How to Calculate the Diffusion Rate

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Diffusion Rate Calculator (Fick's First Law)

This calculator determines the rate of diffusion (flux) based on Fick's First Law. It calculates how fast a substance moves from an area of higher concentration to an area of lower concentration across a specific distance.

Typically in m²/s or cm²/s
e.g., mol/m³ or g/cm³
must be lower than $C_1$
Distance between the points, e.g., meters (m)
function calculateDiffusion() { var D = parseFloat(document.getElementById('diffCoef').value); var C1 = parseFloat(document.getElementById('concHigh').value); var C2 = parseFloat(document.getElementById('concLow').value); var dx = parseFloat(document.getElementById('distance').value); var resultDiv = document.getElementById('diffusionResult'); if (isNaN(D) || isNaN(C1) || isNaN(C2) || isNaN(dx)) { resultDiv.innerHTML = 'Please fill in all numeric fields.'; return; } if (dx <= 0) { resultDiv.innerHTML = 'Distance must be greater than zero.'; return; } if (C2 > C1) { resultDiv.innerHTML = 'Warning: Higher concentration should be greater than lower concentration for standard net flow.'; // We continue calculation but warn. } // Fick's First Law: J = D * (C1 – C2) / dx // The negative sign in the standard formula J = -D * (dC/dx) is accounted for // by taking (High Conc – Low Conc) to get a positive flux magnitude in the direction of flow. var flux = D * (C1 – C2) / dx; var unitHint = ""; // Simple heuristic for unit hints based on common inputs, not exhaustive. if (D < 1e-4 && dx < 1) { // Likely SI units (m²/s, m) unitHint = " (e.g., mol · m⁻² · s⁻¹)"; } else { unitHint = " (units depend on inputs)"; } resultDiv.innerHTML = 'Diffusion Rate (Flux, $J$): ' + flux.toExponential(4) + unitHint; }

Understanding Diffusion and Fick's First Law

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in concentration.

The fundamental equation describing this process in one dimension is Fick's First Law. It states that the flux (rate of flow per unit area) of a substance is proportional to the concentration gradient.

The formula is generally written as:

$J = -D \frac{dC}{dx}$

Where:

  • $J$ is the diffusion flux: This is the amount of substance that flows through a unit area per unit time. Common units are mol/(m²·s) or g/(cm²·s).
  • $D$ is the diffusion coefficient (or diffusivity): A proportionality constant that depends on the diffusing substance, the medium it's moving through, and the temperature. Its units are area per time, such as m²/s or cm²/s.
  • $C$ is the concentration: The amount of substance per unit volume, e.g., mol/m³ or g/cm³.
  • $x$ is the position (distance): The length unit along the direction of diffusion, e.g., m or cm.
  • $dC/dx$ is the concentration gradient: The change in concentration over a distance. The negative sign indicates that diffusion occurs in the direction of decreasing concentration (down the gradient).

For practical calculation between two distinct points, we can simplify the gradient term to $(C_{final} – C_{initial}) / \Delta x$. To find the magnitude of the flow from high to low concentration, we use:

$J = D \frac{(C_{high} – C_{low})}{\Delta x}$

Example Calculation

Let's calculate the diffusion of glucose through a membrane of thickness 0.01 cm.

  • The diffusion coefficient ($D$) of glucose in the membrane is $5 \times 10^{-7}$ cm²/s.
  • The concentration on the high side ($C_1$) is $0.2$ mol/cm³.
  • The concentration on the low side ($C_2$) is $0.05$ mol/cm³.
  • The distance ($dx$) is $0.01$ cm.

Using the formula:

$J = (5 \times 10^{-7} \text{ cm²/s}) \times \frac{(0.2 – 0.05) \text{ mol/cm³}}{0.01 \text{ cm}}$

$J = (5 \times 10^{-7}) \times \frac{0.15}{0.01} \text{ mol} \cdot \text{cm}^{-2} \cdot \text{s}^{-1}$

$J = (5 \times 10^{-7}) \times 15 \text{ mol} \cdot \text{cm}^{-2} \cdot \text{s}^{-1}$

$J = 7.5 \times 10^{-6} \text{ mol}/(\text{cm}^2 \cdot \text{s})$

The rate of glucose diffusion across the membrane is $7.5 \times 10^{-6}$ moles per square centimeter per second.

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