Rate law

A rate law is an equation that relates concentrations of reactants to the reaction rate. For the reaction

<math>aA + bB \rightarrow \; cC + dD<math>

the rate law is

<math>v = k[A]^m[B]^n<math>

where <math>k<math> is the reaction rate constant, and the exponents are reaction orders. The reaction is of order <math>m<math> in <math>A<math> and of order <math>n<math> in <math>B<math>. The overall reaction order is <math>m+n<math>.

Usually, reaction orders are 0, 1, or 2, but they can be fractions or even negative numbers.

First order reactions

A first-order reaction depends on the concentration of only one reactant (a unimolecular reaction). Other reactants can be present, but each will be zero order. The rate law for a first order reaction is

<math> v = -\frac{d[A]}{dt} = k[A]<math>.

When integrating this differential equation, the resulting equation is

<math>\ln[A]_t = -kt + \ln[A]_0<math>

where <math>[A]_t<math> represents the concentration at a particular time and <math>[A]_0<math> represents the initial concentration. A plot of time t vs. -Ln[A] gives a straight line with a slope equal to the reaction rate constant. The half-life of a reaction describes the time needed for half of the reactant to be depleted. (do not confuse this with the half-life involved in nuclear decay.) It can be determined using the equation <math>t_\begin{matrix} \frac{1}{2} \end{matrix} = \begin{matrix} \frac{ln2}{k} \end{matrix}<math>.

Second order reactions

A second-order reaction or bimolecular reaction depends on the concentrations of one second order reactant, or 2 first order reactants.

<math>R = k[A]^2<math> or <math>R=k[A][B]<math>.

When relating these rate laws with time, the results are

<math>\frac{1}{[A]_t} = kt + \frac{1}{[A]_0} <math> or <math> \frac{1} {B_0 – A_0} \ln \frac{A_0 B_t}{B_0 A_t} = kt<math>.

The half life equation for a second order reaction is <math>t_\begin{matrix} \frac{1}{2} \end{matrix} = \frac{1}{k[A]_0}<math>