Measuring the rate of chemical reactions

Chemical reactions occur across an astonishing range of speeds. An explosion is over in a fraction of a second, the souring of milk takes days, and the rusting of an iron railing can take years. Reaction kinetics is the branch of chemistry that studies how fast reactions go and what controls their speed. It answers questions that thermodynamics cannot: not just whether a reaction can happen, but how quickly it does.

Understanding rates matters far beyond the classroom. Chemical engineers tune reaction rates to make fertilizers and fuels economically; food scientists slow the reactions that spoil food; and biochemists study the enzymes that speed the reactions of life. In every case, the starting point is a precise definition of what we mean by 'rate'.

In everyday life a rate is a change divided by the time taken — a car's speed is the change in position per unit time. A reaction rate is defined the same way: it is the change in the concentration of a reactant or product per unit of time. Because concentration is measured in mol/L and time in seconds, the units of reaction rate are typically mol/(L·s), written \text{mol}\,\text{L}^{-1}\,\text{s}^{-1}.

As a reaction proceeds, reactant concentrations fall and product concentrations rise. We can therefore express the rate as the rate of disappearance of a reactant or the rate of appearance of a product. Because a reactant's concentration decreases, \Delta[\text{reactant}] is negative, so a minus sign is inserted to report the rate as a positive number: \text{rate} = -\dfrac{\Delta[\text{reactant}]}{\Delta t} = +\dfrac{\Delta[\text{product}]}{\Delta t}.

To measure a rate, you follow any property of the reaction mixture that changes as the reaction proceeds, and record how it changes with time. Several properties are commonly used: the volume or pressure of a gas that is produced or consumed; the intensity of a colour, measured with a spectrophotometer, if a reactant or product is coloured; the total mass, if a gas escapes; the pH, if hydrogen ions are involved; and the electrical conductivity, if the number of ions changes.

The best property to follow depends on the reaction. A reaction that produces a gas is easily tracked by collecting the gas and reading its volume at regular time intervals. A reaction between coloured species is best followed by the fading or deepening of the colour. The skill is to pick a property that changes clearly and can be measured without disturbing the reaction.

If you plot the concentration of a reactant against time, you get a curve that falls steeply at first and then levels off. From this graph we can define two kinds of rate. The average rate over a time interval is the total change in concentration divided by the length of that interval — the slope of the straight line joining the two points on the curve: \text{rate}_{\text{avg}} = -\dfrac{[A]_2 - [A]_1}{t_2 - t_1}.

The instantaneous rate is the rate at one particular moment. On the concentration-time graph it is the slope of the tangent line drawn at that instant. The instantaneous rate at the very start of the reaction is called the initial rate, and it is often the most useful because at that moment the concentrations are known exactly and no products have yet built up.

Reaction rates are almost never constant. A reaction usually begins fast, when reactant concentrations are highest, and slows down as the reactants are consumed. This is exactly why the concentration-time curve is steepest at the start and flattens toward the end — the instantaneous rate is continually decreasing. Recognising this is the reason chemists so often quote the initial rate rather than a single 'the rate' for the whole reaction.

In a reaction such as N₂ + 3H₂ → 2NH₃, the different species are not consumed and produced at the same numerical rate, because the coefficients differ. Hydrogen is used up three times as fast as nitrogen, and ammonia is produced twice as fast as nitrogen is consumed. To compare rates of different species, you divide each rate by that species' coefficient, which gives a single unique rate for the reaction. For the general reaction aA + bB \rightarrow cC + dD this is -\dfrac{1}{a}\dfrac{\Delta[A]}{\Delta t} = -\dfrac{1}{b}\dfrac{\Delta[B]}{\Delta t} = +\dfrac{1}{c}\dfrac{\Delta[C]}{\Delta t} = +\dfrac{1}{d}\dfrac{\Delta[D]}{\Delta t}.

Let us slow down and really look at what the tangent line is telling us. When you draw a curve of [\text{A}] against time, that curve is a record of the whole history of the reaction in one picture. The instantaneous rate at any moment is the steepness — the slope — of the line that just grazes the curve at that point without crossing it. To measure it, you pick your time of interest, lay a ruler along the curve so it touches at exactly that instant, and then read off the rise over the run of that tangent. A steep tangent means concentration is changing quickly, so the rate is high; a shallow, nearly flat tangent means the reaction has almost stopped. Because the tangent at the start is the steepest tangent of all, the initial rate is always the largest instantaneous rate the reaction ever shows.

It is worth being honest about why we bother with tangents at all. The average rate over an interval, -\dfrac{\Delta[\text{A}]}{\Delta t}, is easy to calculate but it blurs together a fast beginning and a slow end into a single misleading number. If a reaction is nearly finished halfway through your interval, the average badly underestimates how fast things were moving at the start. The instantaneous rate fixes this by shrinking the interval down to a single moment, and the tangent line is simply the visual limit of the straight chords you would draw as the two points on the curve slide closer and closer together. This is exactly the calculus idea of a derivative, dressed in chemical clothing.

Students often rush past the units, but they carry real meaning. A rate of 0.020\ \text{mol}\,\text{L}^{-1}\,\text{s}^{-1} says that in each second, the concentration changes by 0.020 moles per litre. If you ever find yourself unsure whether a number is a rate, a concentration, or a time, check the units: a rate must always be a concentration divided by a time. This also explains why the same reaction can be reported with different-looking numbers — a rate quoted per minute is sixty times the numerically identical rate quoted per second. When you compare two experiments, always confirm the time units match before you draw any conclusion, or you may fool yourself into thinking one run was far faster than another.

Dimensional reasoning is also your safety net in longer problems. Suppose a reaction produces oxygen gas and you measure the rate as a volume of gas per second. That is not yet a reaction rate in the chemist's sense, because it is a volume rate, not a concentration rate. To convert, you use the molar volume of a gas to turn the volume into moles, then divide by the solution volume to reach \text{mol}\,\text{L}^{-1}, and only then divide by time. Keeping a clear eye on units at every stage stops the common mistake of comparing a \text{mol}\,\text{L}^{-1}\,\text{s}^{-1} figure with a raw \text{mL}\,\text{s}^{-1} figure as though they were the same kind of thing.

There are two broad strategies for following a reaction. In a continuous method, an instrument watches a property in real time without disturbing the mixture — a spectrophotometer reading the intensity of a colour, a pressure sensor on a sealed flask, or a pH probe dipped in the solution. Continuous methods are ideal because they give a smooth stream of data from which the whole concentration-time curve can be drawn. In a sampling method, small portions of the mixture are removed at intervals and analysed separately. The catch is that the reaction keeps going inside the sample while you analyse it, so sampling methods usually require you to quench each sample — cooling it in ice, diluting it heavily, or destroying a reactant — to freeze the reaction at the instant of sampling.

A particularly elegant technique is the clock reaction. Here you arrange for a sudden, sharp change — often a dramatic colour appearing — to happen only after a fixed amount of product has formed. You then simply time how long that takes. Because the same fixed amount of product forms each time, a shorter clock time means a faster reaction, and the rate is roughly proportional to 1/t. Clock reactions are wonderful for classroom work because they need no expensive instrument: a stopwatch and a keen eye are enough to compare how temperature or concentration changes the speed of a reaction.

When different species in the same reaction change at different numerical speeds, it is untidy to speak of 'the rate' without saying which substance you mean. Chemists solve this by defining a single unique rate of reaction that every species agrees on. You take each species' own rate and divide by its stoichiometric coefficient. For aA + bB \rightarrow cC + dD this gives one shared value, \text{rate} = -\dfrac{1}{a}\dfrac{\Delta[A]}{\Delta t} = +\dfrac{1}{c}\dfrac{\Delta[C]}{\Delta t}. The beauty of this definition is that it no longer matters which substance you happened to monitor: convert to the unique rate and any two experimenters will report the same number for the same reaction.

Let us walk through why the coefficients must appear. In \text{N}_2 + 3\text{H}_2 \rightarrow 2\text{NH}_3, every time one nitrogen molecule reacts, three hydrogen molecules are used and two ammonia molecules appear. So over any interval hydrogen must disappear three times as fast as nitrogen, and ammonia must appear twice as fast as nitrogen disappears. Dividing each raw rate by its coefficient rescales all three back to the same underlying tempo of the reaction. If you are ever handed one species' rate and asked for another's, this proportionality is the whole trick: multiply or divide by the ratio of coefficients, and mind the sign.

Measuring how the initial rate responds to concentration opens the door to the next big idea in kinetics, the rate law. If you run a reaction several times, changing only the starting concentration of one reactant and holding everything else fixed, you can see exactly how sensitive the rate is to that reactant. Double the concentration and, if the rate also doubles, the rate depends on the first power of that concentration; if the rate quadruples, it depends on the square. Writing this dependence out gives an expression of the form \text{rate} = k[\text{A}]^m[\text{B}]^n, where k is the rate constant and the exponents record the sensitivity. This is why the initial-rate measurements of this section are not busywork — they are the raw data from which rate laws are built.

The exponents m and n are called the orders of the reaction with respect to each reactant, and here is the subtlety that surprises everyone at first: they are not the coefficients from the balanced equation. They can only be found by experiment, because they depend on the hidden mechanism of the reaction, not on the tidy overall stoichiometry. A reactant might have a coefficient of two yet turn out to be first order, or even zero order, meaning its concentration does not affect the rate at all. Keeping this distinction sharp — coefficients come from balancing, orders come from measurement — will save you from one of the most common errors in all of kinetics.

At the very first instant of a reaction, you know the concentrations of every reactant exactly, because they are just the amounts you mixed together, and no products have yet accumulated to run the reverse reaction or to interfere. This is why the initial rate is the gold standard for comparing experiments and for extracting rate laws. Later in the reaction the concentrations are no longer neatly known, products may be building up, and side reactions can muddy the picture. By reading the slope of the tangent right at time zero, you capture the reaction in its cleanest, most controlled moment, which is precisely why the initial-rates method is so widely used.

It helps to see the two graph shapes side by side. A reactant curve starts high and slides downward, steep at first and flattening as the reactant runs low. A product curve starts at zero and climbs upward, steep at first and levelling as the reaction finishes. The two curves are mirror images in a sense, because every bit of reactant consumed becomes product. At any chosen time the magnitude of the slope of the reactant curve, once you account for stoichiometry, matches the slope of the product curve — they are two views of the same underlying event, one counting what is lost and the other counting what is gained.

Everything in this section pays off the moment you leave the lab bench. In industry, engineers deliberately drive reaction rates as high as safety and cost allow, because a faster reaction means more product per hour from the same expensive equipment. In food science the goal is the opposite — spoilage is a chemical reaction, and slowing it by refrigeration or packaging extends shelf life. In your own body, thousands of reactions are tuned to run at just the right pace by enzymes. In every one of these settings, the work begins exactly where this section did: with a careful, quantitative definition of what 'rate' means and a reliable way to measure it.

The rate of reaction is how fast a concentration changes: \text{rate} = -\dfrac{\Delta[\text{reactant}]}{\Delta t} = +\dfrac{\Delta[\text{product}]}{\Delta t} (the minus sign keeps the rate positive as reactants disappear). Crucially, the rates of different species are linked by the coefficients: for 2\text{A} \to \text{B}, A is consumed twice as fast as B forms, so -\tfrac12\dfrac{\Delta[\text{A}]}{\Delta t} = \dfrac{\Delta[\text{B}]}{\Delta t}. Divide each rate by its coefficient to get one common reaction rate.