Half-Life Calculator
Print| Initial Quantity (N0) | 100 |
| Decay Ratio Remains | 9.92% |
Half-life is the duration of time required for a quantity to decrease to exactly half of its initial value. While this metric is most commonly associated with unstable atomic nuclei undergoing radioactive decay, it also describes other forms of exponential and non-exponential decay, such as pharmacological drug elimination, biological clearances, and chemical reactant rates.
Our free Half-Life Calculator solves for any of the four major variables in the exponential decay equation: Initial Quantity (N0), Remaining Quantity (Nt), Elapsed Time (t), and Half-Life (t1/2). It also features an integrated conversion utility to translate between half-life, mean lifetime, and decay constants.
The Science of Half-Life & Carbon Dating
One of the most famous applications of half-life is Carbon-14 (C-14) dating, a technique developed by Willard Libby in the late 1940s. Carbon-14 is a radioactive isotope of carbon that is continuously produced in the Earth’s upper atmosphere by cosmic rays. It is absorbed by plants during photosynthesis and subsequently ingested by animals.
While an organism is alive, the ratio of Carbon-14 to stable Carbon-12 in its body remains in equilibrium with the atmosphere. However, when the organism dies, the replenishment of Carbon-14 stops, and the isotope decays exponentially. With a half-life of approximately 5,730 years, Carbon-14 dating can reliably determine the age of organic materials (like bone, wood, and charcoal) up to about 50,000 years old.
Step-by-Step Archaeology Example
Suppose an archaeologist discovers a fossilized bone containing 25% of the Carbon-14 concentration found in a living sample. We can calculate the time since the organism’s death using the half-life equation:
Nt = N0 × (1/2)^(t / t1/2)
Here, we know:
- Remaining ratio (Nt / N0) =
0.25(25%) - Half-life of Carbon-14 (t1/2) =
5,730 years
Substitute these values into the equation:
0.25 = (1/2)^(t / 5730)
Since 0.25 is (1/2)^2:
(1/2)^2 = (1/2)^(t / 5730)
Equate the exponents:
2 = t / 5730
t = 2 × 5730 = 11,460 years.
The fossil is approximately 11,460 years old.
Equivalent Exponential Decay Formulas
Exponential decay can be written in three mathematically equivalent formats depending on whether you use the half-life, the mean lifetime, or the decay constant as your rate parameter:
| Formula Format | Equation | Key Parameter |
|---|---|---|
| Base-2 Half-Life | Nt = N0 × (1/2)^(t / t1/2) | t1/2 (Half-Life) |
| Base-e Decay Constant | Nt = N0 × e^(-λt) | λ (Decay Constant) |
| Base-e Mean Lifetime | Nt = N0 × e^(-t / τ) | τ (Mean Lifetime) |
Where:
- N0: The initial quantity of the substance.
- Nt: The quantity remaining after elapsed time (t).
- t: The total time elapsed.
- t1/2: The half-life of the substance.
- τ (tau): The mean lifetime, which is the average lifespan of a decaying particle.
- λ (lambda): The decay constant, representing the probability of decay per unit of time.
Derivation of Constant Relationships
By equating the base-e decay constant formula to the base-2 half-life formula at the exact moment of half-life (when t = t1/2 and Nt = N0 / 2), we can derive the relationships between these constants:
N0 × e^(-λ × t1/2) = N0 × (1/2)
Divide both sides by N0:
e^(-λ × t1/2) = 1/2
Take the natural logarithm (ln) of both sides:
-λ × t1/2 = ln(1/2) = -ln(2)
Solve for Half-Life:
t1/2 = ln(2) / λ ≈ 0.693147 / λ
Because mean lifetime (τ) is the reciprocal of the decay constant (τ = 1 / λ):
t1/2 = τ × ln(2) ≈ 0.693147 × τ
Calculate scientific fractions using our Fraction Calculator or determine random distributions with the Random Number Generator.
Frequently Asked Questions (FAQ)
What is the difference between half-life and mean lifetime?
Half-life is the time it takes for 50% of a substance to decay. Mean lifetime (tau) is the average lifespan of an individual atom or particle before it decays. Mean lifetime is longer than half-life; specifically, half-life is approximately 69.3% of the mean lifetime (t1/2 = 0.693 × τ).
What is the decay constant (λ)?
The decay constant (λ) is the fractional rate of decay per unit time. For instance, if the decay constant is 0.05 per year, it means each atom has a 5% probability of decaying in any given year. It is mathematically equal to the reciprocal of the mean lifetime (λ = 1 / τ).
Can half-life be used for things other than radioactive elements?
Yes. The concept of half-life is widely used in pharmacology to measure how quickly a drug is cleared from the bloodstream (elimination half-life). It is also used in ecology to describe the persistence of pesticides, and in finance to model exponential depreciation rates.
Why is Carbon-14 dating only accurate up to 50,000 years?
Because the half-life of Carbon-14 is 5,730 years, after 50,000 years (about 9 half-lives), less than 0.2% of the original Carbon-14 remains. At that point, the concentration of Carbon-14 becomes too tiny to measure accurately, and ambient background radiation interferes with the results.