The Maxwell–Boltzmann distribution, temperature and catalysts

A spread of energies

The molecules of a gas at one temperature do not all have one energy. In the air inside a bicycle tyre each molecule is struck by others billions of times a second, and every collision passes energy from one molecule to another. At any instant a few molecules have just been brought almost to a stop, most are moving at middling speeds, and a very few have been struck several times in a row from behind and are moving very fast. The temperature fixes the average. It does not fix the energy of any one molecule.

The Maxwell–Boltzmann distribution is the graph of that spread. Kinetic energy runs along the horizontal axis, and the vertical axis is the number of molecules with each energy.

The curve has a shape to learn. It starts at the origin, because no molecule has zero energy. It rises steeply to a single peak, then falls away more slowly in a long tail to the right. The tail comes closer and closer to the energy axis without ever reaching it: there is no maximum energy, only energies that fewer and fewer molecules have. A curve that meets the axis on the right, or turns back up, is drawn wrongly.

The peak is at the most probable energy, the energy more molecules have than any other. The mean energy lies to the right of the peak, because the long tail of fast molecules pulls the average up. A label 'average energy' on the peak is therefore wrong.

The area under the whole curve is the total number of molecules in the sample. The area under the part of the curve to the right of the activation energy is the number of molecules with at least that energy, the ones able to react if they collide in the right orientation. For most reactions at room temperature Ea lies far out in the tail, and that area is a thin sliver of the whole.

Maxwell–Boltzmann distribution

The distribution of the energies of the molecules in a gas at a given temperature, shown as the number of molecules with each energy.

Most probable energy

The energy at the peak of the Maxwell–Boltzmann distribution, the energy possessed by the greatest number of molecules.

Mean energy

The average energy of the molecules, which lies to the right of the most probable energy because the distribution is not symmetrical.

Raising the temperature

At a higher temperature the molecules have more energy on average, and the curve changes in three ways. The peak moves to the right, to a higher most probable energy. The peak is lower. And the curve is broader, with more of it in the tail. The number of molecules has not changed, so the area under the new curve equals the area under the old; a peak drawn further right but just as high would have added molecules.

Describe the drawing

Two Maxwell–Boltzmann curves on axes of number of molecules against energy, both starting at the origin. The cyan curve for the lower temperature T₁ rises to a tall peak near the left and falls into a long tail; the amber curve for the higher temperature T₂ peaks lower and further right and is broader, crossing the first curve once and staying above it at high energy. Neither tail reaches the energy axis. A vertical line marks the activation energy, and the area beyond it is shaded for each curve, the amber area visibly larger than the cyan one. Dashed drop lines mark the most probable energy and the mean energy of T₁.

Describe the drawing

The Maxwell–Boltzmann distribution animated through three rising temperatures on fixed axes of number of molecules against energy, with a vertical activation-energy line far out in the tail. At T₁ a tall cyan peak sits near the left and 1.9 percent of the area lies beyond Ea. At 1.2 T₁ the amber curve peaks lower and further right and the shaded tail beyond Ea grows to 4.0 percent; at 1.4 T₁ the coral curve is lower and broader again with 6.7 percent beyond Ea. The last frame overlays all three, each enclosing the same area, and the earlier curves are left as grey dashes while the next one is drawn.

With the activation energy marked on the same axes, the higher-temperature curve lies above the lower one beyond it, so the area to the right of Ea, the number of molecules with enough energy to react, is larger. More collisions have at least the activation energy, so more of them succeed each second, and the rate rises.

There is a second, smaller effect. Faster molecules also collide more often. But the collision frequency rises only with the square root of the absolute temperature, so going from 298 K to 308 K raises it by less than 2%. The rate of a typical reaction nearly doubles over the same ten degrees. Almost all of that increase comes from the larger number of collisions with at least Ea, and that is the point the mark scheme credits first.

A small rise in temperature has a large effect because of where Ea sits. Near the peak, warming the gas moves a modest share of the molecules from one energy to another. Far out in the tail, where Ea usually lies, the number of molecules falls so steeply with energy that a slight stretch of the curve to the right can double the area beyond Ea. The further out Ea lies, the more a given rise in temperature multiplies that area.

Activation energy

The minimum energy that colliding particles must have for a reaction to take place.

Catalysts

A catalyst increases the rate of a reaction without being changed in chemical composition or amount at the end. Manganese(IV) oxide shows both halves of that: a spatula of the black solid makes hydrogen peroxide solution froth within seconds, and once the fizzing stops the solid can be filtered off, dried and weighed at its starting mass.

2H₂O₂(aq) → 2H₂O(l) + O₂(g)Catalysed by manganese(IV) oxide, by iodide ions and by the enzyme catalase. ΔH = −98 kJ per mole of H₂O₂ by every route.

A catalyst works by providing an alternative reaction route with a lower activation energy. The reactants and products are the same, so ΔH is the same; only the barrier between them is lower. A catalyst cannot make more product than the reaction would have made. It lowers the barrier for the reverse reaction by the same amount, so it does not move the position of an equilibrium either: it brings the mixture to equilibrium sooner.

Describe the drawing

An enthalpy profile for hydrogen peroxide decomposing to water and oxygen. Both routes start on the same reactant line and finish on the same product line 98 kilojoules per mole lower. The solid amber uncatalysed route climbs to a peak 75 above the reactants; the dashed cyan route catalysed by iodide climbs to a lower peak only 56 above them. Arrows on the right measure the two activation energies upward from the reactant line and one shared enthalpy change downward.

Describe the drawing

An enthalpy profile for hydrogen peroxide decomposing to water and oxygen, built up in four stages. First the solid amber uncatalysed route climbs to a peak 75 kilojoules per mole above the reactants, with its activation energy arrow. Then a dashed cyan catalysed route appears beneath it between the same reactant and product levels. Then the catalysed activation energy arrow is drawn, 56 kilojoules per mole. Finally one downward arrow marks the enthalpy change of minus 98 kilojoules per mole, the same on both routes.

The catalyst gives no molecule any extra energy, so on the Maxwell–Boltzmann distribution the curve does not move at all. What moves is the activation energy, to the left. The area beyond the new, lower Ea is much larger than the area beyond the old one, so a much larger proportion of collisions have enough energy to react, and the rate rises.

Describe the drawing

A single Maxwell–Boltzmann curve of number of molecules against energy, rising from the origin to a peak and falling into a long tail that never meets the axis. Two vertical lines cross the tail: the amber activation energy without a catalyst far out to the right, and the cyan activation energy with a catalyst further left. The area beyond the catalysed line is shaded lightly and the much smaller area beyond the uncatalysed line more darkly; a bracket under the axis between the two lines marks the extra molecules able to react.

The effect is large because the number of molecules falls so steeply along the tail. Iodide ions lower the activation energy for the decomposition of hydrogen peroxide from about 75 kJ mol⁻¹ to about 56 kJ mol⁻¹. At room temperature that cut of a quarter raises the number of collisions with enough energy about two thousand times. Matching it without the catalyst would take a temperature of about 126 °C, above the boiling point of the solution.

Temperature and a catalyst both increase the number of collisions with at least the activation energy, by different routes. Heating changes the curve and leaves Ea where it was. A catalyst leaves the curve where it was and moves Ea. Draw them that way: one diagram with two curves and one Ea, the other with one curve and two values of Ea.

Catalysts are also classed by phase. A heterogeneous catalyst is in a different phase from the reactants, as solid iron is with the nitrogen and hydrogen of the Haber process, and a homogeneous catalyst is in the same phase, as dissolved iodide ions are with hydrogen peroxide solution.

Catalysts matter to industry because a lower activation energy lets a reaction go at a useful rate at a lower temperature. That saves the fuel burnt to heat the plant and the carbon dioxide the burning releases, and for an exothermic equilibrium it raises the yield as well. In an answer, name both: less energy used, and, where the reaction is an exothermic equilibrium, a better yield.

Catalyst

A substance that increases the rate of a reaction without being changed in chemical composition or amount, by providing an alternative reaction route with a lower activation energy.

Heterogeneous catalyst

A catalyst in a different phase from the reactants, such as solid iron in the Haber process.

Homogeneous catalyst

A catalyst in the same phase as the reactants, such as iodide ions dissolved with hydrogen peroxide.

Other boards: the Boltzmann distribution and sustainability

OCR and Cambridge call the curve the Boltzmann distribution. It is drawn, labelled and marked in the same way, and an answer on those papers can use either name.

OCR's statement on catalysts and sustainability credits two gains from a catalyst: less energy used at a lower temperature, and reactions with a better atom economy and less waste. Ethanoic acid is one example. It is now made by combining methanol with carbon monoxide, with a rhodium or iridium compound dissolved in the mixture as a homogeneous catalyst, and every atom of the reactants ends up in the product.

CH₃OH(l) + CO(g) → CH₃COOH(l)Carbonylation of methanol, catalysed by a rhodium or iridium complex with iodide. One product, so the atom economy is 100%.

The older route, oxidising hydrocarbons from crude oil, gave a mixture of by-products to separate and dispose of. An OCR answer on sustainability names the atom economy and the lower energy demand together.