Half-Life Calculator: Radioactive Decay & Carbon-14 Dating
Free half-life calculator: solve for remaining quantity, time, or half-life. Carbon-14 dating calculator, decay constant to half-life converter. Radioactive decay formula and exponential decay steps.
What This Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:Decay solver: Enter any three of N_t, Nโ, t, tโ/โ; choose which to solve for; get the result plus the formula and step-by-step arithmetic. Constant converter: Enter one of half-life, ฯ, or ฮป; see the other two (relationship below). Carbon-14: Use half-life 5730 years for dating. All calculations client-side; no data stored.
- Ideal Users:Students & teachers: Radioactive decay formula solver, carbon-14 dating calculator, decay constant to half-life converter, exponential decay steps, half-life formula, calculate half-life from remaining amount. Science & archaeology: Age-of-sample and remaining-amount problems. Free online, no sign-up.
- Scope & Limits:Exponential decay only (single half-life). Time and half-life must use the same units. Remaining quantity must be less than initial when solving for time or half-life. Positive values only.
This half-life calculator solves for one of four decay variablesโremaining quantity (N_t), initial quantity (Nโ), time (t), or half-life (tโ/โ)โgiven the other three, and converts between half-life, mean lifetime (ฯ), and decay constant (ฮป).
What Is Half-Life? Radioactive Decay Formula
Half-life (tโ/โ) is the time required for a quantity to decay to half its initial value. The radioactive decay formula is N(t) = Nโ (1/2)^(t / tโ/โ), where Nโ is the initial amount and N(t) is the amount remaining after time t. Equivalently, N(t) = Nโ e^(-t/ฯ) (mean lifetime ฯ) or N(t) = Nโ e^(-ฮปt) (decay constant ฮป), with tโ/โ = ฯ ln 2 = ln 2 / ฮป. This half-life calculator solves for N_t, Nโ, t, or tโ/โ from the other three and converts between tโ/โ, ฯ, and ฮป. For the difference between half-life and mean life and the definition of the decay constant, see the FAQ above.
How to Calculate the Age of a Sample Using Half-Life
To estimate the age (time t) of a sample from remaining and initial quantity, use t = tโ/โ ร ln(Nโ / N_t) / ln 2. For carbon-14 dating, the half-life is about 5,730 years; measure the ratio of remaining to initial C-14 (e.g. by radioactivity or mass spectrometry); then t is the age in years. In the calculator, enter Nโ, N_t, and 5730, choose "Solve for: Time (t)", and read the result; the formula and step-by-step arithmetic are shown. For the derivation and "how do I solve for time?", see the FAQ above.
Decay Constant and Mean Lifetime: Relationship to Half-Life
The decay constant (ฮป) and mean lifetime (ฯ) are related to half-life by ฮป = ln 2 / tโ/โ = 1/ฯ, so tโ/โ = ln 2 / ฮป and ฯ = tโ/โ / ln 2. Definitions of ฮป and ฯ are in the FAQ above. Use the constant converter in the calculator: enter any one of half-life, mean lifetime, or decay constant and get the other two. Common in nuclear chemistry (ฮป) and particle physics (ฯ).
Half-Life Calculator FAQ
? What is the difference between half-life and mean life?
Half-life (tโ/โ) is the time for a quantity to decay to half its initial value. Mean lifetime (ฯ) is the average time until a single nucleus decays. They are related by tโ/โ = ฯ ln 2 (so ฯ = tโ/โ / ln 2 โ 1.443 ร tโ/โ). The decay constant ฮป = 1/ฯ = ln 2 / tโ/โ. Use the constant converter in the calculator to switch between them.
? How is Carbon-14 dating used?
Carbon-14 has a half-life of about 5,730 years. Living organisms take in C-14 from the atmosphere; after death, C-14 decays. By measuring the ratio of remaining C-14 to the initial (atmospheric) level, archaeologists estimate the age of organic remains. Enter half-life 5730 (years), initial and remaining quantities (or time), and solve for the missing value in this calculator.
? What is the decay constant?
The decay constant (ฮป) is the probability per unit time that a nucleus will decay. It appears in N(t) = Nโ e^(-ฮปt). It is related to half-life by ฮป = ln 2 / tโ/โ and to mean lifetime by ฮป = 1/ฯ. Use the constant converter to go from half-life to ฮป or ฯ.
? How do I solve for time given half-life and remaining quantity?
From N_t = Nโ (1/2)^(t / tโ/โ), take logarithms: t = tโ/โ ร ln(Nโ / N_t) / ln 2. Enter Nโ, N_t, and tโ/โ in the calculator, choose "Solve for: Time (t)", and it will compute t. Remaining quantity must be less than initial for decay.
? Can I use any units for time and quantity?
Yes. Time (t) and half-life (tโ/โ) must use the same units (e.g. years, seconds). Quantity (Nโ, N_t) can be in any consistent units (grams, moles, counts). The calculator does not convert unitsโenter values in the units you want and the result will be in those units.