Half-Life Calculator

Created by Davide Borchia
Last updated: Nov 30, 2022

Whether you are considering radioactive atoms or a population of bacteria, the underlying mathematics is the same: our half-life calculator will teach you how to compute the most important quantity of the decay process, the half-life.

Here you will learn:

  • What is the half-life in exponential decays;
  • How to calculate the exponential decay using the half-life;
  • How to calculate the half-life;
  • Examples of half-life in physics and biology.

Decay processes

A decay process is any process that sees a decrease in the measured quantity. Decay processes are usually divided into two categories:

  • Exponential decays; and
  • Non-exponential decays, where we can identify subtypes like:
    • Inverse square decay; or
    • Zipf's law decay

What is the half-life?

In any decay process, we can pinpoint a moment in which the quantity is half the original amount. We call the time elapsed from the starting moment to this point the half-life of the quantity.

We can define the half-life also in terms of probability. This approach best suits discrete quantities or unitary ones. In this case, the half-life is the time after which there is a 50%50\% chance of decay.

In physics, the half-life usually describes stochastic processes as radioactive decay, where an unstable atom emits or absorbs a particle to change species. This process is entirely random, and an atom can go eons without decaying. On the contrary, in biology, a bacteria in a decaying population will die after a given time: no immortal bacteria out there.

How do I calculate the half-life?

Let's learn how to calculate the half-life of an exponential decay (a mathematical model based on the exponential growth).

We define first the law of exponential decay:

N(t)=N(0)etτN(t) = N(0)\cdot e^{-\frac{t}{\tau}}

Where:

  • N(t)N(t) is the quantity at the time tt;
  • N(0)N(0) is the quantity at the initial reference time;
  • tt is the elapsed time; and
  • τ\tau is the average lifetime of each component of the measured quantity.

🔎 τ\tau, the average lifetime, is often expressed as its inverse, the decay constant λ\lambda: λ=1/τ\lambda = 1/\tau.

To find the half-life we can rearrange this expression, knowing that, when the half-life time is reached:

N(t)=0.5N(0)N(t)=0.5\cdot N(0)

We can define the half-life time t0.5t_{0.5} as:

0.5N(0)=N(0)et0.5τ0.5\cdot N(0) = N(0)\cdot e^{-\frac{t_{0.5}}{\tau}}

Which can be easily rewritten to isolate t0.5t_{0.5} using the natural logarithm:

t0.5=ln(2)τt_{0.5} = \ln{(2)}\cdot \tau

Using this relation, we can write the half-life equation using the factor 0.50.5 characteristic for the halving of the quantity:

N(t)=N(0)12tt0.5N(t)= N(0)\cdot\frac{1}{2}^{\frac{t}{t_{0.5}}}

Our half-life calculator works in both directions: you can calculate the half-life of a decay process if you know the initial and final quantities, and the elapsed time, or you can calculate the final (or initial) quantities if you know the half-life.

Examples of how to calculate the half-life

We will calculate the half-life in two situations: radioactive decay and the decline of a bacterial population.

Take a radioactive atom, let's say promethium (were you expecting uranium?). Promethium is the lightest natural radioactive element. Its most common (and stable) isotope, 145Pm^{145}\text{Pm}, has half-life t0.5=17.7 yt_{0.5}=17.7\ \text{y}. Scientists estimate that on Earth, there are about 500 g500\ \text{g} of promethium at a given time. Assuming that no more atoms of this element would form, how much promethium would we have after 100 y100\ \text{y}? Input the known data in the half-life equation:

N(100 y ⁣) ⁣= ⁣500 g12100 y17.7 y ⁣= ⁣9.96 gN(100\ \text{y}\!)\! = \!500\ \text{g}\cdot\frac{1}{2}^{\frac{100\ \text{y}}{17.7\ \text{y}}}\!=\!9.96\ \text{g}

Let's try the other way around. What's the half-life of a bacteria population that starts with 10510^5 individuals, and after a day ends with 51035\cdot 10^3 individuals. Input the data in our half-time calculator:

t0.5=ln(2)tln(N(0)N(t))= ⁣ln(2)1440 minln(1065103) ⁣ ⁣52 min\begin{align*} t_{0.5}& = \frac{\ln{(2)}\cdot t}{\ln{\left(\frac{N(0)}{N(t)}\right)}}\\ \\ & =\! \frac{\ln{(2)}\cdot 1440\ \text{min}}{\ln{\left(\frac{10^6}{5\cdot 10^3}\right)}}\!\simeq\!52\ \text{min}\\ \end{align*}

Now, if you knew the energy released after 52 min52 \ \mathrm{min}, you could calculate the power to estimate how many watts such a decay generates. Then, Omni Calculator's power converter might be the next step to express power in more adequate units. That would be very interesting to check!

Davide Borchia
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)
/
per s
Mean lifetime (τ)
sec
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