#### Learning Objective

- Calculate the half-life of a radioactive element

#### Key Points

- The relationship between time, half-life, and the amount of radionuclide is defined by: [latex]N={N}_{0}{e}^{-\lambda t}[/latex] .
- The relationship between the half-life and the decay constant shows that highly radioactive substances rapidly transform to daughter nuclides, while those that radiate weakly take longer to transform.
- Since the probability of a decay event is constant, scientists can describe the decay process as a constant time period.

#### Term

- half-lifeThe time required for half of the nuclei in a sample of a specific isotope to undergo radioactive decay.

## Decay Rates

Radioactive decay is a random process at the single-atom level; is impossible to predict exactly when a particular atom will decay. However, the chance that a given atom will decay is constant over time. For a large number of atoms, the decay rate for the collection as a whole can be computed from the measured decay constants of the nuclides, or, equivalently, from the half-lives.

Given a sample of a particular radionuclide, the half-life is the time taken for half of its atoms to decay. The following equation is used to predict the number of atoms (N) of a a given radioactive sample that remain after a given time (t):

[latex]N={N}_{0}{e}^{-\lambda t}[/latex]

In this equation, λ, pronounced “lambda,” is the decay constant, which is the inverse of the mean lifetime, and N_{0} is the value of N at t=0. The equation indicates that the decay constant λ has units of t^{-1}_{.}

The half-life is related to the decay constant. If you set N = [latex]\frac{N_0}{2}[/latex] and t = t_{1/2}, you obtain the following:

[latex]{t}_{1/2}=\frac{ln2}{\lambda}[/latex]

This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer. Half-lives vary widely; the half-life of ^{209}Bi is 1019 years, while unstable nuclides can have half-lives that have been measured as short as 10^{−23} seconds.

## Example

What is the half-life of element X if it takes 36 days to decay from 50 grams to 12.5 grams?

50 grams to 25 grams is one half-life.

25 grams to 12.5 grams is another half-life.

So, for 50 grams to decay to 12.5 grams, two half-lives, which would take 36 days total, would need to pass. This means each half-life for element X is 18 days.