The crowd
Picture 1 · particles
Up close there is no voltage to see, only particles. At the zinc plate, atoms keep leaving the metal as Zn²⁺ ions, each leaving two electrons behind, and Zn²⁺ ions keep landing and turning back into metal. At the copper plate the same two-way traffic runs with Cu²⁺.
Set the load to open circuit (slider all the way left). Each plate's traffic now balances: as many ions leave as land. Equilibrium here doesn't mean nothing is happening. It means two opposite processes are tied.
Connect a load and the tie breaks. Zinc dissolves faster than it redeposits; the electrons it leaves behind run through the wire to the copper, where Cu²⁺ ions pick them up and plate out as metal. Inside the cell, ions close the circuit: positive ions drift toward the copper, negative ions toward the zinc, and in the bridge K⁺ and Cl⁻ carry the current across the gap.
Shows
That everything is motion, and that equilibrium is a balance of rates.
Hides
Why the answer is 1.10 volts and not 0.5 or 3. Almost all of it is energetics: how tightly each metal holds its atoms and how strongly water binds each ion, summed up in the standard free energy of the reaction Zn + Cu²⁺ → Zn²⁺ + Cu. The statistics of the crowd (how many ions there are) add only the Nernst concentration term, tens of millivolts at most. Moving dots show neither.
The ledger
Picture 2 · the textbook
A chemistry course turns the cell into bookkeeping. Each half-reaction gets a standard potential, measured against the standard hydrogen electrode (SHE: platinum in hydrogen gas and acid of unit activity, agreed to be zero): −0.76 V for Zn²⁺/Zn, +0.34 V for Cu²⁺/Cu. The Nernst equation corrects each for its actual concentration, and the cell voltage is the difference.
Draw current and the ledger adds line items: the resistance of the electrolyte, an overpotential at each plate to drive its reaction, and a concentration term once the ions next to the plates start to run short (or pile up). It's quick, and given good parameters, what it leaves out is the liquid junction, usually a few millivolts. That's why it's the textbook's picture.
Shows
The numbers, fast, and a clean list of where the voltage goes.
Hides
Position: the ledger never says where in the cell a loss happens. It also carries a quiet assumption, that both electrodes are measured against one and the same hydrogen reference. The "liquid junction" row is the price of that assumption, and textbooks usually set it to zero.
The levels
Picture 3 · electron levels, with the reference drawn in
Put position back, and plot what a voltmeter actually responds to: the energy of the electrons, expressed as a voltage. Up is more positive voltage, which for an electron, being negative, means lower energy. In each metal the electrons' level is the Fermi level, the energy it takes to add one more electron. For each redox couple in solution it's the redox level: the electron voltage at which the couple is balanced, with no net reaction either way.
Then do what the ledger doesn't, and draw the reference itself. The dash-dot line is where a hydrogen electrode's level would sit if unit-activity H⁺ could be placed at that spot without disturbing anything. It's a model quantity: exact for ideal solutions, but inside a junction no instrument can measure it, because it rests on the activity of a single kind of ion. The standard potentials E° are distances measured up or down from that line (the dashed levels).
At open circuit each metal's Fermi level lies exactly on its couple's redox level, and the voltmeter reads the gap between the two metals. Connect a load and gaps open at both plates. Those are the overpotentials, and in this picture you read them straight off as voltage drops. The reference line isn't flat either. It tilts through the electrolyte as current flows, and it steps slightly at each liquid junction. The textbook's E(Cu) − E(Zn) equals the terminal voltage only at open circuit, and then only if that line is level from one plate to the other.
Shows
Where each loss happens, as a voltage you can read, and where the reference would be.
Hides
Every ion that isn't in a redox couple. The sulfate, and the K⁺ and Cl⁻ that carry the current through the bridge, get no line at all.
Every species
Picture 4 · species voltages
The last picture gives every charged species its own line. Take each species' electrochemical potential (its free energy per particle, electrical part included) and divide by its charge. What comes out is a voltage, Vi = μ̃i / ziF, on the same axis as the electrons. Zinc ions, copper ions, sulfate, potassium, chloride and electrons all share one plot.
Read it like a landscape for current. Each species carries current down its own slope, and a flat line means that species carries none. Each ion also has a dashed companion, its standard level; the gap between the two encodes the ion's concentration on a log scale. Don't compare the heights of different species: those depend on a convention. What means something is each line's slope, and the gap between an ion and the electrons where they react.
Connect a load, and several things that picture 3 couldn't show become plain:
- In the thin layer next to each plate, sulfate's line is flat. Sulfate isn't consumed at either plate, so in steady state it carries no current there. Zn²⁺ and Cu²⁺ bend instead.
- In the bulk of each beaker, every line tilts together. That shared tilt is the ohmic drop, the same thing the ledger books as I·R.
- In the bridge, K⁺ and Cl⁻ tilt: they're the ones carrying the current across.
Now compare with picture 3. In the convention used here, neutral elements like zinc metal have zero chemical potential, so the Zn²⁺ line is the zinc redox level. And each ion's standard level sits a tabulated distance from the H⁺ rung of the floating ladder, which is exactly the hydrogen reference line of picture 3, so the dashed V°(Zn²⁺) is its E° line. For zinc and copper, pictures 3 and 4 carry the same information. Picture 4 adds the ions that picture 3 leaves out, and doesn't draw the hydrogen reference, though the dashed V° rungs are placed relative to it.
Shows
Every ion's push, which ions carry the current where, and (next) the slow leak at the junctions.
Hides
Overpotentials read as one clean gap here only because each metal is its own reactant, so its electrons and its ion are both on the plot. For other electrodes, such as hydrogen on platinum or two dissolved ions trading an electron, the overpotential isn't a single gap on this plot. The vertical offsets between different ions depend on conventions, so reading them takes practice.
All four at once
Same cell, same moment, four pictures. Move the load slider and watch one event land four ways: the particles start streaming, the ledger grows loss lines, gaps open in the levels, and every species tilts together.
The crowd particles
The ledger textbook
The levels electron levels
Every species species voltages
Show the numbers
What the Daniell cell is hiding
Textbooks describe an unloaded Daniell cell as being at equilibrium. It isn't, quite. Set the load to open circuit and look at the junctions in picture 4: wherever two solutions meet, lines that are flat on one side plunge or climb on the other. Zn²⁺ dives into the bridge; K⁺ dives into the zinc sulfate. A sloping line is a push, and a push means flow. Even with nothing connected, ions are diffusing out of their own compartment and into the next one; the bridge's K⁺ and Cl⁻ creep into both beakers, and a little zinc and copper sulfate creeps into the bridge.
That leak does two things. The first is a small liquid junction potential: the step in the hydrogen reference line in picture 3, and the "liquid junction" row in the ledger. A concentrated KCl bridge keeps it to a few millivolts, because K⁺ and Cl⁻ move at nearly the same speed and, when the two beakers are alike, the junctions at the two ends of the bridge nearly cancel. Switch to the porous pot, Daniell's original design, and the two sulfate solutions meet directly. Make their concentrations very different and the junction potential grows past ten millivolts.
The second is slower and worse. Copper ions that wander all the way to the zinc plate are reduced right there, no wire needed, and the cell quietly discharges itself. Choose the porous pot, untick Hold steady in picture 1, and watch the zinc plate for copper-coloured flashes. Real Daniell cells were designed to slow this down, never to stop it.
Load makes all of it worse, because under current a junction can't hold still. At the zinc end of the bridge, Zn²⁺ keeps arriving from one side while K⁺ leaves from the other, and Cl⁻ is pushed out into the zinc sulfate. Real junctions drift and smear out. The diagrams here, like the textbook Henderson treatment behind them (which assumes the two solutions blend linearly across the junction), hold each junction in place as a fixed mixing zone. That is a good approximation for the voltage, and a poor description of what the ions are actually doing. Hold steady is that approximation, made visible; untick it to watch the particles do what the diagrams won't.
So "equilibrium" is an approximation with a clock on it, good for minutes to hours but not forever. The particles show the leak happening; picture 4 shows it as slopes. The ledger books it as a correction assumed to be zero, and picture 3 shows only its side effect, the step in the reference.
Which picture, when?
| Picture | Makes easy | Hides |
|---|---|---|
| The crowd | Equilibrium as a balance of rates; what physically moves where. | Where the voltage comes from. |
| The ledger | Quick numbers; designing a cell; a list of losses. | Position, and the single-reference assumption. |
| The levels | Reading overpotentials as voltage drops; seeing where the reference would sit. | Ions outside redox couples, including the ones carrying the current. |
| Every species | Who carries current where; the junction leak; how the other pictures fit together. | Overpotentials in general; it depends on conventions and takes practice to read. |
None of them is the truth about the cell. They're one set of physics in four notations, and it's worth being able to move between them.