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.

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Half-Life:

  seconds

Decay Constant

  seconds

Mean Lifetime

  seconds


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) = N0 × (12)t/t½

Thus, the remaining quantity N(t) is equal to the initial quantity N0 times 1/2 to the power of the elapsed time t divided by the half-life t½

graphic showing the half-life formula

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 = t½ × ln(N(t) ÷ N0)-ln(2)

The time elapsed t for a substance to be reduced is equal to the half-life t½ times the natural log of the remaining quantity N(t) divided by the initial quantity N0, divided by the negative natural log of 2.

half-life formula to solve for elapsed time

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

Table illustrating exponential decay showing the remaining quantity after elapsed half-life time periods in fraction and decimal form.
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

References

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