Half-Life Calculator

Half-life Explanation

The concept of half-life is best understood in the context of radioactivity in general. A substance is radioactive when the atoms that make up the substance have instable nuclei. A nucleus of an atom is made up of protons and neutrons, held together by the strong nuclear force. A nucleus is stable if the strong nuclear force and the electric repulsion between positively charged protons are in balance. When the balance is disrupted, the nucleus becomes unstable and wants to return to a (more) stable state. It does so by releasing energy from the nucleus, which could be in the form of, for example, Helium particles in alpha radiation and gamma ray photons in the case of gamma radiation. This process is called radioactive decay and changes the substance into either a different substance or a different energy state.

Half-life is the the time it takes for some substance to naturally reduce by half. In the context of radioactivity, this means the time it takes for half of a given amount of a radioactive substance to decay.

It is good to note that the exact moment some atom will decay and release energy is a probabilistic event. Thus, the time it takes for a specific amount of a radioactive substance to decay is completely random. In this light, a substance' half-life should be considered an average. A useful parameter is the decay constant (λ) which denotes how likely a radioactive nucleus of a substance is to decay per some unit of time, usually seconds. A higher λ means a faster decay. λ is a "constant" in the sense that each radioactive substance has its own fixed decay rate.

Formulas

A common formula you will see to note the relationship between substance amount and half life is the following:

• N(t) is the amount of the substance remaining
• N₀ is the initial amount of the substance
• t is the elapsed time
• t_1/2 is the half-life

Sometimes you will see this formula:

Where λ is the decay constant.

λ and half-life are mathematically related:

Example

A sealed sample of Radon-222 was placed in a storage container and left alone. Radon-222 has a half-life of 3.8 days. When the sample was first sealed, it contained 80 mg of Radon-222. When a technician returned to check on it, only 10 mg remained. How long has the sample been sitting in storage?

• N(t) = 10 mg
• N₀ = 80 mg
• t_1/2 = 3.8 days

Solving equation 1 for t gives:

Fill in the known values:

t = 11.4 days.