Chain-branching explosion limits

Why can a hydrogen and oxygen mixture sit quietly at one pressure and blow itself apart at a pressure only slightly higher?

Put hydrogen and oxygen in a heated vessel at a few hundred degrees. At a low enough pressure nothing happens. Raise the pressure a little and it detonates. So far, ordinary — more reactant, more reaction.

Then raise the pressure further, and it goes quiet again. Raise it further still and it explodes a second time. The same two gases at the same temperature pass through explosive, quiet, explosive as you do nothing but squeeze them. Any picture in which rate simply rises with concentration has already failed, and the strangeness is worth sitting with: what mechanism can be switched off by adding more of the reagents?

Begin with why anything sharp happens at all. Hydrogen and oxygen do not react molecule against molecule in any useful way; the reaction runs through chain carriers — free radicals such as H, O and OH — each reacting fast and making more carriers. The step that matters produces more than it consumes: a hydrogen atom meeting an oxygen molecule gives OH and O, and that O, meeting hydrogen, gives OH and another H. Count the bookkeeping: one H atom in, three H atoms out. The carriers are reproducing.

Anything that reproduces also dies. Carriers are created at a rate proportional to how many there already are, and destroyed at another rate also proportional to how many there are, so their number grows or shrinks as an exponential whose exponent is the difference. That is the whole explanation of sharpness: nothing need change abruptly for the outcome to change abruptly, since the exponent merely has to change sign. On one side the population decays to nothing; on the other it doubles and redoubles until the gas has burned.

So what destroys carriers — and how does each route depend on pressure?

The first route is the vessel itself. A radical reaching the wall is adsorbed and quenched, and at very low pressure it travels far between collisions, so the wall is easy to reach and carriers die faster than branching makes them. Raise the pressure and the gas becomes a thicket: radicals collide more often, diffuse to the wall more slowly, and branching wins. That is the first limit — and notice its prediction, that the boundary should depend on the size and inner surface of the vessel, because the loss term belongs to the container rather than the gas.

The second route is what makes the whole thing counterintuitive, and it falls out of a comparison you can do on paper. The branching step is a two-body encounter, H with O2, so its rate is proportional to the concentration of each. But those two can instead combine into HO2, and that combination needs a third body to carry away the released energy, so its rate is proportional to H, O2 and whatever bystander takes it. Divide the second by the first and everything cancels except the bystander concentration — which is just pressure. Termination therefore climbs with pressure while branching does not gain on it, and above some pressure the three-body route outruns branching. HO2 is comparatively unreactive here and drifts off to be destroyed, so it is a sink. That is the second limit: near five hundred degrees Celsius it sits around a tenth of an atmosphere, a figure I give as recalled rather than derived. Its signature is that, unlike the first, it is a homogeneous gas-phase competition and should not care about the vessel.

The third limit is the least clean, and it is worth saying so. HO2 is not permanently inert: push pressure and temperature high enough and it attacks hydrogen to give hydrogen peroxide, which decomposes into two OH radicals — branching restored by a slower, roundabout path. Alongside it runs a thermal effect, since a reaction releasing heat faster than the vessel loses it accelerates itself regardless of any radical accounting. How the two contributions divide there is discussed rather than settled.

One thing was constant across the entire map. Hydrogen burning to water is enormously favourable thermodynamically at every pressure on the diagram, in the quiet regions exactly as in the explosive ones — so thermodynamics explains nothing here. Every boundary on the plot is kinetics of a particular kind: two rate terms whose dependence on pressure differs by one power.

Three clarifications, in ascending order of how much they change the picture.

An "explosion limit" is a boundary in pressure and temperature together, not a pressure. The hydrogen-oxygen diagram is a peninsula of explosive conditions poking into a quiet region, and a vertical line through it at fixed temperature gives our three crossings.

This is a branching chain, sharing its arithmetic with fission — and the difference is where the interest lies. A lump of uranium is limited by neutrons leaking through its surface, so its threshold moves one way: more material is never safer. Here the dominant loss above the first limit is itself a chemical reaction, one that gains on branching as pressure rises, which is why this boundary is non-monotonic and a fission threshold is not.

And the account offers a clean falsification test, one the classical work actually ran. Change the vessel: coat its interior with a salt film, or use a narrower tube. If wall quenching sets the first limit and gas-phase three-body termination the second, the first must shift and the second must not. Should the second limit move with a surface coating, the three-body argument is wrong; should the first sit unmoved, wall termination is. Both halves are testable independently, which is what makes it a test rather than a story.

• How two-stage ignition in hydrocarbons arises from degenerate branching, and what octane rating measures.

Every term the collection defines is gathered in the glossary.