THERMODYNAMICS · STORY 07

The potential landscape
inside a battery

There is no magic in 1.10 V. It is the gap between two rungs — computable to the digit.

HOW TO READ

The stage on the right is driven by one cell state: the macroscopic battery shows electrodes, salt bridge, and the voltmeter; the microscopic ions explain why the electrons want this route; and the symbolic panel draws the ladder of electrode potentials — the gap between two rungs is the volts on the meter.

One cell · three representations

E: 1.104 V
Zn²⁺: 1.00 M
Cu²⁺: 1.00 M

Macro · Cell Electrode thinning and plating are schematic; the voltmeter reading is computed from the Nernst equation.

Micro · Interfaces Ion counts scale with concentration; the electron direction follows the sign of the potential gap.

Rung heights are CRC/IUPAC standard potentials; working positions slide by the Nernst equation.

Zinc into zinc sulfate, copper into copper sulfate, a salt bridge across, and the voltmeter lights up:

1.10\ \mathrm{V}

No more, no less. Use stronger solutions — still about 1.10 V. Build the cell ten times bigger — still 1.10 V.

Where does that number come from? Why not 2 V, why not 0.5?

Place your bet.

Predict first: 1.10 V is set by

Taken apart, this is two half-reactions:

Zinc side: $\mathrm{Zn \to Zn^{2+} + 2e^-}$ — zinc is "willing" to shed electrons and dissolve
Copper side: $\mathrm{Cu^{2+} + 2e^- \to Cu}$ — copper ions are "willing" to catch them and deposit

Willingness can be measured: rank every half-reaction against one common ruler (the standard hydrogen electrode) and you get standard electrode potentials:

E^\circ(\mathrm{Cu^{2+}/Cu}) = +0.34\ \mathrm{V} \qquad E^\circ(\mathrm{Zn^{2+}/Zn}) = -0.76\ \mathrm{V}

Electrons jump from the lower rung to the higher one, and the gap is the voltage: $0.34 - (-0.76) = 1.10\ \mathrm{V}$ . The answer is not magic. It is a table.

Hang every common couple on that ruler and you get electrochemistry’s ladder — the standard-potential table of every textbook, turned upright:

At the top, $\mathrm{F_2/F^-}$ (+2.87 V): the hungriest oxidant
Midway, $\mathrm{Cu^{2+}/Cu}$ , $\mathrm{H^+/H_2}$ (defined as 0), $\mathrm{Zn^{2+}/Zn}$
At the bottom, $\mathrm{Li^+/Li}$ (−3.04 V): the most generous reductant

Pick any two rungs; the gap is that pair’s standard voltage. The 3–4 V secret of lithium batteries is simply shopping from both ends of the ladder. The ladder is not memorization material — it is the price list of every battery ever built.

But 1.10 V holds only at standard concentrations (1 mol/L). Move a concentration and the rung slides — the Nernst equation gives the exact slope:

E = E^\circ - \frac{RT}{nF}\ln\frac{[\mathrm{Zn^{2+}}]}{[\mathrm{Cu^{2+}}]}

With $n=2$ at room temperature the slope is 29.6 mV per decade: ten-fold richer zinc drops the voltage 29.6 mV; ten-fold richer copper raises it the same.

Drag the two sliders and watch the meter answer in millivolts. That logarithmic slope is the same term as last story’s $RT\ln Q$ — a battery is the Gibbs valley with an electrical readout: $\Delta G = -nFE$ .

[Zn²⁺] / mol·L⁻¹ 1.00 [Cu²⁺] / mol·L⁻¹ 1.00 E = 1.104 V · ΔG = -213.0 kJ/mol

Put the cell to work. As it discharges, zinc dissolves and copper deposits: $[\mathrm{Zn^{2+}}]$ climbs, $[\mathrm{Cu^{2+}}]$ falls, Q grows, and the voltage slides down the Nernst slope.

Press discharge and watch the curve: for the first 80 % of the journey the voltage barely moves (it takes a four-fold concentration swing to spend one 29.6 mV step) — then the final stretch avalanches, as the logarithm plunges when a concentration nears zero.

That is where the battery "plateau" comes from — and why fuel gauges are hard: on the plateau, voltage carries almost no charge information.

Discharged 0% · E = 1.104 V

The voltage reaches zero: the battery is "dead". Dead where, exactly?

E = 0 \iff \ln Q = \frac{nFE^\circ}{RT} \iff Q = K \approx 10^{37}

A dead battery is the reaction arriving at the floor of its Gibbs valley. This particular valley is just astonishingly deep: K ≈ 10³⁷ puts the floor hard against the "copper ions exhausted" cliff.

Last story said no reaction truly runs to completion — still strictly true, but a floor at 10³⁷ is indistinguishable from "completion" by any instrument ever built. Thermodynamics never lies; it just sometimes writes the margin very small.

One last magic trick: build a cell from two identical half-cells — both copper, one solution concentrated, one dilute.

Any voltage? Yes.

E = \frac{RT}{nF}\ln\frac{c_{\text{conc}}}{c_{\text{dil}}}

A 10:1 ratio gives 29.6 mV — small, but real, and with no chemical reaction at all: just copper ions moving house from rich to poor.

Pure mixing entropy, read out directly in volts. The resting potential of your neurons and salinity-gradient power plants are cousins of this little cell. Entropy is not an abstraction — your every heartbeat pays it a toll.

Concentration ratio c₂/c₁ 10^1.0 Ratio 10:1 → E = 29.6 mV

The bench is yours.

Two concentration sliders, one temperature knob — the voltmeter and the ladder answer live; the discharge button walks the whole curve.

Challenges: push the voltage past 1.20 V touching only concentrations; build a 59 mV concentration cell; then work out how much ladder your phone’s 3.8 V battery must span.

[Zn²⁺] / mol·L⁻¹ 1.00 [Cu²⁺] / mol·L⁻¹ 1.00 E = 1.104 V · ΔG = −nFE = -213.0 kJ/mol

THE PRICE LIST

The ladder’s price list

Standard potentials Each half-reaction’s rung height against the standard hydrogen electrode; the gap between any two rungs is that pair’s standard voltage. CRC/IUPAC values.

The Nernst slope 59.2/n mV per decade of concentration ratio (298 K): 29.6 mV for n = 2. The log term is the Gibbs valley’s RT ln Q wearing volts.

ΔG = −nFE Voltage is molar Gibbs energy with an electrical meter: 1.10 V × 2F = 213 kJ/mol of downhill.

Concentration cells Voltage from pure mixing entropy, zero net chemistry: E = (RT/nF)·ln(c₂/c₁). The prototype of nerve resting potentials and salinity-gradient power.

CONCEPT CHECK · PREDICT, THEN TEST

Dilute the copper half-cell of a Daniell cell ten-fold ([Cu²⁺] from 1 M to 0.1 M). What happens to the voltage?

Nothing — E° depends only on which metals are used It drops about 29.6 mV — Q grows, and the Nernst term spends one step It rises about 29.6 mV — a thinner solution has less resistance It halves — the concentration was cut to a tenth

FREE PLAY · Leave the storyline with a question

The battery bench

Concentrations, temperature, discharge — all yours; the voltmeter, the interfaces, and the ladder answer together.

[Zn²⁺] / mol·L⁻¹1.00 M [Cu²⁺] / mol·L⁻¹1.00 M Temperature T25 °C

E = 1.104 V · ΔG = −nFE = -213.0 kJ/mol

One cell · three representations

E: 1.104 V
Zn²⁺: 1.00 M
Cu²⁺: 1.00 M

Macro · Cell Electrode thinning and plating are schematic; the voltmeter reading is computed from the Nernst equation.

Micro · Interfaces Ion counts scale with concentration; the electron direction follows the sign of the potential gap.

Rung heights are CRC/IUPAC standard potentials; working positions slide by the Nernst equation.

MODEL CARD

Model card · V0.1

System and data

The Daniell cell Zn | Zn²⁺ ‖ Cu²⁺ | Cu, standard potentials from CRC/IUPAC (298.15 K): E°(Cu²⁺/Cu)=+0.3419 V, E°(Zn²⁺/Zn)=−0.7618 V, E°cell=1.1037 V. Unit tests pin the 29.58 mV/decade slope, ΔG°=−213 kJ/mol, and K≈10³⁷.

Teaching approximation

Concentrations stand in for activities (γ=1). Real 1 M ZnSO₄/CuSO₄ have activity coefficients near 0.04–0.05, so measured open-circuit voltages deviate by tens of millivolts; Debye–Hückel/Pitzer corrections are backend work (docs/后端待办.md). Junction and salt-bridge asymmetry potentials are also unmodeled.

Discharge curve

Equal-volume half-cells marched by mass balance ([Zn²⁺]+[Cu²⁺] conserved), E recomputed by Nernst each step down to the cutoff. No internal resistance, polarization, or mass-transport limits — those set how fast a cell can discharge; this page only answers how far.

What cannot be inferred from this page

Electrode kinetics (overpotentials, exchange currents), real-cell capacity fade and self-discharge, and the voltage-curve shapes of intercalation batteries like Li-ion (governed by solid-phase activities, not solution concentrations) are all beyond this model. The ladder reports thermodynamic heights, never arrival speeds.

1.10 V is the gap between two rungs, 29.6 mV is the logarithm’s slope, 10³⁷ is the depth of the floor. A battery creates no energy — it merely puts a meter on the downhill road.

Season 2 · the skeleton of thermodynamics is still going up.

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