The Nernst equation calculates the equilibrium potential for an ion across a membrane based on its concentration gradient.German physical chemist Walther Nernst derived his equation in 1888 while studying galvanic cells. He realized that the electrical voltage at the metal-solution interface arises from the competition between two forces: chemical (the tendency of ions to dissolve) and electrical (the attraction of ions to the charged metal). Later, the equation was adapted for biological membranes. In living cells, the membrane is pierced by ion channels—protein pores that selectively allow certain ions to pass. The Nernst equation calculates the equilibrium potential for each ion: E = (RT/zF) * ln([ion outside]/[ion inside]). Here, R is the gas constant, T is absolute temperature, z is the ion charge, and F is Faraday's constant. At body temperature (37°C), the factor RT/F is approximately 26.7 mV. If the potassium concentration inside a cell is 30 times higher than outside, the potassium equilibrium potential is 26.7 * ln(1/30) ≈ -90 mV. This means the electrical force holding positive potassium ions inside the cell balances their tendency to leak out along the concentration gradient.
Walther Nernst, a student of Ludwig Boltzmann, formulated the equation in 1888 as part of his dissertation on electrochemistry. He studied potentials at the interface between metals and their salt solutions and established the logarithmic dependence of potential on ion concentration. In 1920, Nernst received the Nobel Prize in Chemistry for his work on thermochemistry. In the 1940s, Alan Hodgkin and Bernard Katz applied the Nernst equation to biological membranes, showing that the resting potential of the squid axon is close to the potassium equilibrium potential. This became the foundation for the Hodgkin-Huxley model describing nerve impulse generation.
In electrochemistry, the Nernst equation is used to calculate the EMF of galvanic cells and determine the pH of solutions. In biology, it is used to find equilibrium potentials for K⁺, Na⁺, Cl⁻, and Ca²⁺, which is essential for understanding cell excitability. Limitations: the equation is valid for dilute solutions (concentrations less than 0.1 M) and in the absence of side reactions at the electrodes. In concentrated solutions, activities replace concentrations with correction factors. For biological membranes, it must be considered that the actual membrane potential rarely matches the Nernst potential for a single ion due to permeability to multiple ions.
Calculating voltage across membranes of living cells, essential for understanding nerve and muscle function. In chemistry, determining ion concentrations in solutions using electrodes.
The Nernst equation explains why bananas and potatoes can be used as batteries: different ion concentrations in the flesh and at the electrodes create voltage. School experiments with potato clocks work precisely on this principle.
| M(<r) | variable | mass enclosed within radius r — mass kilogram |
| r | variable | radial distance from the center — radius metre |
| P | variable | pressure — pressure pascal |
| G | constant | constant of proportionality in Newton's law of universal gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| dP/dr | operator | derivative of pressure with respect to radius — radial derivative |
| f | variable | similarity solution function — stream function |
| \beta | variable | pressure gradient parameter — pressure gradient parameter |
| ' | operator | derivative with respect to similarity variable — derivative |
| P_r | variable | Radial pressure — pressure pascal |
| r | variable | Radial coordinate — radius metre |
| ε | variable | Energy density — energy density joule per cubic metre |
| m | variable | Mass enclosed within radius r — mass kilogram |
| P_t | variable | Tangential pressure — pressure pascal |
| G | constant | Fundamental constant governing gravitational strength — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| c | constant | Universal speed limit for massless particles — speed of light = 299792458 metre per second |
| d/dr | operator | Differentiates a function with respect to r — ordinary derivative |
