[A]/t = k[A] How Concentration Varies With Time
A rate law (equation) tells us how the rate of a reaction varies with reactant concentration. But chemists also need an equation to show how reactant concentration changes with time. This equation would allow us to summarize experimental data, which is usually reactant concentrations versus time. We could also answer such questions as, how long will it take for this reaction to be 50% complete or what will be the reactant concentration after 5 hours.
Concentration-Time Equations ( Integrated Rate Law)
First-Order Rate Law
Consider a general reaction such as:
aA ==> products
Suppose that this reaction has a first-order rate law
Rate = -[A]/t = k[A]
Using calculus we can show that such a first-order rate law leads to the following relationship between [A], the concentration of A and time. This equation is sometimes referred to as the integrated rate law.
ln ([A]t / [A]0) = -kt
Where [A]t is the concentration of reactant A at time t, and [A]0 is the initial concentration of reactant A. The ratio [A]t / [A]0 is the fraction of reactant remaining at time t.
Second-Order Rate Law
Consider a general reaction such as:
aA ==> products
Suppose that this reaction has a second-order rate law
Rate = -[A]/t =
k[A]2
Again if we use calculus we can show that such a second-order rate law leads to the following relationship between [A], the concentration of A and time.
1/[A]t = kt + 1/[A]0
Where [A]t is the concentration of reactant A at time t, and [A]0 is the initial concentration of reactant A.
Half-Life of a Reaction
Another measure of the rate of a reaction, involving concentration and time, is the half-life, t1/2 which is defined as the time required for the concentration of a reactant to decrease to one half of its initial concentration. Without going through the derivation, which can be found in your textbook, the equation for the half-life of a first order reaction is.
t1/2 = 0.693/k
Notice that the half-life for a first-order reaction is independent of the initial concentration. So it would take the same time for the concentration of the reactant to decrease from 1.0 M to 0.50 M as it does for a decrease from 0.50 M to 0.25 M. Measuring the half-life of a reaction is another way to determine the rate constant of a first-order reaction.
For the second-order reaction order above we can derive an equation for the half-life which is.
t1/2 = 1/(k[A]0)
Note that the half-life of a second-order reaction is inversely proportional to the initial reactant concentration. Measuring the half-lives at different initial concentrations is one way to distinguish between a first-order and a second-order reaction.
Graphing of Rate Data
Graphing is another way of determining the order of a reaction. The experimental data are plotted in several different ways, first assuming a first-order reaction, then a second-order reaction. The order of the reaction is determined by which graph gives the best fit with the experimental data. For example if a plot of log [A]t versus time gives a straight line then the reaction is first order in A. On the other hand if a plot of 1/[A]t versus time gives a straight line then the reaction is second order in A.