Formula for half-life, given the decay constant and the mean lifetime
Initial quantity (N(0))
Half-life time (T)
sec
Total time
sec
Remaining quantity (N(t))
Decay constant (λ)
/
sec
Mean lifetime (τ)
sec

Half-Life Calculator

By Bogna Haponiuk

The half-life calculator is a tool that helps you understand the principles of radioactive decay. You can use it to not only learn how to calculate half-life, but also as a way of finding the initial and final quantity of a substance or its decay constant. This article will also present you with the half-life definition and the most common half-life formula.

Half-life definition

Each radioactive material contains a stable and an unstable nuclei. Stable nuclei don't change, but unstable nuclei undergo radioactive decay, emitting alpha particles, beta particles or gamma rays and eventually decaying into a stable nuclei. Half-life is defined as the time required for half of the unstable nuclei to undergo their decay process.

Each substance has a different half-life. For example, carbon-10 has a half-life of only 19 seconds, making it impossible for this isotope to be encountered in nature. Uranium-233, on the other hand, has the half-life of about 160 000 years.

This term can also be used more generally to describe any kind of exponential decay - for example, the biological half-life of metabolites.

Half-life is a probabilistic measure - it doesn't mean that exactly half of the substance will have decayed after the time of the half-life has elapsed. Nevertheless, it is an approximation that gets very accurate when a sufficient number of nuclei are present.

Half-life formula

The number of unstable nuclei remaining after time t can be determined according to this equation:

N(t) = N(0) * 0.5(t/T)

where:

  • N(t) is the remaining quantity of a substance after time t has elapsed.
  • N(0) is the initial quantity of this substance.
  • T is the half-life.

It is also possible to determine the remaining quantity of a substance using a few other parameters:

N(t) = N(0) * e(-t/τ)

N(t) = N(0) * e(-λt)

  • τ is the mean lifetime - the average amount of time a nucleus remains intact.
  • λ is the decay constant (rate of decay).

All three of the parameters characterizing a substance's radioactivity are related in the following way:

T = ln(2)/λ = ln(2)*τ

How to calculate the half-life

  1. Determine the initial amount of a substance. For example, N(0) = 2.5 kg.
  2. Determine the final amount of a substance - for instance, N(t) = 2.1 kg.
  3. Measure how long it took for that amount of material to decay. In our experiment, we observed that it took 5 minutes.
  4. Input these values into our half-life calculator. It will compute a result for you instantaneously - in this case, the half-life is equal to 19.88 minutes.
  5. If you are not certain that our calculator returned the correct result, you can always check it using the half-life formula.

Confused by exponential formulas? Try our exponent calculator.

Bogna Haponiuk