## Core Concepts

In this article, you will learn what a half-life is, in 0th. 1st, and 2nd order reactions. You will also learn what the half-life formula is and how to use it to find the age of a substance.

## Topics Covered in Other Articles

- Marie Curie and Radioactivity
- Radioactive Decay
- Alpha, Beta, and Gamma Decay
- Nuclear Processes
- Reaction Order – First & Second Order

## What is a Half-Life?

A half-life (t_{1/2}) is the time it takes half the concentration of a sample to decay. For example, if the starting concentration of a sample is 1M, then the half-life will be the time it takes for the concentration of the sample to decay to 0.5M.

## First Order Half-Life

First-order reactions, including radioactive decay, are reactions where the rate of reaction is only dependent on the concentration of one reactant. First-order reactions have constant half-lives. We can prove this using the integrated first-order rate law:

[A] is the concentration of a reactant at time t, [A]₀ is the initial concentration of the reactant, and k is the rate constant. Here, we can move all the natural logs to the left side of the equation:

Since we are trying to find the half-life of the reactant, we can set t as the half-life and [A] as half of the initial concentration:

Substituting those values into the equation gives us the following:

Finally, dividing the k and the negative gives us an equation for the half-life of a first-order reaction:

## Second Order Half-Life

Second-order reactions are reactions where the rate of reaction is dependent on the concentrations of 2 reactants. The half-lives of second-order reactions depend on the reactant’s initial concentration. The integrated rate law for second-order reactions is as follows:

Since we are trying to find the half-life of the reactant, we can set t as the half-life and [A] as half of the initial concentration:

Substituting those values into the equation gives us the following:

We can then isolate the t to solve for the half-life of a second-order reaction:

## Zero Order Half-Life

In zero-order reactions, the rate of reaction does not depend on the reactant’s concentration. The half-lives of zero-order reactions, however, do depend on the initial concentration of the reactant. The integrated rate law for zero-order reactions is as follows:

Since we are trying to find the half-life of the reactant, we can set t as the half-life and [A] as half of the initial concentration:

Substituting those values into the equation gives us the following:

We can then isolate the t to solve for the half-life of a zero-order reaction:

## Practice Problems

- The half-life for a first-order reaction is 2768 years. Starting with a concentration of 0.345M, what will the concentration be after 11072 years?
- What is the half-life of a compound if 75 percent of a given sample of the compound decomposes in 60 min? Assume first-order kinetics.
- A substance hydrolyzes in water with a rate constant of 2.0 × 10
^{−3}min^{−1}. Calculate the t_{1/2}for the hydrolysis reaction. Assume first-order kinetics.

## Solutions

- 0.0216M. First, find out how many half-lives have passed. By dividing 11072 by 2768, we find that 4 half-lives have passed. Therefore, we can find the final concentration by dividing 0.345M by 2 four times. This leaves us with 0.0216M.
- 30 mins. 25 percent of the sample is left, meaning 2 half-lives have passed. 60 mins divided by 2 is 30 mins, which is the half-life.
- 350 mins. Using the equation for the half-life of a first-order reaction, we can replace k with 2.0 × 10
^{−3}min^{−1}.