What rate means

Rate of reaction tells you how quickly the concentration of a reactant or product changes over time. For an average rate across a chosen time interval:

The modulus signs are there for a reason. A reactant concentration goes down, so $\Delta c$ is negative, but rate is written as a positive value. Unless you have converted to the overall reaction rate, always say which species your rate refers to.

For a reaction such as $2A \to B$, A disappears twice as fast as B appears. To compare the species properly, divide the rate for each species by its stoichiometric coefficient. You then get the same overall rate no matter which species you use.

Average, instantaneous and initial rate

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

Instantaneous rate is the rate at a particular moment, taken from the gradient of a tangent to a concentration–time, volume–time or mass–time curve. The initial rate is the instantaneous rate at

In practice, draw a tangent at the required time, then calculate its gradient using two well-spaced points on that tangent rather than two neighbouring experimental points.

How rate data are collected

During a reaction, concentration is usually not measured directly. Instead, you measure a quantity that changes predictably and convert it if necessary, 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.

Use a gas syringe when a gas product forms and is retained. Use a balance when a gas escapes and the total mass decreases. A colorimeter works 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 is 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. It says reacting particles have to collide with enough energy and with the correct relative orientation before a chemical change can happen. It builds on the kinetic molecular theory: particles move around, collide and exchange energy, but collision theory goes one step further by allowing some collisions to rearrange bonds.

Kinetic energy is the energy a particle has because it is moving. We write it as $E_k$, where $E_k$ is kinetic energy ($\text{J}$). Temperature on the kelvin scale is related to the average kinetic energy of particles; we write it as $T$, where $T$ is thermodynamic temperature ($\text{K}$). A higher $T$ means a higher average $E_k$. Don’t use $^\circ\text{C}$ 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 new bonds. In a reaction such as $AB + CD \to AD + BC$, 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 much on orientation. Large molecules with specific reactive sites are much more selective — useful to remember 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 condition: some collisions fail, while others succeed, because bond rearrangement needs sufficient energy and suitable geometry.

Changing the frequency of collisions

For most reactions, a higher reactant concentration gives a faster rate. There are more particles per unit volume, so they collide more often each second. Gaseous reactants behave in the same way when pressure increases: higher pressure packs the gas particles 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 reacts only on 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 does not work quite like concentration, pressure or surface area. Raising the temperature makes collisions slightly more frequent because particles move faster, but the larger effect is energy: a greater fraction of particles now 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 giving the reaction a different pathway with a lower activation energy. The catalyst factor is developed properly in R2.2.5, but for now keep the collision-theory idea simple: more successful collisions per second means a faster reaction.

Designing fair rate experiments

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

Graphs can help diagnose 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. The molar activation energy is written as $E_a$, where $E_a$ is activation energy per mole of reacting events ($J\ mol^{-1}$, often $kJ\ mol^{-1}$). If the colliding particles have less than this energy, they cannot react, even if they’re lined up well. With at least this energy, a collision may react, as long as the orientation is suitable too.

On an energy profile, activation energy is the gap from the reactants up to the highest point on the pathway. That highest point is the transition state: bonds are partly broken and partly formed, making the arrangement extremely unstable.

Maxwell–Boltzmann distributions

A Maxwell–Boltzmann energy distribution curve shows how kinetic energies are spread 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, broader and shifted towards higher energy, but the total area stays the same.

For rate, the key region is the area to the right of $E_a$. Those particles have enough energy to react if they are correctly oriented. As temperature increases, this area grows noticeably, so successful collisions become more likely. That’s why a modest temperature rise can cause 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 may take part in one or more steps of the reaction pathway, but it is not used up overall.

A catalyst gives the reaction an alternative pathway with a lower activation energy. When $E_a$ is lower, a larger fraction of collisions have enough energy to be successful at the same temperature. It 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. Often, it involves 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 differently. The temperature distribution of particle energies stays the same, but the activation-energy line shifts left. That leaves a larger area to the right of the line, so more particles can react.

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 in a reversible reaction. Equilibrium is reached faster, but the equilibrium composition and the value of the equilibrium constant do not change. 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 that converts reactants into products. An elementary step is one molecular event in a mechanism: it passes through one transition state and contains no intermediate within that 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 used in a later step.

An intermediate is 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 assuming this 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 evaluate a proposed mechanism, check two things every time: the steps must add to the correct overall stoichiometric equation, and the mechanism must match the experimentally determined rate equation.

Treat reaction mechanisms as possible explanations, not final truths. If a mechanism contradicts stoichiometric or kinetic data, it can be ruled out, 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 will 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. The number of transition states matches the number of elementary steps. The number of intermediates is one fewer than the number of elementary steps, as long as the steps form one connected pathway.

For any step, measure the activation energy from the species immediately before that step up to the next transition state. The rate-determining step is usually the step with the largest activation energy under the conditions considered. The overall enthalpy change is still measured from reactants to final products, not 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 you sketch a profile from this kind of 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 make the rate-determining step 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 only assign it to elementary steps, since 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. A bimolecular step is an elementary step where two particles collide and react. A termolecular step is an elementary step where three particles collide together and react.

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 what looks like a three-particle collision by using two or more bimolecular or unimolecular steps with intermediates.

Rate equations are experimental

A rate equation shows, in mathematical form, how the rate of reaction depends on the concentrations of the reactants. For a reaction with reactants A and B, you will often see it written as:

You do not get the exponents in a rate equation by copying coefficients from the balanced equation. They have to be found 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 $[A]$ quadruples the rate, 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 must match the rate equation. For an elementary rate-determining step, the rate law usually follows the particles taking part in that step. With a full multistep mechanism, especially when fast equilibria come before the slow step, the connection may be less direct. That’s why the experimental rate equation takes priority.

Order of reaction

Order with respect to a reactant is the exponent to which that reactant’s concentration is raised in the rate equation. Overall order is found by adding the orders with respect to all reactants in the rate equation. Only integer orders are assessed here.

In 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. But keep an eye on the word “can”: in a complete multistep mechanism, the observed rate equation may also be affected by earlier fast steps, not just the slow step.

Rate–concentration graphs

A rate–concentration graph is drawn from initial rates measured in several experiments. For zero order in A, the rate does not depend on $[A]$, so the graph is horizontal. For first order, the rate is directly proportional to $[A]$, giving a straight line through the origin. For second order, the rate is proportional to $[A]^2$, so the graph curves upwards and becomes steeper as $[A]$ increases.

Concentration–time graphs

A concentration–time graph can come from a single experiment, by following $[A]$ as time passes. In a zero-order reaction, $[A]$ falls linearly with time. First- and second-order curves can look quite alike, so linearized plots are a safer way to identify the order.

For zero order:

For first order: $\ln[A] = -kt + \ln[A]_0$. For second order: $1/[A] = kt + 1/[A]_0$. The plot that gives a straight line identifies the order, and its gradient gives $k$ with the appropriate sign.

To work out the order for a particular reactant, use rate data where that reactant’s concentration changes while the concentrations of the other reactants and the temperature are kept constant. Alternatively, use concentration–time data for that reactant and test it against the zero-, first- and second-order linear forms.

Mechanisms stay as “possible mechanisms” because a matching order graph or rate equation supports a proposal, but doesn’t prove that it’s 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 won’t change it. Changing the temperature will, and so will a catalyst if it changes the reaction 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

If rate is measured in $\mathrm{mol\ dm^{-3}\ s^{-1}}$ and concentration in $\mathrm{mol\ dm^{-3}}$, the units of $k$ are $(\mathrm{mol\ dm^{-3}})^{1-N}\ \mathrm{s^{-1}}$.

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

For zero order, $k$ has units $\mathrm{mol\ dm^{-3}\ s^{-1}}$. For first order, it has units $\mathrm{s^{-1}}$. For second order, it has units $\mathrm{dm^3\ mol^{-1}\ s^{-1}}$. For third order, it has units $\mathrm{dm^6\ mol^{-2}\ s^{-1}}$. A quick check of the units often catches a wrong overall order.

The halogenoalkane link makes a useful comparison. Hydrolysis of a primary halogenoalkane by aqueous hydroxide is typically second order overall, with rate = $k[\text{halogenoalkane}][\mathrm{OH}^-]$, so $k$ has units $\mathrm{dm^3\ mol^{-1}\ s^{-1}}$. Hydrolysis of a tertiary halogenoalkane is typically first order in the halogenoalkane, with rate = $k[\text{halogenoalkane}]$, so $k$ has units $\mathrm{s^{-1}}$.

Temperature and the rate constant

For most reactions, raising the temperature raises $k$. More particles now have enough kinetic energy to get over the activation-energy barrier. The change usually isn’t linear; $k$ increases quickly as temperature goes up.

The Arrhenius equation links the rate constant with temperature:

The linear form is:

$\ln k = -\frac{E_a}{R} \times \frac{1}{T} + \ln A$

If you plot $\ln k$ against $1/T$, this gives a straight line. Its gradient is $-E_a/R$, and its y-intercept is $\ln A$. Since $T$ must be in kelvin, $1/T$ has units $\text{K}^{-1}$.

To calculate $E_a$ from an Arrhenius plot, take the gradient of the best-fit line, multiply it by $-R$, then convert J mol$^{-1}$ to kJ mol$^{-1}$ if needed. The Arrhenius equation and its linear form are in the data booklet, but you still need to read the graph correctly.

Meaning of the Arrhenius factor

The Arrhenius factor describes how often particles collide in a way that could produce a reaction, provided they also meet the energy requirement. It covers both collision frequency and orientation. Simple, symmetrical particles usually have larger values, since many collision angles work. Bulky molecules, or molecules that need a very specific arrangement, usually have smaller values because only a small fraction of collisions have the correct geometry.

To find $A$ from experimental data, start with an Arrhenius plot. The intercept gives $A$, so take the exponential of the intercept to get $A$. Its units match the units of $k$, which means you need the rate equation and the overall order before you can state them correctly.

Worked workflow for extracting Ea and A from an Arrhenius plot.

A strong activation-energy practical uses at least five temperatures, repeats each temperature, and keeps concentrations, volumes and the chosen endpoint constant. You can find the rate from an initial tangent on a volume–time, concentration–time or absorbance–time graph. For a fixed endpoint, it can be approximated by $\frac{1}{t}$. Only plot $\ln k$ or $\ln(\text{rate})$ against $\frac{1}{T}$ when the concentrations are the same at every temperature. Temperature control is often the main limitation, so use a water bath, allow equilibration, and measure the actual reaction temperature instead of assuming the set value.