5.1.1(a)-(c) - Rate equations, orders and rate constants

Kinetic Terms

A reaction rate describes how quickly a reaction changes. For rate equations, OCR usually expresses rate as the change in concentration per unit time, so a common unit is $\,\mathrm{mol} \,\mathrm{dm}^{-3} \,\mathrm{s}^{-1}$.

Two other terms belong in this topic, even though later lessons use them more deeply.

A key boundary for this lesson is that half-life calculations from integrated rate equations are not being taught here. For now, you need the term and the idea. You will use graph evidence and first-order half-life relationships in the next rates lesson.

Worked example: recognising the terms

For the rate equation

the reaction is second order with respect to $\ce{NO}$, first order with respect to $\ce{Cl2}$, and third order overall. The rate constant is $k$; its units cannot be known until the full rate equation and the rate unit are considered.

What a Rate Equation Means

A rate equation shows how the rate depends on reactant concentrations. At A level, the required form is:

$[A]$ and $[B]$ mean concentrations, usually in $\,\mathrm{mol} \,\mathrm{dm}^{-3}$. The powers $m$ and $n$ are the orders with respect to A and B. In this lesson those orders are 0, 1 or 2.

The powers have precise meanings:

This is where students often lose marks: the orders are found from experimental data, not from the coefficients in the overall balanced equation. If a balanced equation says $\ce{2A + B -> {products}}$, that does not prove the rate equation is $\mathrm{rate} = k[A]^{2}[B]$. The reaction may occur by several steps, and the measured rate depends on the slow step and on the experimental conditions.

[DIAGRAM: asset_name: Lesson 5.1.1a-c: Rate equations, orders and rate constants - diagram 01; asset_slug: 05_01_01a_rate_equations_orders_and_rate_constants__diagram_01; recommended_method: drawn_chem; description: A clean 16:9 workflow diagram showing initial-rates analysis: choose two experiments where only one concentration changes, compare concentration ratio to rate ratio, infer order 0/1/2, write rate = k[A]^m[B]^n, then calculate k and derive its units from the rate equation.]

In an initial-rates investigation, each run starts with known initial concentrations. The rate is measured right at the start, or estimated from a fixed early change, so the concentrations have not yet changed much. The fair comparison is to change one initial concentration while keeping the other initial concentrations and temperature constant.

Deducing Orders from Initial Rates

To deduce an order, compare two experiments where only one reactant concentration changes. Keep your reasoning in ratios:

Worked example: finding orders from a table

The initial rates below were measured at the same temperature.

Compare experiments 1 and 2. Only $[A]$ changes. $[A]$ doubles from 0.100 to 0.200, and the rate doubles from $4.00 \times 10^{-5}$ to $8.00 \times 10^{-5}$.

So the order with respect to A is 1.

Compare experiments 1 and 3. Only $[B]$ changes. $[B]$ doubles from 0.200 to 0.400, and the rate becomes four times larger.

So the order with respect to B is 2.

Compare experiments 1 and 4. Only $[C]$ changes. $[C]$ doubles, but the rate is unchanged.

So the order with respect to C is 0. C is not written in the rate equation because $[C]^{0} = 1$.

The rate equation is therefore:

The overall order is:

Notice the careful language: C may still be present in the reaction mixture, but changing its concentration does not affect the measured rate under these conditions.

Worked example: both concentrations change

Sometimes no convenient pair keeps the second concentration fixed. First use a simpler pair to find one order, then remove its known effect from the mixed comparison. Suppose separate evidence establishes first order in A. In another pair, $[A]$ triples, $[B]$ halves and the rate becomes 0.750 times its previous value.

so $n=2$: B is second order. The rate did not simply follow B's change because A changed as well. Check the conclusion: $3\times(0.500)^2=0.750$, exactly the observed factor. This is a ratio calculation; no absolute value of $k$ is needed.