Reaction kinetics

Rate equations: order, rate constant and overall order

A rate equation links the rate of a reaction to the concentrations of the species that affect it, in the form rate = k[A]m[B]n. The order of reaction with respect to a reactant is the power to which its concentration is raised in that equation, so the order tells you how sensitive the rate is to that reactant. At this level the order with respect to any one species is 0, 1 or 2. The orders are found by experiment and have nothing to do with the coefficients in the balanced equation, which is why the same overall equation can have quite different orders for different reactions.

An order of 1 in A (first order in A) means the rate is directly proportional to the concentration of A, so doubling [A] doubles the rate. An order of 2 (second order) means the rate is proportional to [A] squared, so doubling [A] multiplies the rate by 4 and tripling it multiplies the rate by 9. An order of 0 (zero order) means the rate does not depend on the concentration of that reactant at all. Zero order usually means the reactant enters the mechanism after the slow step, or that a solid surface or a catalyst is already working at full capacity and cannot go any faster.

The overall order of reaction is the sum of the individual orders, m + n, and it is the overall order that fixes the units of the rate constant k. The rate constant is the constant of proportionality in the rate equation: it is a property of the reaction at a given temperature, so it changes when the temperature changes or when a catalyst is used, but it does not change when a concentration is changed. Work the units of k out from the rate equation rather than learning a list, by dividing the units of rate (mol dm-3 s-1) by the units of all the concentration terms.

Deducing orders from initial rates

The initial rates method measures the rate at the very start of the reaction, when the concentrations are exactly the values you made up. Run several experiments in which you change the concentration of one reactant at a time, keep every other concentration the same, and keep the temperature constant, because k depends on temperature. Measure the rate as the gradient of the concentration-time curve at t = 0, that is the tangent drawn at the start, or follow the reaction with a colorimeter or a clock method.

You then compare experiments two at a time. A + B -> products gave the results in the table below. Between experiments 1 and 2 the concentration of B is unchanged and [A] doubles, and the initial rate doubles with it, so the order with respect to A is 1. Between experiments 1 and 3 the concentration of A is unchanged and [B] doubles, and this time the rate goes up by a factor of 4, so the order with respect to B is 2. Experiment 4 checks the answer: with [A] = 0.200 mol dm-3 and [B] = 0.200 mol dm-3 the predicted rate is 1.0 x 0.200 x (0.200)2 = 8.0 x 10^-3 mol dm-3 s-1, which is what was measured.

The rate equation is therefore rate = k[A][B]2 and the overall order is 3. Substituting experiment 1 gives k = (1.0 x 10^-3) / (0.100 x (0.100)2) = (1.0 x 10^-3) / (1.0 x 10^-3) = 1.0, in units of dm6 mol-2 s-1. A rate that does not change when a concentration is doubled is just as informative as one that does: it means that reactant is not involved at or before the rate-determining step.

Concentration-time graphs and the half-life of a first-order reaction

Plotting concentration against time gives a curve whose gradient at any point is the rate at that moment. For a zero-order reaction the concentration falls along a straight line, because the rate is constant until the reactant runs out; it is the only order that gives a straight line on a concentration-time graph. For a first-order reaction the curve is steep at the start and flattens as the reaction proceeds, falling by a fixed fraction in each equal interval of time. For a second-order reaction the curve falls even more sharply at the start and then tails off more slowly than a first-order curve.

The half-life is the time taken for the concentration of a reactant to fall to half of its value, and for a first-order reaction it is independent of concentration. This is the key idea: whatever the starting concentration, half of it disappears in the same time. A first-order reaction starting at 0.800 mol dm-3 reaches 0.400, then 0.200, then 0.100 mol dm-3 in equal intervals of time, whereas a second-order reaction has a half-life that doubles each time the concentration halves. Measuring two successive half-lives from a concentration-time graph is therefore a quick way to test whether a reaction is first order.

The half-life and the rate constant of a first-order reaction are linked by k = 0.693 / t1/2. A first-order reaction with a half-life of 120 s has k = 0.693 / 120 = 5.78 x 10^-3 s-1; and a reaction with k = 2.00 x 10^-3 s-1 has a half-life of 0.693 / (2.00 x 10^-3) = 347 s. Notice that the units of k for a first-order reaction are s-1, because the rate equation has one concentration term and the value does not depend on the concentration you start with.

• Zero order: rate is constant, so the concentration-time graph is a straight line.
• First order: constant half-life, concentration falls by half in each successive equal time interval.
• Second order: the half-life doubles each time the concentration halves, so it is not constant.
• First order only: k = 0.693 / t1/2, with k in s-1 when t1/2 is in seconds.

Rate-concentration graphs

A rate-concentration graph shows how the rate changes when you change the concentration of one reactant while everything else, including the temperature, is held constant. It is a more direct picture of order than a concentration-time curve, because the shape of the graph and the order are the same thing. A straight line through the origin means the rate is proportional to the concentration, so the reaction is first order in that reactant. A horizontal line means the rate does not change at all as the concentration changes, so the reaction is zero order. A curve that rises more steeply than a straight line means the order is greater than 1, and plotting the rate against the square of the concentration turns a second-order relationship into a straight line through the origin, which confirms it.

The gradient of a rate-concentration graph carries the rate constant. For a reaction that is first order in A and zero order in everything else, rate = k[A], so the gradient of the straight line is k itself and the intercept is zero. That is exactly how the value of k is obtained from a set of rate measurements, and it is worth checking that the graph really does pass through the origin: a non-zero intercept would mean the rate equation has been written wrongly, or that the measured rate includes a contribution from something else.

The same information can be read from a single concentration-time curve. Take the gradient at several times, convert each time to the concentration at that point, and plot the gradients against the concentrations. The curve of concentration against time is itself strongest evidence about order when you compare successive half-lives, which is why examiners accept a half-life calculation, an initial-rates table or a linear rate-concentration graph as three different routes to the same order.

Temperature and the rate constant

Raising the temperature increases the rate of almost every reaction, and for a reaction with a typical activation energy a rise of 10 K roughly doubles the rate. Two things change. The molecules move faster, so they collide more often, but this is a small effect: the frequency of collisions rises by only a few per cent for every 10 K. The important effect is on the Boltzmann distribution. At a higher temperature the distribution of molecular energies flattens and spreads out to the right, and the number of molecules whose energy is at least equal to the activation energy increases sharply.

Only collisions with energy greater than or equal to the activation energy can lead to reaction, so a large increase in the fraction of such collisions converts into a large increase in rate. Because that fraction depends on temperature exponentially rather than in a straight-line way, the rate constant does the same: k = A x e^(-Ea/RT) is the relationship between them, and it is why a small change in temperature produces a large change in k. This is also why temperature control matters so much in kinetics experiments: the whole point of the initial rates method is to change one concentration at a time, and if the temperature drifts, k drifts with it and the orders come out wrong.

The effect on the rate of a reaction has to be separated from the effect on its equilibrium position. Changing the temperature changes k for the forward and the reverse reaction by different amounts, so it can shift an equilibrium, but the effect on the rate is the change in the rate constant itself. Changing a concentration, on the other hand, changes the rate without changing k at all, and adding a catalyst changes k because it provides an alternative route with a lower activation energy, without changing the position of equilibrium.

From the rate equation to the mechanism

Most reactions take place in several steps, and the rate-determining step (the slow step) is the one that controls the rate of the whole reaction. The rate equation lists the species that are involved in that slow step, and in any faster steps that come before it: a step that is much faster than the slow one does not affect the overall rate, so species that only appear after the slow step never appear in the rate equation. This gives you a two-way link between mechanism and rate equation, and examiners test it in both directions.

To deduce a rate equation from a mechanism, count the species in the slow step. If the slow step is 2NO2 -> NO3 + NO, the rate depends on the concentration of NO2 twice, so the reaction is second order in NO2 and rate = k[NO2]2. The fast step NO3 + CO -> NO2 + CO2 is irrelevant to the rate, which is why the overall equation for this mechanism is NO2 + CO -> NO + CO2 yet carbon monoxide does not appear in the rate equation at all. To suggest a mechanism from a rate equation, do the reverse: put the species named in the rate equation into the slow step, with the same number of each species as its order, and put everything else in a fast step afterwards.

The species that appear in a mechanism but not in the overall equation must be identified correctly. An intermediate is produced in one step and used up in a later step, so it appears in the middle of the mechanism and cancels out of the overall equation: in the NO2 / CO mechanism NO3 is an intermediate. A catalyst is used up in one step and re-formed in a later step, so it too cancels from the overall equation, but it is present at the start and at the end: that is exactly what happens to Fe2+ ions in the reaction between iodide ions and peroxodisulfate ions, and to NO and NO2 in the atmospheric oxidation of sulfur dioxide.

Homogeneous and heterogeneous catalysts

A catalyst increases the rate of a reaction by providing an alternative route with a lower activation energy, so a greater fraction of collisions is effective at the same temperature. It is not used up in the overall reaction and it does not change the position of equilibrium, only the time taken to reach it. A homogeneous catalyst is in the same phase as the reactants, as when an aqueous Fe2+ salt catalyses a reaction between two aqueous ions. A heterogeneous catalyst is in a different phase from the reactants, which in practice almost always means a solid catalyst with gaseous or liquid reactants passing over it.

A heterogeneous catalyst works at its surface. Reactant molecules are adsorbed onto active sites on the surface, which holds them close together in a favourable orientation; the bonds in the adsorbed molecules are weakened by the interaction with the surface, which lowers the activation energy; the reaction then takes place on the surface; and the product molecules are desorbed, freeing the active sites for the next reactants. In the Haber process nitrogen and hydrogen are adsorbed onto the surface of the iron catalyst, the strong triple bond in the nitrogen molecule is weakened, and ammonia is desorbed. In a catalytic converter, platinum, palladium and rhodium adsorb NO and CO from the exhaust and convert them into N2 and CO2, which are then desorbed. A solid catalyst can be poisoned by impurities that stick to the active sites and never leave, which is why the iron in a Haber plant has to be protected from sulfur compounds.

A homogeneous catalyst works by taking part in the mechanism itself. It is used up in one step and re-formed in a later step, so it cancels out of the overall equation. In the Fe2+ / Fe3+ catalysis of the iodide-peroxodisulfate reaction, the Fe2+ ion is oxidised by the peroxodisulfate ion in one step and the Fe3+ ion is then reduced back to Fe2+ by iodide ions in a later step, so the iron is used and re-made. Here the uncatalysed reaction is very slow because two negatively charged ions, I- and S2O8^2-, repel each other, so the catalysed route through the positively charged iron ion is much easier. Atmospheric nitrogen oxides work in the same way: NO2 oxidises SO2 to SO3 and is reduced to NO, and the NO is then re-oxidised to NO2 by oxygen in the air, so a small amount of NOx can convert a large amount of sulfur dioxide.