Nernst equation
The Nernst equation gives the equilibrium potential for a single ion. It answers how concentration ratio, ion charge, and temperature determine the membrane voltage at which the chemical and electrical forces on that ion are exactly balanced.
Core definition and formula
The full equation uses the natural logarithm and absolute temperature. For one ion at a time, the equilibrium potential depends on the outside-to-inside concentration ratio and the ion valence \(z\).
\[
\begin{aligned}
E_{ion} &= \frac{R \cdot T}{z \cdot F}\ln\!\left(\frac{[ion]_{out}}{[ion]_{in}}\right)\cdot 1000
\end{aligned}
\]
At body temperature, a common classroom form for monovalent ions is:
\[
\begin{aligned}
E_{ion} &= \frac{61.5}{z}\log_{10}\!\left(\frac{[ion]_{out}}{[ion]_{in}}\right)
\end{aligned}
\]
Symbols: \(E_{ion}\) is the equilibrium potential in mV, \(R\) is the gas constant, \(T\) is temperature in kelvin, \(F\) is Faraday’s constant, and \(z\) is the ionic charge. A positive \(z\) is used for cations such as K+ and Na+, while a negative \(z\) is used for anions such as Cl−.
How to interpret the result
A positive equilibrium potential means the membrane would need to be positive inside, relative to outside, to balance that ion. A negative equilibrium potential means the membrane would need to be negative inside. The result is for one ion only, not the full resting membrane potential of a real cell.
If a membrane potential \(V_m\) is also known, the driving force can be estimated with \(V_m - E_{ion}\). When \(V_m = E_{ion}\), there is no net electrochemical driving force for that ion. When they differ, the ion tends to move in the direction that brings the membrane potential closer to its own equilibrium potential.
- Using concentrations in the wrong order inside the logarithm.
- Forgetting to use a negative charge for Cl−.
- Using °C directly instead of kelvin in the full equation.
- Interpreting \(E_{ion}\) as the total membrane potential for the whole cell.
Micro example: for K+ with \([K^+]_{in} = 140\) mM and \([K^+]_{out} = 5\) mM at body temperature, the equilibrium potential is strongly negative. That explains why K+ is often associated with negative voltages in neurophysiology.
This tool is useful when the focus is one ion at a time: K+, Na+, Cl−, or Ca2+. It is not the correct endpoint when multiple ions and membrane permeability must be combined; the next step after the Nernst equation is the Goldman-Hodgkin-Katz equation and driving-force analysis in membrane physiology.