◄ WORLD V · SONNY 5DART 590 · a helldive into the living

THE NERNST EQUATION the voltage across a membrane

Every living cell holds a battery. Ion pumps build up concentration differences — potassium high inside, sodium high outside — and each ion’s equilibrium voltage follows the Nernst equation E = (RT/zF) ln([out]/[in]), the point where the electrical pull exactly cancels the chemical push. Potassium’s sits near −90 mV, sodium’s near +60 mV; the resting membrane lives between them. The equation under every heartbeat and thought.

THE TECHNIQUE E = (RT/zF) ln([out]/[in])

The demo computes the equilibrium potential for potassium and sodium (ln via series): live demo







  
HISTORY & CREDIT Walther Nernst · 1888

  
“The membrane voltage is the average of the ions.” — the resting potential is a weighted balance (Goldman) dominated by whichever ions the membrane is most permeable to, not a plain average. cited

  
two forces · the chemical gradient pushes an ion one way; the voltage it builds pulls back.
equilibrium · where they cancel: E = (RT/zF) ln([out]/[in]).
1888 · Walther Nernst (Nobel 1920); potassium −90 mV, sodium +60 mV.

  
A battery, written in a logarithm. biophysics






  
RECOMMEND FOR I-13 the potentials, on the compiler

  
On i-13 (range-reduced ln), Eᵤ ≈ −87 mV (in 140 / out 5) and Eₘₐ ≈ +65 mV (out 145 / in 12) — textbook −90 / +60:

  
$ i13 run lf_nernst-equation.i13
RUN OK · 2368 step(s) · call depth 23
e_k_mv = -86.6   k_negative = 1
e_na_mv = +64.8  na_positive = 1

  
Recommend as a NULL — a theorem (B39). The equilibrium potential is a pinned function of the concentrations and charge; every correct evaluation agrees. NULL — the voltage across a membrane.