Diffusion Coefficient Calculator
Compute diffusion coefficient using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
D = kB * T / (6 * pi * eta * r)
The Stokes-Einstein equation calculates the diffusion coefficient D from Boltzmann constant kB, temperature T, viscosity eta, and particle radius r. For liquid-phase molecular diffusion, the Wilke-Chang correlation provides an empirical estimate.
Worked Examples
Example 1: Protein in Water
Problem:Calculate the diffusion coefficient of a protein with hydrodynamic radius 3.5 nm in water (viscosity 0.001 Pa.s) at 298 K.
Solution:D = kB*T / (6*pi*eta*r)\nD = (1.381e-23 * 298) / (6 * 3.1416 * 0.001 * 3.5e-9)\nD = 4.115e-21 / 6.597e-11\nD = 6.24e-11 m\u00B2/s
Result:D = 6.24e-11 m\u00B2/s
Example 2: Small Molecule in Water
Problem:Estimate the diffusion coefficient of a molecule with radius 0.15 nm in water at 310 K (body temperature).
Solution:D = kB*T / (6*pi*eta*r)\nD = (1.381e-23 * 310) / (6 * 3.1416 * 0.001 * 1.5e-10)\nD = 4.281e-21 / 2.827e-12\nD = 1.51e-9 m\u00B2/s
Result:D = 1.51e-9 m\u00B2/s
Frequently Asked Questions
What is the diffusion coefficient?
The diffusion coefficient (D) quantifies how fast a substance spreads through another medium due to random molecular motion. It is measured in units of area per time, typically m²/s or cm²/s. A higher diffusion coefficient means molecules move through the medium more quickly. The value depends on temperature, the size of the diffusing particle, and the viscosity of the surrounding medium. For small molecules in water at room temperature, D is typically on the order of 1e-9 m²/s.
How does temperature affect diffusion?
Temperature has a direct proportional effect on the diffusion coefficient. As temperature increases, molecules gain more kinetic energy and move faster, leading to higher diffusion rates. The Stokes-Einstein equation shows D is linearly proportional to T. However, temperature also affects viscosity, which typically decreases as temperature rises, further increasing diffusion. A rule of thumb is that the diffusion coefficient roughly doubles for every 20-25 K increase in temperature for aqueous solutions.
What are typical diffusion coefficient values?
Diffusion coefficients vary enormously depending on the phase. In gases at atmospheric pressure, D is typically 1e-5 to 1e-4 m²/s. In liquids, D ranges from about 1e-10 to 1e-9 m²/s for small molecules and drops to 1e-11 m²/s or lower for large proteins. In solids, diffusion is extremely slow, with D values of 1e-14 m²/s or smaller. These values are critical for designing chemical reactors, drug delivery systems, and separation processes.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy