Reaction Rates: Average and Instantaneous
The rate of a chemical reaction is the change in the concentration of a reactant or product per unit time; the average rate is calculated over a time interval, and the instantaneous rate is the rate at one moment, equal to the slope of the tangent to the concentration–time curve. In brief: rate = −Δ[A]/Δt for a reactant A, so if [A] falls from 1.000 M to 0.607 M in 50 s, the average rate is 0.0079 M/s.
Chemical kinetics asks how fast reactions go. Thermodynamics says whether a reaction can occur, while kinetics tells us how quickly it will do so. A reaction rate depends on the concentrations of the reactants and on the conditions, and measuring it is the first step in working out the mechanism of a reaction. This article defines the rate, shows how to calculate average and instantaneous rates from data and explains how stoichiometry links the rates of different species.
What is a reaction rate?
For a reaction A → B, the rate can be expressed as the decrease of the reactant concentration or the increase of the product concentration:
rate = −Δ[A]/Δt = +Δ[B]/Δt
Square brackets denote the molar concentration, so the unit of rate is mol/(L·s), written M/s. The minus sign for a reactant makes the rate a positive number, since Δ[A] is negative. Rates for gases can also be expressed in terms of pressure, and for heterogeneous systems in terms of area.
How does stoichiometry affect the rate?
When the coefficients differ, the species change at different speeds. To define one rate for the entire reaction, divide each rate of change by its stoichiometric coefficient. For the reaction aA + bB → cC + dD:
rate = −(1/a) Δ[A]/Δt = −(1/b) Δ[B]/Δt = (1/c) Δ[C]/Δt = (1/d) Δ[D]/Δt
Example: decomposition of dinitrogen pentoxide
For 2N2O5 → 4NO2 + O2, the rate is −½ Δ[N2O5]/Δt = ¼ Δ[NO2]/Δt = Δ[O2]/Δt. If NO2 is formed at 0.0080 M/s, then N2O5 is consumed at 0.0040 M/s (half the rate of NO2 formation, since 2 mol of N2O5 give 4 mol of NO2) and O2 is formed at 0.0020 M/s. The overall reaction rate is 0.0020 M/s.
How do you calculate an average rate?
The average rate over a time interval Δt is the change in concentration divided by the time elapsed. Because most reactions slow down as the reactants are used up, the average rate depends on the interval selected: it is highest at the beginning of the reaction and falls later.
Rate data for a first-order reaction
The table gives the concentration of a reactant A that decomposes according to A → products, with a rate constant of 0.0100 s−1.
| Time interval (s) | [A] at start (M) | [A] at end (M) | Δ[A] (M) | Average rate (M/s) |
|---|---|---|---|---|
| 0–50 | 1.000 | 0.607 | −0.393 | 0.00786 |
| 50–100 | 0.607 | 0.368 | −0.239 | 0.00478 |
| 100–150 | 0.368 | 0.223 | −0.145 | 0.00290 |
| 150–200 | 0.223 | 0.135 | −0.088 | 0.00176 |
The average rate falls steadily from 0.00786 M/s to 0.00176 M/s, since the concentration of A decreases. Each value was found by dividing the loss of concentration by 50 s, for example 0.393 / 50 = 0.00786 M/s.
How do you find an instantaneous rate?
The instantaneous rate is the rate at a specific moment. It is the limit of the average rate as the time interval approaches zero, and it is equal to the negative slope of the tangent line to the concentration–time curve at that time:
- Plot [A] against time.
- Draw a tangent to the curve at the point of interest.
- Calculate the slope of the tangent, Δ[A]/Δt, using two points on the line.
- The rate is the negative of the slope.
For the data above, the instantaneous rate at t = 100 s is k[A] = 0.0100 × 0.368 = 0.00368 M/s. This lies between the average rates of the neighboring intervals (0.00478 and 0.00290 M/s), as it should. The initial rate is the instantaneous rate at t = 0, which in this case is 0.0100 × 1.000 = 0.0100 M/s. Initial rates are important because the reverse reaction and the effect of products are still negligible then.
How are rates measured?
- Spectrophotometry: following the absorbance of a colored reactant or product, for example iodine or permanganate.
- Gas collection or pressure: measuring the volume or pressure of the gas produced, as in the reaction of magnesium with acid.
- Conductivity or pH: following changes in ion concentration.
- Titration of samples: removing small samples at times and quenching the reaction.
- Mass change: weighing a flask that loses a gas.
Who started the study of reaction rates?
Ludwig Wilhelmy studied in 1850 how quickly cane sugar is converted into glucose and fructose in the presence of acid, which is regarded as the first quantitative study of a reaction rate. In 1864 Cato Guldberg and Peter Waage proposed the law of mass action, which relates the rate to the concentrations of the reactants. Jacobus van ‘t Hoff’s book Études de dynamique chimique of 1884 laid the foundation for modern chemical kinetics.
Which mistakes are common?
- Reporting a negative rate: the minus sign for reactants makes the rate positive.
- Forgetting the stoichiometric coefficients: species with different coefficients change at different speeds.
- Mixing up average and instantaneous rate: they are equal only for a straight-line graph.
- Using the wrong interval: the average rate depends on the interval.
- Equating rate and rate constant: the rate changes during the reaction, and k does not.