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# The Nernst and Goldman equations

- Measurement
- Exam priority: medium

Two formulas that work out a cell's voltage from the ion concentrations inside and outside it.

Updated 2026-11-01

Also called: Equilibrium potential, Goldman-Hodgkin-Katz equation

## Key points

- Nernst at 37 °C: E = (61.5/z) × log(C_{out}/C_{in}); a tenfold change in the ratio shifts ≈61 mV for a monovalent ion.
- Key values: E_{K} ≈ −90 to −95 mV; E_{Na} ≈ +60 to +65 mV; E_{Ca} ≈ +120 mV; E_{Cl} ≈ −65 to −90 mV.
- Goldman: the membrane potential approaches the equilibrium potential of whichever ion is most permeant at that moment.
- An ion's driving force is V_{m} − E_{ion}: the farther the membrane is from its equilibrium, the more current flows when its channels open.

## What it is

The Nernst equation gives the equilibrium potential of a single ion: the voltage that exactly balances its chemical drive to diffuse. The Goldman-Hodgkin-Katz equation combines several ions weighted by the membrane's permeability to each and predicts the actual potential. Together they explain why rest is negative, why the action potential peak nears the Na⁺ equilibrium and how these values shift with serum K⁺.

## Key facts

- The sign depends on charge: for anions (z = −1) the log is inverted.
- Only a tiny number of ions need to cross to charge the membrane; bulk concentrations barely change.
- At rest P_{K} is far greater than P_{Na}; at the action potential peak the ratio briefly reverses.

### Action potential

**Rest** −62 mV **Threshold** −55 mV **Peak** +25 mV **Duration** ≈ 1–2 ms

Hyperkalemia: with more K⁺ outside the cell the K⁺ gradient falls and the resting potential becomes less negative (closer to threshold). The sustained depolarization inactivates some Na⁺ channels: the upstroke is slower and lower (in the heart, a wide QRS) and the myocyte repolarizes sooner (peaked T wave). Excitability first rises, then falls (weakness, arrhythmias).

Tap a phase (the pills above or its band) or a refractory period (ARP, RRP) at the base of the plot.

- Mainly Na⁺
- Mainly K⁺
- Rest
- Threshold
- ARP: absolute refractory period
- RRP: relative refractory period
- Curve at normal K⁺

With high extracellular K⁺, watch the resting level move towards zero and the action potential change shape.

## Equations

**Nernst**Eion = (61.5 / z) × log(Co / Ci)

**z**
ion charge

**C o**
extracellular concentration

**C i**
intracellular concentration

Valid at 37 °C with the result in mV. Example: Na⁺ 142/14 → ≈ +62 mV.

**Goldman-Hodgkin-Katz**Vm = 61.5 × log[(PK[K]o + PNa[Na]o + PCl[Cl]i) / (PK[K]i + PNa[Na]i + PCl[Cl]o)]

**P**
membrane permeability to each ion

Cl⁻ is inverted (inside on top) because it is an anion.

## Approximate equilibrium potentials (37 °C)

| Parameter | Value | Note |
| --- | --- | --- |
| E_{K} | ≈ −90 to −95 mV | K⁺ 4 out / 140 in |
| E_{Na} | ≈ +60 to +65 mV | Na⁺ 142 out / 10–14 in |
| E_{Cl} | ≈ −65 to −90 mV | varies by cell |
| E_{Ca} | ≈ +120 to +130 mV | free intracellular Ca²⁺ ≈0.0001 mM |

Approximate values for a healthy adult at rest; each laboratory sets its own ranges.

## Clinical notes

- Hyperkalaemia: as extracellular K⁺ rises, E_{K} becomes less negative and the resting membrane depolarises; hence the arrhythmia risk.
- Hypokalaemia: E_{K} more negative; in the myocardium it disturbs repolarisation and favours arrhythmias, especially with digoxin.

## Memory aid

Sixty-one per tenfold: each time a monovalent ion's ratio is multiplied by ten, its equilibrium moves about 61 mV.

## Test yourself

Think of the answer, then tap the question to see it.

## If extracellular K⁺ rises from 4 to 8 mEq/L, how much does E_{K} change, and which way?

61.5 × log 2 ≈ 18 mV more positive: from ≈ −94 to ≈ −76 mV. The resting membrane depolarises.

## Why does the action potential peak not reach E_{Na} exactly?

Some K⁺ permeability remains and Na⁺ channels inactivate quickly; by Goldman, the voltage approaches E_{Na} without reaching it.

## What does it mean for an ion to be 'at equilibrium' by Nernst?

Its chemical and electrical gradients are equal and opposite: there is no net flux even with its channels open.

## Why does Ca²⁺ rush in so strongly when its channels open?

E_{Ca} is ≈ +120 mV and the membrane rests near −80 mV: the driving force is huge, and the ion is divalent too.

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## Related topics

- [The resting membrane potential](https://durgical.com/en/physiology/the-resting-membrane-potential/)
- [The nerve action potential](https://durgical.com/en/physiology/the-nerve-action-potential/)
- [Ion channels and gating](https://durgical.com/en/physiology/ion-channels-and-gating/)
- [The ECG in potassium and calcium disorders](https://durgical.com/en/physiology/the-ecg-in-potassium-and-calcium-disorders/)

## Related conditions

- [Hyperkalaemia](https://durgical.com/en/conditions/hyperkalaemia/)
- [Hypokalaemia](https://durgical.com/en/conditions/hypokalaemia/)

[← All of physiology](https://durgical.com/en/physiology/)

Educational physiology content; not a diagnostic or treatment guide and no substitute for a health professional's advice. Values are approximate references. No doses are given.

Sources: [12.4 The Action Potential](https://openstax.org/books/anatomy-and-physiology-2e/pages/12-4-the-action-potential) (OpenStax (Rice University), 2022); [3.1 The Cell Membrane](https://openstax.org/books/anatomy-and-physiology-2e/pages/3-1-the-cell-membrane) (OpenStax (Rice University), 2022); Hall JE. Guyton and Hall Textbook of Medical Physiology, 13th ed. (ch. 5). Reference book used to check facts; its text and figures are not reproduced (Elsevier, 2016).

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Representative model, for educational purposes only.

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