How fast is Fast? The Mathematics of Change

Consider a reaction of Red molecules (A) to make Blue molecules (B), i.e. A -> B. If we were able to see the reaction on a molecular scale, the reaction of each individual molecule of occurs very rapidly, but the overall color of the vessel changes more slowly. Snapshots of the reaction in progress might look like this:

The number of reactants and products in the reaction vessel changes with time, with the relative number of reactant molecules destroyed and number of products formed per reaction event determined by the reaction stoichiometry. Each reactant molecule is identical to every other one, but they all don't react at the same instant. At each point in time, the probability of reaction per unit time is the same for each molecule in the sample, and that probability influences the overall reaction rate. But that isn't the only thing that determines the overall reaction rate. The total number of reactions at any instant is the probability of reaction per unit time multiplied by the number of reactants remaining in the vessel. Thus the reaction proceeds quickly at first, when there are lots of reactants around, and appears to slow as the reactants are consumed. A simulation of a similar reaction involving reactive collisions between molecules can be run on you browser. A plot of the time dependence of the number of molecules of each type looks 'smooth' when there are lots and lots of molecules in the sample, so individual reaction events get 'averaged' out. The concentrations of the reactants and products change in time like this:

First Order Rate Law
The simplist reaction mechanism is that of unimolecular decomposition (or isomerisation). In such a process, a single reactant undergoes a transformation at a constant probability per unit time. Such a mechanism leasds to a first-order reaction rate law. Examples of reactions such as these are radioactive decay, bacterial growth, and compound interest. Let's assume the reaction has a simple stoichimetry:

A First-Order Rate Law is called such because the rate of product formation ( or reactant depletion ) is proportional to the first power of the number of available reactants (or reactant concentration):

where [A] represents the concentration (number density) of species A in the sample.

Second Order Rate Law
If two molecules undergo a bimolecular reaction such as a reaction that involves a collisional encounter to produce products, and has a stoichiometry like this:

we expect a reaction rate law that is Second Order in the concentration of [A] (Collision probability to the number of each of the collision partners in the collision volume and hence proportional to square of the density)

Zeroth Order Rate Law
If a reaction is catalysed by a surface and has enough (excess) reactant, the rate of the reaction depends on the area of the catalyst, not on how much reactant is present. This is an unusual circumstance outside of the realm of catalyzed reactions and is described by a Zeroth Order rate law:

Remember that the rate of a reaction is a time derivative of concentration, so the rate law is a differential equation.

As an example, let's consider a first order reaction:

The rate constant, k, represents somehow the probability that a given reaction event will occur per unit time. Note the rate constants, k, have units, and the units are different for different forms of the rate law

As an equation that relates time and concentration, separation of those variables to different sides of the equation may be possible, as for a first order reaction:

This equation may be integrated from the initial conditions: