The Nernst equation is used extensively in the discussion of resting membrane potential and action potentials in this book. The derivation presented here is necessarily mathematical and requires some knowledge of differential and integral calculus to understand thoroughly. However, I have tried to explain the meaning of each step in words; hopefully, this will allow those without the necessary background to follow the logic qualitatively.

This derivation of the Nernst equation uses equations for the movement of ions down concentration and electrical gradients to arrive at a quantitative description of the equilibrium condition. The starting point is the realization that at equilibrium there will be no net movement of the ion across the membrane. In the presence of both concentration and electrical gradients, this means that the rate of movement of the ion down the concentration gradient is equal and opposite to the rate of movement of the ion down the electrical gradient. For a charged substance (an ion), movement across the membrane constitutes a transmembrane electrical current, I. Thus, at equilibrium

where IC and IE are the currents due to the concentrational and electrical gradients, respectively.

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