Introduction to Rate Laws

A rate law is a means by which we can relate the rate of a chemical reaction to concentrations of the reactants.  The rate law for a reaction is dependent on the specifics of how a reaction proceeds called the mechanism (what bonds break first, what bonds form first, any intermediate chemical species).  By determining the rate law for a reaction we gain insight into potential mechanisms.  This allows us to connect the macroscopic rate we observe in the lab with the microscopic or molecular ideas of the what is controlling the rate.

Overview of Chemical Kinetics

Reaction Rates

The rate of a chemical reaction is a measure of how fast the reaction is proceeding.  Specifically it is a measure of the change in the concentration of the chemical species as a function of time.

Because the concentrations of the reactants and products are related by the balanced chemical equation, we can generally discuss the "rate of the reaction" and then relate this rate to the changes for any given species issuing the stoichiometric coefficients from the balanced chemical equation.

For example, let's look at the reaction

\[\rm{N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)}\]

The rate for the consumption of N2 gas is related to the rate of formation of ammonia gas.  In particular, we would note that as the reaction proceeds as written, the number of moles of N2 and thus the concentration will be decreasing and the number of moles of NH3 will be increasing.  Since for each mole of N2 consumed 2 moles of NH3 are formed, the rate of formation of NH3 will be twice the rate of loss of N2.

We can write this formally as

\[{-\Delta [{\rm N_2}] \over \Delta t} = {+\Delta [{\rm NH_3}] \over 2\Delta t} ={-\Delta [{\rm H_2}] \over 3\Delta t} = rate\]

That is the change in the concentration of N2 for some period of time is equal to the change in concentration of NH3 for the same period of time divided by two (since this one is changing twice as fast).  To make reactant and products equal, we need to multiply the change of the reactants by a negative sign since they are decreasing while the products are increasing.

Rather than looking at finite changes in time, the rate is better related as a derivative such that

\[{-{\rm d}[{\rm N_2}] \over {\rm d}t} = {+{\rm d}[{\rm NH_3}] \over 2{\rm d}t} ={-{\rm d}[{\rm H_2}] \over 3{\rm d}t} = rate\]

Because the rate is defined as a change in concentration per time, it is generally given the units M s-1 (Molarity per second).

While it is generally possible to relate the changes in concentration of the reactants and products, it is important to note that this does not always work.  In some reactions there are several steps that occur along the reaction.  Therefore, sometimes the reactants are converted to an intermediate chemical species before forming products.  In these cases the rates cannot be directly tied to each other.

Factors Affecting Rates

The factors affecting the rates of a chemical reaction can be roughly broken down into four categories.

  1. Medium or Nature of the Reactants   The physical state of the reactants has an affect on the rate of reaction.   Since the rate depends on the details of the reaction mechanism, the physical state of the reactants is important.  It is fairly straight forward to see this for reactions involving solids.  Small iron particles will react with oxygen to form iron oxide (rust) faster than large particles since the iron must come into contact with the oxygen to react. The "inside" of the iron particle is much slower to react than the surface since it can take a long time for the oxygen to penetrate. Another way of stating this is that when the surface area of a given amount of substance is increased, the reaction rate will increase. All of these arguments are really just stating that when more material is immediately available (not "buried" in the bulk), then the reaction will go faster.
  2. Concentration  This is one of the factors that we spend a great deal of time on, since the way the concentration affects the rate tells us a lot about how the reaction is actually happening (the mechanism).  The relationship between the rate and concentration is called the rate law and needs to be experimentally measured since it cannot be determined by simply looking at the balanced reaction. Concentration is synonymous with pressure when you have gas reactants.
  3. Temperature  Temperature is another key factor in determining rate since typically there is an energy barrier between the reactants and the products.  That means that before you can form any new bonds of the product molecules, you have to break bonds in the reactants or form some intermediate compounds that are usually higher in energy.  Only molecules with sufficient energy will react.  The higher the temperature, the more molecules with sufficient energy. Thus all elementary reactions have a faster rate at higher temperatures.
  4. A catalyst  A catalyst is a compound that essentially changes the reaction mechanism for a reaction allowing it to proceed at a faster rate.  A catalyst is part of the reaction mechanism but it is not consumed during the chemical reaction and thus does not appear as a reactant or a product.  The typical role of a catalyst is to lower the energy barrier between reactants and products (activation energy) allowing the reaction to proceed at a faster rate.
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    Empirical Rate Laws

    Rate laws for an overall chemical reaction cannot be deduced from the written reaction but instead must be determined from experiments (thus the name "empirical" meaning derived from experience of observation).

    A rate law relates the rate of a reaction to the concentration of the reactants.

    For example the reaction

    \[\rm{NO_2(g) + CO(g) \rightarrow NO(g) + CO_2(g)}\]

    has been found to have the following rate law

    rate = k[NO2]2

    That is the rate of the reaction is proportional to the square of the concentration of NO2 gas.  The proportionality constant, k, is called the rate constant.  We would say this reaction is 2nd order in NO2 since it depends on the concentration of NO2 to the 2nd power.  We might also say this reaction is zeroth order in CO since it depends on the concentration of CO raised to the power zero (which is another way to say it doesn't depend at all on the concentration of CO).  The overall order of this reaction is 2nd order which is the sum of all the orders of the reactants (2 + 0 =2).

    Rate laws will always have the same form.  The rate will be equal to the rate constant times the concentrations of the reactants raised to some power.  The units of the rate constant will depend on the overall order of the reaction.  The rate always has units of M s-1, so the rate constant will have whatever units are needed to make this happen.

    For example, if a reaction is overall first order, rate = k times a concentration, the rate constant will have units of s-1.  If it is overall second order, rate = k times concentrations2, then the rate constant units are M-1 s-1.

    Generally reactions are either zeroth order, first order, or second order, but a wide array of possibilities exist for higher order reactions or reactions that are fractional orders.

    Determining Rate Laws

    You can't simply look at an overall reaction and know the associated rate law.  Instead you must measure it in the lab.

    The easiest way to do this is to run a series of experiments with different initial conditions.  Since the rate of a reaction can vary with time, we compare the very initial rate of the reaction.  This also avoids any complications with changes in rate due to backwards reactions.

    By changing the initial concentrations, we can see what concentrations affect the rate and in what way.  For example for the reaction

    \[\rm{NO_2(g) + CO(g) \rightarrow NO(g) + CO_2(g)}\]

    we might run three different experiments with different concentrations of the reactant gases.

    Experiment Initial NO2 conc. (M) Initial CO conc. (M) Initial Rate (M s-1)
    1 5.0 x 10-4 1.6 x 10-2 2.8 x 10-9
    2 5.0 x 10-4 3.2 x 10-2 2.8 x 10-9
    3 1.5 x 10-3 3.2 x 10-2 2.5 x 10-8

    The concentration dependence of the rate can be determined by comparing the different experiments.  For example the difference between experiments #1 and #2 is that the CO concentration has been doubled in the 2nd experiment.  Looking at the initial rate measured, it is clear that it is the same in both experiments.  Thus the rate is independent of CO concentration.  We would say the rate is zeroth order in CO.  The NO2 dependence can be determined by comparing experiments #2 and #3.  Here the NO2 concentration has been increased by a factor of 3 in the third experiment compared to the second.  This increases the initial rate.  If we compare the initial rate from experiment 3 to experiment 2, we see that the rate has increased by a factor of nine.  Since we increase the concentration by 3 and the rate increase by 9 we know the rate is dependent on the NO2 concentration squared.  Or we would say the reaction is second order in NO2.  Finally, we could also say this rate is second order overall.

    We could also determine a value for the rate constant since we now know the rate law

    \[\rm{rate = k[NO_2]^2}\]

    We know the initial rate and the concentration of NO2, so we could choose the data points from any of our experiments and plug them in to solve for k.  Using the data from the first experiment, we see that

    \[k = {2.8 x 10^{-9} M s^{-1} \over (5.0 x 10^{-4} M)^2} = 0.011 M^{-1} s^{-1}\]

    Knowing this reaction is second order in NO2 gives us some insight into the mechanism. Specifically we know that it must involve some bimolecular step that involves a collision between two NO2 molecules.  We care about rate laws because they give us insight into the mechanism of the reaction.


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