diffusion coefficient calculator

What is the diffusion coefficient?

The diffusion coefficient, usually written as D, quantifies how fast molecules, ions, or particles spread through a medium due to random molecular motion. A larger diffusion coefficient means faster spreading. A smaller value means slower transport.

In practical terms, D helps answer questions like: How quickly does oxygen move through tissue? How fast does salt diffuse through water? How rapidly does a drug molecule migrate through a polymer membrane?

Formulas used in this diffusion coefficient calculator

1) Fick's First Law

D = -J / (dC/dx)

• J = diffusive flux (mol·m⁻²·s⁻¹)
• dC/dx = concentration gradient (mol·m⁻⁴)
• D = diffusion coefficient (m²/s)

The negative sign reflects diffusion from higher concentration to lower concentration. If your sign convention is inconsistent, you may get a negative D; in that case, double-check signs for flux and gradient.

2) Mean Squared Displacement (MSD) Relationship

D = x² / (2nt)

• x = root-mean-square displacement (m)
• n = number of dimensions (1, 2, or 3)
• t = elapsed time (s)

This approach is common when diffusion is measured by particle tracking or microscopy.

How to use this calculator

For Fick's Law data

• Select Fick's First Law in the dropdown.
• Enter measured flux and concentration gradient.
• Click Calculate D to get the coefficient in m²/s.

For particle-tracking data

• Select Mean Squared Displacement.
• Enter displacement, time, and dimensionality.
• Click Calculate D.

Worked examples

Example A: Membrane transport (Fick's Law)

Suppose measured flux is J = -2.5×10⁻⁶ mol·m⁻²·s⁻¹ and concentration gradient is dC/dx = -0.015 mol·m⁻⁴.

D = -J/(dC/dx) = -(-2.5×10⁻⁶)/(-0.015) = 1.67×10⁻⁴ m²/s in magnitude (sign depends on your convention).

Example B: Brownian motion in 3D

A tracked particle has RMS displacement x = 8×10⁻⁴ m after t = 120 s in 3D.

D = x²/(2nt) = (8×10⁻⁴)² / (2×3×120) = 8.89×10⁻¹⁰ m²/s.

Typical diffusion coefficient ranges

• Gases in gases: approximately 10⁻⁵ to 10⁻⁴ m²/s
• Small molecules in liquids: approximately 10⁻⁹ to 10⁻¹⁰ m²/s
• Ions in water: roughly around 10⁻⁹ m²/s
• Solids in solids: often 10⁻²⁰ to 10⁻¹⁴ m²/s (strongly temperature-dependent)

What affects diffusion coefficient values?

• Temperature: higher temperature usually increases D.
• Viscosity: more viscous fluids generally decrease D.
• Molecule size: larger particles tend to diffuse more slowly.
• Medium structure: porous, crowded, or polymeric media can reduce effective diffusion.
• Interactions: electrostatic and chemical binding effects alter apparent transport rates.

Unit tips for accurate results

• Always convert length to meters.
• Use seconds for time.
• Ensure concentration gradients are in mol·m⁻⁴, not mol/L per cm.
• Check scientific notation inputs (for example, 2.5e-6).

FAQ

Can D be negative?

Physically, the diffusion coefficient magnitude is non-negative. A negative computed value usually indicates sign-convention inconsistency in input values.

Is this calculator suitable for non-ideal systems?

It is best for ideal or near-ideal diffusion estimates. In reactive systems, porous media, or multicomponent transport, more advanced models may be required.

What if my value looks unrealistic?

Re-check units first. Most large errors come from mixing cm and m, or minutes and seconds.

Final note

This diffusion coefficient calculator is a practical tool for students, researchers, and engineers who need quick, transparent calculations. For publication-grade analysis, pair these estimates with uncertainty analysis and experimental calibration.