The Nernst Equation and Equilibrium Potentials

Consider a membrane that separates two compartments containing different concentrations of a single permeant ion, say potassium (K⁺). If the membrane has channels open to K⁺, ions will diffuse down their concentration gradient, from the high-concentration side to the low-concentration side, driven by ordinary thermal diffusion — the same process that spreads perfume through a room. But K⁺ carries charge. As positive charge leaves the high-concentration compartment, that compartment becomes slightly negative relative to the other, and this emerging electrical potential difference pulls K⁺ back the way it came. Diffusion moves the ion one way; the resulting electric field pushes it back. These two forces reach a balance point, a specific voltage at which they are precisely equal and opposite. At that voltage, K⁺ ions still cross the membrane in both directions, but the flux in each direction is equal, so the net flux is zero. This voltage is the equilibrium potential for potassium, written E_K. The equilibrium potential depends on exactly one property of the system: the ratio of the ion's concentration outside to its concentration inside (and its valence, z). It does not depend on how many channels are open, how permeable the membrane is, or what other ions are doing. A membrane completely impermeable to K⁺ still has a well-defined E_K — it is simply not the membrane's actual voltage unless K⁺ channels happen to be open. This is the crucial conceptual separation: the equilibrium potential is a property of the concentration gradient (thermodynamics), while the actual membrane potential is a property of which channels are open and how permeable each is (kinetics/electrodynamics, formalized later by the Goldman-Hodgkin-Katz equation). Walther Nernst derived the quantitative relationship for general electrochemical half-cells using classical thermodynamics: at equilibrium, the electrochemical potential of the ion must be equal on both sides of the membrane. The electrochemical potential has a chemical term (dependent on concentration) and an electrical term (dependent on voltage and charge). Setting the two sides equal and solving for the voltage difference gives the Nernst equation.

Step 1 — Define the electrochemical potential. For an ion species with concentration [ion] and charge number z, in a region at electrical potential V, its electrochemical potential per mole is: μ = μ° + RT ln[ion] + zFV where μ° is a reference chemical potential, R is the gas constant (8.314 J/(mol·K)), T is absolute temperature in kelvin, and F is Faraday's constant (96,485 coulombs/mol, the charge on one mole of monovalent ions).

Step 2 — Apply the equilibrium condition. At equilibrium, no net work is done moving the ion from outside to inside, so the electrochemical potential is equal on both sides: μ°+ RT ln[ion]ₒᵤₜ + zFVₒᵤₜ = μ° + RT ln[ion]ᵢₙ + zFVᵢₙ

Step 3 — Solve for the voltage difference. Rearranging, and defining the membrane potential as V_m = Vᵢₙ − Vₒᵤₜ (the convention in neuroscience — inside relative to outside): zF(Vᵢₙ − Vₒᵤₜ) = RT ln([ion]ₒᵤₜ/[ion]ᵢₙ) E_ion = (RT / zF) · ln([ion]ₒᵤₜ / [ion]ᵢₙ) This is the Nernst equation. Notice the logic embedded in its structure: if the outside concentration is higher than inside ([out]/[in] > 1), the logarithm is positive, so a positive ion (z > 0) has a positive equilibrium potential — the inside must become positive to stop that ion's continued inward diffusion. This matches intuition: to repel further entry of a cation that "wants" to diffuse in, the inside must build up positive charge.

Step 4 — Convert to working units. At mammalian body temperature (37°C = 310 K), converting natural log to log base 10 (ln x = 2.303 log₁₀x) and plugging in constants: RT/F = (8.314 × 310) / 96,485 ≈ 26.7 mV (RT/F) × 2.303 ≈ 61.5 mV So for a monovalent ion (z = ±1) at body temperature: E_ion = 61.5 mV × log₁₀([ion]ₒᵤₜ / [ion]ᵢₙ) / z At room temperature (20°C, common in some textbook problems and experiments), the constant is closer to 58 mV; at 6.3°C (Hodgkin and Huxley's classic squid axon experiments), it is closer to 50 mV. For divalent ions like Ca²⁺, z = 2, so the coefficient halves, to about 30.7 mV per decade of concentration ratio at 37°C.

Step 5 — Plug in real physiological concentrations. Typical values for a mammalian neuron:

For K⁺: E_K = 61.5 × log₁₀(5/140) = 61.5 × log₁₀(0.0357) = 61.5 × (-1.447) ≈ -89 mV. For Na⁺: E_Na = 61.5 × log₁₀(145/15) = 61.5 × log₁₀(9.67) = 61.5 × 0.985 ≈ +60.6 mV. Note the huge gradient needed to produce Ca²⁺'s equilibrium potential — intracellular free Ca²⁺ is kept extraordinarily low (roughly 100 nanomolar, i.e., 0.0001–0.0002 mM) by pumps and buffers, which is why even small Ca²⁺ influxes carry large signaling weight and why E_Ca is so far positive.

Step 6 — Interpret physically what "equilibrium" means at the channel level. Imagine K⁺ leak channels open in the membrane. At any given instant, ions flow through open channels in both directions, driven independently by diffusion and by the electric field. When V_m = E_K, the number of K⁺ ions diffusing outward per second through the channels exactly equals the number pushed inward by the electric field per second — a dynamic, not static, balance. Ions are still moving; net flow is zero. If V_m is more positive than E_K, the net electrical force is insufficiently negative to hold K⁺ in, and net K⁺ efflux occurs, pushing V_m toward E_K. If V_m is more negative than E_K, net influx occurs. The membrane is always relaxing toward whichever ion's equilibrium potential dominates its permeability — this self-correcting behavior is why resting potential sits close to (but not exactly at) E_K, since resting membranes are mostly, but not exclusively, K⁺-permeable.

Long before anyone could patch-clamp a single channel, the central prediction of the Nernst equation — that a membrane's potential should track a permeant ion's concentration gradient in a precise, logarithmic way — needed direct experimental test. The definitive test came from Alan Hodgkin and Bernard Katz in 1949, working on the squid giant axon at the Plymouth Marine Laboratory, building on groundwork from Hodgkin and Andrew Huxley's earlier intracellular recordings.

The problem. By the late 1930s and 1940s, physiologists knew resting potentials were negative inside and that action potentials involved a rapid, large depolarization, even briefly reversing polarity ("overshoot"). Bernstein's older membrane hypothesis (1902) proposed that resting potential was simply the K⁺ equilibrium potential, and that the action potential was caused by a generalized breakdown of membrane selectivity (all ions equally permeable, driving V_m toward zero). But Bernstein's hypothesis could not explain the overshoot — if the membrane became generally leaky, V_m should approach zero, not cross zero and become positive. Hodgkin and Katz suspected instead that the action potential's rising phase and overshoot were driven by a selective, transient increase in sodium permeability, which should push V_m toward E_Na — a large positive value, not toward zero.

The experiment. Hodgkin and Katz systematically varied the concentration of Na⁺ in the seawater bathing the squid giant axon, replacing NaCl with an impermeant substitute (choline chloride) to lower external [Na⁺] while keeping osmolarity constant, and recorded the action potential intracellularly with the glass microelectrode technique (itself pioneered by Hodgkin, Huxley, and Katz in 1939). They measured how the peak (overshoot) of the action potential changed as external Na⁺ was reduced.

The result. The overshoot voltage fell in an orderly, quantitatively predictable way as external Na⁺ was lowered — approximately following the ~58 mV per tenfold change in concentration predicted by the Nernst equation at their experimental temperature (~20°C used in some trials; ~6–11°C for the squid axon experiments proper, giving a somewhat smaller slope, around 50 mV per decade at that temperature). When external [Na⁺] was reduced to about a third of normal seawater concentration, the action potential overshoot dropped by an amount closely matching the Nernst prediction; at very low external Na⁺, action potentials failed to fire altogether, since there was no longer an inward driving force for Na⁺. Critically, the resting potential was barely affected by these manipulations, because the resting membrane's permeability to Na⁺ is low — resting V_m stayed close to E_K regardless of external Na⁺.

What it proved. This dissociation — action potential peak highly sensitive to external Na⁺, resting potential largely insensitive — was the direct experimental signature of a membrane whose permeability changes selectively and transiently during the action potential, shifting the dominant ion from K⁺ (at rest) to Na⁺ (at the peak). It confirmed that membrane potential at any instant is not a fixed cellular property but tracks whichever ion's equilibrium potential the membrane happens to be most permeable to — precisely the concept the Nernst equation quantifies for a single ion, and which the Goldman-Hodgkin-Katz equation later formalized for the mixture of permeabilities present at any moment. This work, together with the voltage-clamp experiments of 1952, earned Hodgkin and Huxley (with John Eccles) the 1963 Nobel Prize in Physiology or Medicine; Katz shared a separate Nobel in 1970 for later work on synaptic transmission. A second, independent confirmation came from simple ion-substitution experiments on frog and crab muscle and nerve fibers in the same era: raising external [K⁺] depolarizes resting cells in a manner that, plotted as V_m against log[K⁺]ₒᵤₜ, gives a straight line with a slope close to the Nernst prediction (~58–61 mV/decade) over a wide range, deviating only at very low external K⁺ — a deviation later explained by the small but nonzero Na⁺ and Cl⁻ permeabilities that the Nernst equation for K⁺ alone does not capture, foreshadowing the need for GHK.

Example 1 — A neuron with altered chloride handling (clinical/physiological reasoning). A cortical interneuron normally has:

• [Cl⁻]ₒᵤₜ = 110 mM
• [Cl⁻]ᵢₙ = 8 mM (kept low by the KCC2 transporter, which extrudes Cl⁻)
• Temperature: 37°C (coefficient 61.5 mV)

Compute E_Cl normally: E_Cl = (61.5 mV / -1) × log₁₀(110/8) = -61.5 × log₁₀(13.75) = -61.5 × 1.138 ≈ -70 mV Now suppose a pathological condition (as occurs in some forms of neuropathic pain and in certain epilepsies) downregulates KCC2, causing intracellular Cl⁻ to rise to 25 mM. Recompute: E_Cl = -61.5 × log₁₀(110/25) = -61.5 × log₁₀(4.4) = -61.5 × 0.643 ≈ -39.5 mV

Interpretation: GABA_A receptors are chloride channels; opening them drives V_m toward E_Cl. Normally E_Cl (-70 mV) sits below a typical resting potential (~-65 mV) and well below threshold, so GABA_A activation is hyperpolarizing/shunting — inhibitory, as expected. After KCC2 loss, E_Cl rises to about -39.5 mV, now above resting potential and possibly above threshold in some neurons. GABA_A activation can then depolarize the cell — a shift from inhibitory to excitatory signaling caused entirely by a change in the transmembrane Cl⁻ gradient, with no change to the receptor itself. This is a real proposed mechanism in some models of neuropathic pain and neonatal seizure susceptibility (immature neurons also have higher internal Cl⁻ due to low KCC2 expression, making GABA depolarizing early in development).

Example 2 — Calcium equilibrium potential and a quantitative sanity check. A cardiac myocyte has:

• Extracellular free Ca²⁺: 1.2 mM
• Intracellular free Ca²⁺ (resting, cytosolic): 100 nM = 0.0001 mM
• z = +2
• Temperature: 37°C

E_Ca = (61.5 mV / 2) × log₁₀(1.2 / 0.0001) = 30.75 × log₁₀(12,000) = 30.75 × 4.079 ≈ +125 mV Now imagine a pharmacological or pathological perturbation: a Ca²⁺-ATPase (SERCA/PMCA) inhibitor causes cytosolic free Ca²⁺ to rise five-fold, to 500 nM = 0.0005 mM, while extracellular Ca²⁺ is unchanged. E_Ca(new) = 30.75 × log₁₀(1.2/0.0005) = 30.75 × log₁₀(2400) = 30.75 × 3.380 ≈ +104 mV

Interpretation: Even a five-fold rise in intracellular Ca²⁺ — physiologically enormous, easily enough to trigger widespread Ca²⁺-dependent signaling and contraction — barely dents E_Ca, because the logarithmic relationship compresses large concentration changes into modest voltage changes. This illustrates a general lesson: equilibrium potentials for ions with steep gradients and low absolute concentrations (like Ca²⁺) stay robustly far from resting potential, which is precisely why Ca²⁺ influx through open channels always provides a strong, reliable inward (depolarizing/excitatory or signal-triggering) driving force under nearly all physiological conditions — a key design feature exploited throughout synaptic transmission and excitation-contraction coupling.

Walther Nernst (1864–1941) was a German physical chemist, not a physiologist — he never studied a neuron. He developed the equation that bears his name in 1888 as a general description of electrode potentials in electrochemical cells, part of his broader program applying thermodynamics to chemistry. Nernst's other major legacy is the third law of thermodynamics (the Nernst heat theorem), for which he won the 1920 Nobel Prize in Chemistry — notably, not for the equation neuroscientists use daily. The equation's application to biological membranes came decades later, once physiologists like Julius Bernstein (who had actually studied under Nernst's contemporaries and absorbed the thermodynamic framework) recognized that a cell membrane behaves, for a single permeant ion, exactly like one half of Nernst's electrochemical cell. The subsequent physiological validation — showing that real membranes actually obey this law for real ions — is credited to Hodgkin, Huxley, and Katz's mid-20th-century work on the squid giant axon.

Clinical electrophysiology and pharmacology. Local anesthetics, antiarrhythmics, and antiepileptic drugs act on Na⁺, K⁺, and Ca²⁺ channels; predicting a drug's effect requires knowing each ion's equilibrium potential and driving force. Long QT syndrome, a cardiac arrhythmia disorder, is understood through mutations in K⁺ or Na⁺ channels that alter how closely membrane potential can approach E_K or E_Na during the cardiac action potential.

Neonatal and pain medicine. The developmental shift in GABA_A signaling (from depolarizing to hyperpolarizing) as KCC2 expression rises postnatally, and its reversal in some models of chronic pain and epilepsy, is explained entirely through changes in E_Cl — directly computed via the Nernst equation from measured intracellular chloride concentrations, an active target for drug development (KCC2 enhancers).

Diagnostic and research electrophysiology. Patch-clamp and two-electrode voltage-clamp experiments routinely use the Nernst equation to set "reversal potential" predictions for specific channels — if a channel's measured reversal potential matches the predicted E_ion for a candidate permeant species, that is strong evidence for the channel's ion selectivity (used to characterize novel channels, including in Ion Channels research and disease-mutant channel studies).

Biotechnology and toxicology. Engineered optogenetic and chemogenetic ion channels are designed and validated partly by predicting their reversal potentials from known ionic selectivity and concentration gradients, using the same Nernst logic, before use in circuit-mapping experiments.

Agriculture and plant biology. Nernst-based reasoning extends beyond neurons to plant cell membranes and guard cells, where K⁺ and Cl⁻ equilibrium potentials govern stomatal opening and closing, relevant to engineering drought-tolerant crops.

Conservation physiology. Comparative studies of ion regulation in osmoconformers versus osmoregulators (e.g., how marine versus freshwater fish maintain viable neuronal function despite very different extracellular ion concentrations) use the Nernst framework to predict how environmental salinity changes shift neuronal excitability — relevant to climate-driven salinity change in estuarine ecosystems.

:::quiz[A neuron has [Na⁺]out = 150 mM and [Na⁺]in = 15 mM at 37°C. Using 61.5 mV as the Nernst coefficient, what is E_Na?]

• +15 mV
• +61.5 mV ✓
• -61.5 mV
• +123 mV

:::quiz[Why does E_Cl come out negative even though [Cl⁻] is much higher outside the cell than inside?]

• Because chloride channels are always closed
• Because the Nernst equation only applies to cations
• Because chloride's negative charge (z = -1) flips the sign of the log ratio ✓
• Because chloride never reaches equilibrium

Ask the tutor: How do I combine multiple ions' equilibrium potentials to predict the actual resting membrane potential using the Goldman-Hodgkin-Katz equation?