundreds.
Does the Stokes-Einstein equation work for any shape of particle?
The equation assumes a spherical particle. For elongated or irregularly shaped macromolecules, the true diffusion coefficient deviates from the spherical prediction, and researchers either measure the hydrodynamic radius directly or apply shape-correction factors (such as the Perrin friction factor) to account for the non-spherical geometry.
How is the diffusion coefficient measured experimentally?
Common experimental methods include dynamic light scattering (DLS), fluorescence recovery after photobleaching (FRAP), and analytical ultracentrifugation, all of which infer D from how quickly particles spread out or how their scattered/fluorescent signal decorrelates over time.
What is a typical diffusion coefficient range for small molecules versus large protein complexes?
Small molecules and ions typically diffuse in the range of several hundred to over a thousand µm²/s, small globular proteins fall around 50 to 150 µm²/s, and large protein complexes or virus particles can drop to just a few µm²/s or lower, all consistent with the D∝1/r scaling this calculator's chart illustrates.