Diffusion Coefficient
Calculate the diffusion coefficient of particles in a solvent using the Einstein-Stokes equation. Supports spheres, disks, and ellipsoid shapes with friction coefficient analysis.
About This Calculator
The diffusion coefficient calculator helps you compute the diffusion coefficient (D) of particles in a solvent using the Einstein-Smoluchowski relation, D = kBT / ξ, where kB is Boltzmann's constant, T is the absolute temperature, and ξ is the friction coefficient determined by the particle's shape, size, and the solvent's dynamic viscosity.
This calculator supports seven particle shapes: spheres (using the classic Stokes-Einstein relation), disks in three orientations (face on, edge on, and random rotational motion), and ellipsoids in three orientations (lengthways, sideways, and tumbling). Each shape has a specific friction coefficient formula derived from fluid dynamics principles. The Stokes-Einstein equation for spheres, D = kBT / (6πηr), is widely used in chemistry, biology, and materials science to estimate the size of nanoparticles and macromolecules from diffusion measurements.
The diffusion coefficient describes how quickly particles spread from regions of higher concentration to regions of lower concentration due to Brownian motion. It is a critical parameter in fields including drug delivery (predicting how quickly drug molecules diffuse through biological tissues), protein characterization (determining hydrodynamic radius via dynamic light scattering), materials science (understanding dopant diffusion in semiconductors), and environmental science (modeling pollutant dispersion in groundwater).
Diffusion is a universal physical process governed by Fick's laws. The Einstein-Smoluchowski relation was independently derived by Albert Einstein and Marian Smoluchowski in the early 1900s, providing a theoretical foundation for Brownian motion and linking microscopic particle properties to macroscopic diffusion behavior.
Regional Notes
Diffusion coefficient calculations are based on universal physical constants and equations that apply worldwide. The calculator uses SI units (m²/s for diffusion coefficient, Pa·s for viscosity, nm for particle size, and °C for temperature). The same inputs will produce identical results regardless of geographic region.
Frequently Asked Questions
What is the diffusion coefficient?
The diffusion coefficient (D) is a physical constant that measures how quickly particles spread from high-concentration regions to low-concentration regions due to random Brownian motion. It depends on temperature, particle size and shape, and the viscosity of the surrounding medium. It is measured in square meters per second (m²/s).
How is the diffusion coefficient calculated?
The diffusion coefficient is calculated using the Einstein-Smoluchowski relation: D = k_B × T / ξ, where k_B is Boltzmann's constant (1.380649 × 10⁻²³ J/K), T is the absolute temperature in Kelvin, and ξ is the friction coefficient of the particle in the solvent. The friction coefficient depends on the particle's shape, size, and the solvent's dynamic viscosity.
What is the Stokes-Einstein equation?
The Stokes-Einstein equation is a special case of the Einstein-Smoluchowski relation for spherical particles. It states D = k_B × T / (6π × η × r), where η is the solvent viscosity and r is the particle radius. This is the most commonly used form for estimating diffusion coefficients of spherical nanoparticles and macromolecules in solution.
How does temperature affect the diffusion coefficient?
The diffusion coefficient increases with temperature because higher temperatures give particles more kinetic energy, increasing their random motion. The relationship is directly proportional: doubling the absolute temperature (in Kelvin) will approximately double the diffusion coefficient, assuming the viscosity and friction coefficient remain unchanged.
How does particle shape affect the diffusion coefficient?
Particle shape significantly affects the diffusion coefficient through the friction coefficient. Spherical particles experience the lowest friction and therefore diffuse fastest. Disks and ellipsoids experience higher friction due to greater surface area, reducing their diffusion coefficient. For ellipsoids, the orientation of movement (lengthways vs sideways) changes the friction coefficient by roughly a factor of two.
What units is the diffusion coefficient measured in?
The diffusion coefficient is measured in square meters per second (m²/s) in SI units. For nanoparticles and biomolecules, it is commonly expressed in smaller units such as cm²/s or µm²/s. Typical values for small molecules in water at room temperature range from 10⁻⁹ to 10⁻¹⁰ m²/s, while nanoparticles have values around 10⁻¹¹ to 10⁻¹³ m²/s.
How does solvent viscosity affect particle diffusion?
Solvent viscosity is inversely proportional to the diffusion coefficient. Higher viscosity solvents create more resistance to particle movement, reducing the diffusion rate. For example, particles in honey (high viscosity) diffuse much slower than in water (low viscosity). The friction coefficient is directly proportional to viscosity in all shape models.
What is the friction coefficient in diffusion?
The friction coefficient (ξ) quantifies the resistance a particle experiences when moving through a solvent. It depends on the particle's shape and size, and the solvent's dynamic viscosity. For a sphere, ξ = 6π × η × r (Stokes' law). For non-spherical shapes, geometric correction factors apply. The friction coefficient has units of s/kg or kg/s.