Topic 9.11: Electrolysis and Faraday’s Law

Electrolysis and Faraday’s law: counting electrons

Back in Hall’s woodshed, he wired his battery jars to a carbon-lined crucible of aluminum oxide dissolved in a molten mineral called cryolite, with a carbon rod dipped into it. Aluminum oxide will never fall apart into aluminum on its own, because aluminum holds on to oxygen far too tightly, so Hall’s batteries had to force electrons onto the aluminum. Using electricity to drive a reaction like that is called electrolysis, a battery run backward. When he broke the crucible open, there they were, little silvery beads of aluminum, and in the bookkeeping we use, every atom in them was an aluminum ion that took in three electrons.

Suppose his batteries pushed three moles of electrons through that crucible. At most, how many moles of aluminum could he get? Electrons are an ingredient we can count. We count them with current. Charge comes in coulombs.

One coulomb is about six billion billion electrons. One amp is one coulomb every second. Two amps for ten seconds is twenty coulombs. Total charge is current times time. We call it little q, to keep it apart from big Q.

A mole of electrons carries ninety-six thousand, four hundred eighty-five coulombs. That’s Faraday’s constant, F, the same F that sat in the Nernst equation. Remember the copper cell where the two blues matched? Its strong jar lost half a mole of copper. Two electrons each makes one mole of electrons.

That gives us a chain. Current times time gives coulombs. Divide by F to get moles of electrons. The half-reaction turns those into moles of metal. The molar mass turns that into grams. That whole chain is Faraday’s law.

Here’s a copper plating cell on our bench. The batteries push electrons into a black carbon rod and pull them out of a copper plate, so copper builds up on the rod while the plate dissolves to replace it.

Say we run it at two amps for thirty minutes. Current is in coulombs per second, so first I turn thirty minutes into eighteen hundred seconds, and two coulombs every second for eighteen hundred seconds is thirty-six hundred coulombs.

Dividing by Faraday’s constant cancels the coulombs and leaves moles of electrons, and since each copper ion takes two electrons, it’s one mole of copper for every two moles of electrons. The molar mass of copper takes us to one point one nine grams, three significant figures like the data. And that’s sensible, since thirty-six hundred coulombs is only about four hundredths of a mole of electrons, which makes about two hundredths of a mole of copper, a little over a gram.

The copper plate loses exactly the same one point one nine grams. And none of that is the mass of electrons. It’s copper atoms moving from one electrode to the other, and the electrons are just how we keep count.

Running the chain backward

The chain runs both ways, so from the metal you made you can work back to the current. Let’s do one together. In a classic lab you zinc-plate a penny for ten minutes, and it gains zero point two five oh grams. Grams over the molar mass gives moles of zinc, two electrons each doubles it, and Faraday’s constant turns that into about seven hundred thirty-eight coulombs. The last step, from charge to current, is yours.

So what was the current? Give it in amps. Now let’s push one mole of electrons into three different metals. Silver needs one electron per atom, so you get a whole mole, about a hundred and eight grams.

Copper needs two, so half a mole, about thirty-two grams, and aluminum needs three, so a third of a mole, less than nine grams. That’s why aluminum is so hungry for electricity, three electrons for every light little atom.

The chain can even tell you an ion’s charge. Gold forms gold plus one and gold plus three, so which one is in a plating bath? You find out by measuring. Say half an amp for twenty minutes deposits one point two two grams of gold, so find the moles of electrons and the moles of gold, and their ratio is the number of electrons per ion. How many electrons did each gold ion take?

Now one straight off the 2021 AP exam, how long five amps through molten magnesium chloride takes to make two grams of magnesium. The first point on this question is for the moles of electrons, and the half-reaction is where they come from, two electrons for every magnesium. The rest is all yours, from grams to seconds. So how long does it take? Give it in seconds.

The 72-second tip

Look at that last row again. That’s aluminum. Now we’re going to make a lot of it. Today, factories make Hall’s metal in rooms like this. It’s a potroom, a line of cells hundreds of meters long.

Carbon blocks hang down into the same molten cryolite and aluminum oxide Hall used, and liquid aluminum pools on the bottom. The carbon gives up electrons as it slowly turns into carbon dioxide, and the power supply pushes them up into the liquid aluminum. Let’s zoom in to the surface of that metal. Here comes an aluminum ion, and here come its electrons, one, two, three, and now it’s an aluminum atom, part of the pool, like every other atom in it.

This cell is running at two hundred ten thousand amps. Turn the current up so it makes aluminum twice as fast. So let’s keep the promise with a replacement tip, a pure aluminum pyramid with the original’s hundred-ounce mass, two point eight three kilograms, because the real one’s metal came from an older chemical method, before Hall’s process existed.

Two thousand eight hundred thirty grams is about a hundred and five moles of aluminum, three electrons each makes about three hundred fifteen moles of electrons, and times Faraday’s constant that’s three point oh four times ten to the seventh coulombs, about thirty million. Through a hundred amps, which is already a lot, that would take about eighty-four hours. But this cell runs at four hundred twenty thousand.

So how many seconds does this cell need? That’s the time to pass the charge for our tip. So let’s watch the charge for a whole Washington Monument tip go through, filling the pyramid as the coulombs go by. Halfway there, thirty-six seconds, half the charge and half the metal. Seventy-two seconds, and there’s the metal for our whole tip.

That’s with every electron doing its job, in a cell that’s already running, and the metal still has to be cast. So it pours into a pyramid mold, glowing orange, and cools to silver. In 1884 a pyramid like this went on display at Tiffany’s. Less than two years later, a twenty-two-year-old with homemade batteries figured out how to make the metal with electricity, and today one cell makes it in a little over a minute.

So a cell’s voltage is its push toward equilibrium. Compare Q to one to compare E to E naught, and when Q reaches K the push is gone. And the charge a cell moves is a count of electrons, current times time over Faraday’s constant, turned into atoms by the half-reaction. Hall’s batteries pushed because their chemicals were far from equilibrium, and the aluminum they made was counted in electrons, three for every atom, all the way up to the top of the Washington Monument.