In electrochemistry, the Nernst equation is an equation which can be used (in conjunction with other information) to determine the equilibrium reduction potential of a half-cell in an electrochemical cell. It can also be used to determine the total voltage (electromotive force) for a full electrochemical cell. It is named after the German physical chemist who first formulated it, Walther Nernst.
The two (ultimately equivalent) equations for these two cases (half-cell, full cell) are as follows:
At room temperature (25 °C), RT/F may be treated like a constant and replaced by .025679 V or 25.679 mV for cells.
It can also be written in terms of millivolts using 59.1 mV instead of .0591 V.
The Nernst potential, of a ion of charge , across a membrane, is determined by the concentration ratio in and outside the cell:
When the membrane is in thermodynamic equilibrium, i.e. no net flux of ions, the membrane potential must be equal to the Nernst potential. However, in physiology, due to active ion pumps, the inside and outside of a cell are not in equilibrium. In this case the resting potential can be determined from e.g. the Goldman equation.
The potential across the cell membrane that exactly opposes net diffusion of a particular ion through the membrane is called the Nernst potential for that ion. As seen above, the magnitude of the Nernst potential is determined by the ratio of the concentrations of that specific ion on the two sides of the membrane. The greater this ratio, the greater the tendency for the ion to diffuse in one direction, and therefore the greater the Nernst potential required to prevent the diffusion.
The Nernst Equation may be derived in several different ways. Chemistry textbooks frequently give the derivation in terms of entropy and the Gibbs free energy, but there is a more intuitive method for anyone familiar with Boltzmann factors.
For simplicity, we will consider a solution of redox-active molecules that undergo a one electron reversible reaction
The ratio of oxidized to reduced molecules, [Ox]/[Red], is equivalent to the probability of being oxidized (giving electrons) over the probability of being reduced (taking electrons), which we can write in terms of the Boltzmann factors for these processes:
Quantities here are given per molecule, not per mole, and so Boltzmann's constant k and the electron charge e are used instead of the gas constant R and Faraday's constant F. To convert to the molar quantities given in most chemistry textbooks, it is simply necessary to multiply by Avogadro's number: and .
The entropy of a molecule is defined as
In an electrochemical cell, the cell potential E is the chemical potential available from redox reactions (). E is related to the Gibbs free energy change only by a constant: , where n is the number of electrons transferred. (There is a negative sign because a spontaneous reaction has a negative and a positive E.) The Gibbs free energy is related to the entropy by , where H is the enthalpy and T is the temperature of the system. Using these relations, we can now write the change in Gibbs free energy,
At equilibrium, E = 0 and Q = K. Therefore
Or at standard temperature,
In dilute solutions, the Nernst equation can be expressed directly in terms of concentrations (since activity coefficients are close to unity). But at higher concentrations, the true activities of the ions must be used. This complicates the use of the Nernst equation, since estimation of non-ideal activities of ions generally requires experimental measurements.
The Nernst equation also only applies when there is no net current flow through the electrode. The activity of ions at the electrode surface changes when there is current flow, and there are additional overpotential and resistive loss terms which contribute to the measured potential.
At very low concentrations of the potential determining ions, the potential predicted by Nernst equation tends to ±infinity. This is physically meaningless because, under such conditions, the exchange current density becomes very low, and then other effects tend to take control of the electrochemical behavior of the system.
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