Finding the Order of Reaction and Rate Equation

The rate equation links the rate of a reaction to the concentrations of the reactants that affect it. For a general reaction where A and B react, the rate equation has the form:

\( \)rate\( = k[A]^m[B]^n \)

Here \(k\) is the rate constant, \([A]\) and \([B]\) are the concentrations of the reactants, and \(m\) and \(n\) are the orders of reaction with respect to A and B. Crucially, these orders cannot be predicted from the balanced equation, they must be found from experimental data. This builds directly on the ideas covered in introduction to kinetics, so make sure you are comfortable with what "rate of reaction" actually measures before working through orders.

The order of reaction with respect to a particular reactant tells you how the rate changes when that reactant's concentration changes, while everything else is held constant.

• Zero order: changing \([A]\) has no effect on rate. Doubling \([A]\) leaves the rate unchanged.
• First order: rate is directly proportional to \([A]\). Doubling \([A]\) doubles the rate.
• Second order: rate is proportional to \([A]^2\). Doubling \([A]\) quadruples the rate.

These orders are almost always found by measuring initial rates while systematically varying one concentration at a time, a method covered in detail on experiments to find the order of reaction.

The overall order of reaction is simply the sum of the individual orders in the rate equation:

\( \)overall order\( = m + n \)

For example, if a reaction is first order in A and second order in B, the rate equation is \( \)rate\( = k[A][B]^2 \), and the overall order is \(1 + 2 = 3\). The overall order tells you the total power to which concentrations are raised, which also determines the units of \(k\).

Since rate always has units of \(\)mol\(\,\)dm\(^{-3}\,\)s\(^{-1}\), the units of \(k\) depend on the overall order, because they must cancel out the concentration terms to leave rate in its usual units. You can always find the units of \(k\) by rearranging the rate equation and substituting units in directly, rather than memorising a table.

Plotting concentration against time gives different shaped curves depending on the order of reaction. For a first order reaction, the integrated rate law is \( [A] = [A]_0 e^{-kt} \), so concentration decays exponentially. These curve shapes, alongside rate against concentration graphs, are explored fully in reaction rate graphs.

The initial rate of a reaction between A and B was measured for three experiments:

Comparing experiments 1 and 2, \([A]\) doubles while \([B]\) is constant, and the rate quadruples, so the reaction is second order with respect to A. Comparing experiments 2 and 3, \([B]\) doubles while \([A]\) is constant, and the rate stays the same, so the reaction is zero order with respect to B. The rate equation is therefore:

\( \)rate\( = k[A]^2 \)

and the overall order of reaction is 2. Substituting any row of data lets you calculate \(k\): using experiment 1, \( 0.002 = k(0.10)^2 \), so \( k = 0.2\ \)dm\(^{3}\,\)mol\(^{-1}\,\)s\(^{-1} \).

A reaction mechanism is made up of several elementary steps, and the slowest of these is called the rate-determining step. The overall rate equation is determined only by the species involved in the rate-determining step (and any earlier steps that produce them), which is why a reactant can appear in the balanced equation with a large coefficient yet not appear in the rate equation at all, or appear with a power that does not match its stoichiometric coefficient. Once you have found the order of reaction with respect to each reactant experimentally, you can use it to suggest which species take part in the rate-determining step, and therefore propose a plausible mechanism consistent with the kinetics.

Related lessons

• Uk A Level Chemistry