# How to Calculate Diffusion Rate ••• oksix/iStock/GettyImages

Diffusion takes place because of particle motion. Particles in random motion, like gas molecules, bump into one another, following Brownian motion, until they disperse evenly in a given area. Diffusion then is the flow of molecules from an area of high concentration to that of low concentration, until equilibrium is reached. In short, diffusion describes a gas, liquid or solid dispersing throughout a particular space or throughout a second substance. Diffusion examples include a perfume aroma spreading throughout a room, or a drop of green food coloring dispersing throughout a cup of water. There a number of ways to calculate diffusion rates.

#### TL;DR (Too Long; Didn't Read)

Remember that the term "rate" refers to the change in a quantity over time.

## Graham’s Law of Diffusion

In the early 19th century, Scottish chemist Thomas Graham (1805-1869) discovered the quantitative relationship that now bears his name. Graham’s law states that the diffusion rate of two gaseous substances is inversely proportional to the square root of their molar masses. This relationship was arrived at, given that all gases found at the same temperature exhibit the same average kinetic energy, as understood in the Kinetic Theory of Gases. In other words, Graham’s law is a direct consequence of the gaseous molecules having the same average kinetic energy when they are at the same temperature. For Graham’s law, diffusion describes gases mixing, and the diffusion rate is the rate of that mixing. Note that Graham’s Law of Diffusion is also called Graham’s Law of Effusion, because effusion is a special case of diffusion. Effusion is the phenomenon when gaseous molecules escape through a tiny hole into a vacuum, evacuated space or chamber. The effusion rate measures the speed by which that gas is transferred into that vacuum, evacuated space or chamber. So one way of calculating diffusion rate or effusion rate in a word problem is to make calculations based on Graham’s law, which expresses the relationship between molar masses of gases and their diffusion or effusion rates.

## Fick’s Laws of Diffusion

In the mid-19th century, German-born physician and physiologist Adolf Fick (1829-1901) formulated a set of laws governing the behavior of a gas diffusing across a fluid membrane. Fick’s First Law of Diffusion states that flux, or the net movement of particles in a specific area within a specific period of time, is directly proportional to the gradient’s steepness. Fick’s First Law can be written as:

flux = -D(dC ÷ dx)

where (D) refers to the diffusion coefficient and (dC/dx) is the gradient (and is a derivative in calculus). So Fick’s First Law fundamentally states that random particle movement from Brownian motion leads to the drift or dispersal of particles from regions of high concentration to low concentrations – and that drift rate, or diffusion rate, is proportional to the gradient of density, but in the opposite direction to that gradient (which accounts for the negative sign in front of the diffusion constant). While Fick’s First Law of Diffusion describes how much flux there is, it is in fact Fick’s Second Law of Diffusion that further describes the rate of diffusion, and it takes the form of a partial differential equation. Fick’s Second Law is described by the formula:

T = (1 ÷ [2D])x2

which means that the time to diffuse increases with the square of the distance, x. Essentially, Fick’s First and Second Laws of Diffusion provide information on how concentration gradients affect diffusion rates. Interestingly enough, the University of Washington devised a ditty as a mnemonic to help remember how Fick’s equations assist in calculating diffusion rate: “Fick says how quick a molecule will diffuse. Delta P times A times k over D is the law to use…. Pressure difference, surface area and the constant k are multiplied together. They’re divided by diffusion barrier to determine the exact rate of diffusion.”

## Other Interesting Facts About Diffusion Rates

Diffusion can occur in solids, liquids or gases. Of course, diffusion takes place fastest in gases and slowest in solids. Diffusion rates can likewise be affected by several factors. Increased temperature, for instance, speeds up diffusion rates. Similarly, the particle being diffused and the material it is diffusing into can influence diffusion rates. Notice, for example, that polar molecules diffuse faster in polar media, like water, whereas nonpolar molecules are immiscible and thereby have a hard time diffusing in water. Density of the material is yet another factor affecting diffusion rates. Understandably, heavier gases diffuse far more slowly compared to their lighter counterparts. Moreover, the size of the area of interaction can impact diffusion rates, evidenced by the aroma of home cooking dispersing through a small area faster than it would in a larger area.

Also, if diffusion takes place against a concentration gradient, there must be some form of energy that facilitates the diffusion. Consider how water, carbon dioxide and oxygen can easily cross cell membranes by passive diffusion (or osmosis, in the case of water). But if a large, non-lipid soluble molecule has to pass through the cell membrane, then active transport is required, which is where the high-energy molecule of adenosine triphosphate (ATP) steps in to facilitate the diffusion across cellular membranes.

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