Chemical Kinetics

Differential Rate Laws

In many reactions, the rate of reaction changes as the reaction progresses. Initially the rate of reaction is relatively large, while at very long times the rate of reaction decreases to zero (at which point the reaction is complete). In order to characterize the kinetic behavior of a reaction, it is desirable to determine how the rate of reaction varies as the reaction progresses.

A rate law is a mathematical equation that describes the progress of the reaction. In general, rate laws must be determined experimentally. Unless a reaction is an elementary reaction, it is not possible to predict the rate law from the overall chemical equation. There are two forms of a rate law for chemical kinetics: the differential rate law and the integrated rate law.

The differential rate law provides a relationship between the rate of reaction and the concentrations of the various species in the system.

Differential rate laws can take on many different forms, especially for complicated chemical reactions. However, many chemical reactions obey one of three differential rate laws. Each rate law contains a constant, k, called the rate constant. The units for the rate constant depend upon the rate law, because the rate always has units of mole L^-1 sec^-1 and the concentration always has units of mole L^-1.

Consider the chemical reaction: A → B
There are three commonly encountered differential rate laws.

Zero-Order Reaction

For a zero-order reaction, the rate of reaction is a constant. When the limiting reactant is completely consumed, the reaction abrupts stops.

Differential Rate Law: r = k

The rate constant, k, has units of mole L^-1 sec^-1.

First-Order Reaction

For a first-order reaction, the rate of reaction is directly proportional to the concentration of one of the reactants.

Differential Rate Law: r = k [A]

The rate constant, k, has units of sec^-1.

Second-Order Reaction

For a second-order reaction, the rate of reaction is directly proportional to the square of the concentration of one of the reactants.

Differential Rate Law: r = k [A]²

The rate constant, k, has units of L mole^-1 sec^-1.

These three behaviors are illustrated in the following plots. The graph at the left shows concentration-time plots for zero-order (wine red line), first-order (gold line), and second-order (blue line) reactions. The corresponding rate-concentration plots are shown at the right.

In examining the plots, bear in mind that as the reaction progresses, the concentration of a reactant decreases. This corresponds to moving from right to left on the plot of reaction rate vs concentration. In this example, reactant A has a stoichiometric coefficient of one, so the reaction rate (plotted in the graph at the right) corresponds with the negative value of the slope of the concentration-time curve (plotted in the graph at the left). Carefully examine the graphs and take note of the following points:

For a zero-order reaction (wine red line), the rate of reaction is constant as the reaction progresses.
For a first-order reaction (gold line), the rate of reaction is directly proportional to the concentration. As the reactant is consumed during the reaction, the concentration of A drops and so does the rate of reaction.
For a second-order reaction (blue line), the rate of reaction increases with the square of the concentration, producing an upward curving line in the rate-concentration plot. For this type of reaction, the rate of reaction decreases rapidly as the concentration of the reactant decreases.

Concentration vs Time

Rate vs Concentration

Experiment

Objectives

Determine the differential rate law for a chemical reaction.
Determine the rate constant for a chemical reaction.

Consider the following reaction between formic acid (HCOOH) and bromine in aqueous solution:

HCOOH (aq) + Br₂ (aq) → 2 H⁺ (aq) + 2 Br^- (aq) + CO₂ (aq)

In this chemical system, the only species that absorbs visible light is bromine; thus spectrophotometry can be employed to determine how the concentration of bromine varies with time during the reaction.

One syringe contains a 2.00 mM solution of bromine and the other syringe contains a 0.200 M solution of formic acid. The two liquids are mixed in equal portions. As the reaction is occurring, the concentration of bromine is plotted versus time in the left graph and the rate of reaction is plotted versus the bromine concentration in the right graph.

Note that the concentration of formic acid is 100 times greater than the concentration of bromine. The consequence of having such a large excess of formic acid is that the formic acid concentration is essentially constant (within 1%) throughout the reaction. For this reason, a change in rate of reaction must be attributable to a change in the concentration of bromine. This experimental approach is called the isolation method and it allows one to determine the order of the reaction with respect to bromine. Additional experiments are necessary to determine the order of the reaction with respect to formic acid.

In general, one would expect a differential rate law of the form shown below. The constants a and b are the orders of the reaction with respect to Br₂ and HCOOH, respectively.

r = - k [Br₂]^a [HCOOH]^b

Because [HCOOH] is essentially constant, the differential rate law can be re-written as

r = - k_exp [Br₂]^a
k_exp = k [HCOOH]^b

The quantity k_exp is the pseudo-rate constant which will be measured in this experiment. Because the value of b is not known, it will not be possible to determine k itself in this experiment. However, it is possible to determine k_exp as well as the value of a ( = 0, 1 or 2), which is the order of the reaction with respect to bromine.

Questions

What is the order of this reaction with respect to bromine?
That is, is the reaction zero-, first- or second-order with respect to bromine?
Calculate the rate constant k_exp for this reaction using values read from the graph.
Click on either graph to see the position of the cursor tip.

Note the units on the graph. The concentration is millimolar (mmole/L) rather than molar (mole/L), and the rate of the reaction is mmole L^-1 sec^-1 rather than mole L^-1 sec^-1. You will need to convert from mmole to mole to obtain the value for k_exp.

[Br₂] vs Time

Rate vs [Br₂]