D29.3 Relative Rates of Reaction

When experimentally studying the rate of a reaction, it is not necessary to measure the concentrations of all the reactants and products—it is sufficient to measure just one concentration, preferably the concentration of the species that is easiest to measure. This is because the rate of a reaction can be expressed in terms of the change in concentration of any reactant or product. These changes in concentration are related by the stoichiometry of the reaction. Let’s use the ammonia decomposition reaction as an example:

2 NH3(g) ⟶ N2(g) + 3 H2(g)

From the balanced reaction, we can see that one N2 molecule is produced for every two NH3 molecules that have reacted. Therefore, the formation of N2 is half as fast as disappearance of ammonia:

\text{rate =} \dfrac{{\Delta}[\text{N}_2]}{{\Delta}t} = -\dfrac{1}{2}\dfrac{{\Delta}[\text{NH}_3]}{{\Delta}t}

The negative sign accounts for the fact that NH3 (reactant) concentration is decreasing while N2 (product) concentration is increasing. The fraction ½ accounts for the stoichiometry.

Similarly, because 3 mol H2 forms during the time required for formation of 1 mol N2, we have:

\text{rate =} \dfrac{{\Delta}[\text{N}_2]}{{\Delta}t} = \dfrac{1}{3}\dfrac{{\Delta}[\text{H}_2]}{{\Delta}t}

The concentrations vs. time graph for this reaction is shown below. At any time, the instantaneous rates for reactants and products are related by the reaction stoichiometry. For example, at 500 s, the rate of H2 production is three times greater than that for N2 production.

Figure: relative rates of reaction. Concentrations of reactants and products during the reaction 2NH3 → N2 + 3H2 as a function of time at 1100 °C is plotted. The rates of change of the three concentrations are related by the reciprocals of their stoichiometric coefficients. An example of this is shown by the different slopes (click on the “+” signs) of the tangents at t = 500 s.

The rate of a reaction is therefore defined by taking the change in concentration per unit time of a reactant or a product and multiplying by the reciprocal of the stoichiometric coefficient for that reactant or product. The reaction rate determined this way is the same regardless of which reactant or product is measured during an experiment. For a generic reaction:

a A + b B → c C + d D

where lower-case letters are stoichiometric coefficients and upper-case letters represent chemical formulas, the rate of the reaction is:

\text{rate =} -\dfrac{1}{a}\;\dfrac{{\Delta}[\text{A}]}{{\Delta}t} = -\dfrac{1}{b}\;\dfrac{{\Delta}[\text{B}]}{{\Delta}t} = \dfrac{1}{c}\;\dfrac{{\Delta}[\text{C}]}{{\Delta}t} = \dfrac{1}{d}\;\dfrac{{\Delta}[\text{D}]}{{\Delta}t}

 

Exercise: Definition of Reaction Rate

Exercise: Reaction Rate and Stoichiometry

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Chem 109 Fall 2023 Copyright © by Jia Zhou; John Moore; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.