## The Nernst Equation

The diffusion potential example of Figure 4-1 does not describe an equilibrium condition, but rather a transient situation that occurs only as long as there is a net diffusion of ions across the barrier. Equilibrium would be achieved in Figure 4-1 only when [Na+] and [Cl-] are the same in compartments 1 and 2. At that point, there would be no concentrational force to support net diffusion of either Na+ or Cl- across the membrane and there would be no electrical potential across the barrier. Under what conditions might there be a steady electrical potential at equilibrium? To see this, consider a small modification to the previous example, shown in Figure 4-2. In the new example, everything is as before, except that the barrier between the two compartments of the box is selectively permeable to Cl-: Na+ cannot cross. Once again, we assume that the box has rigid walls so that we can neglect movement of water for the present.

The analysis of the situation in Figure 4-2 is similar to that of the diffusion potential, except that now the "mobility" of Na+ is reduced effectively to zero by the permeability characteristics of the barrier. Chloride ions will move down their concentration gradient from compartment 1 to compartment 2, but now no positive charges accompany them and negative charges will quickly build up in compartment 2. Thus, the voltmeter will record an electrical potential across the barrier, with side 2 being negative with respect to side 1. Because only Cl- can cross the barrier, equilibrium will be reached when there is no further net movement of chloride across the barrier. This happens when the electrical force driving Cl- out of compartment 2 exactly balances the concen-trational force driving Cl- out of compartment 1. Thus, at equilibrium a chloride ion moves from side 1 to side 2 down its concentration gradient for every chloride ion that moves from side 2 to side 1 down its electrical gradient. There will be no further change in [Cl-] in the two compartments, and no further change in the electrical potential, once this equilibrium has been reached.

Equilibrium for an ion is determined not only by concentrational forces but also by electrical forces. Movement of an ion across a cell membrane is determined both by the concentration gradient for that ion across the membrane and by the electrical potential difference across the membrane. We will use these ideas extensively in this book, so the remainder of this chapter will be spent examining how these principles apply in simple model situations and in real cells.

What would be the measured value of the voltage across the barrier at equilibrium in Figure 4-2? This is a quantitative question, and the answer is provided by Equation (4-1), which is called the Nernst equation after the physical chemist who derived it. The Nernst equation for Figure 4-2 can be written as

fRT^

kzfJ

Here, ECl is the voltage difference between sides 1 and 2 at equilibrium, R is the gas constant, T is the absolute temperature, Z is the valence of the ion in question (-1 for chloride), F is Faraday's constant, ln is the symbol for the natural, or base e, logarithm, and [Cl-]1 and [Cl-]2 are the chloride concentrations in compartments 1 and 2.

The value of electrical potential given by Equation (4-1) is called the equilibrium potential, or Nernst potential, for the ion in question. For example, in Figure 4-2 the permeant ion is chloride and the electrical potential, ECl, across the barrier is called the chloride equilibrium potential. If the barrier in

Figure 4-2 allowed Na+ to cross rather than Cl-, Equation (4-1) would again apply, except that [Na+]1 and [Na+]2 would be used instead of [Cl-], and the valence would be +1 instead of -1. If sodium were the permeant ion, the resulting potential, ENa, would be the sodium equilibrium potential. The Nernst equation applies only to one ion at a time and only to ions that can cross the barrier.

A derivation of Equation (4-1) is given in Appendix A. The Nernst equation comes from the realization that at equilibrium the total change in energy encountered by an ion in crossing the barrier must be zero. If the change in energy were not zero, there would be a net force driving the ion in one direction or the other, and the ion would not be at equilibrium. There are two important sources of energy change involved in crossing the barrier shown in Figure 4-2: the electric field and the concentration gradient. Nernst arrived at his equation by setting the sum of the concentrational and electrical energy changes across the barrier to zero.

In biology, we usually work with a simplified form of Equation (4-1):

'58 mVA

The simplification arises from converting from base e to base 10 logarithms, evaluating (RT/F) at standard room temperature (20°C), and expressing the result in millivolts (mV). That is where the constant 58 mV comes from in Equation (4-2). From the simplified Nernst equation, it can be seen that ECl in Figure 4-2 would be -58 mV. That is, in crossing the barrier from side 1 to side 2, we would encounter a potential change of 58 mV, with side 2 being negative with respect to side 1. This is as expected from the fact that chloride ions, and therefore negative charges, are accumulating on side 2. If the barrier were selectively permeable to Na+ rather than Cl-, the voltage across the barrier would be given by ENa, which would be +58 mV given the values in Figure 4-2. What would be the equilibrium potential for chloride in Figure 4-2 if the concentration of NaCl was 1.0 M on both sides of the barrier? (Hint: in that case the concentration gradient would be zero.)

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