Reaction kinetics

📚The chemistry

26.1 Rate equations

The rate equation links the rate of a reaction to the concentrations of the reactants:

• m and n are the orders of reaction with respect to A and B — in this syllabus 0, 1 or 2. The overall order is m + n.
• k is the rate constant. It is constant at a fixed temperature, and it increases with temperature (more particles with energy ≥ E_a), which is how temperature raises the rate.
• Orders are found by experiment. They are not the coefficients in the balanced equation, except by coincidence.

What an order means: zero order in A — changing [A] has no effect on the rate; first order — doubling [A] doubles the rate; second order — doubling [A] quadruples the rate.

Units of k

Rearrange the rate equation and cancel. Rate is mol dm⁻³ s⁻¹:

Finding orders from graphs

Concentration–time graphs (concentration of a reactant against time):

• Zero order: a straight line sloping down — constant rate until the reactant runs out.
• First order: a curve with a constant half-life — the concentration halves in the same time whatever it starts at.
• Second order: a curve that is steep at first and levels off more, with successive half-lives getting longer.

Rate–concentration graphs (rate against concentration of one reactant):

• Zero order: a horizontal line.
• First order: a straight line through the origin; its gradient is k (if only one reactant is varied and others are constant).
• Second order: an upward curve through the origin (a parabola).

To get a rate from a concentration–time curve, draw a tangent and measure its gradient; the tangent at t = 0 gives the initial rate.

The initial rates method

Run the reaction several times, changing the starting concentration of one reactant at a time while keeping the others constant, and measure the initial rate each time. Compare pairs of experiments in which only one concentration changes: the factor by which the rate changes, compared with the factor by which the concentration changed, gives the order.

Half-life

The half-life, t_½, is the time for the concentration of a reactant to fall to half its value. For a first-order reaction it is independent of the starting concentration — the defining feature of first order — and linked to k by

(0.693 is ln 2.) Measure two or three successive half-lives from a graph: if they are equal, the reaction is first order.

Mechanisms and the rate-determining step

Most reactions happen in several steps. The rate-determining step is the slowest step, and it controls the overall rate. The rate equation contains the species that take part in the rate-determining step (and in any steps before it), each to the power of the number of that species involved. Species that react only after the slow step do not appear in the rate equation — they are zero order.

• An intermediate is formed in one step and used up in a later one; it does not appear in the overall equation.
• A catalyst is used in one step and regenerated in a later one; it does not appear in the overall equation either, but it can appear in the rate equation.

The classic example is the S_N1 hydrolysis of 2-bromo-2-methylpropane (topic 15): rate = k[(CH₃)₃CBr]. The OH⁻ is zero order because it only reacts in the fast second step, after the slow ionisation. For primary halogenoalkanes (S_N2), rate = k[RBr][OH⁻], because both are in the single step.

26.2 How catalysts work

Heterogeneous catalysts

The catalyst is in a different phase, usually a solid with gaseous reactants. Its action has three stages:

1. Adsorption: reactant molecules form weak bonds to active sites on the catalyst surface. This holds them close together and in a favourable orientation.
2. Bond weakening: bonding to the surface weakens bonds within the reactant molecules, lowering the activation energy, and the reaction takes place on the surface.
3. Desorption: the product molecules break away from the surface, freeing the active sites for more reactant.

Examples: iron in the Haber process (N₂ and H₂ adsorb on iron, their bonds are weakened, NH₃ desorbs); palladium, platinum and rhodium in catalytic converters, removing oxides of nitrogen (2CO + 2NO → 2CO₂ + N₂). A good catalyst adsorbs strongly enough to weaken bonds, but weakly enough for the products to desorb.

Homogeneous catalysts

The catalyst is in the same phase and works by being used in one step and reformed in a later step, providing a route with a lower activation energy. Two named examples:

• Oxides of nitrogen in the oxidation of atmospheric SO₂ (topic 12): \[ \mathrm{SO_2 + NO_2 \rightarrow SO_3 + NO} \] \[ \mathrm{NO + \tfrac{1}{2}O_2 \rightarrow NO_2} \]
• Fe²⁺ or Fe³⁺ in the reaction of I⁻ with S₂O₈²⁻. The direct reaction, S₂O₈²⁻ + 2I⁻ → 2SO₄²⁻ + I₂, is slow because both reactants are negative ions, which repel, giving a high activation energy. Iron ions provide a route in which every step is between oppositely charged ions: \[ \mathrm{S_2O_8^{2-} + 2Fe^{2+} \rightarrow 2SO_4^{2-} + 2Fe^{3+}} \] \[ \mathrm{2Fe^{3+} + 2I^{-} \rightarrow 2Fe^{2+} + I_2} \] Either Fe²⁺ or Fe³⁺ works, because each is regenerated in the other step. E^⦵ values confirm both steps are feasible: +2.01 − 0.77 = +1.24 V and +0.77 − 0.54 = +0.23 V (topic 24).