Topic 5.10: Multistep Reaction Energy Profile

Multistep energy profiles: the part thieves steal

It’s the middle of the night, it’s raining on a quiet street, and for about two seconds, under one of the parked cars, something throws sparks. The next morning, the owner gets in and turns the key, and the car sounds like this. If you get down on the wet ground and look underneath, there’s a gap in the exhaust pipe, with a clean cut on either side.

Someone slid under this car in the night, sawed out one part in a few minutes, and left everything else. Across the United States, insurance claims for this exact theft went from more than sixteen thousand in 2020 to nearly sixty-five thousand in 2022.

So what’s so valuable about this one part? It’s called a catalytic converter, and if we open it up, there’s a ceramic honeycomb inside, thousands of tiny channels that all of the exhaust has to squeeze through. The walls of those channels are dusted with a few grams of three metals, platinum, palladium and rhodium, and back in March 2021, an ounce of rhodium sold for nearly thirty thousand dollars, more than fifteen times the price of gold.

One of the biggest jobs those metals have is turning the carbon monoxide coming out of the engine, which is a poison, into carbon dioxide. And the strange thing is that once the converter is hot, the metal keeps doing that for years of driving, and it’s still there, because whatever the reaction does with it, it never uses it up.

So the thief stole the one part of the car’s chemistry that never gets used up, which makes you wonder what that metal is actually doing in there. By the end of this lesson, you’ll be able to draw the energy path of a reaction that happens in steps, read the real one that a Nobel Prize winner drew for this exact metal, and explain how a few grams of metal like platinum can clean a poison out of a car’s exhaust for years without being used up, which is exactly why somebody crawled under this car to steal it.

To see what that metal is doing, we first need a way to draw the path a reaction takes when it happens in more than one step. So let’s start with two of the gases in this exhaust, on their own, before they ever reach the metal.

Exhaust carries a mix of gases, and two of them, nitrogen dioxide and carbon monoxide, can react with each other all on their own, with no metal anywhere, and chemists have studied that reaction closely in the lab. They found that below about two hundred twenty-five degrees Celsius, this reaction doesn’t happen in a single collision, it happens in two steps, one after the other.

And a list of steps like that is called a mechanism, and being handed one and asked to sketch its energy profile is a classic AP question. In step one, two nitrogen dioxide molecules collide and an oxygen atom hops from one to the other, making NO three and NO, and that’s the slow step.

Then in step two, the NO three runs into a carbon monoxide and hands it that extra oxygen, which turns the carbon monoxide into carbon dioxide and gives back an NO two, and that step is fast. Now let’s draw the energy of this whole group of molecules as the reaction goes, starting here, with two NO twos and a CO.

As the two NO twos collide, the energy climbs to a peak, where the oxygen is halfway between the two nitrogens, and that’s a transition state, an arrangement that lasts for an instant, not something you could ever put in a bottle. Then the energy drops into a valley, and down here we have real molecules, NO three and NO, with the CO still waiting its turn.

NO three is the intermediate, made in step one and used up in step two, and intermediates always sit in the valleys. Notice that this valley sits higher than where we started, because step one is uphill, by about ninety-five kilojoules per mole. Step two gets its own hump, a much smaller climb, because step two is the fast one, and then the energy falls all the way down to the products, far below where we started.

So every step gets its own hump, and every hump has its own activation energy, the climb from the level where that step starts up to its peak. The second step’s climb starts down in the intermediate’s valley, not back at the very beginning. And the overall energy change only cares about the start and the finish, about minus two hundred twenty-six kilojoules per mole, no matter what happens in between.

That’s how a mechanism becomes an energy profile, and since you’ve just watched it happen, let’s put the picture in your hands. Here’s one moment from the middle of this reaction, NO three sitting next to NO with the CO still waiting, so drag it to the spot on the curve where it belongs.

Measuring every climb

Here’s a made-up two-step reaction, drawn the way the AP exam usually hands it to you, with an energy in kilojoules per mole on every level that matters. Step one starts at zero and climbs to eighty, so its activation energy is eighty minus zero, eighty kilojoules per mole. For step two it’s tempting to read off fifty, because the second peak sits at fifty, but fifty is measured from zero, and step two doesn’t start at zero.

Step two starts down here, in the intermediate’s valley at twenty, so its climb is fifty minus twenty, thirty kilojoules per mole. This is the one people miss most often, so let me put it another way. An activation energy is the climb a step makes from wherever that step starts, so the bottom of the arrow always goes on that step’s own starting level.

The overall energy change is easier, because it’s just finish minus start, minus sixty minus zero, which is minus sixty kilojoules per mole. And we can check that, because step one went up twenty and step two came down eighty, and plus twenty plus minus eighty is minus sixty, the same answer. One valley in the middle means one intermediate, and two peaks mean two transition states, one for each step.

Let’s try a harder one together, a made-up reaction with three steps, which means three humps and two intermediates sitting in the valleys. Step one starts at twenty and peaks at ninety-five, so its climb is seventy-five kilojoules per mole. The second step is where people slip, so I want you to place that arrow yourself.

Step two starts somewhere on this diagram, so drag the bottom of its arrow to that spot and read off its activation energy. Here’s the same diagram on the board, with both of those arrows drawn in.

Now you try one on your own, so what’s the overall energy change for this whole three-step reaction, in kilojoules per mole? And step three climbs from the second valley at ten up to fifty, so that’s forty kilojoules per mole.

The highest peak on this whole diagram belongs to step two, so does step two have the biggest climb of the three? Now watch what happens if I move every single level up by forty, and notice that none of these answers change at all, because every one of them is a difference between two levels.

Sketching a profile from a mechanism

When a question gives you a mechanism in words and asks you to sketch the profile yourself, it helps to go back to our exhaust reaction and pull out the rules it followed. First, one hump for every step, so two steps means two humps. Second, each step ends higher than it started if it’s endothermic and lower if it’s exothermic, which is why our valley sits above the start.

Third, the slow step gets the biggest climb, and fourth, the finish sits below the start when the whole reaction is exothermic, and above it when it’s endothermic. Here’s a mechanism you haven’t seen before, along with four sketches, and only one of them follows all four rules. Step one is fast and exothermic, step two is slow and endothermic, and the reaction is exothermic overall, so which of these four sketches matches it?

There’s one more thing these pictures can trick you into, so watch what happens if I stretch this diagram out to twice its width. Every single energy stays exactly where it was. That’s because the horizontal axis shows reaction progress, just the order in which the arrangements of atoms happen, and it doesn’t measure time at all, so a wide hump doesn’t mean a slow step, and each point on the curve is one arrangement of all the atoms, never how much of the reaction is done.

So now you can read the route a reaction takes, one step at a time. And that’s exactly what we need to understand the metal in that converter, because what the metal does is give the molecules a completely different route to take.