3.1.9 Rate Equations

In rate = k[A]ᵐ[B]ⁿ, m and n are experimentally determined reaction orders. The overall order is m + n, and the powers do not have to match coefficients in the balanced equation.

Read the data before the equation. If doubling [A] doubles the rate while other concentrations stay constant, the reaction is first order in A; if the rate quadruples, it is second order.

For AQA, keep the allowed individual reaction orders very simple: 0, 1 or 2. Order 0 means rate is independent of that reactant concentration, order 1 gives direct proportionality and order 2 gives a squared dependence.

A zero-order term can be omitted from the written rate equation because [A]⁰ = 1. That does not mean A is absent from the chemistry; only that changing its concentration does not affect rate under those conditions.

In k = Ae^(−Eₐ/RT), A is the Arrhenius constant, Eₐ the activation energy, R the gas constant and T the absolute temperature. The exponent must be dimensionless, so match the energy units of Eₐ and R.

Raising T makes −Eₐ/RT less negative, so the exponential factor and k increase. The Arrhenius equations and R are supplied when required; you need to recognise the quantities and use the equations correctly.

The order with respect to a reactant is the power of its concentration in the experimentally determined rate equation. The rate constant k is the proportionality constant for that reaction at a stated temperature.

The units of k depend on overall order. You do not need to memorise a separate list if you rearrange the rate equation and derive the units from rate ÷ concentration powers.

Once the orders are known, substitute one complete experiment into the rate equation to calculate k. Then derive its units by dividing the rate units by the concentration terms and their powers.

If the data give a second experiment, we can use it as a check. The same k should emerge at the same temperature; if it does not, revisit the orders or the arithmetic.

If you raise the temperature, k increases for a given reaction. Keep that separate from concentration: k can change even though the reactant concentrations have not, and a catalyst changes the effective rate constant by providing a different pathway.

Higher temperature increases the fraction of collisions able to surmount Eₐ, so k increases. The concentrations have not changed merely because k has changed.

Use k = A e^(−Ea/RT), or ln k = ln A − Ea/(RT). In an ln k against 1/T plot, gradient = −Ea/R and intercept = ln A.

Take care with the exponential and unit conversion for Eₐ. Enter the entire exponent −Eₐ/(RT) with brackets; a missing sign or factor of 1000 can change k by many orders of magnitude.

Linearise the Arrhenius equation as ln k = −Eₐ/R(1/T) + ln A. That has the form y = mx + c, so an ln k against 1/T graph has gradient −Eₐ/R and intercept ln A.

Compare ln k = −Eₐ/R(1/T) + ln A with y = mx + c. The gradient should be negative, and Eₐ = −gradient × R after accounting for any scale factor on the 1/T axis.

Reaction orders come from experiment. A balanced chemical equation gives overall stoichiometry but does not normally tell you the powers in the rate equation because the overall equation need not be an elementary step.

So do not copy equation coefficients into rate = k[A]ᵐ[B]ⁿ unless the experimental data actually support those orders.

Use the experimental rate equation as a constraint on a proposed mechanism. Species in or before the slow step must account for the observed orders, but a balanced overall equation alone cannot reveal the mechanism.

A proposed rate-determining step must be compatible with the observed rate equation, possibly after using a preceding fast equilibrium to eliminate an intermediate concentration. Reject mechanisms that predict the wrong concentration dependence.

For the instantaneous rate from a concentration–time graph, draw a tangent at the required time and find its gradient. Use well-separated points on the tangent itself, not two points from the original curve.

A reactant concentration falls, so its concentration gradient is negative. If the question asks for the positive reaction-rate magnitude, take the appropriate sign convention into account.

Estimate initial rate from the tangent at t = 0, or from suitably early data if that is what the question supplies. A long time interval gives an average rate after concentrations have already changed.

When you draw the tangent, make it follow the initial direction of the curve and use a large triangle for the gradient. Tiny coordinate differences magnify reading error.

When you use rate–concentration data to find an order, hold the other reactants constant and compare factors. If doubling [A] leaves the rate unchanged the order is 0; if the rate doubles it is 1; if the rate quadruples it is 2.

Choose experiment pairs where only the concentration of the reactant you are testing changes. If two reactant concentrations change at once, use another pair or account for one known order before deducing the other.

Once each individual order is known, place it as the power on that reactant concentration in the rate equation. A zero-order concentration term can be omitted because it equals 1.

Write the complete rate equation before calculating k. That keeps the order evidence and the numerical substitution as two separate steps, which makes errors easier to spot.

Use the observed rate equation as a constraint on a proposed mechanism. The rate-determining step, together with any preceding fast equilibrium needed to express intermediates, must reproduce the observed concentration dependence.

A mechanism that balances chemically can still be wrong kinetically. If it predicts the wrong reaction orders, reject it.