How fast? The rate of chemical change

📚The chemistry

Measuring rate

Rate of reaction is the change in concentration of a reactant or product per unit time, with units mol dm⁻³ s⁻¹.

Concentration is rarely measured directly. In practice you follow something proportional to it and plot it against time:

• Volume of gas collected in a syringe — for any reaction producing a gas;
• Loss of mass on a balance — also for gas-producing reactions, provided the gas is dense enough for the change to be measurable;
• Change in colour, measured with a colorimeter or spectrophotometer — if a reactant or product is coloured;
• Change in pH, using a probe;
• Time to reach a fixed point — the classic “disappearing cross” sulfur precipitation, where rate is taken as \( 1/t \);
• Change in electrical conductivity, if the number or charge of ions changes.

From the graph, the rate at any instant is the gradient of the tangent at that point. The rate is greatest at \( t = 0 \), when the reactants are at their highest concentration, and falls as they are consumed — which is why the initial rate, the gradient of the tangent at time zero, is the only fair value to quote when comparing different starting concentrations.

Collision theory

Particles react only when they collide, and a collision leads to reaction only if both of these hold:

The temperature link comes from S1.1: absolute temperature is proportional to the average kinetic energy of the particles.

The factors that change rate

• Concentration — more particles per unit volume, so a greater collision frequency, so more successful collisions per second.
• Pressure (for gases) — the same argument. Increasing pressure squeezes the same number of molecules into a smaller volume, increasing concentration and hence collision frequency.
• Surface area (for a solid) — more particles are exposed at the surface where reaction can occur, so collision frequency rises. Powdering a solid can increase the rate dramatically, which is why flour and custard powder can cause dust explosions.
• Temperature — two effects, and both should be mentioned. Particles move faster, so collisions are more frequent; and, much more importantly, a far greater proportion of collisions have energy at least equal to \( E_{\mathrm{a}} \). The second effect dominates by a wide margin, which is why a rise of only 10 K can roughly double a reaction rate.
• Catalyst — below.

The Maxwell–Boltzmann distribution

At any temperature, particles do not all have the same energy: they are spread over a distribution. The Maxwell–Boltzmann curve plots the number (or fraction) of particles against kinetic energy. Its features, all of which earn marks:

• it starts at the origin — no particle has zero energy;
• it rises to a peak (the most probable energy) then falls;
• it is asymmetric, with a long tail that approaches but never touches the energy axis — there is no upper limit on energy;
• the total area under the curve equals the total number of particles, so it must not change when you redraw it at a different temperature;
• \( E_{\mathrm{a}} \) is marked as a vertical line, and the area to the right of it represents the particles able to react.

At higher temperature the curve flattens and shifts to the right: the peak is lower and further along, and the tail is raised. The area to the right of \( E_{\mathrm{a}} \) increases markedly — that is the picture of why heating speeds up a reaction. A catalyst does not change the curve at all; it moves the \( E_{\mathrm{a}} \) line to the left, which again increases the area to its right.

Catalysts

A catalyst increases the rate of reaction by providing an alternative reaction pathway with a lower activation energy. It is not consumed and does not appear in the overall equation.

Three consequences worth stating explicitly:

• On an energy profile, the catalysed route is drawn as a lower peak — sometimes with two smaller humps if the catalysed mechanism has more steps. The reactant and product levels are unchanged.
• A catalyst therefore does not change \( \Delta H \), and it does not change the position of equilibrium or the value of \( K \) — it speeds up the forward and reverse reactions equally, so equilibrium is simply reached sooner (R2.3).
• Biological catalysts are called enzymes. The different mechanisms of homogeneous and heterogeneous catalysts will not be assessed.

AHL Mechanisms, molecularity and the rate-determining step

Most reactions occur as a series of elementary steps, and the overall equation is just their sum. The slowest step determines the overall rate and is called the rate-determining step — and it need not be the first step.

Two species to distinguish carefully:

Molecularity is the number of particles taking part in an elementary step: unimolecular (one), bimolecular (two), termolecular (three). Termolecular steps are rare, because a simultaneous three-particle collision with correct geometry is very improbable — which is itself a good reason to be suspicious of any proposed mechanism containing one.

A proposed mechanism is acceptable only if it is consistent with both the stoichiometry and the kinetic data: the steps must add to the overall equation, and the rate equation predicted by the rate-determining step must match the one measured.

AHL Rate equations and order

For a reaction of A and B:

\( m \) and \( n \) are the orders with respect to A and B, and \( m + n \) is the overall order. Only integer values are assessed.

Orders can only be determined experimentally. They are not the stoichiometric coefficients, and assuming they are is the central error of this topic. They reflect the composition of the rate-determining step, not the overall equation, which is exactly why kinetics is evidence about mechanism.

Finding orders from a table of initial rates: compare two experiments in which only one concentration changes. If doubling \( [\mathrm{A}] \) leaves the rate unchanged the order is zero; if it doubles the rate the order is one; if it quadruples the rate the order is two.

The constant half-life of a first-order reaction is the quickest diagnostic from a concentration–time graph: read off the time to fall from 1.0 to 0.5, then from 0.5 to 0.25, and if they are equal the reaction is first order.

AHL The rate constant and its units

\( k \) is constant for a given reaction at a given temperature and increases sharply as temperature rises. Its units depend on the overall order, and are deduced by rearranging the rate equation:

The pattern: each order above the first adds dm³ mol⁻¹. Derive them rather than memorising, by writing \( k = \dfrac{\text{rate}}{[\mathrm{A}]^m[\mathrm{B}]^n} \) and cancelling.

AHL The Arrhenius equation

Both forms are in the data booklet. \( A \) is the Arrhenius factor, which accounts for the frequency of collisions with the correct orientation — the geometry half of collision theory made quantitative.

Qualitatively: raising \( T \), or lowering \( E_{\mathrm{a}} \), makes the exponent less negative and so increases \( k \). Because the dependence is exponential, a small change in either produces a large change in rate.

The linear form is what you use with data. Plot \( \ln k \) on the y-axis against \( 1/T \) on the x-axis:

• the gradient is \( -E_{\mathrm{a}}/R \), so \( E_{\mathrm{a}} = -\text{gradient} \times R \) — and because the gradient is negative, \( E_{\mathrm{a}} \) comes out positive, as it must;
• the intercept on the \( \ln k \) axis is \( \ln A \), so \( A = e^{\text{intercept}} \).

\( R \) is in J K⁻¹ mol⁻¹, so \( E_{\mathrm{a}} \) comes out in J mol⁻¹ — divide by 1000 for kJ mol⁻¹.