Stokes-Einstein equation, abbreviated as \(SE\), also called Stokes-Einstein-Sutherland equation, is used in fluid mechanics and statistical physics and quantifies the diffusion coefficient of a small particle suspended in a viscous fluid. It establishes a direct connection between the particle’s thermal motion (Brownian motion), the temperature of the fluid, and the fluid’s resistance to flow (viscosity). In practical terms, it explains how rapidly particles such as colloids, fine sediments, or macromolecules spread out due to random thermal energy in a fluid medium.
Stokes-Einstein Equation |
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\( D \;=\; \dfrac{ k_b \cdot T }{ 6 \cdot \pi \cdot \mu \cdot r }\) (Stokes-Einstein) \( k_b \;=\; \dfrac{ 6 \cdot \pi \cdot \mu \cdot r \cdot D }{ T }\) \( T \;=\; \dfrac{ D \cdot 6 \cdot \pi \cdot \mu \cdot r }{ k_b }\) \( \mu \;=\; \dfrac{ k_b \cdot T }{ D \cdot 6 \cdot \pi \cdot r }\) \( r \;=\; \dfrac{ k_b \cdot T }{ D \cdot 6 \cdot \pi \cdot \mu }\) |
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| Symbol | English | Metric |
| \( D \) = Diffusion Coefficient | \(lbf\) | \(N\) |
| \( k_b \) = Boltzmann Constant | \(lbm-ft^2 \;/\; sec^2\) | \(kJ \;/\; molecule-K\) |
| \( T_a \) = Absolute Temperature | \(^\circ R\) | \(^\circ K\) |
| \( \pi \) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( \mu \) (Greek symbol mu) = Dynamic Viscosity | \(lbf-sec \;/\; ft^2\) | \(Pa-s\) |
| \( r \) = radius of spherical parcticle | \(in\) | \(mm\) |
Physically, the equation reflects a balance between two competing effects. Thermal energy, governed by absolute temperature, drives random motion and enhances diffusion. In contrast, viscous drag described by Stokes’ law, resists particle movement. The Stokes–Einstein equation combines these effects to show that diffusion increases with temperature and decreases with both fluid viscosity and particle size. For a spherical particle, the diffusion coefficient is inversely proportional to the particle’s radius, meaning larger particles diffuse more slowly than smaller ones in the same fluid under identical conditions.
In civil and environmental engineering, the equation is particularly relevant in the analysis of fine particle transport in water and air, including colloidal suspensions, contaminant migration in groundwater, and sedimentation processes at very small scales where Brownian motion is significant. It also underpins the interpretation of laboratory techniques such as dynamic light scattering used to determine particle size distributions in suspensions.
