Diffusion Coefficient Calculator

What Is the Diffusion Coefficient?

The diffusion coefficient $D$ measures how readily particles spread from regions of higher concentration to lower concentration through random Brownian motion. It appears in Fick's laws of diffusion and has SI units of $\mathrm{m^2/s}$. Larger $D$ means faster mixing; smaller $D$ means slower transport.

For colloidal and molecular particles in a viscous solvent, the Einstein–Smoluchowski relation connects $D$ to absolute temperature $T$ and the particle friction coefficient $\xi$:

$$D = \frac{k_B T}{\xi}$$

where $k_B = 1.380649 \times 10^{-23}\,\mathrm{J/K}$ is Boltzmann's constant. Higher temperature increases diffusion. Higher friction (from viscosity, size, or shape) decreases it.

How to Use This Calculator

Choose whether to solve for the diffusion coefficient $D$ or the friction coefficient $\xi$. When solving for $D$, pick a particle shape, enter temperature, size, and solvent viscosity. The tool updates results in real time and shows a step-by-step breakdown.

• Solve for D: Uses shape-specific Stokes-type friction formulas, then $D = k_B T / \xi$.
• Solve for ξ: Rearranges the same relation as $\xi = k_B T / D$ from a measured diffusion coefficient. Shape is not required.
• Units: Temperature supports K, °C, and °F. Length supports m, µm, nm, and Å. Viscosity supports Pa·s, mPa·s, cP, and P.

Friction Coefficient by Particle Shape

The friction coefficient $\xi$ depends on solvent dynamic viscosity $\eta$ and particle geometry. For a sphere at low Reynolds number, Stokes' law gives $\xi = 6\pi\eta a$, which recovers the Einstein–Stokes formula:

$$D = \frac{k_B T}{6\pi\eta a}$$

This calculator also supports disks and elongated ellipsoids:

• Sphere (low $Re$): $\xi = 6\pi\eta a$
• Disk, face on: $\xi = 16\eta a$
• Disk, edge on: $\xi = (32/3)\eta a$
• Disk, random / rotating: $\xi = 12\eta a$
• Ellipsoid, lengthways: $\xi = 4\pi\eta a / (\ln(2a/b) + 1/2)$
• Ellipsoid, sideways: $\xi = 8\pi\eta a / (\ln(2a/b) + 1/2)$
• Ellipsoid, tumbling: $\xi = 6\pi\eta a / \ln(2a/b)$

Here $a$ is the radius (sphere or disk) or semi-major axis (ellipsoid), and $b$ is the semi-minor axis for ellipsoids with $a > b$. These expressions assume creeping flow, similar to the regime checked with a Reynolds number calculator and related to viscous drag in the Stokes law calculator.

Worked Example: Sphere in Water

Consider a spherical nanoparticle with radius $a = 2\,\mathrm{nm}$ in water at $25^\circ\mathrm{C}$ with $\eta = 0.00089\,\mathrm{Pa\cdot s}$.

Friction: $\xi = 6\pi\eta a = 6\pi(0.00089)(2\times 10^{-9}) = 3.355\times 10^{-11}\,\mathrm{kg/s}$.

Diffusion: $D = k_B T / \xi$ with $T = 298.15\,\mathrm{K}$ gives $D \approx 1.23\times 10^{-10}\,\mathrm{m^2/s}$.

When Diffusion Coefficients Matter

Diffusion coefficients appear throughout physical chemistry and transport engineering: estimating mixing times, interpreting dynamic light scattering, sizing colloids from measured $D$, and relating mass-transfer dimensionless groups such as the Sherwood number and Peclet number, both of which include $D$ explicitly.

Radioactive decay and molecular diffusion are different processes. For nuclear half-life calculations, use the half-life calculator instead.

Assumptions and Limitations

• Formulas assume Newtonian solvent and low Reynolds number (creeping flow).
• Particles are treated as rigid continuum objects; molecular-scale slip and solvent structure can matter for very small solutes.
• Ellipsoid formulas require $a > b$ and are approximate hydrodynamic models.
• Viscosity must match the solvent and temperature you intend; water viscosity changes strongly with temperature.