Reaction Rate Laws

In order to introduce the idea of rate, let's suppose we wish to know how fast the following reaction is going:

One way to do this is to define rate as the change in concentration of some species with respect to time, and then measure the concentrations of all species at multiple times to determine the rate. The results of such a hypothetical experiment is given in the for the reaction of hydrogen and iodine. The initial concentrations of H₂ and I₂ are equal at all times and the initial concentration of product is zero:

As you can see, the rate of formation of HI is twice the rate of disappearance of H₂ or I₂ at any given time. Also, note that the rate slows in time due to decreasing concentrations of the reactants. Stated mathematically, the relationship between the formation of products and the disappearance of reactants for this reaction is:

In general, for the reaction below:

The rate is expressed as follows:

Another expression for a rate is called the differential rate law, or simply, the rate law. It expresses the rate of a reaction in terms of the concentrations of the reactants raised to an experimentally determined power. The exponent on each concentration term is called the order of the reaction in that particular reactant. The sum of the exponents in the rate law is called the order of the reaction. The powers on the concentration terms in a rate law are NOT the stoichiometric coefficients from the balanced equation! For example, the rate law for the will have the following form:

Note that the exponents are not a and b but some experimentally determined powers p and q which may or may not equal a and b. The order of the is, therefore, p + q. We will discuss in Determining Rate Laws how those exponents can be determined.

Also notice in the the presence of the rate constant k. Students often have trouble distinguishing between the rate of a reaction and its rate constant. The rate of a reaction is the total rate of a reaction and is the "rate" in the rate law. The units of rate are always M / s. The rate constant, k, is an experimentally determined proportionality constant that gives some measure of the intrinsic "reactivity" of the reaction. The units on the rate constant depend on the order of the reaction, n, such that k has units M^{1 - n} / s. In that way, the units of the rate constant, k, are chosen to make the units of rate always M / s regardless of order. For example, if the order of the reaction is 3, the rate law could be rate = k [A]²[B]. The units of rate are M / s and the rate constant must, therefore, have units of M^-2 / s.

The attentive reader may have noticed that we have only considered the rate of the forward reaction, neglecting any sort of reverse reaction in the rate law. To make our math easier, we have intentionally ignored the reverse reaction and we will continue to do so. This is a justified practice for reactions with negligible reverse rates, such as those with equilibrium constants, K, much greater than 1. For reactions with K's around 1, this is also a valid approximation because they are usually carried out such that products are removed from the reaction mixture as they are formed, keeping concentrations of products low.

Rate laws can also be expressed to relate the concentration of reactants to the time of the reaction. Such an expression is called an integrated rate law because it is the integral of the differential rate law. For those of you who have had some calculus, we present a derivation of an integrated rate law from the differential rate law below for a first order reaction. If you have not had calculus, don't worry. Just pay attention to the end result of the derivation.

Rearranging this equation gives the following:

Using a similar technique, one can derive the integrated rate law for any differential rate law. As we will see in Determining Rate Laws, the fact that the 1st order integrated rate law is linear allows us to test if a reaction is first order by plotting ln [A] versus time.