The spontaneous movement of molecules from an area of higher concentration to an area of lower concentration is known as diffusion. The rate of diffusion, or flux, quantifies this movement, representing the amount of substance that moves across a specific area over a given period of time. Understanding how to calculate this rate is fundamental to fields ranging from cell biology to chemical engineering.
Understanding Fick’s First Law
The standard method for calculating the rate of diffusion in a steady state—where the concentration gradient remains constant over time—is through Fick’s First Law. This law states that the diffusion flux is directly proportional to the concentration gradient.
The law is commonly expressed by the formula: \(J = -D \cdot \frac{dC}{dx}\). Here, \(J\) represents the diffusion flux (rate per unit area, e.g., \(\text{mol} \cdot \text{m}^{-2} \cdot \text{s}^{-1}\)). \(D\) is the diffusion coefficient, a proportionality factor describing how easily a substance moves through a medium (e.g., \(\text{m}^2/\text{s}\)).
The term \(\frac{dC}{dx}\) is the concentration gradient, representing the change in concentration (\(C\)) over a change in distance (\(x\)). A steeper gradient results in a greater diffusion flux. The negative sign indicates that the net flux is always from high concentration to low concentration.
For simplified calculations across a defined distance, the differential term \(\frac{dC}{dx}\) is often replaced by \(\frac{\Delta C}{\Delta x}\). This discrete form uses the difference in concentration (\(\Delta C\)) divided by the distance the substance must travel (\(\Delta x\)). This expression is useful when analyzing diffusion across a layer of known thickness, allowing direct calculation of the gradient term.
Key Variables Affecting the Rate
The concentration gradient is the most direct factor affecting the diffusion rate. A greater difference in the amount of substance between two regions results in a faster rate of diffusion. As the system approaches equilibrium, the gradient lessens, and the rate of diffusion slows down significantly.
The distance, or thickness, that a substance must travel is inversely related to the rate of diffusion. A substance diffusing across a thin barrier will move much faster than one diffusing across a thick layer, because the distance (\(\Delta x\)) is in the denominator of the gradient term.
Temperature has a significant impact because it directly affects the kinetic energy of the molecules. Higher temperatures cause particles to move more rapidly, which increases the diffusion coefficient (\(D\)) and raises the rate of diffusion. Conversely, cooling a system reduces molecular movement and slows the diffusion rate.
The physical properties of the diffusing molecule and the medium itself also determine the value of \(D\). Molecules with a smaller mass or size tend to move faster and have a larger diffusion coefficient compared to heavier molecules.
The viscosity of the medium, which is its resistance to flow, also affects the diffusion coefficient. A substance diffusing through a thick, gel-like medium, such as cytoplasm, will experience greater resistance and a lower \(D\) value than the same substance diffusing through a less viscous medium, like air or pure water.
Step-by-Step Calculation Examples
Consider a solute diffusing across a membrane of \(1 \times 10^{-5}\) meters (\(10\) micrometers) thickness. Suppose the concentration of the solute is \(0.10 \text{ mol}/\text{m}^3\) on one side (\(C_1\)) and \(0.02 \text{ mol}/\text{m}^3\) on the other side (\(C_2\)). The diffusion coefficient (\(D\)) for this solute in the membrane material is \(5 \times 10^{-10} \text{ m}^2/\text{s}\).
The first step is to determine the concentration gradient, \(\frac{\Delta C}{\Delta x}\). This calculation yields a gradient of \(\frac{(0.02 \text{ mol}/\text{m}^3 – 0.10 \text{ mol}/\text{m}^3)}{1 \times 10^{-5} \text{ m}}\), resulting in \(-8,000 \text{ mol}/\text{m}^4\). The negative value confirms the direction of movement from \(C_1\) to \(C_2\).
Next, we apply Fick’s First Law in its simplified form: \(J = -D \cdot \frac{\Delta C}{\Delta x}\). Plugging in the values gives \(J = -(5 \times 10^{-10} \text{ m}^2/\text{s}) \cdot (-8,000 \text{ mol}/\text{m}^4)\). The two negative signs cancel out, yielding a positive flux.
The final calculated flux is \(4 \times 10^{-6} \text{ mol} \cdot \text{m}^{-2} \cdot \text{s}^{-1}\). This result means that \(4 \times 10^{-6}\) moles of the solute are moving across every square meter of the membrane each second.