Topic 9.9: Cell Potential and Free Energy

Cell potential and free energy: reading the table

Chemists have measured how strongly each metal’s ions pull on electrons and collected the results in one table, called a table of standard reduction potentials. Every entry reads as a reduction, and they all share one reference, hydrogen, set at zero, at standard conditions, which means one molar solutions at twenty-five degrees Celsius and gases at one atmosphere.

The higher a half-reaction sits, the harder its ion pulls on electrons, so when a pair runs on its own, as a galvanic cell, the higher one is reduced at the cathode, and the lower one runs backward, as an oxidation, at the anode. On our bench, copper sits at plus zero point three four volts and zinc way down at negative zero point seven six, so copper is the cathode and zinc is the anode, just like we saw.

The cell potential is the gap between the two rungs, E naught of the cathode minus E naught of the anode. And we take both numbers straight from the table, signs and all, so it’s zero point three four minus negative zero point seven six, which is positive one point one zero volts, exactly the number on the meter.

Now let’s pair copper with silver, which sits even higher, at plus zero point eight zero, so this time the silver is reduced and the copper is oxidized. Each silver ion takes only one electron but each copper atom gives up two, so we need two silver ions for every copper atom. This is where a lot of people double the silver’s voltage, because they doubled its half-reaction, but you never multiply a reduction potential, because a volt measures the push on each electron, and twice as many electrons doesn’t push any one of them harder.

So go ahead and work it out, what’s the standard cell potential for copper reacting with silver ions, in volts? Now let’s look at the cell from that 2025 exam question, aluminum and zinc, where both potentials are negative.

Using the table, what’s the standard cell potential for the aluminum and zinc cell? So a cathode can have a negative reduction potential, as long as the other half sits even lower. Each aluminum atom gives up three electrons and each zinc ion takes two, so two aluminums and three zinc ions trade six electrons.

Another part of that exam question was about mass, so for one mole of this reaction, which electrode’s mass changes more? Back at the bench, there’s one more part of that question. It asked which metal you’d pair with zinc to get the biggest voltage. Here are four metals, so tap the one you’d pair with zinc to get the biggest voltage.

Free energy from cell potential: ΔG° = −nFE°

So what does one point one zero volts actually mean? A volt is a joule of energy for every coulomb of charge, so every coulomb of electrons that flows from zinc to copper releases one point one zero joules.

If we flip those two rungs over and draw the energy of an electron instead, the electrons start up high at the zinc, which holds them loosely. Then they drop down to the copper, which pulls on them harder.

So I picture it as a waterfall, where the voltage is the height of the drop. The total energy depends on that height, and on how much charge goes over the edge. And that’s exactly what the equation connecting cell potential to free energy says, delta G naught equals negative n F E naught, where n is the moles of electrons transferred in the balanced reaction, and F is Faraday’s constant, ninety-six thousand four hundred eighty-five coulombs per mole of electrons.

For zinc and copper, n is two, so delta G is negative two moles of electrons, times ninety-six thousand four hundred eighty-five coulombs per mole of electrons, times one point one zero joules per coulomb. The moles of electrons cancel and the coulombs cancel, leaving about negative two hundred twelve thousand joules, which is negative two hundred twelve kilojoules per mole of reaction.

The minus sign is there because the energy the electrons carry off is energy the reaction gives up, so this negative delta G means the reaction is thermodynamically favored, exactly what the positive cell potential told us. So the signs always line up, a positive cell potential means a negative delta G, a favored reaction and a galvanic cell, and a negative cell potential means a positive delta G, an unfavored reaction that needs an outside power source, which makes it electrolytic.

Now suppose we double every coefficient in the copper and silver reaction, so what happens to E naught and to delta G naught? Running our bench cell backward flips the reaction, so its cell potential is negative one point one zero volts and delta G is positive two hundred twelve kilojoules, which is why the power supply had to push with more than one point one zero volts.

The 2024 AP exam tested this with a plating job. You plate a thin layer of rhodium onto silver. The question gave you these two half-reactions. Plating lays down rhodium metal, so rhodium ions get reduced. That’s the cathode.

At the anode, water gets oxidized to oxygen gas. Those are the same bubbles we saw on the platinum. Here the higher half, at one point two three volts, gets oxidized. On its own, it would be reduced. The power supply forces it backward, like our bench cell.

Four rhodium ions take three electrons each. That’s twelve. Making three oxygen molecules frees four electrons each. Twelve again. Add the two halves. The twelve electrons cancel out. Twelve electrons move, so n is twelve.

Using the two reduction potentials, what’s the standard cell potential for this plating reaction? Next they asked why this plating needs an outside power source. One sentence earns that point. E naught is negative, so delta G naught is positive. The reaction isn’t thermodynamically favored, and the power source has to supply the energy.

Back to the plating equation. Twelve electrons move, so n is twelve. What’s delta G naught, in kilojoules per mole of reaction?

Back to Liberty: cutting the circuit

So let’s go back to the statue, because now we can explain her whole problem. Iron sits at negative zero point four four on our table, well below copper at plus zero point three four. That means when the two touch, iron is the one that gives up its electrons. The copper just passes them along to oxygen.

And iron rusting in wet, salty air is a thermodynamically favored reaction, so there’s no switch you can flip to turn it off, and the cleanest way to stop this battery is to break its circuit. So imagine you’re one of the engineers in 1984, and choose the spot where you’d cut Liberty’s circuit.

During the restoration, workers took out her old iron armature bars, and on one of them, near an arm, they found the names of the forty ironworkers who had put her framework together in the 1880s. They replaced the bars with stainless steel, an alloy of iron with chromium, nickel and molybdenum that resists corrosion, and they put Teflon between the new bars and the copper. Teflon doesn’t conduct electricity, and it lets the metals slide without wearing each other down.

At the particle level, the Teflon sits where the metals used to touch, so even with salt water all around, no electrons can cross between them, and the stainless steel itself resists corrosion. Workers finished the restoration in 1986, and on the Fourth of July the country celebrated her centennial with fireworks over the harbor, and when inspectors checked her again ten years later, they found no new corrosion problems.

So everything you learned today was hiding inside her, oxidation at the anode and reduction at the cathode, electrons through metal and ions through solution, and a cell potential that tells you which way the electrons go, and, through delta G, whether the reaction runs on its own. Every number we used today assumed standard conditions, fresh one molar solutions, but as a real battery runs, those concentrations change, and so does its voltage, and that’s where we’re headed next.