D34.6 Half-Life of a Reaction

The half-life (t½) of a reaction is the time required for the concentration of a reactant to be reduced to half of its initial value. In each succeeding half-life, the remaining concentration of the reactant is again halved. The half-life of a reaction can be derived from the integrated rate law. Hence, there is a general equation for half-life for zeroth-order, first-order, and second-order reaction.

First-Order Reaction

The integrated rate law gives:

When t = t½:

[A]t½ = ½[A]0

Therefore:

The half-life of a first-order reaction is inversely proportional to the rate constant k: a larger k (a faster reaction) has a shorter half-life; a smaller k (a slower reaction) has a longer half-life. Moreover, the half-life is conveniently independent of the concentration of the reactant. Therefore, you do not need to know the initial concentration to calculate the rate constant from the half-life, or vice versa.

Exercise: Half-Life, Rate, and Concentration

Second-Order Reactions

The integrated rate law is:

When t = t½, [A]t½ = ½[A]0, therefore:

and:

For a second-order reaction, t½ is inversely proportional to the rate constant and the concentration of the reactant. Therefore, t½ is not constant throughout the reaction. Each successive half-life increases as the reaction proceeds due to decreasing concentration of reactant. Consequently, unlike the situation with first-order reactions, the rate constant of a second-order reaction cannot be calculated directly from the half-life unless the initial concentration relating to that half-life is known.

Zeroth-Order Reactions

For a zeroth-order reaction:

When t = t½, [A]t½ = ½[A]0, therefore:

The half-life of a zeroth-order reaction is inversely proportional to the rate constant and directly proportional to the concentration of the reactant. Therefore, each successive t½ decreases as the reaction progresses and the reactant concentration decreases.

Activity: Half-life and Order from Concentration-Time Data