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From Voltage to Concentration: The Nernst Equation

A potentiometer hands you a voltage, but you wanted a concentration. The bridge between them is one of the most useful equations in chemistry — the Nernst equation — and it says something surprisingly intuitive: every tenfold change in concentration nudges the voltage by a fixed, tiny step. This guide builds that equation from feeling, not formulas.

The Core Puzzle: Voltage Is Not Concentration

In the last guide we built a cell that reports a voltage. But a voltage in millivolts is not the answer anyone wants; people want "there are 3 milligrams of silver per litre." So we need a rule connecting the electrode potential of the indicator electrode to the concentration of the analyte it is sensing. That rule is the Nernst equation.

Here is the single most important thing to feel before any algebra: the relationship is not straight, it is logarithmic. Doubling the concentration does not double the voltage. Instead, the voltage cares about the concentration the way your ears care about loudness — by ratios, not by raw amounts. Going from 1 to 10 changes the voltage by the same step as going from 10 to 100, or 100 to 1000.

The Magic Number: 59 Millivolts Per Decade

Once you accept that the voltage moves in equal steps for each tenfold (each "decade") of concentration, the only question left is: how big is one step? For a simple ion carrying a single charge, at room temperature, the answer is about 59 millivolts per tenfold change. That number — 59 mV — is worth memorising, because it turns the Nernst equation from a formula into a feeling.

So if you dilute a silver solution tenfold and the voltage drops by 59 mV, your electrode is behaving exactly as theory predicts. Dilute it a hundredfold and you expect a 118 mV drop — two steps of 59. This regular, predictable march is what makes potentiometry trustworthy: the electrode is not improvising, it is obeying a law.

Two refinements make the picture honest. First, the step size depends on the ion's charge: an ion carrying two charges (like calcium, Ca²⁺) moves the voltage only about half as much per decade, roughly 30 mV, because each unit of voltage now carries twice the chemical "weight." Second, the 59 mV figure is for room temperature; warm the solution and the step grows slightly, which is why precise meters ask you to enter the temperature.

Reading the Equation Without Fear

Now the formula will look friendly. The Nernst equation says the measured potential equals a fixed starting value plus the step-per-decade multiplied by the logarithm of the concentration:

E  =  E°  +  (0.059 / n) · log[ion]

E     = the potential you measure (volts)
E°    = the fixed starting value for this electrode
n     = the ion's charge (1 for Ag+, 2 for Ca2+)
log   = base-10 logarithm of the concentration
0.059 = the room-temperature step, in volts (59 mV)
The Nernst equation in plain pieces: a fixed offset plus a logarithmic term that does all the responding.

Read it left to right as a story. E° is where the voltage would sit at a reference concentration — the constant part. The log term is the only part that moves, and it moves by 0.059/n volts for every factor of ten in concentration. Everything we felt in the previous section is sitting right there in the symbols.

Turning the Law into a Measurement

In practice you almost never trust E° from a textbook, because real electrodes drift and each one is a little different. Instead you let the electrode tell you its own behaviour. This is just potentiometry joined to a calibration curve: measure a few standards of known concentration, plot voltage against the logarithm of concentration, and you should get a straight line whose slope is that magic 59-mV step.

  1. Measure the cell voltage for several standards of known concentration.
  2. Plot voltage on the vertical axis against log(concentration) on the horizontal — expect a straight line.
  3. Check the slope: near 59 mV per decade for a single-charged ion means the electrode is healthy.
  4. Measure your unknown's voltage and read its concentration straight off the calibration line.

If your measured slope comes out at, say, 48 mV per decade instead of 59, the electrode is tired or contaminated and should not be trusted. This is a quiet superpower of the Nernst equation: it does not just convert voltage to concentration, it lets the instrument grade its own honesty before you believe a single result.