Diffusion and permeability

Diffusion is the migration of molecules down a concentration gradient

Experience tells us that molecules always move spontaneously from a region where they are more concentrated to a region where they are less concentrated. As a result, concentration differences between regions gradually diminish as the movement proceeds. Diffusion always transports molecules from a region of high concentration to a region of low concentration, because the underlying molecular movements are completely random . That is, any given molecule has no preference for moving in any particular direction. The effect is easy to illustrate. Imagine two adjacent regions of comparable volume in a solution ( Fig. 2.1 ). There are 5200 molecules in the left-hand region and 5000 molecules in the right-hand region. For simplicity, assume that the molecules may move only to the left or to the right. Because the movements are random, at any given moment approximately half of all molecules would move to the right and approximately half would move to the left. This means that, on average, roughly 2600 would leave the left side and enter the right side, whereas 2500 would leave the right and enter the left. Therefore a net movement of approximately 100 molecules would occur across the boundary going from left to right. This net transfer of molecules caused by random movements is indeed from a region of higher concentration into a region of lower concentration.

Fick’s first law of diffusion summarizes our intuitive understanding of diffusion

The preceding discussion indicates that the larger the difference in the number of molecules between adjacent compartments, the greater the net movement of molecules from one compartment into the next. In other words, the rate at which molecules move from one region to the next depends on the concentration difference between the two regions. The following definitions can be used to obtain a more explicit and quantitative representation of this observation:

• 1.

  Concentration gradient is the change of concentration, Δ C , with a change in distance, Δ x (i.e., Δ C/ Δ x ).

• 2.

  Flux (symbol J ) is the amount of material passing through a certain cross-sectional area in a certain amount of time.

With these definitions, the earlier observation can be simply restated as “flux is proportional to concentration gradient,” or

By inserting a proportionality constant, D , we can write the foregoing expression as an equation:

The proportionality constant, D , is referred to as the diffusion coefficient or diffusion constant . The minus sign accounts for the fact that the diffusional flux, or movement of molecules, is always down the concentration gradient (i.e., flux is from a region of high concentration to a region of low concentration). The graphs in Fig. 2.2 illustrate this sign convention.

Equation 2.2 applies to the case in which the concentration gradient is linear, that is, a change in concentration, Δ C, for a given change in distance, Δ x. For cases in which the concentration gradient may not be linear, the equation can be generalized by replacing the linear concentration gradient, Δ C /Δ x, with the more general expression for concentration gradient, dC / dx (a derivative). The diffusion equation now takes the form

This equation is also known as Fick’s First Law of Diffusion . It is named after Adolf Fick, a physician who first analyzed this problem in 1855.

To complete the discussion of Fick’s First Law, we should examine the dimensions (or units) associated with each parameter appearing in Equation 2.3 . Because flux, J , is the quantity of molecules passing through unit area per unit time, it has the dimensions of “moles per square centimeter per second” (= [mol/cm ² ]/sec = mol·cm ⁻² ·sec ⁻¹ ). Similarly, the concentration gradient, dC / dx , being the rate of change of concentration with distance, has dimensions of “moles per cubic centimeter per centimeter” (= [mol/cm ³ ]/cm = mol·cm ⁻⁴ ). For all the units to work out correctly in Equation 2.3 , the diffusion coefficient, D , must have dimensions of cm ² /sec (= cm ² ·sec ⁻¹ ).

Essential aspects of diffusion are revealed by examining random, microscopic movements of molecules