Calculating Half-Times of Diffusion
Diffusion Is Rapid over Short Distances but Extremely Slow over Long Distances
From Fick's law, one can derive an expression for the time it takes for a substance to diffuse a particular distance. If one defines conditions such that all the solute molecules are concentrated at the starting position (see textbook Figure 3.8A), then the concentration front moves away from the starting position, as shown for a later time point in textbook Figure 3.8B. As the substance diffuses away from the starting point, the concentration gradient becomes less steep (Δcs decreases) and thus net movement becomes slower.
The time it takes for the substance at any given distance from the starting point to reach one-half of the concentration at the starting point (tc = ½) is given by the following equation:
where K is a constant and Ds is the diffusion coefficient. The above equation shows that the time required for a substance to diffuse a given distance increases in proportion to the square of that distance. Let us consider two numerical examples. First, how long it would take a small molecule to diffuse across a typical cell? The diffusion coefficient for a small molecule like glucose is about 10–9 m2 s–1, and the cell length may be 50 µm. Thus, for this example:
This calculation shows that small molecules diffuse over cellular dimensions rapidly. What about diffusion over longer distances? Calculating the time needed for the same substance to move a distance of 1 m (e.g., the length of a corn leaf), we find:
a value that exceeds by orders of magnitude the life span of a corn plant, which lives only a few months. This shows that diffusion in solutions can be effective within cellular dimensions but is far too slow for mass transport over long distances. As you will learn in Chapter 4 of the text, diffusion is of great importance as a driving force for the water vapor lost from leaves, because the diffusion coefficient for a molecule in air is much greater than in aqueous solutions.