What rate means

Rate of reaction measures how fast the concentration of a reactant or product changes with time. For an average rate over a chosen time interval:

r = |Δc| / Δt, where r is the average rate of reaction (mol m⁻³ s⁻¹ in SI; commonly mol dm⁻³ s⁻¹ in IB calculations), Δc is the change in concentration of a specified species (mol m⁻³ in SI; commonly mol dm⁻³), and Δt is the time interval (s).

The modulus signs are there for a reason. A reactant’s concentration falls, so Δc is negative, but the rate is reported as a positive quantity. Always state which species the rate refers to unless you have converted it to the overall reaction rate.

For a reaction such as 2A → B, A disappears twice as fast as B appears. To compare species fairly, divide the rate measured for each species by its stoichiometric coefficient. You then get the same overall rate whichever species you use.

Average, instantaneous and initial rate

Average rate is calculated over a finite time interval. It’s useful, but it can hide the way most reactions slow down as the reactants are used up.

Instantaneous rate is the rate at one moment, found from the gradient of a tangent to a concentration–time, volume–time or mass–time curve. The initial rate is the instantaneous rate at t = 0, where t is time (s). In practice, draw a tangent at the required time and calculate its gradient using two well-spaced points on the tangent, not two neighbouring experimental points.

How rate data are collected

Concentration is not usually measured directly during a reaction. Instead, you measure something that changes in a predictable way and convert it if needed, using stoichiometry, gas laws or a calibration curve. Useful measurements include gas volume, gas pressure, mass loss, pH, conductivity, absorbance or colour intensity, and titration of samples removed at known times. Data loggers help when readings change quickly or when you need many points.

A gas syringe works when a gas product forms and is retained. A balance works when a gas escapes and the total mass falls. A colorimeter is suitable when the concentration of a coloured reactant or product changes. A pH probe suits acid–base reactions, while conductivity can help when the number or mobility of ions changes.

Time can be used in two quite different ways. In a continuous method, time is the independent variable: you record volume, mass or concentration at many times. In a clock method, time is the dependent variable: you change concentration or temperature and measure the time taken to reach a fixed visible event, such as a precipitate obscuring a mark. For clock reactions, rate is often taken as proportional to 1/t, provided the same fixed extent of reaction is used each time.

Collision theory

Collision theory is a model of reaction rate: reacting particles have to collide with enough energy and with the correct relative orientation for chemical change to happen. It builds on the kinetic molecular theory. Particles move, collide and exchange energy, and collision theory adds that some of those collisions can rearrange bonds.

Kinetic energy is the energy a particle has because it is moving. We write it as Eₖ, where Eₖ is kinetic energy (J). Temperature on the kelvin scale is related to the average kinetic energy of particles; we write it as T, where T is thermodynamic temperature (K). A higher T gives a higher average Eₖ. Don’t use °C in kinetic arguments about proportionality; the collision theory link is with kelvin.

A collision only succeeds if it gets the reacting particles over the energy barrier and brings the right atoms close enough to form the new bonds. In a reaction such as AB + CD → AD + BC, a collision where A meets D and B meets C can work. If the wrong ends face each other, the particles may simply rebound, even when they are moving quickly.

Collision geometry is the relative arrangement of particles during a collision that determines whether the required bonds can break and form. Small symmetrical particles, such as monatomic ions, do not depend as strongly on orientation. Large molecules with specific reactive sites are much pickier — a useful detail when you meet enzymes and organic mechanisms.

The kinetic molecular theory explains why particles collide and why their kinetic energies are spread out. Collision theory adds the chemical requirement: some collisions fail, while others succeed, because bond rearrangement needs enough energy and suitable geometry.

Changing the frequency of collisions

For most reactions, a higher reactant concentration makes the reaction faster. There are more particles per unit volume, so they collide more often each second. The same reasoning works for gaseous reactants when pressure increases: at higher pressure, gas particles sit closer together, which makes collisions more frequent. Pressure has little direct effect on reactions involving only liquids and solids because they are not easily compressed.

Surface area matters in heterogeneous reactions, where reactants are in different phases. Phase is a region of matter with uniform chemical composition and physical properties. A solid lump can react only at its exposed surface. Grind it into a powder, and far more particles are exposed, so collisions at the interface happen more often. This is why powdered fuels and fine metal powders can be surprisingly reactive.

Temperature behaves differently from concentration, pressure and surface area. Raising temperature does increase collision frequency a little, because particles move faster. Its larger effect, though, is on energy: a bigger fraction of particles have enough kinetic energy for successful collisions. Lowering temperature usually slows chemical and biochemical reactions; refrigeration works for this reason.

A catalyst increases rate by providing a different pathway with a lower activation energy. The catalyst factor is developed properly in R2.2.5, but for now keep the simple collision-theory message: more successful collisions per second means a faster reaction.

Designing fair rate experiments

When you study one factor, keep the others constant. If you investigate concentration, keep temperature, total volume, surface area or particle size, catalyst amount, pressure for gases, stirring and the fixed endpoint constant. If you investigate temperature, keep concentrations and volumes constant and give the mixture time to reach the chosen temperature before starting.

Graphs are useful for spotting errors. Random error is scatter caused by unpredictable variation, such as reading uncertainty or slight timing differences; it appears as points scattered above and below a best-fit line or curve. Systematic error is a consistent bias, such as a gas leak or a miscalibrated probe; it shifts results in one direction or gives an intercept that should not be there. Repeating measurements helps reveal random error; calibration and improved apparatus help reduce systematic error.

The energy barrier

Activation energy is the minimum energy colliding particles need for a successful collision that leads to reaction. We write the molar activation energy as Eₐ; this is the activation energy per mole of reacting events (J mol⁻¹, often kJ mol⁻¹). If the particles collide with less than this energy, they cannot react, even if their alignment is good. With at least this energy, the collision may react, as long as the orientation is suitable.

On an energy profile, activation energy is the gap between the reactants and the highest point on the pathway. That highest point is the transition state, where bonds are partly broken and partly formed. The arrangement is extremely unstable.

Maxwell–Boltzmann distributions

A Maxwell–Boltzmann energy distribution curve shows the spread of kinetic energies among particles in a sample at a particular temperature. The area under the curve represents the total number of particles. At a higher temperature, the curve is lower and broader, and it shifts to higher energy. The total area stays the same.

For rate, the key region is the area to the right of Eₐ. Those are the particles with enough energy to react if they are correctly oriented. As temperature rises, this area grows noticeably, so successful collisions become more likely. A modest temperature rise can therefore give a large increase in rate.

What a catalyst does

A catalyst is a substance that increases the rate of a chemical reaction and is regenerated by the end of the reaction. It is not used up overall, although it may take part in one or more steps of the reaction pathway.

A catalyst gives the reaction an alternative pathway with a lower activation energy. With a lower Eₐ, a larger fraction of collisions have enough energy to succeed at the same temperature. What doesn’t change? The catalyst does not increase the average kinetic energy of the particles, and it does not change the overall enthalpy change of the reaction.

In both exothermic and endothermic reactions, the catalysed pathway starts at the same reactant energy and finishes at the same product energy as the uncatalysed pathway. The catalysed route has lower peaks. It often has more than one step, with intermediates between transition states, but the detailed mechanisms of homogeneous and heterogeneous catalysis are not assessed here.

A Maxwell–Boltzmann diagram shows the same idea another way. The temperature distribution of particle energies stays the same, but the activation-energy line shifts left. That makes the area to its right larger, so more particles can react.

An Enzyme is a biological catalyst, usually a protein, that increases the rate of a biochemical reaction with high specificity. Enzymes are central to living systems because many biological reactions would be far too slow without catalysis.

Catalysts speed up both the forward and backward reactions of a reversible reaction. They get the system to equilibrium faster, but they do not change the equilibrium composition or the value of the equilibrium constant. Transition elements often work well as catalysts because they can form variable oxidation states, bind reactants at surfaces or active sites, and provide orbitals that help weaken bonds during reaction.

Mechanisms and elementary steps

A reaction mechanism is a proposed sequence of elementary steps showing how reactants are converted into products. An elementary step is one molecular event in that mechanism: it passes through one transition state and has no intermediate within the step.

A multistep reaction occurs through two or more elementary steps. To get the overall equation, add the steps and cancel any species that is made in one step and then used in a later step.

An intermediate is a species formed in one elementary step and consumed in a later elementary step, so it does not appear in the overall equation. A transition state is the maximum-energy arrangement along a reaction pathway, existing only fleetingly while bonds are partly broken and partly formed. On energy profiles, intermediates sit in the valleys between peaks; transition states are the peaks.

The rate-determining step

The rate-determining step is the slowest elementary step in a mechanism, so it controls the overall rate under the stated conditions. It is often the step with the largest activation energy, but use the kinetic evidence rather than just assuming it from a drawing.

The rate-determining step does not have to come first. For example, a fast first step can form an intermediate, then a later slow step can control how quickly products appear. When you test a proposed mechanism, check two things each time: adding the steps must give the correct overall stoichiometric equation, and the mechanism must match the experimentally determined rate equation.

Treat reaction mechanisms as possible explanations, not final truths. A mechanism can be ruled out if it contradicts stoichiometric or kinetic data, but several mechanisms may fit the same data until further evidence is collected. In organic chemistry, the hydrolysis of tertiary halogenoalkanes proceeds through a carbocation intermediate; that is the kind of mechanistic evidence you’ll meet again in Reactivity 3.4.

Reading multistep energy profiles

In a multistep energy profile, every peak represents a transition state. Each valley between two peaks is an intermediate. So the number of transition states matches the number of elementary steps. The number of intermediates is one fewer than the number of elementary steps, provided the steps are connected in a single pathway.

For each step, measure the activation energy from the species just before that step up to the following transition state. The rate-determining step is usually the one with the largest activation energy under the conditions considered. The overall enthalpy change is still taken from reactants to final products; you don’t find it by adding peak heights.

Constructing from kinetic data

Kinetic data can show which step is slow. If changing the concentration of a species changes the rate, that species is involved in, or affects, the rate-determining step. If changing a species has no effect on rate, it is not present in the rate-determining step of the rate law under those conditions.

When sketching from such data, label the axes as energy and reaction coordinate. Put products lower than reactants for an exothermic reaction and higher for an endothermic reaction. Include intermediates and transition states, and draw the rate-determining step as the one with the largest relevant activation-energy barrier.

Molecularity applies to a step, not usually to the overall reaction

Molecularity means the number of reacting particles involved in one elementary step. We assign it only to elementary steps, because an elementary step is a single molecular event.

A unimolecular step is an elementary step where one particle changes, for example by breaking apart or rearranging. In a bimolecular step, two particles collide and react. A termolecular step involves three particles colliding together and reacting.

Comparison of elementary step molecularity by particles involved and likelihood.

Termolecular steps are rare. Three particles have to meet at the same time, with suitable geometry and enough energy. In practice, many mechanisms avoid an apparent three-particle collision by using two or more bimolecular or unimolecular steps with intermediates.

Rate equations are experimental

A rate equation gives the mathematical link between reaction rate and the concentrations of reactants. For a reaction with reactants A and B, it often looks like this:

r = k[A]^m[B]^n, where k is the rate constant (units depend on the overall order), [A] is the concentration of A (mol dm⁻³), [B] is the concentration of B (mol dm⁻³), m is the order with respect to A (dimensionless), and n is the order with respect to B (dimensionless).

The exponents in a rate equation do not come from the coefficients in the balanced equation. Chemists find them experimentally, usually by measuring initial rates while changing one reactant concentration at a time and keeping the others constant.

Initial-rate data showing second order in A and zero order in B.

To work out a rate equation from initial-rate data, compare two experiments where only one concentration changes. If doubling [A] doubles the rate, the reaction is first order in A. If doubling [A] makes the rate four times larger, it is second order in A. If changing [A] has no effect on the rate, it is zero order in A. Do the same for each reactant, then write the rate equation.

Any proposed mechanism has to fit the rate equation. For an elementary rate-determining step, the rate law usually matches the particles taking part in that step. In a full multistep mechanism, especially one with fast equilibria before the slow step, the connection may be less direct. That’s why the experimental rate equation comes first.

Order of reaction

Order with respect to a reactant means the exponent used for that reactant’s concentration in the rate equation. Overall order is found by adding the orders with respect to all reactants in the rate equation. Here, only integer orders are assessed.

For a simple elementary rate-determining step, the order with respect to a reactant can show how many particles of that reactant are involved in the slow step. Watch the word “can”, though: in a full multistep mechanism, earlier fast steps may also affect the observed rate equation, not just the slow step.

Rate–concentration graphs

A rate–concentration graph is drawn from initial rates measured in several experiments. If the reaction is zero order in A, the rate does not depend on [A], so the graph is horizontal. If it is first order, the rate is directly proportional to [A], giving a straight line through the origin. If it is second order, the rate is proportional to [A]², so the curve gets steeper as [A] increases.

Concentration–time graphs

A concentration–time graph can come from a single experiment, by measuring [A] as time passes. In a zero-order reaction, [A] decreases linearly with time. First- and second-order curves can look quite similar, which is why linearized plots are more dependable.

For zero order: [A] = −k**t + [A]₀, where [A]₀ is the initial concentration of A (mol dm⁻³). For first order: ln[A] = −k**t + ln[A]₀. For second order: 1/[A] = k**t + 1/[A]₀. The plot that gives a straight line tells you the order, and its gradient gives k with the appropriate sign.

To work out the order for one reactant, you need rate data collected while changing that reactant’s concentration and keeping the other reactant concentrations and the temperature constant. Alternatively, use concentration–time data for that reactant and test it against the zero-, first- and second-order linear forms.

Mechanisms are still only “possible mechanisms”. A matching order graph or rate equation supports a proposed mechanism, but it does not prove that it is the only one. Another mechanism may produce the same rate law, so chemists look for extra evidence such as intermediates, isotope effects or product distributions.

What the rate constant tells you

The rate constant, k, belongs to a particular reaction at a stated temperature. Changing concentrations doesn’t change it, but changing the temperature does, and a catalyst can change k by providing a different pathway.

Once you know the rate equation, find k by rearranging it and putting in one set of experimental data. For r = k[A]^m[B]^n, the overall order is N = m + n, where N is the overall order (dimensionless). If rate is measured in mol dm⁻³ s⁻¹ and concentration in mol dm⁻³, the units of k are (mol dm⁻³)^(1−N) s⁻¹.

How overall reaction order determines the units of the rate constant k.

For zero order, k has units mol dm⁻³ s⁻¹. For first order, k has units s⁻¹. For second order, k has units dm³ mol⁻¹ s⁻¹. For third order, k has units dm⁶ mol⁻² s⁻¹. A quick check of the units often catches a wrong overall order.

The halogenoalkane comparison is handy. Hydrolysis of a primary halogenoalkane by aqueous hydroxide is typically second order overall, with rate = k[halogenoalkane][OH⁻], so k has units dm³ mol⁻¹ s⁻¹. Hydrolysis of a tertiary halogenoalkane is typically first order in the halogenoalkane, with rate = k[halogenoalkane], so k has units s⁻¹.

Temperature and the rate constant

For most reactions, raising the temperature increases k. A greater fraction of particles then has enough kinetic energy to overcome the activation-energy barrier. The change is usually not linear; k climbs quickly as temperature increases.

The Arrhenius equation links the rate constant to temperature:

k = Ae^(−Eₐ / R**T), where A is the Arrhenius factor (same units as k), e is the base of natural logarithms (dimensionless), R is the gas constant (8.31 J mol⁻¹ K⁻¹), and the other symbols have their meanings already given.

The linear form is:

ln k = −Eₐ/R × 1/T + ln A

If ln k is plotted against 1/T, the graph is a straight line. Its gradient is −Eₐ/R, and its y-intercept is ln A. Since T must be in kelvin, 1/T has units K⁻¹.

To find Eₐ from an Arrhenius plot, take the gradient of the best-fit line, multiply it by −R, and convert J mol⁻¹ to kJ mol⁻¹ if needed. The data booklet gives both the Arrhenius equation and its linear form, but you still need to be able to interpret the graph.