where D+ is the diffusion coefficient of the monovalent cation; D_ is the diffusion coefficient of the monovalent anion; Co and C\ are the concentrations of the salt in compartments o and i, respectively; R is the gas constant; T is the absolute temperature; and F is the Faraday constant. At 37°C (T = 310K), 2.3 RT/F ' 60 mV. Thus,

Equation 8 discloses the following characteristics of diffusion potentials:

Clearly, when Co = Cii, there can be no net flow due to diffusion. Diffusion potentials can arise only in a system containing ion gradients:

Clearly, if both ions have equal mobilities (or sizes), there is no inherent leader and no inherent lagger, so dipoles are not formed. Another way of viewing this condition is that, when both ions have the same mobility, a one-to-one flow across the disk (i.e., bulk electroneutrality) is assured by the fact that both ions are driven by the same AC; a diffusion potential is not necessary for the preservation of bulk electroneutrality.

When Cio is not equal to Cii, the magnitude of V is directly proportional to the difference between the individual ionic diffusion coefficients. The orientation of V is such that it retards the more mobile ion and accelerates the less mobile ion. Thus, if the mobility of the anion exceeds that of the cation, the more dilute solution will be electrically negative with respect to the more concentrated solution. Conversely, if the cation has a greater mobility than the anion, the more dilute solution will be electrically positive with respect to the concentrated solution.

A particularly interesting situation arises when the barrier is impermeant to one of the ionic species—say,

The same expression, with the opposite sign, is obtained if we set D_ = 0.

Equation 9 states that, when one of the ionic species of a dissociated salt cannot penetrate the barrier, V is dependent only on the concentration ratio across the membrane and is independent of the permeability (or diffusion coefficient) of the ion that can penetrate the barrier. The reason for this independence becomes evident when we recall the law of electroneutrality. If the membrane is impermeant to one of the ions, there can be no net flow of the other, permeant, ion across the membrane, or bulk electroneutrality would be violated. Thus, despite the presence of a concentration difference, there can be no net diffusion of salt across the barrier, and the system is in a state of equilibrium. Because the net flow of the permeant ion is prohibited, the mobility of this ion is of no importance.

The following example may serve to clarify this point and the roles played by concentration differences and electrical potential differences in the overall driving force for the diffusion of ions. Let us place a 0.1-M solution of K+ proteinate in compartment o and a 0.01-M solution of the same salt in compartment i. If the barrier is impermeant to the large proteinate anion, the electrical potential difference across the barrier at 37°C will be:

with the dilute solution electrically positive compared to the concentrated solution. Now we may ask, "Why is there no diffusion of K+ from compartment o to compartment i down a tenfold concentration difference despite the fact that the membrane is highly permeable to K+?'' The answer derives from the fact that there are two forces acting on the K+ ion. There is a chemical force arising from the fact that the concentration of K+ in compartment o is ten times that in compartment i; this force tends to drive K+ from compartment o to compartment i. In addition, Eq. 9 tells us that there is a 60-mV electrical potential difference between compartments o and i, with compartment i electrically positive. This electrical potential difference tends to drive the positively charged K+ ion from compartment i to compartment o. These two oppositely directed driving forces (the chemical force and the electrical force) exactly balance each other so that there is no net driving force for the diffusion of K+ and, hence, no net movement.

Equation 9 was derived by the great German physical chemist Walther Hermann Nernst (1864-1941) from thermodynamic considerations and is referred to as the Nernst equation (also under these conditions it is often referred to as the Nernst potential or the Nernst equilibrium potential). In essence, it embodies the fact that (under conditions of uniform temperature and pressure) there are only two driving forces that influence the diffusion of charged particles: a force arising from concentration differences and a force arising from electrical potential differences. When solutions having different concentrations of the same dissociable salt are placed on opposite sides of a barrier that is impermeant to one of the dissociation products, there will be no net movement of salt across the barrier despite the concentration difference. Net movement of the permeant ion is prevented by the development of an electrical potential difference across the barrier whose magnitude and orientation are such that they exactly cancel the driving force arising from the chemical concentration difference across the barrier. (A more detailed discussion of this point is beyond the scope of this presentation. In essence, the two forces that influence the movements of charged particles—concentration differences and electrical potential differences—can be converted into the same units of force per ion or force per mole [e.g., dynes/ mol], and a tenfold concentration ratio at 37°C exerts the same force on a monovalent ion as does a 60-mV electrical potential difference. When, as in the above example, the two forces are oriented in opposite directions, the net force is zero, and there can be no net flow.)

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