Rate Laws and Determining Reaction Orders

The last three chapters built a qualitative and semi-quantitative picture: rate depends on collision frequency, the fraction of energetic collisions, and the pathway's activation energy. This chapter converts that picture into precise mathematics. The rate law is the experimentally determined equation linking reaction rate to reactant concentrations, and the order of a reaction with respect to each species is the exponent that equation assigns to it — a number that, crucially, cannot be read off the balanced equation and must be found from data.

You should already be fluent with the initial rates method of measuring rate (Chapter 1 of this block) and the qualitative concentration dependence developed in Chapter 3; this chapter makes both precise.

1. The Rate Law and Reaction Order

For a general reaction $a\text{A} + b\text{B} \rightarrow \text{products}$, the rate law expresses the rate as a product of concentration terms raised to powers:

Here $k$ is the rate constant (Section 4), and $m$ and $n$ are the orders of reaction with respect to $\text{A}$ and $\text{B}$ respectively — found experimentally, and in general unrelated to the stoichiometric coefficients $a$ and $b$. The overall order of the reaction is the sum $m+n$.

Non-negotiable warning. Do not assume $m=a$ and $n=b$. Many reactions do coincidentally show this match for an elementary single-step reaction (a mechanistic idea developed in the next chapter), but a reaction's rate law reflects its actual mechanism, not merely its overall balanced equation, and must be determined from data.

Orders are most often small non-negative integers ($0$, $1$, $2$), though fractional and even negative orders occur in more complex mechanisms. Each individual order describes how sensitively the rate responds to that one species' concentration, holding all others fixed:

• Zero order ($m=0$): rate is independent of $[\text{A}]$; doubling $[\text{A}]$ leaves the rate unchanged.
• First order ($m=1$): rate is directly proportional to $[\text{A}]$; doubling $[\text{A}]$ doubles the rate.
• Second order ($m=2$): rate is proportional to $[\text{A}]^2$; doubling $[\text{A}]$ quadruples the rate.

2. Determining Order from Initial Rates Data

The standard experimental technique, introduced as the initial rates method in Chapter 1, runs several experiments with different starting concentrations and compares their initial rates. Comparing two experiments in which only one concentration differs isolates that species' order directly.

Worked example 1. The following initial rates are measured for $\text{A} + \text{B} \rightarrow \text{products}$:

| Experiment | $[\text{A}]/\text{mol}\,\text{dm}^{-3}$ | $[\text{B}]/\text{mol}\,\text{dm}^{-3}$ | Initial rate$/\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}$ | |---|---|---|---| | 1 | $0.10$ | $0.10$ | $2.0\times 10^{-3}$ | | 2 | $0.20$ | $0.10$ | $8.0\times 10^{-3}$ | | 3 | $0.20$ | $0.20$ | $8.0\times 10^{-3}$ |

Order with respect to A: comparing Experiments 1 and 2, $[\text{B}]$ is held fixed while $[\text{A}]$ doubles. The rate rises from $2.0\times 10^{-3}$ to $8.0\times 10^{-3}$, a factor of $4$:

The reaction is second order in A. Order with respect to B: comparing Experiments 2 and 3, $[\text{A}]$ is held fixed while $[\text{B}]$ doubles, yet the rate is unchanged ($8.0\times 10^{-3}$ in both):

The reaction is zero order in B. The rate law is therefore $\text{rate} = k[\text{A}]^2$, and the overall order is $2+0=2$, second order overall — despite $\text{B}$ appearing in the balanced equation.

Method. To find each order, select the pair of experiments in which only the concentration of interest changes. Find the factor by which that concentration changed and the factor by which the rate changed, then solve $(\text{concentration factor})^{\text{order}} = \text{rate factor}$. Common factor pairs are worth recognising on sight: doubling with no rate change is order $0$; doubling with a doubled rate is order $1$; doubling with a quadrupled rate is order $2$; doubling with an eightfold rate is order $3$.

3. Determining Order When No Clean Pair Exists

Real data does not always offer two experiments differing in only one concentration. When concentrations change by an awkward, non-integer factor, the order is found by solving $(\text{factor}_1)^{m} = \text{factor}_2$ directly using logarithms:

Worked example 2. Two experiments show $[\text{A}]$ changing from $0.15$ to $0.45~\text{mol}\,\text{dm}^{-3}$ (factor $3.0$), with the initial rate changing from $4.0\times 10^{-4}$ to $3.6\times 10^{-3}~\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}$ (factor $9.0$), all other concentrations held fixed:

confirming second order in $\text{A}$, consistent with the clean-ratio shortcut of Section 2 but obtained here by a method that works for any concentration ratio, not only convenient doublings.

4. The Rate Constant, $k$

Once the rate law's form is established, the rate constant $k$ is found by substituting any one experiment's data back into the rate law and solving.

Worked example 3. Using Experiment 1 from Worked example 1, where $\text{rate} = k[\text{A}]^2$:

The units of $k$ depend on the overall order of the reaction, since rate always carries units of $\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}$ and the concentration terms must supply whatever combination of $\text{mol}\,\text{dm}^{-3}$ is needed to cancel correctly:

| Overall order | Units of $k$ | |---|---| | $0$ | $\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}$ | | $1$ | $\text{s}^{-1}$ | | $2$ | $\text{dm}^3\,\text{mol}^{-1}\,\text{s}^{-1}$ | | $3$ | $\text{dm}^6\,\text{mol}^{-2}\,\text{s}^{-1}$ |

Key distinction. The rate constant $k$ is genuinely constant for a given reaction at a fixed temperature — it does not depend on concentration (that dependence is already captured by the $[\text{A}]^m[\text{B}]^n$ factors). $k$ does depend on temperature, a relationship made precise by the Arrhenius equation in a later chapter.

5. Zero-, First- and Second-Order Concentration-Time Graphs

Each order produces a characteristically shaped concentration-time graph, a useful visual diagnostic alongside the numerical initial-rates method:

• Zero order: $[\text{A}]$ falls linearly with time (constant gradient, since the rate itself never changes while $\text{A}$ remains present), reaching zero and then staying there.
• First order: $[\text{A}]$ falls exponentially, with a curve that becomes progressively shallower; a first-order reaction has a constant half-life (the time for $[\text{A}]$ to halve), independent of the starting concentration — a diagnostic test used again in the exercises of this chapter.
• Second order: $[\text{A}]$ also falls in a curve, but compared with a first-order curve having the same initial concentration and initial rate it starts with the same gradient, then flattens out sooner (the rate falls as $[\text{A}]^2$) and stays above the first-order curve, leaving a long tail; its half-life is not constant, but instead increases as the reaction proceeds (each successive halving takes longer than the last).

A plot of rate against concentration, obtained from the gradient of several points on a concentration-time curve (or from several initial-rate experiments), gives a second, complementary diagnostic: a horizontal line indicates zero order, a straight line through the origin indicates first order, and an upward-curving parabola indicates second order.

6. A Worked Determination from a Concentration-Time Curve

Order can also be found without the initial-rates method, by measuring the instantaneous rate at several points along a single concentration-time curve (via tangents, as in Chapter 1) and examining how that rate depends on the concentration at each point.

Worked example 4. Tangents to a single concentration-time curve give:

| $[\text{A}]/\text{mol}\,\text{dm}^{-3}$ | $0.80$ | $0.40$ | $0.20$ | |---|---|---|---| | rate$/\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}$ | $6.4\times 10^{-3}$ | $1.6\times 10^{-3}$ | $4.0\times 10^{-4}$ |

Each halving of $[\text{A}]$ (from $0.80\to0.40$, and from $0.40\to0.20$) quarters the rate: $1.6\times 10^{-3}/6.4\times 10^{-3} = 0.25 = (0.5)^2$. The reaction is second order in A, found entirely from one experimental run rather than several separate ones — a technique that trades the need for multiple experiments for the need to read several accurate tangents off a single curve.

Common Exam Pitfalls

• Reading reaction order off the stoichiometric coefficients. Order must come from experimental data; it coincides with the coefficients only for an elementary single-step reaction, and this cannot be assumed without justification.
• Comparing the wrong pair of experiments. Only a pair in which exactly one concentration changes (all others fixed) isolates that species' order directly; comparing a pair where two concentrations change at once conflates their effects.
• Forgetting that zero order means "no effect," not "no participation." A species can be zero order in the rate law while still appearing in the balanced equation and being chemically essential to the mechanism.
• Quoting $k$ without units, or with the wrong units for the overall order. The units of $k$ are not fixed; they follow from the overall order and must be derived, not memorised as a single fixed unit.
• Treating $k$ as dependent on concentration. $k$ is a constant at fixed temperature; apparent changes in a calculated $k$ across experiments signal an error in the assumed rate law or arithmetic, not a genuine property of $k$.
• Confusing the shapes of first- and second-order concentration-time curves, or assuming every reaction has a constant half-life (only true for first order).

Summary

• The rate law, $\text{rate}=k[\text{A}]^m[\text{B}]^n$, gives the experimentally determined dependence of rate on concentration; $m$, $n$ (the orders) and their sum (the overall order) are found from data, not from the balanced equation.
• Comparing initial rates between experiments where only one concentration changes isolates that species' order via $(\text{concentration factor})^{\text{order}} = \text{rate factor}$; non-integer ratios are solved using logarithms, $m = \ln(\text{rate factor})/\ln(\text{concentration factor})$.
• The rate constant $k$ is found by substituting one experiment's data into the established rate law; its units depend on the overall order ($\text{s}^{-1}$ for first order, $\text{dm}^3\,\text{mol}^{-1}\,\text{s}^{-1}$ for second order, and so on).
• $k$ is constant at fixed temperature, independent of concentration; its temperature dependence is quantified later by the Arrhenius equation.
• Zero order gives a linear concentration-time graph; first order gives an exponential decay with constant half-life; second order gives a curve with a long tail (it stays above the first-order curve of equal initial rate) and an increasing half-life.
• Order can also be determined from the instantaneous rate (via tangents) at several points on a single concentration-time curve, without needing multiple separate experiments.