# Half-Life Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding and Jack Bowater
Last updated: Jan 11, 2023

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 stable and unstable nuclei. Stable nuclei don't change, but unstable nuclei undergo a type of radioactive decay, emitting alpha particles, beta particles, or gamma rays and eventually decaying into 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 a 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

We can determine the number of unstable nuclei remaining after time $t$ using this equation:

$N(t) = N(0)\times0.5^{(t/T)},$

Where:

• $N(t)$ – Remaining quantity of a substance after time $t$ has elapsed;
• $N(0)$ – Initial quantity of this substance; and
• $T$ – half-life.

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

$N(t) = N(0)\times e^{(-t/\tau)}\\[1.0em] N(t) = N(0)\times e^{(-\lambda t)},$

Where:

• $\tau$ – mean lifetime - the average amount of time a nucleus remains intact; and
• $\lambda$ – decay constant (rate of decay).

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

$T = \frac{\ln(2)}{\lambda} = \ln(2)\times \tau$

## How to calculate the half-life

1. Determine the initial amount of a substance. For example, $\small N(0) = 2.5\ \text{kg}$.
2. Determine the final amount of a substance - for instance, $\small N(t) = 2.1\ \text{kg}$.
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 $\small 19.88\ \text{minutes}$.
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.

Half-life is a similar concept to doubling time in biology. Check our generation time calculator to learn how exponential growth is both useful and a problem in laboratories! Also, we use a similar concept in pharmacology, and we call it the "drug half-life". Find out more about that in our drug half-life calculator.

## FAQ

### What is half life?

Half-life is defined as the time taken by a substance to lose half of its quantity.

### How to calculate half life?

To find half-life:

1. Find the substance's decay constant.
2. Divide the natural logarithm of 2 or ln(2) by the decay constant of the substance.

### What is the half life of radium?

The half-life of radium-218 is 25.2 x 10-6 seconds.

### What is the half life of carbon?

The half-life of carbon-20 is 16 x 10-3 seconds.

### What is the half life of uranium?

The half-life of uranium-235 is 22.21 x 1015 seconds.

Bogna Szyk
Initial quantity (N(0))
Half-life time (T)
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Total time
sec
Remaining quantity (N(t))
Decay constant (λ)
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