Solve for remaining (N_t)
Use when you know initial amount, time elapsed, and half-life. Answers "how much is left after time t?"
Half-Life Calculator
Determine decay rates and time intervals.
Decay solver
Enter any three of the four values; choose which one to solve for. N_t = quantity remaining, N_0 = initial quantity, t = time, t1/2 = half-life.
Constant converter
Interconvert half-life (tโ/โ), mean lifetime (ฯ), and decay constant (ฮป). Relationship: tโ/โ = ฯ ln 2 = ln 2 / ฮป.
Result
t5730
t = t_{1/2} ร ln(N_0 / N_t) / ln 2
t = 5730 ร ln(100/50) / ln 2 = 5730 ร 0.6931 / 0.6931 โ 5730.0000
Converted constants
tโ/โ = ฯ ln 2 = ln 2 / ฮป
- Half-life tโ/โ: 5730.000000
- Mean lifetime ฯ: 8266.642584
- Decay constant ฮป: 0.00012097
Carbon-14 dating
The half-life of Carbon-14 is about 5,730 years. Archaeologists use the ratio of remaining C-14 to initial (from atmosphere) to estimate the age of organic remains. Enter half-life 5730 (years), initial and remaining quantities (or time), and solve for the missing value.
When to Use the Decay Solver and Constant Converter
Use the decay solver when you have three of the four decay variables; choose which to solve for. Use the constant converter when you need ฯ or ฮป from half-life (or vice versa). Formulas and relationships are in the article below.
Workflow Tips
Solve for time (t)
Use when you know initial, remaining, and half-life. Essential for carbon-14 dating and "how old is this sample?"
Solve for half-life
Use when you know initial, remaining, and time. Infers half-life from two measurements.
Constant converter
Enter one of half-life, mean lifetime (ฯ), or decay constant (ฮป); the other two are computed. Relationship derivation is in the article below.
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
This half-life calculator solves for one of four decay variables, remaining quantity (N_t), initial quantity (Nโ), time (t), or half-life (tโ/โ), given the other three, and converts between half-life, mean lifetime (ฯ), and decay constant (ฮป).
- 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.
How to Use This Calculator
Select the variable you want to solve for from the dropdown: Remaining quantity (N_t), Time elapsed (t), Initial quantity (Nโ), or Half-life (tโ/โ). Enter the three known values in the labeled fields, making sure time and half-life share the same unit (years, seconds, days, etc.). Quantity fields accept any consistent unit, grams, moles, or particle counts. The calculator returns the solved value along with the formula and step-by-step arithmetic used to reach it. Below the main solver, the Constant Converter lets you enter any one of half-life, mean lifetime (ฯ), or decay constant (ฮป) to instantly see the other two. For carbon-14 dating, set the half-life to 5,730 years, enter the initial and remaining C-14 amounts, choose "Solve for Time," and the result is the estimated age of the sample in years.
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 units, enter values in the units you want and the result will be in those units.
Science & Lab Reference Note
Educational Use: These tools use standard scientific formulas and accepted constants. Results are intended for learning, homework, and general reference, not for regulated lab work, industrial processes, or clinical applications.
Verification Recommended: Real-world conditions (purity, temperature, pressure, humidity) affect outcomes. For research, manufacturing, or safety-critical work, confirm with a qualified professional or calibrated lab equipment.
Not Professional Advice: This site does not provide chemical, medical, or engineering advice. All calculations run locally in your browser; no data is stored or transmitted.
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