Reaction Rate Laws

The goal of a kinetics experiment is to measure the concentration of a species at a particular time during a reaction so that a rate law can be determined. However, it is exceedingly difficult to get an accurate measurement of a concentration at a known time because the techniques used to measure concentrations don't work instantaneously, but rather take time to perform. One of the earliest methods used to measure concentrations at specified times is to quench the reaction either by flash freezing it or by adding a substance that severely inhibits the reaction. Both of these techniques are problematic because one can't be sure that the reaction has completely stopped. The reaction may still be going on during the analysis. Additionally, the reaction mixture is destroyed for the purposes of kinetic experiments, so the chemist must make multiple trial runs and waste a large amount of reagents to observe the concentrations at multiple points in time.

A more modern technique to measure concentration is absorbance spectroscopy. This experiment may be used when a product or reactant has an absorbance frequency unique to those of other components of the reaction mixture. By measuring the absorbance of a particular product or reactant at a variety of known concentrations, you can construct a plot of absorbance versus concentration called a Beer's Law plot. This calibration chart allows you to calculate the unknown concentration given the reaction solution's absorbance. The advantage of this method is that a large number of data points with well known times can be quickly collected using only one reaction mixture.

When looking at the expression for the , you should notice that the variables in the equation are the concentration terms and the powers p and q:

Because we can measure the concentrations in the rate law using the techniques described above, the unknowns we wish to measure are k, p, and q. One method of directly measuring k, p, and q is called the method of initial rates. By measuring the initial rate (the rate near reaction time zero) for a series of reactions with varying concentrations, we can deduce to what power the rate depends on the concentration of each reagent. For example, let's use the method of initial rates to determine the rate law for the following reaction:

whose rate law has the form:

Using the following initial rates data, it is possible to calculate the order of the reaction for both bromine and acetone:

To calculate the order of the reaction for bromine, notice that experiments 1 and 2 hold the concentration of acetone constant while doubling the concentration of bromine. The initial rate of the reaction is unaffected by the increase in bromine concentration, so the reaction is zero order in bromine. We can prove this mathematically by taking the ratio of the rates from experiments 1 and two:

As you can see in the above equations, by holding the concentrations of all but one species constant between two experiments, you can calculate the order of the reaction in a single reactant at a time. By similar reasoning, we can conclude that because the rate of reaction doubled when the concentration of acetone was doubled (cf. experiments 1 and 3) the reaction must be first order in acetone. However, had the rate quadrupled or octupled with a doubling of the acetone concentration, the reaction would have been second or third order in acetone, respectively. In practice, you will likely never see a reaction with an order higher than 3. If you calculate an order higher than 3 for a reaction, double check your math because that is highly unusual. If you compute a fractional power for a reactant's order, do not be discouraged; they are quite common (especially half-order reactions).

To calculate the value of k, the rate constant, you simply plug into the rate law the values of the concentrations, the orders, and the rate of the reaction from any one of the three experiments. All three experiments should give a value of 1.64 x 10^-4 s^-1. You should prove this to yourself.

One major disadvantage to using the method of initial rates is the need to perform multiple experiments. Another disadvantage is that it works only for relatively slow reactions. If your reaction proceeds too fast, the rate that you measure will have a large range of uncertainty. To combat these problems, chemists have developed a method that uses data on concentration versus time during a reaction to infer the order of a reaction. The strategy works by comparing concentration versus time information to the mathematical predictions made by integrated rate laws. This is usually accomplished by assuming that a reaction has a certain order and making a plot out of the data that should be linear if the assumption about the rate law is correct. If we guess correctly, the graph is linear. If we are wrong, then the graph is curved and we need to choose another order against which to plot our data.

To practice this method, we will first need to know the forms of the integrated rate laws for some common reaction orders. Luckily, only orders zero through two are common so we need to only consider reactions with these three orders. Due to the mathematical complexities of the problem, we will only consider rate laws of the form rate = [A]ⁿ. However, with some experimental tricks I will explain below, that treatment does allow us to use the method of integrated rates for a reaction with any arbitrary rate law.

By integrating the differential rate law (the simple rate law from Heading ), we derive the integrated rate law for a reaction. Without proof, we present below the forms of the integrated rate laws for reaction orders zero through two:

As you can see, each order of reaction has a unique input and output variable that produces a straight line. For example, if we graph the following rate data for the decomposition of H₂O₂ assuming that it could be zero, first, or second order, we find that only the graph for a 2nd order reaction (1/[A] versus t) gives a straight line. Therefore, the reaction has the rate law rate = k [H₂O₂]².

Assuming that we did not know that the decomposition is second order, we will make a series of three graphs to determine the order of the reaction:

If the graph was linear, we would conclude that the rate law is zero Order, but it is not. Therefore, let's see if it is first order by plotting ln [H₂O₂] versus time in the following graph:

Seeing that the reaction is not first order due to the non-linearity of the above graph, we move to plotting 1/[H₂O₂] versus time to test whether the decomposition reaction is second order.

Because the above graph is linear, we know that the reaction is second order. The slope of the line is twice the rate constant, k from the rate law.

What you should learn from the above discussion is that you can use integrated rate laws to determine both the rate constant for a reaction and the form of the rate law. It may seem more complicated to use integrated rate laws rather than the method of initial rates to determine the rate law, but it really does take much longer to perform and analyze the several reactions needed for the method of initial rates than it does to produce the necessary graphs for the integrated rate law method--especially with a good graphing program.

You may have noticed that we only list three integrated rate laws, ignoring rate laws like rate = k [A] [B]. That doesn't mean we can't use integrated rate laws to determine the rate law for those types of reactions. We simply must be more clever about how we do it. For a two- component second order reaction with a rate law rate = k [A] [B], we can make the concentration of B so large as compared to A that the concentration of B is almost constant. Assuming that the concentration of B is constant, the reaction becomes pseudo-first order--that is, the reaction will behave as if it was first order. The kinetics data for this reaction will give a graph of ln [A] versus time that is linear, telling us that the reaction is first order in A. If the reaction happened to be second order in A, in the present example, then a graph of 1/[A] versus time would be linear. We can, similarly, determine the order of B by making the concentration of A large. For multiple component rate laws, you can simply make the concentration of all reagents but the one of interest large to iteratively determine the order of the reaction in each component. (Chemists are fond of using the word 'iterative' to say 'repetitive'.)

An earlier and less common way to measure rate is by the half-life of a reaction. A half-life is the time it takes for one half of the starting material to be transformed into its products. Often you will hear half-life associated with radioactive decay phenomena (which follow first order kinetics), but the term can be applied to any reaction.

The half-life of a reaction not only depends on the rate constant of the reaction (those with larger k's have shorter half-lives) but also on the integrated rate law for the reaction. To derive the form of the half-life expression for a first order reaction, we start with its integrated rate law, and then substitute the value 0.5 for the ratio of [A] to [A]_o:

Using the same techniques with the different integrated rate laws, you can derive the half-life expression for a reaction of any order. Summarized below are the half-lives for reactions of orders zero through two.