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Diffusion Coefficient Calculator

Chemistry

Calculate the diffusion coefficient of a spherical particle in solution using the Stokes-Einstein equation. Enter temperature, viscosity, and radius.

Reviewed by the thecalcu.com team · Last updated June 21, 2025

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Diffusion Coefficient D (m²/s)

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Diffusion Coefficient D (cm²/s)
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Mean Square Displacement at 1 s (nm²)
1,472,238,657.27

This calculator computes your Diffusion Coefficient D (m²/s), Diffusion Coefficient D (cm²/s), Mean Square Displacement at 1 s (nm²) from the values you enter.

Inputs
Temperature (°C)Solvent Viscosity (mPa·s)Particle/Molecule Radius (nm)
Outputs
Diffusion Coefficient D (m²/s)Diffusion Coefficient D (cm²/s)Mean Square Displacement at 1 s (nm²)

What is a Diffusion Coeff.?

The Diffusion Coefficient Calculator computes the translational diffusion coefficient D of a spherical particle or molecule in solution using the Stokes-Einstein equation: D = kT/(6πηr), where k is Boltzmann's constant (1.381 × 10⁻²³ J/K), T is temperature in Kelvin, η is solvent viscosity in Pa·s, and r is the particle radius in metres. Results are returned in m²/s and cm²/s, along with the mean square displacement in 3D at 1 second.

The Stokes-Einstein equation is fundamental to physical chemistry, biophysics, and materials science. It connects thermal energy (kT, the energy scale of molecular motion) to viscous drag (6πηr, the Stokes drag on a sphere), yielding the diffusion coefficient that governs how quickly molecules spread by Brownian motion. Published by Albert Einstein in his 1905 annus mirabilis papers alongside special relativity, it provided the first quantitative connection between macroscopic diffusion and atomic-scale thermal motion.

For molecular weight determination from diffusion: rearrange to r = kT/(6πηD), then use protein density approximations or the Mark-Houwink relation to convert r to molecular weight. This approach is used in dynamic light scattering (DLS) instruments, which report hydrodynamic radius and from it estimate molecular weight.

Why Use a Diffusion Coefficient Calculator?

The Stokes-Einstein equation spans many orders of magnitude in D depending on the particle size and viscosity: D ranges from ~10⁻⁹ m²/s for small molecules to ~10⁻¹² m²/s for nanoparticles. Tracking the unit conversions (nm → m for r, mPa·s → Pa·s for η) and computing Boltzmann's constant × temperature / (6π × viscosity × radius) manually is error-prone. This calculator handles all conversions internally.

For biochemistry, biophysics, and materials science students, the Stokes-Einstein equation appears in transport phenomena, nanoparticle characterisation, and protein dynamics calculations. The connection between D and hydrodynamic radius r ties diffusion theory to experimental DLS measurements.

Who Should Use This Calculator?

Physical chemistry and biophysics students studying transport properties, diffusion, and Brownian motion in the context of solution kinetics and macromolecular dynamics.

Biochemists and structural biologists estimating diffusion coefficients of proteins, nucleic acids, and cellular components from their hydrodynamic radii, comparing to DLS experimental data.

Pharmaceutical scientists and formulation chemists computing diffusion coefficients of drug molecules, liposomes, and nanoparticles through biological fluids and tissue matrices.

Materials scientists and nanotechnologists characterising colloidal particles, quantum dots, and nanoparticles by their diffusion behaviour in various media.

Chemical engineers designing membrane separation, drug delivery systems, and chemical reactors where mass transfer by diffusion is rate-limiting.

What Insights Does the Diffusion Coefficient Calculator Give You?

Diffusion Coefficient D (m²/s) is the primary output in SI units, used in Fick's law calculations, mass transfer equations, and comparison with tabulated literature values.

Diffusion Coefficient D (cm²/s) is the CGS unit widely used in older literature and in American physical chemistry textbooks. Many tabulated D values are in cm²/s (e.g., D(O₂ in water, 25°C) = 2.1 × 10⁻⁵ cm²/s).

Mean Square Displacement at 1 s (nm²) quantifies how far the particle diffuses in 1 second by Brownian motion: <r²> = 6D × 1 s. The square root gives the RMS displacement, the length scale of diffusional motion per second, relevant for assessing whether a molecule can cross a biological barrier by diffusion alone.

How to use this Diffusion Coeff. calculator

  1. Enter the Temperature in °C. Physiological temperature = 37°C; laboratory standard = 25°C.
  2. Enter the Solvent Viscosity in mPa·s. Water at 25°C = 0.89 mPa·s; at 37°C = 0.69 mPa·s; for other solvents, look up in CRC Handbook or viscosity tables.
  3. Enter the Particle/Molecule Radius in nanometres. For proteins, use the hydrodynamic radius (from DLS or SAXS). For nanoparticles, use the core radius or hydrodynamic diameter/2.
  4. Read D (m²/s), compare to literature values for similar-sized molecules to verify your radius estimate.
  5. Use Mean Square Displacement to estimate the timescale for a particle to diffuse a specific distance: t = <r²> / (6D), where <r²> is the target displacement squared.
Show formula & methodology ↓Show less ↑

Formula & Methodology

Stokes-Einstein equation:

D = kT / (6πηr) k = 1.380649 × 10⁻²³ J/K  (Boltzmann constant) η in Pa·s  (1 mPa·s = 10⁻³ Pa·s) r in metres  (1 nm = 10⁻⁹ m)

Mean square displacement in 3D:

<r²> = 6Dt    (at time t in seconds) RMS displacement = √(6Dt)

Worked example, albumin protein at 37°C:

Serum albumin, hydrodynamic radius r ≈ 3.6 nm, in water at 37°C (η = 0.69 mPa·s = 6.9 × 10⁻⁴ Pa·s):

D = (1.381 × 10⁻²³ × 310.15) / (6π × 6.9 × 10⁻⁴ × 3.6 × 10⁻⁹)   = 4.283 × 10⁻²¹ / (4.692 × 10⁻¹¹)   = 9.13 × 10⁻¹¹ m²/s  <r²> at 1s = 6 × 9.13 × 10⁻¹¹ = 5.48 × 10⁻¹⁰ m² = 548 nm² RMS = 23.4 nm

Albumin diffuses ~23 nm in 1 second by Brownian motion, over an hour, it diffuses ~√(6 × 9.13 × 10⁻¹¹ × 3600) = 1.4 mm. This sets the timescale for albumin distribution within small tissue volumes, relevant for pharmacokinetic modelling.

Frequently Asked Questions

What is the diffusion coefficient?
The diffusion coefficient (D) is a transport property that characterises how quickly molecules or particles spread through a medium by random thermal motion (Brownian motion). It appears in Fick's first law: J = −D × (dC/dx), where J is the diffusion flux (mol/m²/s), D is in m²/s, and dC/dx is the concentration gradient. A larger D means faster diffusion. D depends on the size of the diffusing particle, the viscosity of the medium, and temperature.
What is the Stokes-Einstein equation?
The Stokes-Einstein equation is D = kT/(6πηr), where k is Boltzmann's constant (1.381 × 10⁻²³ J/K), T is absolute temperature in Kelvin, η is the dynamic viscosity of the solvent in Pa·s, and r is the hydrodynamic radius of the diffusing particle in metres. It was derived by Albert Einstein in 1905 using the balance between thermal energy (driving diffusion) and viscous drag (opposing motion). It is valid for spherical particles in a continuum medium where r >> solvent molecule size.
What is the hydrodynamic radius (Stokes radius)?
The hydrodynamic (Stokes) radius is the effective spherical radius of a particle that gives the correct diffusion coefficient in the Stokes-Einstein equation. For a rigid sphere, it equals the actual geometric radius. For non-spherical or flexible molecules (polymers, proteins), the hydrodynamic radius is larger than the crystallographic radius and depends on molecular shape, solvation shell, and conformational flexibility. Protein hydrodynamic radii typically range from 1–10 nm; DNA from 5–100 nm; small molecules from 0.1–0.5 nm.
What are typical diffusion coefficients for molecules in water at 25°C?
Typical D values in water at 25°C: Small molecules (MW ~100): 5–10 × 10⁻¹⁰ m²/s. Proteins (~50 kDa, r ≈ 3 nm): ~7 × 10⁻¹¹ m²/s. Antibodies (~150 kDa, r ≈ 5 nm): ~4 × 10⁻¹¹ m²/s. Large protein complexes (~1 MDa): ~1 × 10⁻¹¹ m²/s. 100 nm nanoparticles: ~4 × 10⁻¹² m²/s. Colloidal gold (d = 20 nm): ~2 × 10⁻¹¹ m²/s. By comparison, diffusion in air is 10,000× faster: O₂ in air has D ≈ 2 × 10⁻⁵ m²/s.
How does temperature affect diffusion coefficient?
The Stokes-Einstein equation shows that D ∝ T/η. Both T and viscosity change with temperature. For water: η(25°C) = 0.89 mPa·s; η(37°C) = 0.69 mPa·s. At 37°C: D(37°C)/D(25°C) = (310.15/0.69) / (298.15/0.89) = 449.5/334.9 = 1.34. Physiological temperature (37°C) gives about 34% higher diffusion coefficients than 25°C room temperature, significant for calculating drug diffusion kinetics and intracellular transport rates in vivo.
How do I use the Diffusion Coefficient Calculator?
Enter the temperature in °C, the solvent viscosity in mPa·s (water at 25°C = 0.89 mPa·s; at 37°C = 0.69 mPa·s), and the particle/molecule radius in nanometres. The calculator applies D = kT/(6πηr) and returns D in m²/s and cm²/s, plus the mean square displacement after 1 second in 3D.
What is mean square displacement (MSD) and why is it useful?
Mean square displacement (MSD) quantifies how far a particle travels from its starting point by diffusion in time t: <r²> = 6Dt in 3D (or 4Dt in 2D, or 2Dt in 1D). The square root √(6Dt) gives the root mean square displacement, the typical distance travelled by diffusion. For a protein with D = 7 × 10⁻¹¹ m²/s in 1 second: √(6 × 7×10⁻¹¹ × 1) = 20.5 μm. This tells you the length scale over which diffusion is significant on a given timescale, critical for cell biology and drug delivery design.
Is the Stokes-Einstein equation accurate for small molecules?
The Stokes-Einstein equation assumes the particle is much larger than the solvent molecules, so the solvent can be treated as a continuum. For small molecules (molecular weight < 200 Da) where r is comparable to solvent molecule size, the equation overestimates D by 20–50%. Corrections exist for small molecules using the microviscosity or slip boundary conditions instead of stick boundary conditions (6πηr is for stick; 4πηr for perfect slip), but for practical purposes the Stokes-Einstein equation is a good order-of-magnitude estimate even for small solutes.
How is diffusion coefficient used in drug delivery and pharmaceutical formulation?
Diffusion coefficient governs how quickly drugs move through biological barriers: across cell membranes, through mucus, within tumor interstitium, and through solid dosage form matrices. For sustained-release tablets, the drug diffuses through a polymer matrix with D typically 10⁻¹⁴ to 10⁻¹⁰ m²/s. For targeted nanoparticle drug delivery (relevant to many cancer treatments developed by Indian pharmaceutical companies like Sun Pharma), D of the nanoparticle determines how far it can penetrate into tumor tissue. The FDA and CDSCO both require pharmacokinetic data that implicitly depends on diffusion coefficients.
What is dynamic light scattering (DLS) and how does it measure diffusion coefficient?
Dynamic light scattering (DLS) is the primary experimental technique for measuring diffusion coefficients of particles and macromolecules in solution. It measures the time-dependent fluctuations in scattered laser light caused by Brownian motion. The diffusion coefficient is extracted from the autocorrelation function of the scattered light. From D, the hydrodynamic radius is calculated via the Stokes-Einstein equation: r = kT/(6πηD). DLS particle size analysers are standard equipment in pharmaceutical formulation labs in India (required by CDSCO for nanoparticle characterisation).