Part C · How stuff changes

10 · Speed and catalysts

A mixture of hydrogen and oxygen can sit in a balloon for years. Add a spark and it explodes. A diamond would be slightly more stable as graphite, yet no diamond has ever been seen turning into pencil lead. Whether a reaction is downhill and how fast it goes are two separate questions.

Chapters 7 and 9 answered the first question: where things end up. This chapter is about the second: how long it takes them to get there. The answer is again the Boltzmann factor, applied this time to the top of a hill rather than the bottom of a well.

Only the hard collisions count

In chapter 9, molecules reacted only when they collided hard enough: with at least a certain energy, the activation energy Ea. Bonds have to be stretched and partly broken before new ones can form, and that costs energy up front. Most collisions are too gentle and the molecules just bounce off.

What fraction of collisions are hard enough? Chapter 7's answer: the chance of having energy E falls off like e−E/kT. So the fraction above the barrier is about e−Ea/RT (written per mole, with R instead of k). On a log scale that's a straight line, and heating makes the line shallower.

A reaction has a barrier of 50 kJ/mol. You warm it from 25 °C to 35 °C. That's only 3% hotter in kelvin (298 K to 308 K). How much faster does it go?

About twice as fast. The barrier sits far out in the tail of the distribution, about 20 RT away, and out there a small change in T makes a big relative change. That's the old rule of thumb that +10 °C roughly doubles a reaction rate. It holds for barriers near 50 kJ/mol; higher barriers are even more sensitive. It's why food keeps in the fridge, why a fever affects your whole body chemistry, and why lizards are sluggish in the morning.

The fraction of collisions with at least energy E, e−E/RT, on a logarithmic scale (each grid line is a factor of 10⁵). Dashed: 25 °C for reference. The dot marks the fraction that clears the barrier.

This is the Boltzmann factor alone. Real rates also depend on how often molecules collide and whether they hit in the right orientation; those fold into the prefactor A below and change only slowly with temperature.

Put together, this gives the Arrhenius equation, found by fitting data in 1889 before anyone could derive it:

k = A e−Ea/RT

Here k is the rate constant and A collects how often molecules try. Take logs and ln k is a straight line in 1/T with slope −Ea/R. Chemists measure barriers exactly that way: rates at a few temperatures, a straight-line fit, and read off the slope.

Valleys and passes

A more complete picture: the energy of a reacting system is a landscape over all the atoms' positions. Reactants sit in one valley and products in another. To get from one to the other the system must cross a ridge, and it will almost always go over the lowest point of the ridge, a mountain pass. Mathematically the pass is a saddle point, a minimum in every direction except one, and chemists call it the transition state.

Below is a toy landscape with 300 independent walkers, each jiggled by temperature. They all start in the left valley (reactants). The right valley (products) is lower. Watch how long it takes them to find the pass.

Top: the energy landscape (darker = lower energy; bands are contour steps of 0.5) with walkers in orange. Bottom: an Arrhenius plot. Each time the walkers have made 30 crossings at one temperature, a point is added: ln of the measured rate against 1/T. Each point is noisy, like real data; leave it running at one temperature and further measurements are averaged into the same point. Spread the temperatures out for a better slope. The fitted slope gives an effective barrier, close to the height of the pass shown in the readout.

A real reaction's landscape has one dimension per atomic coordinate (3N of them), not two. The walkers here move like dust in syrup (overdamped Brownian motion), not like molecules in a gas. Neither change affects the main point: the rate is set by the height of the lowest pass.

Try the lowest temperature. The walkers stay in the left valley for a long time, even though the right one is lower. They are trapped, not by anything holding them there, but because a big barrier makes the downhill trip extremely rare. Most of the world around you is like this. Wood, sugar and your own body are all thermodynamically unstable in air: they "want" to become CO2 and water, but the barriers are high enough that it takes a flame to get them over. Diamond is less stable than graphite at room conditions by about 3 kJ/mol, but the barrier is so high that diamonds really are forever.

Catalysts: a different pass

Now press Catalyst. A second, lower pass opens on the ridge. The walkers stream through it, the Arrhenius points jump up, and the fitted slope (the effective barrier) is smaller. But compare the readouts: the share of walkers that ends up in the products valley is the same as without the catalyst. A catalyst changes how fast the system reaches equilibrium, never where it ends up.

In this toy, the catalyst only reshapes the ridge and leaves both valleys untouched, which is why the equilibrium split doesn't move. Walkers sitting on the ridge are counted in neither valley. The fitted barriers come out somewhat above the true pass heights (about 3.7 against 3.5, and 1.8 against 1.4). That's because the rate's prefactor also depends on temperature and on how wide the pass is; the catalyst's pass is narrower. Real Arrhenius plots show the same effect, which is why chemists call the slope an effective activation energy.

This is how a catalyst works physically. It takes part in the reaction (holding molecules on a metal surface, or in an enzyme's pocket, in a shape where the bonds are easier to rearrange) and is given back unchanged at the end. It doesn't "lower the barrier" of the original route. It provides a different route with a lower pass. And because it's the same pass in both directions, it speeds the reverse reaction by exactly the same factor.

Here's the same thing with molecules. The grey discs are surfaces. Switch them from inert to catalytic and run the box again: near them, the barrier for A2 + B2 ⇌ 2 AB is much lower. The reverse reaction is switched on here, so watch where the AB count levels off.

80 molecules at fixed temperature, reacting both ways: A2 + B2 ⇌ 2 AB. Orange flashes are forward reactions, blue are reverse. The chart shows AB over time for this run, and the previous run faded, so you can compare inert and catalytic surfaces. The dotted line is where the AB count should level off according to the Boltzmann distribution.

Catalysts run the modern world. Car exhaust passes over platinum-group metals that convert carbon monoxide and unburnt fuel to CO2. The Haber–Bosch process makes ammonia for fertiliser on iron catalysts. Life runs on enzymes, protein catalysts. Catalase, in the yeast experiment below and in your liver, breaks down millions of hydrogen peroxide molecules per second per enzyme.

Common trap

"A catalyst lowers the activation energy" and "a catalyst shifts the reaction toward products". The first is shorthand: the catalyst offers a different path with its own, lower pass, while the old path is still there. The second is false: equilibrium depends only on the two valleys (the free energies of reactants and products), and the catalyst doesn't touch either. It speeds forward and reverse reactions equally.

Physics / ML bridge

Transition state theory gives the rate as an attempt frequency times the Boltzmann probability of being at the saddle point. In the Eyring form:

k = (kBT/h) e−ΔG‡/RT

Here ΔG‡ is the free-energy height of the pass. If you've used simulated annealing, parallel tempering or MCMC on a multimodal posterior, this is the same physics: the time to hop between modes grows like eΔE/T (Kramers' law). Annealing works by starting hot, so barriers are cheap, and cooling slowly so the walker settles into a deep minimum. A catalyst is a good proposal move: a new transition that keeps the stationary distribution unchanged but mixes much faster.

The Arrhenius form is an empirical fit, and the prefactor depends weakly on temperature. Light particles, especially hydrogen, can quantum-tunnel through barriers rather than climb them, which makes some rates faster than any classical estimate, particularly when cold. And nature doesn't anneal: it mostly sits wherever the last fast process left it, which is why kinetic trapping (diamond, wood in air, protein misfolding) is everywhere.

Tell it to a 6-year-old
Look
Two glow sticks, a cup of warm tap water and a cup of ice water. Make the room as dark as you can.
Wonder
"If we put one glow stick in warm water and one in icy water, which one will glow brighter?"
Try
Snap both glow sticks, drop one in each cup, wait a minute, and compare. Then leave them overnight: which one is still glowing in the morning?
True sentence
"Inside the glow stick, tiny pieces bump into each other and make light. In warm water they jiggle faster and bump harder, so more of them make light at once, and it's brighter. In cold water they're slower, so it's dimmer, but it lasts longer."

This is exactly the Boltzmann tail: warmer means more collisions clear the barrier. Use warm tap water, never boiling, and never cut a glow stick open. Next time, try a helper (catalyst): with 3% hydrogen peroxide, a spoon of dry yeast in warm water and a squirt of dish soap, the yeast's enzyme turns the peroxide into water and oxygen in seconds instead of months. That's "elephant toothpaste"; wear glasses and do it on a tray.

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