Diffusion Of Electrolytes

The previous section dealt with the diffusion of uncharged particles, but many of the fundamental properties of the diffusional process described therein also apply to the diffusion of charged particles. In both instances, net flow due to diffusion is the result of random thermal movements, and the diffusion coefficients of the particles are inversely proportional to their molecular or hydrated ionic radii. However, because ions bear a net electrical charge, the diffusion of a salt such as NaCl, which exists in aqueous solution in the form of oppositely charged dissociated ions, is somewhat more complicated for two reasons. First, because of the attractive force between particles bearing opposite electrical charges, the ions resulting from the dissociation of a salt do not diffuse independently. Second, under conditions of uniform temperature and pressure, only a difference in concentration can provide the driving force for the diffusion of uncharged particles; net flows do not occur in the absence of concentration differences. For the case of a charged particle, the driving force for diffusion is made up of two components: (1) a difference in concentration, and (2) the presence of an electrical field. The effect of an electrical field on the movement of charged particles is readily illustrated by the familiar phenomenon of electropho-resis. If an anode and a cathode are placed in a beaker that contains a homogeneous solution of NaCl, Cl-will migrate toward the anode, and Na+ will migrate toward the cathode. These net movements are due to attractive and repulsive forces that act on charged particles within an electrical field and take place despite the initial absence of a concentration difference. Each of these aspects of ionic diffusion is separately discussed below.

Origin of Diffusion Potentials

Consider the system illustrated in Fig. 3. If a solution containing equal concentrations of urea and sucrose is placed in compartment o and distilled water is placed in compartment i, both urea and sucrose will diffuse across the sintered glass disk independently. Because urea is smaller than sucrose, its diffusion coefficient will be greater than that of sucrose (i.e., Durea> Dsucrose), and it will move more rapidly in response to the same driving force (ACurea = ACsucrose) so that, after sufficient time has elapsed, the solution in compartment i will contain more urea than sucrose. In essence, we have employed differential rates of diffusion as a separatory method. If the barrier is permeable to urea but impermeable to sucrose, only urea will appear in compartment i; this is the principle of separation by dialysis. Now, let us replace the solution in compartment o with an aqueous solution of potassium acetate (KAc). This salt dissociates into a relatively small cation (K+) and a larger anion (Ac-). If the diffusion of these ions were independent (as in the example with urea and sucrose), it would be possible, in time, to withdraw from compartment i a solution that contains more cation than anion—that is, a solution that has excess K+. This, in fact, has never been observed. Indeed, the failure to observe a separation of charge in solutions, as well as compelling theoretical arguments, are the bases of a universally accepted postulate known as the law of electroneutrality. In essence, this law states that any macroscopic or bulk portion of a solution must contain an equal number of opposite charges; that is, it must be electrically neutral. This is a very important statement that must be kept in mind whenever we consider the bulk movements of ions. Returning to our example of the diffusion of KAc, we may ask, "Why does K+ not outdistance Ac-? What is responsible for the maintenance of electroneutrality under these conditions?'' The answer is that the tendency for K+ to outdistance Ac-because of the difference in ionic size (and, hence, diffusion coefficients) is counteracted by the electrostatic attractive forces that exist between these two oppositely charged particles. This mutual attraction, which tends to draw the two ions closer together, exactly balances the effect of the difference in ionic size, which would tend to draw them apart.

One may conceptualize the physical process as follows. In Fig. 6, we see the small ion (K+) and the larger ion (Ac-) lined up along a partition (the starting line). When the partition is removed, the race begins. Each ion initially moves from left to right at a rate determined by its ionic size; in other words, each ion initially diffuses from left to right at a rate determined by its inherent diffusion coefficient (denoted by DK and DAc). Because DK > DAc, there will be an initial separation of these oppositely charged ions, which, if unopposed, would eventually lead to a violation of the law of electroneutrality. However, the electrostatic attraction between these particles tends to hold them together. Thus, the attraction of Ac- for K+ tends to slow down the rate of diffusion of K+, and, conversely, the attraction of K+ for Ac- tends to speed up the rate of diffusion of Ac-. The net result is that the two ions move from left to right at the same rate but in an oriented fashion (K+ in front of Ac-) and thus form a small diffusing dipole (or ion pair). The diffusion coefficient of this ion pair, or dipole, is greater than that of Ac- alone but less than that of K+ (DK > DKAc > DAc).

Initial Transient

State State

FIGURE 6 Illustration of dipole (ion pair) formation as a result of the differential rates of diffusion of K+ and Ac".

(The situation resembles what would happen if a fast swimmer and a slow swimmer were connected by an elastic cord; there would be an initial rapid separation of the two swimmers, but soon the pair would move together at a rate intermediate to the rate at which each could swim independently.)

Now let us apply this ''dipole'' concept to diffusion of KAc across a sintered glass disk. Consider the system illustrated in Fig. 7. We place a well-stirred solution of KAc in compartment o and a more dilute solution of KAc in compartment i; diffusion from o to i is initially prevented by the presence of a solid partition adjacent to the sintered glass disk (Fig. 7A). If we now pair up a K+ ion with a neighboring Ac— ion so as to form hypothetical dipoles (using the convention that the head of the arrow represents the positive ion), we find, as shown in Fig. 7A, that the dipoles are randomly oriented simply because the distribution of ions in a homogeneous solution is random. Thus, for every dipole

FIGURE 7 Generation of a diffusion potential as a result of oriented ionic diffusion through a uniform membrane separating two well-stirred compartments. The head of the arrow represents the positive ion. (A) With the partition in place, the dipoles are randomly oriented, and there is no electrical potential difference between compartments. (B) When the partition is suddenly removed, the concentration difference across the membrane causes the dipoles to orient within the membrane and diffuse into the compartment with the lower concentration.

FIGURE 7 Generation of a diffusion potential as a result of oriented ionic diffusion through a uniform membrane separating two well-stirred compartments. The head of the arrow represents the positive ion. (A) With the partition in place, the dipoles are randomly oriented, and there is no electrical potential difference between compartments. (B) When the partition is suddenly removed, the concentration difference across the membrane causes the dipoles to orient within the membrane and diffuse into the compartment with the lower concentration.

pointed in a given direction there will be another dipole of equal magnitude oriented in the exactly opposite direction. The total dipole moments in compartments o and i, as well as within the membrane, are therefore equal to zero, and if electrodes are inserted into the two compartments the electrical potential difference between these electrodes will be zero.

If we suddenly remove the solid partition, K+ and Ac— will diffuse from the higher concentration in compartment o to the lower concentration in compartment i (Fig. 7B). For the reasons discussed above, this diffusional process can be represented as a series of dipoles crossing the sintered glass disk in an oriented fashion. The random orientations of the dipoles, characteristic of the two well-stirred homogeneous solutions in compartments o and i, have been converted within the disk to an oriented distribution by the presence of a concentration difference across the disk. The sum of all these oriented dipoles can be represented by a single dipole whose positive end is pointed toward compartment i and whose negative end is pointed toward compartment o. If electrodes are inserted into compartments o and i, compartment i will be found to be electrically positive with respect to compartment o.

It is important to emphasize that there is no bulk separation of charges because the distance between the leading K+ ion and the lagging Ac— ion averages only a few (~10) angstrom units. The electrical potential difference across the membrane is not due to a bulk (chemically detectable) separation of charge but to the fact that the ions cross the disk in an oriented fashion rather than in a random fashion. In essence, the orientation of the dipoles within the disk can be viewed as having converted the disk into a battery with the positive pole facing compartment i. (Technically speaking, there is a small separation of charges sufficient to charge the capacitance of the membrane, which is completed very rapidly; thereafter, the movement of anions and cations across the membrane is one to one. Furthermore, the amount of charge separation is minute compared to the number of ions present in the two solutions so that bulk electroneutrality is preserved; see also Chapter 9.)

The electrical potential difference arising from the diffusion of ions derived from a dissociable salt from a region of higher concentration to one of lower concentration is referred to as a diffusion potential. It arises whenever the ions resulting from the dissociation of the salt differ with respect to their mobilities or diffusion coefficients. The orientation of the diffusion potential is such as to retard the diffusion of the ion having the greater mobility and to accelerate the diffusion of the ion with the lower mobility so as to maintain electro-neutrality. Thus, the magnitude of a diffusion potential will be directly dependent on the difference between the mobilities of the anion and cation.

It can be shown that the diffusion potential (V) arising from the diffusion of a salt that dissociates into a monovalent anion and a monovalent cation is given by:

for example, the cation. When D+ = 0, Eq. 8 reduces to:

r iogl C

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