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# Chemical Kinetics

## Differential Rate Laws

### Concepts

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 relates the rate of reaction to the concentrations of the various species in the system.

Differential rate laws can take on many different forms, especially for complicated chemical reactions. However, most 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.

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 (red line), first-order (green 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 reactant decreases. This corresponds to moving from right to left on the plot of reaction rate vs concentration. In this example, the reactant 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 (red line), the rate of reaction is constant as the reaction progresses.
• For a first-order reaction (green line), the rate of reaction is directly proportional to the concentration. As the reactant is consumed during the reaction, the concentration 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 (faster than linearly) as the concentration of the reactant decreases.

### 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:

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.

Examine the two graphs and determine the differential rate law for this reaction. Calculate the rate constant using values read from the graph(s).

Rate of Reaction                     Integrated Rate Laws