Rate Laws

We can calculate the instantaneous rate for any point in a concentrations vs. time experiment, but the only really unique and important one is the initial instantaneous rate.  The initial rate of a reaction is described by a rate law.  The initial rate of a chemical reaction is proportional to the initial concentration of all the reactants raised to some power.  To remove the proportionality, we can add a constant, the rate law constant.

Rate₀ = k [reactant]₀^x

We can write a rate law expression for any chemical reaction as long as we know the reactants.  For example, for the generic reaction:

aA + bB → cC + dD

The rate law expression is

Rate₀ = k[A]₀^x[B]₀^y

If we think about this rate law expression, there seem to be a LOT of variables present.  We can determine the value of a number of those variables if we design our experiments thoughtfully.  Let's look at a specific example.
For the reaction of NO₂(g) with Cl₂(g), we have performed the following experiment: [NO₂]₀ = 1.228M, [Cl₂]₀ = 1.316M, Rate₀ = 2.881x10^{-3 M}/_min.  The rate law expression for this reaction is:

Rate₀ = k [NO₂]₀^x[Cl₂]₀^y

Notice that we don't really need to know the products or the balanced chemical equation to write out the rate law expression.  That doesn't mean we don't have to practice balancing chemical equation, keep on practicing!!  Plugging the numbers in to the rate law expression:

(2.881x10^{-3 M}/_min) = k (1.228M)^x(1.316M)^y

That's still 3 variables, so we need (mathematically) more equations to help us solve them.  We could just randomly start mixing reactants together, but if we're deliberate in our planning, we can make our jobs a little easier.  As good scientists, we try to change one 1 variable at a time whenever we're doing an experiment.  Why?  Because then if the result changes, we know it has to be caused by the variable we changed.  In this case, we can set up another experiment, and let's change the initial concentration of NO₂ but hold the initial concentration of Cl₂ constant.  For our second run: [NO₂]₀ = 2.456M, [Cl₂]₀ = 1.316M, Rate₀ = 1.152x10^{-2 M}/_min.  Plugging in to the rate law expression again, we get:

(1.152x10^{-2 M}/_min) = k (2.456M)^x(1.316M)^y

Let's get a little mathematical here... if these two equalities are valid (which they are), then their ratio is also a valid equality.  Bam.  Cancelling out everything that can cancel, we're left with the simplified expression:

(1/4) = (1/2)^x

x = 2

Therefore, the reaction is second order with respect to [NO₂]₀.