# How can I compute the half-life of a first order reaction?

I am confused about integrated rate reactions and half life problems. For example: The first order rate constant for the radioactive decay of radium-223 is 0.0606/day. what is the half life of radium-223?
Tara 02/05/99

 Vocabulary half life first order reaction integrated rate law rate constant
The rate law for this first order rate equation is

where [A] is the concentration of radium-223 at time t and k is the rate constant (0.0606 day-1). You want to calculate the time it takes for the concentration to fall from [A]0 to ½[A]0. The rate law will have to be integrated.

Rearrange the equation so that all concentration dependent terms are collected on one side, and all time dependent terms are on the other:

Integrating from time 0 to time t,

the first order integrated rate equation is

The half life is the time required for the concentration to drop to one-half its original value. We can write

and solving for the half life,

Author: Fred Senese senese@antoine.frostburg.edu

General Chemistry Online! How can I compute the half-life of a first order reaction?