Definition and Measurement of Reaction Rate

Chemical kinetics is the study of how fast reactions happen and what factors control that speed — a question thermodynamics never answers, since a reaction can be energetically favourable (negative $\Delta G$ ) and yet proceed too slowly to be observed at room temperature. The rusting of iron and the combustion of petrol are both thermodynamically spontaneous, but one takes years and the other takes milliseconds; kinetics, not thermodynamics, explains the difference. This chapter opens Block C by defining reaction rate precisely, relating it to the stoichiometry of the equation, and surveying the experimental techniques used to measure it. Everything in the five chapters that follow — collision theory, rate laws, mechanisms and the Arrhenius equation — is built on the quantity defined here.

You should already be comfortable with molar concentration ( $\text{mol}\,\text{dm}^{-3}$ ), balancing equations, and reading a gradient from a graph; all three are used immediately below.

1. Defining the Rate of a Chemical Reaction

The rate of reaction is the change in concentration of a reactant or product per unit time:

\text{rate} = \frac{\Delta[\text{species}]}{\Delta t}

with SI-compatible units of $\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}$ (sometimes $\text{mol}\,\text{dm}^{-3}\,\text{min}^{-1}$ for slower reactions). Because a reactant is consumed, its concentration falls with time and $\Delta[\text{reactant}]$ is negative; by convention the rate itself is always reported as a positive quantity, so a minus sign is inserted when the rate is tracked through a reactant:

\text{rate} = -\frac{\Delta[\text{reactant}]}{\Delta t} = +\frac{\Delta[\text{product}]}{\Delta t}

Definition. Reaction rate is the rate at which reactants are consumed, or equivalently the rate at which products form, expressed as a concentration change per unit time. It is always quoted as a non-negative quantity.

A rate defined this way is still ambiguous until one further distinction is made: is it the rate over a finite time interval, or the rate at one particular instant? Section 2 makes this precise.

2. Average Rate and Instantaneous Rate from a Concentration-Time Graph

The average rate over an interval $[t_1, t_2]$ is simply the total concentration change divided by the elapsed time — a single number that discards everything about how the rate varied within that interval:

\text{rate}_{\text{avg}} = \frac{[\text{A}]_2 - [\text{A}]_1}{t_2 - t_1}

Because almost every reaction slows down as reactants are used up, this average is never the whole story. The instantaneous rate at a single time $t$ is the gradient of the tangent to the concentration-time curve at that point:

\text{rate}(t) = -\left(\frac{d[\text{A}]}{dt}\right)_{t}

Graphically, this means drawing the best-fit tangent line by eye at the point of interest and measuring its gradient from two points comfortably spaced along it — never from the curve itself, and never from a chord joining two points on the curve, which gives an average rate over that chord, not an instantaneous one.

Key distinction. A chord drawn between two points on a concentration-time curve gives an average rate over that interval. Only a tangent drawn at a single point gives the instantaneous rate at that instant. The two coincide only for a straight-line (zero-order) plot, where the rate never changes.

Worked example 1. A reactant $\text{A}$ decomposes, and its concentration is tracked over one minute:

| $t/\text{s}$ | $0$ | $10$ | $20$ | $30$ | $40$ | $50$ | |---|---|---|---|---|---|---| | $[\text{A}]/\text{mol}\,\text{dm}^{-3}$ | $1.00$ | $0.80$ | $0.65$ | $0.54$ | $0.46$ | $0.40$ |

The average rate over the first $30~\text{s}$ is found from the end points of that interval only:

\text{rate}_{\text{avg}} = \frac{1.00 - 0.54}{30 - 0} = \frac{0.46}{30} = 1.5\times 10^{-2}~\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}

To estimate the instantaneous rate at $t = 20~\text{s}$ , a tangent is drawn at the point $(20, 0.65)$ of the plotted curve. Suppose the tangent, extended, passes through $(10, 0.77)$ and $(30, 0.53)$ — two points comfortably either side of $t=20~\text{s}$ , chosen to make the gradient easy to read:

\text{rate}(20) = -\frac{0.53 - 0.77}{30 - 10} = -\frac{-0.24}{20} = 1.2\times 10^{-2}~\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}

The instantaneous rate at $t=20~\text{s}$ ( $1.2\times10^{-2}$ ) is smaller than the average rate over $0$ – $30~\text{s}$ computed above ( $1.5\times10^{-2}$ ) because the reaction slows as $[\text{A}]$ falls; in general the two numbers measure genuinely different things and need not be close.

3. The Stoichiometric Relationship Between Rates

For a reaction with unequal coefficients, every species does not appear or disappear at the same rate, even though all these rates describe the same single reaction. Consider the decomposition

2\text{N}_2\text{O}_5(g) \rightarrow 4\text{NO}_2(g) + \text{O}_2(g)

For every two moles of $\text{N}_2\text{O}_5$ consumed, four moles of $\text{NO}_2$ and one mole of $\text{O}_2$ appear, so $\text{NO}_2$ is produced twice as fast as $\text{N}_2\text{O}_5$ is consumed, and $\text{O}_2$ only one half as fast (one quarter as fast as $\text{NO}_2$ ):

-\frac{1}{2}\frac{d[\text{N}_2\text{O}_5]}{dt} = \frac{1}{4}\frac{d[\text{NO}_2]}{dt} = \frac{d[\text{O}_2]}{dt}

Dividing each rate of change by its own stoichiometric coefficient gives a single quantity — the rate of reaction proper — common to every species. When a question asks simply for "the rate," always check which species' concentration generated the number, and convert using the balanced equation before comparing rates of different species, or before comparing your answer with another student's who tracked a different species.

Warning. "The rate of formation of $\text{NO}_2$ " and "the rate of the reaction" are not interchangeable unless $\text{NO}_2$ has a stoichiometric coefficient of $1$ . Always divide by the coefficient before calling a number "the rate of reaction."

4. Measuring Rate by Mass Loss (Gas-Evolving Reactions)

When a reaction releases a gas that is allowed to escape, the total mass of the reaction vessel and its contents falls continuously, and this mass loss can be logged automatically on an electronic balance — a simple, high-precision, continuous-monitoring method well suited to reactions such as

\text{CaCO}_3(s) + 2\text{HCl}(aq) \rightarrow \text{CaCl}_2(aq) + \text{H}_2\text{O}(l) + \text{CO}_2(g)

Worked example 2. A small, fixed mass of marble chips (the limiting reagent) is added to excess dilute hydrochloric acid in an open flask standing on a balance; cotton wool plugs the neck to stop acid spray escaping while still allowing $\text{CO}_2$ out. The total mass loss is logged:

| $t/\text{s}$ | $0$ | $20$ | $40$ | $60$ | $80$ | $100$ | $120$ | |---|---|---|---|---|---|---|---| | mass loss $/\text{g}$ | $0$ | $0.14$ | $0.24$ | $0.30$ | $0.33$ | $0.34$ | $0.34$ |

The average rate of $\text{CO}_2$ production over the first $20~\text{s}$ , in mass terms, is $\dfrac{0.14}{20} = 7.0\times 10^{-3}~\text{g}\,\text{s}^{-1}$ . Converting to a molar rate using $M(\text{CO}_2) = 44.0~\text{g}\,\text{mol}^{-1}$ :

\text{rate} = \frac{7.0\times 10^{-3}~\text{g}\,\text{s}^{-1}}{44.0~\text{g}\,\text{mol}^{-1}} = 1.6\times 10^{-4}~\text{mol}\,\text{s}^{-1}

The curve flattens after about $100~\text{s}$ : the mass loss stops increasing because the marble chips, present in a fixed finite quantity, are the limiting reagent (the acid is in excess) and have been used up, so no more gas can be produced. A flat plateau on a mass-loss or gas-volume graph of a reaction that goes to completion signals that the limiting reagent is exhausted, not that the reaction has merely slowed down.

This method fails whenever the gas evolved is too soluble to escape completely (e.g. $\text{CO}_2$ in cold, dilute solution loses some of its mass change to dissolution) or when the reaction releases no gas at all, in which case an entirely different observable is needed.

5. Measuring Rate by the Volume of Gas Collected

An alternative for gas-evolving reactions is to channel the gas into a gas syringe (or an inverted measuring cylinder over water, for gases that do not dissolve appreciably) and record the volume collected against time. The resulting volume-time graph has exactly the same shape and the same interpretation as the mass-loss graph of Section 4: its initial gradient is the initial rate, and the volume at which the curve plateaus corresponds to the total gas produced once the limiting reagent is exhausted.

\text{rate (in gas terms)} = \frac{\Delta V_{\text{gas}}}{\Delta t}

A gas syringe gives a rate directly in $\text{cm}^3\,\text{s}^{-1}$ , which can be converted to a molar rate using the molar gas volume ( $24\,000~\text{cm}^3\,\text{mol}^{-1}$ at room temperature and pressure, $293~\text{K}$ and $101~\text{kPa}$ ) exactly as the mass loss was converted using molar mass in Section 4. This method is preferred over mass loss whenever the gas product is valuable to isolate, when the balance available lacks sufficient resolution, or simply as the more direct and intuitive classroom technique — but it shares the same two failure modes: it cannot be used for a gas that dissolves significantly in the reaction mixture, nor for a reaction that produces no gas.

6. Measuring Rate Without a Gas: Titration and Colorimetry

Many reactions of interest — esterification, the acid-catalysed iodination of propanone, the reaction between bromine water and an alkene — evolve no gas at all, and a different observable must be tracked.

The quenching (titration) technique withdraws a small, identical-volume sample from the reaction mixture at regular time intervals, immediately quenches it (stops the reaction in that sample instantly, typically by adding it to ice-cold water or a reagent that consumes a key species) and titrates it against a suitable reagent to find the concentration of a reactant or product at that moment. Plotting the titration results against sampling time gives the familiar concentration-time curve of Section 2, from which both average and instantaneous rates follow exactly as before. Because each titration destroys its sample, this method is discontinuous — it gives a curve built from separate snapshots, each requiring a fresh quench, rather than one continuous trace.

Colorimetry exploits a reaction in which one species absorbs visible light and the others do not — the reaction of bromine water with an alkene, or the classic iodine-starch "clock" reactions, are standard examples. A colorimeter continuously records the absorbance of the mixture, which (via a pre-constructed calibration curve, since absorbance is proportional to concentration by the Beer-Lambert law) converts directly into a concentration-time graph without the need to quench or withdraw any sample at all — a genuinely continuous method, unlike titration.

Electrical conductivity is a further continuous technique, useful whenever the total number or identity of ions in solution changes measurably during the reaction (for example, as a neutral molecule ionises, or as one ionic species is replaced by another of different mobility).

7. Continuous Monitoring Versus the Initial Rates Method

The methods above fall into two broad families. Continuous monitoring (mass loss, gas volume, colorimetry, conductivity) records one physical quantity throughout the entire reaction and yields a full concentration-time curve, from which the instantaneous rate at any chosen moment can be read by the tangent method of Section 2. The initial rates method, used especially when a full curve is laborious or when the question is specifically how the rate depends on a starting concentration, instead runs several separate experiments, each with a different known starting concentration, and measures only the initial rate of each — the gradient of the tangent at $t=0$ , before any reactant concentration has changed enough to complicate the comparison. This is the technique that underlies the next chapters' determination of reaction order, where several initial rates at different starting concentrations are compared directly.

Why the initial rate specifically? At $t=0$ every concentration is known exactly (it is whatever was measured out), with no need to infer it from a curve or a titration partway through the reaction. Comparing rates later in the reaction would require knowing every concentration at that later time, which is far harder to pin down reliably.

Common Exam Pitfalls

Reporting a negative rate. A reactant's concentration genuinely falls, so $\Delta[\text{reactant}]/\Delta t$ is negative, but "rate" by convention is always quoted as a positive number — the minus sign in $\text{rate} = -d[\text{A}]/dt$ exists precisely to correct for this, and it must not then be carried into the final numerical answer.
Drawing a chord instead of a tangent. A straight line joining two points on the curve gives the average rate over that interval, not the instantaneous rate at either point. Only a line touching the curve at a single point, extended through a wide reading baseline, gives an instantaneous rate.
Ignoring stoichiometric coefficients when comparing rates of different species. "Rate of formation of $\text{NO}_2$ " in the Section 3 example is four times "the rate of the reaction" (coefficient $4$ ), and twice the rate of consumption of $\text{N}_2\text{O}_5$ (coefficients $4$ versus $2$ ) — a question that asks for "the rate" expects the coefficient-corrected value, not whichever species happened to be measured.
Mistaking a plateau for the reaction merely slowing down. A genuinely flat, zero-gradient section of a mass-loss or gas-volume graph means the limiting reagent is exhausted and the reaction has finished, not that it is proceeding very slowly.
Using the wrong method for a soluble gas or for a reaction that evolves no gas. $\text{CO}_2$ collected over water loses accuracy because some of it dissolves; a reaction producing no gas at all cannot be studied by mass loss or gas-syringe methods regardless of how slow or fast it is.
Confusing a discontinuous (titration/quenching) method with a continuous one. Titration gives only as many data points as separate samples were quenched and titrated; colorimetry, mass loss and gas volume give an unbroken trace, and only the latter are properly called "continuous monitoring."

Summary

Reaction rate is the rate of change of a reactant's or product's concentration with time, $\text{rate} = -\dfrac{d[\text{reactant}]}{dt} = +\dfrac{d[\text{product}]}{dt}$ , always quoted as a positive quantity in $\text{mol}\,\text{dm}^{-3}\,\text{s}^{-1}$ .
The average rate over an interval uses the two end-point concentrations only; the instantaneous rate at a point is the gradient of the tangent to the concentration-time curve at that point, found from a chord only in the special case of a straight-line graph.
Rates of different species in the same reaction are related by their stoichiometric coefficients: dividing each species' rate of change by its own coefficient gives the single, common rate of reaction.
Mass loss and gas volume methods suit gas-evolving reactions and give continuous mass/volume-time curves (convertible to concentration-time data); both fail for very soluble gases and for reactions that evolve no gas.
Titration (quenching) is a discontinuous method giving separated snapshots; colorimetry and conductivity are continuous methods that track an optical or electrical property proportional to concentration.
The initial rates method measures only the tangent gradient at $t=0$ across several separate experiments with different starting concentrations, avoiding the need to know concentrations at any later, less certain time — the technique on which the determination of reaction order (next chapters) is built.