Topic 4.5: Stoichiometry

Square canisters, round holes

In April 1970, about two hundred thousand miles from Earth, an oxygen tank exploded on Apollo 13, and the three astronauts had to climb into their lunar module, Aquarius, and use it as a lifeboat. Aquarius was built to keep two people alive for two days, and now it had to keep three people alive for about four, which caused a problem nobody could see.

Every time they breathed out, they added carbon dioxide to the air in the cabin. You can’t see carbon dioxide in real life, so I’ve made it glow, and you can watch it build up.

By their second day in the lifeboat, a warning light came on. The carbon dioxide was climbing toward a dangerous level. The lunar module cleaned its air with round canisters of a chemical called lithium hydroxide, and it didn’t have enough of them. The command module had plenty of canisters, but they were square, and the lunar module’s openings were round.

So a team of engineers in Houston, led by Ed Smylie, worked out a fix built only from things the crew had on board, plastic bags, a cardboard cue card from the flight plan, a hose from a spacesuit and a lot of gray tape. Swigert and Lovell built two of them, and within minutes of hooking up the first one, the carbon dioxide started to fall.

NASA’s photo caption calls it the Mail Box. The tape got the square canister connected, but what cleaned the air was the chemistry inside it, where lithium hydroxide grabs carbon dioxide and locks it into a solid.

And that chemistry has a limit, because every gram of lithium hydroxide can only take so much carbon dioxide before it’s used up. By the end of this lesson, you’ll be able to do the math behind that box. How much lithium hydroxide does it need? You’ll work it out from one balanced equation. The same map works when you weigh a solid, measure a gas or pour a solution.

At the very end, you’ll load a canister of your own. Then you’ll find out whether it keeps the air clean. So let’s open up one of those square canisters. Then we can see what’s going on inside.

Inside, the canister holds a bed of little white granules of lithium hydroxide, and if we keep going, down into one of the tiny pores in a granule, we land on the surface of a crystal. Lithium hydroxide is an ionic solid, so this surface is a pattern of lithium ions and hydroxide ions, and the gas drifting past is the cabin air, which on Apollo was almost pure oxygen.

Now watch a carbon dioxide molecule land. The ions from two formula units of lithium hydroxide rearrange around it into lithium carbonate, which stays right here as part of the solid, and the two hydrogens and one oxygen that are left over float away as a molecule of water. It happens again and again, always two lithium hydroxides for every carbon dioxide.

Chemists write that as one balanced equation, two lithium hydroxides plus one carbon dioxide make one lithium carbonate and one water. On the AP exam, nearly every stoichiometry question starts with these coefficients. Sometimes it hands you a picture of molecules instead of numbers. We’ll try one in a few minutes.

Let’s count the atoms on each side. There are two lithiums on the left and two on the right. There are four oxygens on the left, two in the hydroxides and two in the CO₂, and on the right there are three in the carbonate and one in the water, so four again.

The two hydrogens end up in the water. And the one carbon ends up in the carbonate. Every atom that goes in comes out, just with new partners. That lets us predict the amount of one substance from the amount of another.

The coefficients work like a recipe. Two lithium hydroxides for every one carbon dioxide. That also means two dozen for every dozen, and two moles for every mole. A mole is just a much bigger count.

That recipe turns into a conversion factor, one mole of CO₂ for every two moles of lithium hydroxide, which you can write right side up or upside down, depending on which way you’re going. So try this one. Which conversion factor turns moles of lithium hydroxide into moles of carbon dioxide captured?

Grams in, grams out

Here’s a map that works for every one of these problems. Lithium hydroxide lives on one island and carbon dioxide on the other, and the only bridge between them is the mole ratio, because the coefficients only talk about particles and moles, never grams. But nobody scoops out moles in a lab, they weigh grams, so each island gets an on-ramp, the molar mass, which turns grams into moles and back.

Let’s try a training problem, a load of one hundred point zero grams of pure lithium hydroxide. How many grams of carbon dioxide can it capture? First I get from grams onto the lithium hydroxide island, so I divide by lithium hydroxide’s molar mass, twenty-three point nine five grams per mole, and grams cancel grams.

Then I cross the bridge, one mole of CO₂ for every two moles of lithium hydroxide, so moles of lithium hydroxide cancel, and the molar mass of CO₂, forty-four point oh one grams per mole, takes me back to grams. That comes to ninety-one point eight eight grams of carbon dioxide, four significant figures, because our hundred point zero grams had four.

Now hold on, because that looks strange at first. The recipe is two to one, and yet the masses come out almost equal. That’s because two moles of lithium hydroxide weigh forty-seven point nine grams, and one mole of carbon dioxide weighs forty-four point oh one, so two light things weigh about as much as one heavy thing.

So that two to one is about particles, and it never means two grams for every gram. The coefficients count the particles, the molar masses weigh them, and every one of these problems needs both. Now let’s go the other way, because the same map works backwards. Say a job needs the canister to capture two hundred fifty grams of carbon dioxide. How much lithium hydroxide do we need?

I’ll start you off with the two hundred fifty grams of CO₂ and divide by its molar mass to get onto the CO₂ island. Then it’s your turn to cross the bridge and get off on the other side. How many grams of lithium hydroxide does it take to capture two hundred fifty grams of carbon dioxide?

When the lithium hydroxide runs out

So a hundred grams of lithium hydroxide can take about ninety-two grams of CO₂, and then what? Let’s cut the canister open and watch. The white lithium hydroxide gets used up starting at the end where the air comes in. The used part, which I’ve shaded so you can see it, has turned into lithium carbonate.

When the used part reaches the far end, there’s no lithium hydroxide left to grab anything, so the carbon dioxide goes straight through, and the warning light comes back on. The lithium hydroxide ran out first. Chemists call it the limiting reactant, because it limits how much product you can make. The carbon dioxide kept on coming, so it was in excess.

Let’s look at that with a handful of particles, so we can count every one. Here are eight lithium hydroxides’ worth of ions in one pore. Set the number of carbon dioxide molecules so that everything reacts and nothing is left over, then press React.

Now let’s send in six carbon dioxides instead of four. The first four each find two lithium hydroxides, and then the lithium hydroxide is gone, so two carbon dioxides are left floating there with nothing to react with.

So the pore ends up with four lithium carbonates and four waters, plus two leftover carbon dioxides, which are part of the answer too. An AP question can hand you a picture like this and ask what’s in the box afterward, and the classic mistake is drawing only the products.

Now let’s do the same thing in moles, the way you’ll usually see it. Say we start with four point oh oh moles of lithium hydroxide and three point oh oh moles of carbon dioxide. It’s tempting to say there’s more lithium hydroxide, so the CO₂ must run out first, and that’s backwards, because the recipe uses lithium hydroxide twice as fast. Here is the safest way to decide. Ask how much of one product each supply could make.

Four moles of lithium hydroxide could make two moles of lithium carbonate, three moles of CO₂ could make three moles, and the smaller number wins, so lithium hydroxide is the limiting reactant, even though we started with more of it. So having more moles of something doesn’t mean it lasts longer, because what matters is how fast the recipe uses it up.

Let’s keep track in a table with three rows, before, change and after. Before, we have four moles of lithium hydroxide, three of CO₂ and no products yet. The change row is where the coefficients live, and only there. Lithium hydroxide is the limit, so all four moles get used up, minus four, the recipe uses half as much CO₂, so minus two, and we make plus two of lithium carbonate and plus two of water.

So afterward we have no lithium hydroxide, one mole of CO₂ left over, and two moles each of lithium carbonate and water, which is the same picture we saw in the pore, just with a lot more particles. Now let’s change the amounts. This graph shows how much lithium carbonate we make as we add more and more lithium hydroxide, with the CO₂ fixed at three moles.

At first, every two moles of lithium hydroxide make one more mole of product. Three moles of CO₂ need six moles of lithium hydroxide. Past six, the CO₂ runs out and the line goes flat. Now the CO₂ is the limiting reactant. So if we doubled the CO₂ and kept four moles of lithium hydroxide, we’d still make two moles, because the extra CO₂ has nothing to react with.

Now what if we double the lithium hydroxide from four moles to eight and keep three moles of CO₂? How much lithium carbonate can form? And if we double both, the product doubles too, to four moles.

Counting a gas you can’t weigh

So far I’ve been handing you carbon dioxide in grams and moles, but nobody on Aquarius could put the cabin air on a scale, so you measure a gas by its volume, its pressure and its temperature. And that gives our map a second on-ramp, the ideal gas law, P V equals n R T, which you can solve for n, the number of moles.

For this R, the volume has to be in liters, the pressure in atmospheres and the temperature in kelvin, and R is zero point zero eight two oh six. Let’s set up a test like the ones engineers run on the ground, pushing a hundred liters of carbon dioxide, measured at one atmosphere and 298 kelvin, through a canister. How much lithium hydroxide does that use up?

First, P V over R T gives four point oh nine moles of carbon dioxide. Then the bridge doubles that to eight point one eight moles of lithium hydroxide, and times twenty-three point nine five grams per mole, that’s a hundred ninety-six grams.

A hundred liters is about a big trash bag full of gas, and those four moles of CO₂ weigh about a hundred eighty grams, so needing a little more than that in lithium hydroxide is just what we’d expect. A classmate solves this same problem with twenty-two point four liters per mole. What’s wrong with that?

Paris, 1783: hydrogen by the barrel

Apollo’s canister has no solution in it. For one reaction with all three on-ramps, we have to go back almost two hundred years. On the first of December, 1783, a reported four hundred thousand people crowded around the Tuileries garden in Paris to watch two men go up in a balloon filled with hydrogen, and Benjamin Franklin watched from his carriage.

Nobody sold hydrogen, so Jacques Charles and the Robert brothers had to make it, in a ring of oak barrels full of iron nails, with sulfuric acid poured into one barrel after another. The gas bubbled up through a tub of water that washed out the acid, and then pipes carried it up into a silk balloon that held three hundred eighty cubic meters. The barrels got so hot that the iron pipes were too hot to touch.

Let’s turn one barrel to glass. Iron plus sulfuric acid gives iron two sulfate and hydrogen gas, one of each, and that one equation has a solid, a solution and a gas in it, all three on-ramps at once. Down at the surface of a nail, every iron atom that leaves as an ion sends one hydrogen molecule bubbling away.

Their first hydrogen balloon, three months earlier, held thirty-five cubic meters, and to fill it they poured nearly a quarter of a tonne of sulfuric acid onto half a tonne of scrap iron. So which one runs out? Half a tonne of iron is about nine thousand moles, and a quarter of a tonne of sulfuric acid, at ninety-eight point oh eight grams per mole, is at most about twenty-five hundred moles.

The recipe is one to one, so the acid runs out first. It’s the limiting reactant, and the iron was way in excess. On a mild day, say twenty degrees Celsius, that’s two hundred ninety-three kelvin, at one atmosphere. The gas law turns twenty-five hundred moles of hydrogen into about sixty-one thousand liters. That’s sixty-one cubic meters.

That’s the most they could get, and they needed only thirty-five cubic meters. Good thing. A real reaction gives less than the maximum, and the reports say filling that balloon was a struggle. Mixing an acid with water makes a solution, and for solutions we measure concentration as molarity, moles of solute per liter of solution, so the third on-ramp is moles equals molarity times liters. So two liters of three molar acid hold six moles of it, and going the other way, dividing the moles by the molarity tells you how many liters you need.

A book from 1784 shows how to fill a small balloon from a single barrel, so let’s try it. This barrel has twenty kilograms of nails, which is plenty, so the acid decides how much hydrogen we get. The acid is three point oh oh molar, and let’s say it’s a freezing afternoon, zero degrees Celsius and one atmosphere, exactly where twenty-two point four liters per mole works. The test balloon holds two cubic meters, which is two thousand liters.

How many liters of the acid will fill the balloon exactly? Work it out, then pour it and see. And here’s a barrel set up the other way around. It has only a couple of kilograms of nails and plenty of acid. The nails dissolve away completely and the bubbling stops, but there’s still acid in the barrel, because this time the iron was the limiting reactant.

The blue cloud that stops growing

Let’s do one more solution problem, and this time both reactants come dissolved in water. Here are fifty milliliters of zero point two oh oh molar copper two sulfate, which is that clear blue, and we pour in thirty milliliters of zero point five oh oh molar sodium hydroxide. Copper ions and hydroxide ions grab each other and make this pale blue solid, copper two hydroxide, and then it stops growing, and the liquid on top is still blue.

Only two ions really react, one copper ion for every two hydroxides, and the sodium and sulfate ions just stay in solution. This one’s yours, so I’ll only give you the plan. Use molarity times liters to get the moles of each ion, ask how much solid each one could make, take the smaller amount, and turn it into grams. How many grams of copper two hydroxide can form?

That blue liquid on top is copper ions that never found enough hydroxide. Now start over with fifty milliliters of hydroxide. That’s zero point zero two five moles, enough for zero point zero one two five moles of solid. The copper only has enough for zero point zero one, so this time the copper runs out first. The liquid on top is colorless.

Will the canister hold?

Now we have every tool that box needs. Back to Aquarius. This is a training problem, so the numbers are ours and not NASA’s. The canister has to take up three hundred liters of carbon dioxide, measured at one atmosphere and 298 kelvin, and we have two loads of pure lithium hydroxide to choose from, four hundred grams and six hundred grams. Load the canister you think will hold all three hundred liters, and run it.

Let’s prove it with the map. Three hundred liters at one atmosphere and 298 kelvin is twelve point two seven moles of CO₂, the bridge doubles that to twenty-four point five four moles of lithium hydroxide, and times twenty-three point nine five, we need five hundred eighty-eight grams. On the AP exam, you’d answer it in one sentence. The six-hundred-gram load is enough, because capturing three hundred liters of CO₂ takes only five hundred eighty-eight grams of lithium hydroxide. Four hundred grams falls short.

For the real thing, engineers in Houston tested the setup in an altitude chamber before the crew ever built it. The Mail Box kept the air clean for the rest of the flight. On the seventeenth of April, 1970, Lovell, Haise and Swigert splashed down in the Pacific, and a lot of gray tape and two lithium hydroxides for every carbon dioxide were part of the reason they made it home. The math behind that box? Now you can do it yourself.