Half-Life Calculator
The following tools can generate any one of the values from the other three in the half-life formula for a substance undergoing decay to decrease by half.
Half-Life Calculator
Please provide any three of the following to calculate the fourth value.
Half-Life, Mean Lifetime, and Decay Constant Conversion
Please provide any one of the following to get the other two.
Definition and Formula
Half-life is defined as the amount of time it takes a given quantity to decrease to half of its initial value. The term is most commonly used for atoms undergoing radioactive decay, but it describes any process of exponential decay. One of its best-known applications is carbon-14 dating. The half-life of carbon-14 is approximately 5,730 years, and it can be used reliably to date samples up to around 50,000 years old. The method, developed by Willard Libby, rests on the fact that carbon-14 is constantly produced in the atmosphere: it is incorporated into plants through photosynthesis and into animals when they eat the plants. Once the plant or animal dies, its carbon-14 decays away, so measuring how much remains in a sample reveals when it died.
Three equivalent formulas describe exponential decay:
N(t) = N0 × (1/2)t / t1/2 = N0 × e−t / τ = N0 × e−λt
- N0 is the initial quantity
- Nt is the quantity remaining after time t
- t1/2 is the half-life
- τ is the mean lifetime
- λ is the decay constant
If an archaeologist found a fossil sample containing 25% of the carbon-14 level of a living sample, the age of the fossil could be found by rearranging the first formula, since Nt, N0, and t1/2 are known: 25% is two halvings (100% → 50% → 25%), so t = 2 × 5,730 = 11,460 years.
Derivation of the Relationship Between Half-Life Constants
Using the equations above, a relationship can be derived between t1/2, τ, and λ, which makes it possible to determine all three as long as at least one is known. Set t = t1/2 in N(t) = N0e−λt: the remaining quantity is by definition N0/2, so e−λ·t1/2 = 1/2. Taking logarithms gives λ·t1/2 = ln(2), hence:
λ = ln(2) / t1/2 and τ = 1 / λ = t1/2 / ln(2)
The mean lifetime τ is the average time a particle survives before decaying; because some particles live much longer than one half-life, τ is always larger than t1/2, by the factor 1/ln(2) ≈ 1.4427.
Frequently Asked Questions
What is half-life?
Half-life is the time it takes for a quantity to decrease to half of its initial value. After one half-life 50% remains, after two half-lives 25%, after three 12.5%, and so on. The term is most often used for radioactive decay, but it applies to any exponential decay.
How is carbon-14 dating calculated?
Carbon-14 has a half-life of about 5,730 years. Living material holds a steady level of carbon-14; once the organism dies the carbon-14 decays, so the fraction remaining in a sample gives its age: t = t1/2 × log2(N0/Nt). A sample with 25% of the living level of carbon-14 is two half-lives old, about 11,460 years.
How are half-life, mean lifetime, and decay constant related?
They describe the same decay in three ways: decay constant λ = ln(2) / t1/2, mean lifetime τ = 1 / λ = t1/2 / ln(2). A larger decay constant means a shorter half-life and a faster decay.
Can the remaining quantity be larger than the initial quantity?
Not in a decay calculation — the remaining quantity must be smaller than the initial quantity. A growing quantity follows exponential growth instead, which uses the same mathematics with a negative decay constant.