Nernst equation

The Nernst equation is a simple formula that answers one focused question about a single kind of charged particle (an ion, like sodium or potassium): given how much of it sits inside the cell versus outside, what voltage across the cell's thin outer skin (the membrane) would make that ion stop moving on average? Picture a crowd squeezed into one room and almost empty in the next, with a door between them. The packed crowd naturally pushes toward the emptier room. But if these people carry electric charge, you can set up an electrical pull at the door that pushes back just hard enough to cancel the crowding. The voltage that exactly balances those two opposing pushes — the chemistry pull from a difference in concentration, and the electrical pull from a difference in charge — is called that ion's equilibrium potential, and the Nernst equation is how you calculate it.

In its usual form the equation is E = (RT / zF) × ln([ion]outside / [ion]inside). Here E is the equilibrium voltage you are solving for; R is a fixed gas constant and T is temperature, so RT measures how vigorously particles jostle from heat; F is a fixed number that converts amounts of charge; z is the ion's charge (for example +1 for sodium or potassium, +2 for calcium, −1 for chloride); and ln is the natural logarithm, a way of turning a ratio of the two concentrations into a number. The whole thing says: the bigger the imbalance between inside and outside, the larger the voltage needed to hold that ion still — and the sign of the charge decides which way the voltage points. This single-ion answer is the building block for understanding the resting voltage of neurons and the rapid voltage swings that carry their signals.