# Half-Life Calculator

Calculate half-life, elapsed time, initial, or remaining quantity using this easy calculator. Read below to learn the half-life formula and how to solve it.

## Half-Life:

### Decay Constant

### Mean Lifetime

### Decay Constant

### Mean Lifetime

## What is Half-Life?

*Half-life* is the time needed to reduce the quantity of a substance to half its initial amount. It’s is commonly used to define the decay of radioactive material, where half-life is the amount of time needed for half the atoms of a particular isotope to decay.^{[1]}

In the medical world, half-life is also used to describe the time it takes for the concentration of a substance to be reduced to half of the initial dose in your body.^{[2]} This is known as *elimination half-life*.

## How to Calculate Half-Life

You can calculate the amount of substance that will remain after an elapsed time period, given the half-life of the substance and the equation for half-life.

### Half-Life Formula

The half-life formula states:

N(t) = N_{0} × (12)^{t/t½}

Thus, the remaining quantity *N(t)* is equal to the initial quantity *N _{0}* times 1/2 to the power of the elapsed time

*t*divided by the half-life

*t*

_{½}## How to Calculate Time using Radioactive Decay

Using a technique known as *carbon dating*, it’s possible to estimate the age of an organic compound. This is done by measuring the amount of carbon-14 present in a sample of organic (or previously organic) material and comparing it to the amount that should be present.

Because the half-life of carbon-14 is known to be 5,730 years, we can use the half-life formula to calculate the elapsed time, or age of the material.

### Half-Life Formula for Time

By rearranging the half-life formula, we can solve for the amount of time elapsed for an initial quantity to reduce to the remaining quantity.

t = t_{½} × ln(N(t) ÷ N_{0})-ln(2)

The time elapsed *t* for a substance to be reduced is equal to the half-life *t _{½}* times the natural log of the remaining quantity

*N(t)*divided by the initial quantity

*N*, divided by the negative natural log of 2.

_{0}
**For example,** using the formula above let’s use the carbon dating methodology to calculate the age of a sample of organic material that had 15% of the amount of carbon-14 that it should contain if the sample were currently alive. Recall that the half-life of carbon-14 is 5,730 years.

t = 5,730 × ln(15 ÷ 100)-ln(2)

t = 5,730 × ln(0.15)-ln(2)

t = 5,730 × -1.897-0.693

t = 5,730 × -1.897-0.693

t = -10,870.498-0.693

t = 15,682.81 years

So, using the half-life formula, we can deduce that this sample is roughly 15,682.81 years old.

## Half-Life Decay Table

Number of Half-Lives | Remaining Quantity (fraction) | Remaining Quantity (decimal) |
---|---|---|

0 | 1 | 1.0 |

1 | 1/2 | 0.5 |

2 | 1/4 | 0.25 |

3 | 1/8 | 0.125 |

4 | 1/16 | 0.0625 |

5 | 1/32 | 0.03125 |

6 | 1/64 | 0.015625 |

7 | 1/128 | 0.0078125 |

8 | 1/256 | 0.00390625 |

9 | 1/512 | 0.001953125 |

10 | 1/1,028 | 0.000972762645914 |

## Recommended

## References

- United States Nuclear Regulatory Committee, Half-life (radiological), https://www.nrc.gov/reading-rm/basic-ref/glossary/half-life-radiological.html
- Hallare J, Gerriets V., Half Life,
*StatPearls*, 2020 Oct 6, https://www.ncbi.nlm.nih.gov/books/NBK554498/