The term diffusion refers to the net displacement (transport) of matter from one region to another due to random thermal motion. It is classically illustrated by an experiment in which an iodine solution is placed at the bottom of a cylinder and pure water is carefully layered above this colored solution. Initially, there is a sharp demarcation between the two solutions; however, as time progresses, the upper solution becomes increasingly colored, and the lower solution becomes progressively paler. Ultimately, the column of fluid achieves a uniform color, and the diffusion of iodine ceases. This state of maximum homogeneity, or uniformity, will persist as long as the cylinder is undisturbed; a perceptible deviation from homogeneity on the part of the entire column or any bulk (macroscopic) portion of the column will never be observed. The properties of the system, at this point, are said to be time independent, or, stated in another way, the system is said to have achieved a state of equilibrium.
The kinetic characteristics of diffusion can be readily developed by considering the simplified case illustrated in Fig. 3. Compartment o and compartment i both contain aqueous solutions of some uncharged solute (i) at concentrations C(o and Ci, respectively, where the superscripts designate the compartment and the subscript designates the solute. (An uncharged solute refers to a molecule that bears no net charge and thus includes solutes that have an equal number of oppositely charged groups such as zwitterions.) These compartments are separated by a sintered glass disk, which, because it possesses many large pores, may be treated as if it were an aqueous layer having a cross-sectional area A (the total area of the pores) and a thickness Ax; the disk simply serves as a barrier to prevent bulk mixing of the two solutions. We assume that each compartment is well stirred so that the concentrations Cio and Cii are uniform. Furthermore, for the sake of simplicity, we will also assume that both compartments have sufficiently large volumes so that their concentrations remain essentially constant during the period of observation, despite the fact that solute i may leave or enter these compartments across the disk.
Because the solute molecules are in continual random motion, due to the thermal energy of the system, there is a continual migration of these molecules across the disk in both directions. Thus, some of the molecules originally in compartment o will randomly wander across the barrier and enter compartment i. The rate of this process, because it is the result of random motions, is proportional to the likelihood that a molecule in compartment o will enter the opening of a pore in the
disk and is therefore proportional to Co. Thus, we may write:
Rate of molecular migration (diffusion) from o to i = kCo where k is a proportionality constant. Similarly, we may write:
Rate of molecular migration (diffusion) from i to o = kCii
The rate of net movement of molecules across the barrier is the difference between the rates of these two unidirectional movements so that:
Rate of net molecular migration (diffusion)
Thus, the net flow of an uncharged solute across a permeable barrier due to diffusion is directly proportional to the concentration difference across the barrier. In addition, the rate of transfer is directly proportional to the cross-sectional area A and inversely proportional to the thickness of the barrier Ax; that is, for a given concentration difference ACi, doubling the area of the disk will result in a doubling of the rate of transfer from one compartment to the other, whereas doubling the thickness of the disk will halve the rate of transfer. Therefore, we may replace k with a more explicit expression that contains a new proportionality constant, Di, where k = ADi/Ax, and we obtain:
Rate of net migration (diffusion) across the barrier = ADtACi/Ax
Dividing both sides of this expression by A, we obtain the rate of net flow per unit area, which is often referred to as the flux (or net flux). This is commonly symbolized by J, which is expressed in units of amount of substance per unit area per unit time (e.g., mol/cm2 hr). Thus,
where Di is the diffusion coefficient of the solute i and is a measure of the rate at which i can move across a barrier having an area of 1 cm2 and a thickness of 1 cm when the concentration difference across this barrier is 1 mol/L. The coefficient Di is dependent on the nature of the diffusing solute and the nature of the barrier or the medium in which it is moving (interacting); we shall examine this dependence in further detail below. If Ji is expressed in mol/cm2 hr, Ax in cm, and AC,- in mol/1000 cm3, then Di emerges with the (somewhat uninformative) units of cm2/hr.
Equation 1 describes diffusion across a flat or planar barrier having a finite thickness. Such systems are often called discontinuous systems, inasmuch as the barrier introduces a sharp and well-defined demarcation between the two surrounding solutions. Diffusion of iodine into water, in the experiment described above, represents a continuous system, as there is no discrete boundary between the two solutions. The general expression for diffusion in a continuous system can be derived from Eq. 1 by simply making the thickness of the disk vanish mathematically. Thus, as Ax approaches 0, ACi/Ax approaches dCi/dx so that:
Equation 2 was derived in 1855 by Fick, a physician, and is often referred to as Fick's (first) law of diffusion. It simply states that the rate of flow of an uncharged solute due to diffusion is directly proportional to the rate of change of concentration with distance in the direction of flow. The derivative dCi/dx is referred to as the concentration gradient and is the driving force for the diffusion of uncharged particles. Thus, Eq. 2 states that there is a linear relation between the diffusional flow of i and its driving force, where Di is the proportionality constant. Equation 2 is but one example of many linear relations between flows and driving forces observed in physical systems. For example, Ohm's law states that there is a linear relation between electrical current (flow) and its driving force, the voltage (electrical potential difference), where the proportionality constant is the electrical conductance g (recall that g = 1/R, where R is resistance). Thus, we can write:
We will see below that the diffusion of charged particles (ions) can be described by a combination of Eqs. 2 and 3, where the driving force is a combination of the concentration difference (or gradient) (chemical force or potential) and the electrical potential difference (or gradient).
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