Why Chemical Reactions Happen

A match can sit in its box for years, surrounded by the oxygen in the air, and nothing happens to it. Drag its head across the strip on the box and it bursts into flame, and the flame creeps down the stick on its own. The wood and the air were touching all along. The strike added a little heat for a fraction of a second, and the flame gives back thousands of times more.

Below is a simple model of a match: a stick of wood 46 mm long with a 5 mm head, cut into half-millimeter slices. Each slice burns faster the hotter it gets, gives off heat as it burns, warms its neighbors, and loses heat to the air. Choose how hot the strike makes the head, , press Strike, and watch the temperature along the stick:

Strike gently, so the head reaches only 200 °C, and the warmth leaks away within a couple of seconds. Strike a little harder, to 260 °C, and the head flares past 1,000 °C, then the hot zone walks down the stick at about 3 mm a second with no more help. Between the two lies a sharp line, near 230 °C in this model1A real safety match crosses it with chemistry: friction turns a trace of red phosphorus on the striking strip into white phosphorus, which ignites in air at about 30 °C and sets off the potassium chlorate and sulfur in the head..

Each burning slice heats the next until that one makes heat faster than it loses it, and then it burns too: the fire pays for its own next step. That leaves two questions. Why does the wood need the strike at all, when burning gives out so much heat? And why does a little more heat make such a sharp difference? The answers apply to every chemical reaction, from rusting to cooking to the chemistry in your cells.

Bonds#

Wood is mostly cellulose, chains of carbon, hydrogen, and oxygen atoms, and when it burns those atoms end up in carbon dioxide and water. The atoms don't change; they swap partners. A chemical reaction is that swap, so first we need to know what holds two atoms together.

The simplest bond is the one in a hydrogen molecule, H₂: two hydrogen atoms sharing two electrons. Drag the right-hand atom, or set the distance to , and watch the energy of the pair; the plot's distance scale lines up with the atoms:

At 0.74 Å the curve bottoms out and there's no force: that's the bond's length2An ångström (Å) is 10⁻¹⁰ m, so a hydrogen molecule is about a ten-millionth of a millimeter long.. Pull the atoms apart and they tug back, like the ends of a spring. Push them together and they shove apart far more steeply. Pull far enough and the curve flattens: the atoms no longer feel each other, and the bond is gone.

The well's depth, 458 kJ/mol, is what the bond is worth[1]. A kilojoule per mole (kJ/mol) counts energy for a mole of molecules, 6×10²³ of them, here about 2 grams of hydrogen. So a bond is a spring with a cliff at one end: stretch it hard enough and it doesn't pull back. To break it, something has to put that energy in.

Atoms in a molecule never sit still at the bottom of the well; they shake. Give the bond of vibration and watch the energy line and the atoms:

With little energy the atoms swing over a short stretch, about 130 trillion times a second for real hydrogen. Add energy and the swing widens, mostly outward, because the inner wall is so steep. At 458 kJ/mol the line clears the top of the well, and the atoms fly apart on their first outward swing3The slider stops at 26 kJ/mol because quantum mechanics keeps every bond shaking with at least this much, the zero-point energy. So breaking H₂ from its lowest state takes 432 kJ/mol; tables quote 436 kJ/mol at room temperature..

Every reaction is built from bonds breaking and forming. Natural gas is mostly methane, CH₄, and burning it with oxygen rearranges nine atoms. Drag the slider to run the swap, and turn the model to see the molecules' shapes:

One carbon, four hydrogens, and four oxygens go in, and the same come out. Only the bonds change: four C–H and two O=O bonds break, and two C=O and four O–H bonds form. Yet methane and oxygen can sit mixed for years without a single swap, like the match and the air. Real reactions never break every bond at once, as the middle of the slider does; they go one small step at a time, and each step still has to pay for some bond-breaking before it gets anything back. That payment comes from heat.

Heat is motion#

The molecules in a gas never stop moving. The nitrogen and oxygen in the air around you fly at about 470 m/s on average, faster than an airliner, and each molecule hits another about seven billion times a second4From the kinetic theory of gases: a nitrogen molecule about 0.37 nm across, moving at an average 475 m/s through 2.5×10²⁵ molecules per cubic meter (air at 25 °C), sweeps out enough room to hit about 7×10⁹ others a second, traveling roughly 70 nm between hits.. What we feel as temperature is that motion. Here is a box of gas, drawn flat and slowed down enormously. Set the temperature of its walls to and watch the molecules:

Heat the walls to 1,200 K and molecules bounce off them faster, so within a few seconds the whole box warms. The average speed grows as the square root of the absolute temperature: four times hotter, twice as fast. But the average hides a lot. At any moment a few molecules crawl and a few, the orange ones, move far faster than the rest, and every collision changes which ones those are.

Before the next figure, a question about where that spread comes from.

In this box the walls don't add or remove energy, so the total stays fixed. Every molecule starts at the same speed. Watch the histogram of speeds, and press same speed again to rerun it:

The spike melts within a few collisions per molecule. In a glancing hit one molecule can leave with most of the pair's energy and the other with little, and the speeds smear out until they match the dashed curve, the Maxwell–Boltzmann distribution[2]. From then on it holds steady. Collisions keep changing which molecules are fast, but the shape stays, and the same shape appears however the gas started. A gas at a temperature always has this spread.

Those rare fast molecules are the ones that matter for chemistry. The next plot is the real distribution of energies in a gas, now in three dimensions, at . The orange line marks , about what the first step of a typical reaction needs. The shaded tail beyond it is too thin to see, so switch to a logarithmic scale, where each step up the axis is ten times more:

At room temperature the average molecule carries 3.7 kJ/mol of energy of motion, and about one in 150 million carries 50 kJ/mol or more. On the log scale the tail is a line sloping steadily down: each extra kJ/mol makes a molecule rarer by the same factor. Now warm the gas by 10 °C. The average energy rises 3% and the average speed under 2%, yet the share above 50 kJ/mol nearly doubles, to 1 in 80 million.

For energies well above the average, the share of molecules with at least energy $E$ falls off roughly as $e^{-E/RT}$, called the Boltzmann factor, where $R$ is the gas constant, 8.314 J/(mol·K), and $T$ the absolute temperature[3]. Raise the needed energy to 75 kJ/mol and the doubling becomes nearly a tripling: the steeper the climb, the more a little warmth helps.

Collisions#

To react, two molecules have to meet, and in a gas that means colliding. At seven billion collisions a second, if every collision reacted, every reaction in air would be over in a nanosecond. Most are bounces, and the first reason is that a collision's energy depends on how squarely the molecules meet. Send a blue molecule at a red one with an offset of of their combined width, carrying , and watch the head-on part:

A square hit, no offset, puts all 70 kJ/mol into the impact. At a 60% offset only 45 kJ/mol goes in along the line of centers; the rest carries the molecules past each other. At 80% a third is left, too little, and they glance off. Only the head-on part can squeeze the molecules hard enough to start rearranging bonds. In a gas, collisions come at every offset and energy, and the share whose head-on energy reaches a threshold follows the Boltzmann factor exactly.

Real molecules aren't smooth balls, so the direction they come from matters too. A textbook case is a hydroxide ion, OH⁻, swapping places with the chlorine in chloromethane, CH₃Cl. Turn the knob to send the hydroxide in from a different direction, starting straight from the back, opposite the chlorine:

From behind, the oxygen's spare electrons reach the back of the carbon. As the C–O bond forms, the C–Cl bond stretches and breaks, and the hydrogens flip over like an umbrella blown inside out, a change called a Walden inversion. From the side or the chlorine's end, say 120°, the hydroxide runs into hydrogens or the chlorine and bounces away. For most reactions only some directions work, cutting the useful collisions further.

Now put it all together in a box: 60 blue and 60 red molecules that turn into two green products when a blue and a red collide with enough head-on energy. The walls hold the gas at 300 K. Choose how much energy a reaction needs, , and watch the count of blue molecules fall:

With no energy needed, every blue–red collision reacts, and the blue molecules are gone in a couple of seconds. At 7.5 kJ/mol only about one collision in 20 has enough head-on energy, and the count falls more slowly. At 15 kJ/mol it's about one in 400, and the reaction crawls. The readout compares the share the box measures with the Boltzmann factor, $e^{-E/RT}$, and they agree: doubling the energy needed squares the odds against it5I've used energies a few times smaller than real ones so the box reacts while you watch. At 50 kJ/mol, one collision in about 500 million would count..

The barrier#

The energy a collision needs is called the activation energy, and it comes from the bonds themselves. The simplest reaction there is shows where. A hydrogen atom hits a hydrogen molecule end-on and takes one of its atoms, setting the other free: H + H₂ → H₂ + H. Drag the slider to move the reaction along its easiest path and watch the two distances and the energy:

The old bond has to stretch before the new one can form, and stretching costs energy. At the top of the hill both distances are 0.93 Å: the middle atom is half bonded to each neighbor. That arrangement is the transition state, and from there the new bond wins. Accurate quantum calculations put the barrier at 40 to 41 kJ/mol and the transition state at 0.93 Å[4][5], and the model surface I've used reproduces both6It's the London–Eyring–Polanyi–Sato surface[6], a classic formula built from the H₂ bond curve, with its one adjustable constant set to give the 40 kJ/mol barrier..

The atoms could take other paths, since both distances can be anything. Plot the energy for every pair of distances and you get a surface, like a map of hills and valleys. Turn it, and move the ball along the same path as before:

Each valley is a molecule with a far-off atom; its walls are the bond being squeezed or stretched. The plateau in the far corner is three separate atoms, 458 kJ/mol up, what breaking the old bond outright would cost. The easiest route between the valleys crosses the pass, less than a tenth as high, because the bonds trade off instead of breaking first. Every reaction has a surface like this, with more dimensions, and its activation energy is the height of its lowest pass.

Arrhenius's idea can be tested on our box of gas. The figure below measures, at one temperature after another, the share of collisions that arrive with at least the barrier's energy head-on, and plots it against the inverse of the temperature on a log scale. Choose a barrier and watch the points appear:

The points fall on a straight line whose slope is the barrier divided by the gas constant, so a chemist can measure a rate at a few temperatures, plot it this way, and read off the activation energy. Pick the 10 kJ/mol barrier and the line tilts more steeply: a higher barrier makes a reaction more sensitive to temperature7The measured points run a few percent below the line, partly because in crowded moments the box counts some repeat contacts as extra gentle collisions.. Written as a rate, this is the Arrhenius equation:

The rate constant $k$ says how fast the reaction goes. $A$ counts collisions that meet in a workable direction, and the exponential is the share energetic enough to cross the barrier $E_a$. Warming makes collisions only a few percent more frequent; nearly all the effect is in the exponential.

A common rule of thumb says many reactions run about twice as fast for every 10 °C. First, a prediction.

The plot shows a reaction's rate against temperature, relative to its rate at the colder temperature. Set the barrier, , and compare a fridge and a kitchen counter:

With a 50 kJ/mol barrier, 10 °C warmer gives 1.9 times the rate, and the counter at 22 °C runs 3.8 times faster than the fridge at 4 °C, a large part of why food keeps longer cold. A 20 kJ/mol barrier gains only 30% from ten degrees; a 100 kJ/mol one, almost four times. "Twice as fast per 10 °C" is a rule for barriers near 50 kJ/mol, not a law. The same steepness explains the match: a few hundred degrees turns a hopeless barrier into an easy one.

Catalysts#

Heating gets more molecules over a barrier. The other way is to find a lower pass. A catalyst offers one: the reactants bond to it for a while, and while they're held their bonds are stretched and lined up, so less has to break at once. It comes out unchanged at the end. Lower the catalyzed barrier by and compare the two routes:

Every kJ/mol taken off the barrier multiplies the rate by the same factor, so the gains compound. At body temperature, a barrier 10 kJ/mol lower makes a reaction about 50 times faster; 40 kJ/mol lower, about 5 million times. The reverse trip, from products back to reactants, climbs the same highest point, so it speeds up by exactly the same factor. A catalyst can't change where a reaction ends up, only how fast it gets there[3].

Many catalysts are solid surfaces. A car's catalytic converter spreads a few grams of platinum, palladium, and rhodium over a ceramic honeycomb, and exhaust gases react on the metal far faster than in the gas[9]. Here are two boxes of blue and red molecules needing 17.5 kJ/mol to react. On the right box's catalyst patches a blue molecule sticks for a while, and a red one hitting it there needs only . Watch the products pile up:

In the plain box a pair reacts a couple of times every ten seconds; on the catalyst, about seven times as often, until the blue molecules run low. Raise the catalyst's barrier to match the gas and the advantage vanishes; the box even does a little worse, since the stuck molecules are out of the gas. Holding molecules doesn't help by itself: the lower barrier does the work. The patches are never used up. A real converter works only once its metal is a few hundred degrees hot, which is why a car pollutes most in its first minutes.

Crowding#

Your cells are full of catalysts called enzymes, proteins with a pocket shaped to hold one reaction's molecules. Carbonic anhydrase, which turns carbon dioxide into bicarbonate in your blood, handles about a million molecules a second[10]. Another enzyme does in milliseconds a step in making DNA's letters that would take 78 million years, on average, by itself in water[11]. But a catalyst can only work on what reaches it, which brings us to crowding.

In this box, each blue and red pair that reacts is swapped for a fresh pair, so the numbers stay fixed, like a reactor fed at a steady rate. Change the number of blue molecules, , and red ones, , and watch the measured rate:

Double the blue molecules and the rate doubles; double both and it quadruples8Each measurement wanders by 10 to 20% in a box this small, so let it run a while before comparing.. Each blue molecule meets red ones at a rate set by how many red ones there are, so the rate follows blue × red. With concentrations in square brackets, $\text{rate}=k[\mathrm{A}][\mathrm{B}]$ is a rate law, with the Arrhenius constant $k$ in front. It's why a glowing splint bursts into flame in pure oxygen, nearly five times as concentrated as the oxygen in air.

Enzymes add a twist: there are far fewer of them than molecules to work on, and each handles one at a time. Set the amount of substrate, the molecule the enzyme works on, to and watch the six enzymes:

With little substrate the pockets sit empty most of the time, and the rate rises in step with the substrate, like any collision. With ten times more, almost every pocket is full whenever you look, and more substrate can't help. The level that fills half the pockets is called $K$, and the curve is the Michaelis–Menten equation, first measured in 1913[12].

Halfway#

So far every reaction has run one way. But the hydrogen surface looked the same from either valley: a collision that pushes atoms over the pass one way can push them back. Many reactions stop partway for this reason. Nitrogen dioxide, NO₂, is a brown gas; two NO₂ molecules can join into one colorless N₂O₄, which can split again. Here are two boxes at 300 K, one started with only N₂O₄, the other with only NO₂. Watch the share of NO₂ in each:

The boxes start at opposite ends and meet in the middle, each holding a bit under half its nitrogen as NO₂. The reactions don't stop: molecules keep splitting and joining about twenty times a second each way, at the same pace. That balance is chemical equilibrium. With too much N₂O₄, splitting outpaces joining; with too much NO₂, joining wins, until the two rates match. The flat model's bond is weak, a few kJ/mol; in real N₂O₄ it's about 57 kJ/mol, and at room conditions about a fifth of the molecules are split.

Squeeze the box and something interesting happens. Before trying it, a prediction.

The box below starts at equilibrium. Squeeze it to of its width and watch the brown of the gas, plotted as the amount of NO₂ per unit of space:

Squeeze the box to half its width and the brown jumps to twice as dark, then fades over a few seconds to about 1.4 times the original, as the NO₂ share drops from about 46% to 33%. Crowded together, NO₂ molecules meet more often, so joining outpaces splitting, and joining turns two molecules into one, which eases the crowding. Henry Le Chatelier stated the general rule in 1884: disturb a mixture at equilibrium and it shifts to partly undo the disturbance[13].

Temperature shifts the balance too. Splitting N₂O₄ takes in about 57 kJ/mol, and joining gives it back[14], so a sealed tube of the gas is pale in ice water and dark brown in boiling water. Set the tube's temperature, :

In ice water only about 7% of the N₂O₄ is split; at room temperature, 19%; in boiling water, 82%, with eleven times as much NO₂ as in ice water. Heat favors the direction that takes heat in, Le Chatelier's rule again. Underneath is the Arrhenius equation: splitting has the higher barrier, higher than joining's by the 57 kJ/mol it takes in, so warming speeds it up more.

Heat out, heat in#

Now, where the match's heat comes from. Breaking bonds costs energy and forming them releases it, and which side wins decides whether a reaction gives out heat or takes it in. Choose a reaction and compare the energy to break every bond in the reactants with what the products' bonds give back:

Burning a mole of methane breaks bonds worth 2,648 kJ and forms bonds worth 3,450 kJ. The 802 kJ difference leaves as heat, matching the measured 802 kJ per mole, about 50 kJ per gram[15]. The C=O and O–H bonds in carbon dioxide and water are among the strongest that carbon, hydrogen, and oxygen can form, which is why burning wood pays so well. For nitrogen and oxygen, N₂'s triple bond is so strong that making nitric oxide takes in 181 kJ/mol, so it happens only in very hot places like lightning and engines9The bond energies are averages over many molecules, except for the diatomic molecules, whose bonds are measured directly. They don't always land this close to the measured heats, but they're rarely far off.. This heat is the reaction's enthalpy change, $\mathrm{\Delta }H$.

That heat shows up as faster molecules. In this insulated box, a blue and a red molecule react when they hit with at least 7.5 kJ/mol head-on, and the products leave with more energy than the pair brought, or less. Choose which, and watch the thermometer:

The reaction that gives out heat warms the box to about 650 K, and the warming feeds back: more reacts in the third and fourth seconds than in the first two, though fewer molecules are left to meet. That self-acceleration is the match's secret. The other reaction takes energy from the molecules' motion, so the box cools below 200 K, and the colder it gets the fewer collisions clear the barrier. It crawls with about 40% left, which is why cooking needs a stove.

Back to the match#

Now every piece of the match is in place. Burning gives out heat, heat speeds burning steeply, and the wood loses heat to the air roughly in proportion to how much hotter it is. In 1928 Nikolay Semenov put the two on one plot, and the comparison explains ignition[16]. Set the strike's temperature, press Strike, and watch the dot on the temperature axis:

Where the orange curve is above the grey line the wood warms, and where it's below it cools, so the dot runs to the nearest crossing. Near room temperature they cross at once, which is why a match keeps for years. The middle crossing, near 320 °C, is the ignition point, a knife edge: strike to 300 °C and the dot slides back, strike to 330 °C and it runs away to steady burning above 1,000 °C. That's the sharp line from the first figure.

Blowing steepens the grey line. Push the cooling to 0.25 and the upper crossing vanishes: no temperature makes heat as fast as the wind removes it, so the flame dies. A gentle breeze can feed a fire more air than it cools it, which this simple picture leaves out.

Last, the real thing. A match flame isn't burning wood: the flame's heat splits the wood into gases, and the gases burn where they mix with air. Turn the burning match, then blow it out:

The flame stands clear of the wood because the solid never burns. Heat breaks the cellulose into carbon monoxide, hydrogen, methane, and heavier vapors, which flow out and burn in the air. The flame's heat makes more gas and the gas makes more heat: the loop that kept the dot at the top of Semenov's plot. Blown out, the wood still gives off gas for a few seconds, unburnt, as white smoke. Hold another flame in that smoke and it can catch and run back down to the wood; it's easiest to see with a candle.

The same few ideas set how long food keeps, how much a fever speeds your metabolism, why a car's exhaust needs a hot catalyst, and why iron rusts faster in warm, wet air. What they don't do is predict a barrier from scratch. That means solving quantum mechanics for every electron along the reaction's path, which is possible for three hydrogen atoms and still hard for most reactions chemists care about.

References#

1. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV: Constants of Diatomic Molecules, Van Nostrand Reinhold (1979). The H₂ bond length and well depth behind the bond figures.
2. J. C. Maxwell, “Illustrations of the dynamical theory of gases,” Philosophical Magazine 19 (1860), 19–32. The distribution of molecular speeds.
3. P. Atkins, J. de Paula, and J. Keeler, Atkins' Physical Chemistry, 11th edition, Oxford University Press (2018). Collision theory, the Arrhenius equation, catalysis, and equilibrium.
4. P. Siegbahn and B. Liu, “An accurate three-dimensional potential energy surface for H₃,” Journal of Chemical Physics 68 (1978), 2457–2465. The barrier and transition state of H + H₂.
5. D. G. Truhlar and C. J. Horowitz, “Functional representation of Liu and Siegbahn's accurate ab initio potential energy calculations for H + H₂,” Journal of Chemical Physics 68 (1978), 2466–2476.
6. S. Sato, “On a new method of drawing the potential energy surface,” Journal of Chemical Physics 23 (1955), 592. The LEPS surface used in the barrier figures.
7. S. Arrhenius, “Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren,” Zeitschrift für physikalische Chemie 4 (1889), 226–248. doi:10.1515/zpch-1889-0116.
8. H. Eyring, “The activated complex in chemical reactions,” Journal of Chemical Physics 3 (1935), 107–115. doi:10.1063/1.1749604.
9. R. M. Heck and R. J. Farrauto, “Automobile exhaust catalysts,” Applied Catalysis A: General 221 (2001), 443–457. doi:10.1016/S0926-860X(01)00818-3.
10. S. Lindskog and J. E. Coleman, “The catalytic mechanism of carbonic anhydrase,” Proceedings of the National Academy of Sciences 70 (1973), 2505–2508. doi:10.1073/pnas.70.9.2505.
11. R. Wolfenden and M. J. Snider, “The depth of chemical time and the power of enzymes as catalysts,” Accounts of Chemical Research 34 (2001), 938–945. doi:10.1021/ar000058i.
12. L. Michaelis and M. L. Menten, “Die Kinetik der Invertinwirkung,” Biochemische Zeitschrift 49 (1913), 333–369. English translation by K. A. Johnson and R. S. Goody, Biochemistry 50 (2011), 8264–8269.
13. H. Le Chatelier, “Sur un énoncé général des lois des équilibres chimiques,” Comptes rendus de l'Académie des sciences 99 (1884), 786–789.
14. M. W. Chase, Jr., NIST-JANAF Thermochemical Tables, 4th edition, Journal of Physical and Chemical Reference Data, Monograph 9 (1998). Enthalpies and entropies of NO₂ and N₂O₄.
15. P. J. Linstrom and W. G. Mallard, editors, NIST Chemistry WebBook, NIST Standard Reference Database 69, National Institute of Standards and Technology. Heats of formation and combustion used for the measured values.
16. N. N. Semenov, “Some problems relating to chain reactions and to the theory of combustion,” Nobel Lecture, 11 December 1956.