Calculate radioactive decay, remaining quantity, initial amount, half-life period, or decay time with nuclear isotope presets.

About this tool

Half-Life Calculator evaluates exponential decay processes commonly encountered in nuclear physics, radiometric archaeology, geochronology, and pharmacokinetics. It solves for any of the four principal variables in first-order radioactive decay: remaining quantity, initial quantity, half-life period, or elapsed decay duration.

Calculations run in real time locally within your browser with automatic time unit conversions (ranging from seconds up to billions of years) and pre-configured nuclear isotope references.

When to use it

Radiocarbon dating: Calculate the archaeological age of organic artifacts using Carbon-14 decay.
Nuclear medicine: Determine residual dosage and required activity for diagnostic radioisotopes such as Technetium-99m and Iodine-131.
Environmental physics: Assess residual environmental contamination from fission byproducts including Cesium-137 and Strontium-90.
Pharmacology: Estimate plasma elimination clearance and biological half-life of pharmaceutical compounds.
Geological age dating: Model deep-time rock crystallization ages using Uranium-238 and Uranium-235 decay series.

How to use it

Select what variable you want to solve for under What would you like to calculate?:
- Remaining quantity N(t): Determines the amount left after a known elapsed time.
- Initial quantity N(0): Determines the starting mass or count required to leave a specified residue.
- Half-life period (t 1/2): Solves the characteristic half-life from measured initial and final quantities.
- Elapsed time / Age (t): Calculates how long decay has been active (radiometric dating).
Optionally select a Radionuclide preset (such as Carbon-14 or Uranium-235) to automatically load its scientific half-life constant.
Enter your values into the numerical inputs. You can mix and match different time units (e.g. days for half-life, years for elapsed duration).
Review the calculated result, kinetics breakdown, and the first 6 half-life cycles in the progression table below.
Click Copy result to copy the formatted number to your clipboard.

Mathematical formulas & decay kinetics

Radioactive decay is a first-order exponential process governed by the differential rate equation:

\frac{dN}{dt} = -\lambda N

Integrating this equation gives the standard exponential decay law:

N(t) = N_0 e^{-\lambda t} = N_0 \left(\frac{1}{2}\right)^{t / t_{1/2}}

Where:

$N(t)$ = Quantity remaining after time $t$
$N_0$ = Initial quantity at time $t = 0$
$t_{1/2}$ = Half-life of the substance
$t$ = Total elapsed time
$\lambda$ = Radioactive decay constant
$e$ = Base of the natural logarithm ( $\approx 2.71828$ )

Solving for each variable

1. Remaining quantity ( $N(t)$ )

N(t) = N_0 \times (0.5)^{t / t_{1/2}}

2. Initial quantity ( $N_0$ )

N_0 = \frac{N(t)}{(0.5)^{t / t_{1/2}}} = N(t) \times 2^{t / t_{1/2}}

3. Half-life ( $t_{1/2}$ )

t_{1/2} = \frac{t \times \ln(0.5)}{\ln(N(t) / N_0)}

4. Elapsed time / Age ( $t$ )

t = t_{1/2} \times \frac{\ln(N(t) / N_0)}{\ln(0.5)}

Decay constant and mean lifetime

The radioactive decay constant ( $\lambda$ ) represents the probability per unit time that a single nucleus will undergo decay:

\lambda = \frac{\ln(2)}{t_{1/2}} \approx \frac{0.693147}{t_{1/2}}

The mean lifetime ( $\tau$ ), or average lifespan of a nucleus before decay, is the reciprocal of the decay constant:

\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{\ln(2)} \approx 1.4427 \times t_{1/2}

Reference table: common radioactive isotopes

Isotope	Decay Mode	Half-Life	Primary Application
Technetium-99m ( $^{99m}\text{Tc}$ )	Isomeric transition ( $\gamma$ )	6.01 hours	Diagnostic medical imaging
Iodine-131 ( $^{131}\text{I}$ )	Beta minus ( $\beta^-$ )	8.02 days	Thyroid cancer therapy
Radon-222 ( $^{222}\text{Rn}$ )	Alpha ( $\alpha$ )	3.82 days	Indoor radiation hazard monitoring
Cobalt-60 ( $^{60}\text{Co}$ )	Beta minus ( $\beta^-$ )	5.27 years	Industrial radiography, sterilizing
Tritium ( $^3\text{H}$ )	Beta minus ( $\beta^-$ )	12.32 years	Self-powered lighting, fusion fuel
Strontium-90 ( $^{90}\text{Sr}$ )	Beta minus ( $\beta^-$ )	28.79 years	Fission product waste management
Cesium-137 ( $^{137}\text{Cs}$ )	Beta minus ( $\beta^-$ )	30.17 years	Industrial gauges, calibration
Carbon-14 ( $^{14}\text{C}$ )	Beta minus ( $\beta^-$ )	5,730 years	Radiocarbon dating of organics
Plutonium-239 ( $^{239}\text{Pu}$ )	Alpha ( $\alpha$ )	24,110 years	Nuclear power reactor fuel
Uranium-235 ( $^{235}\text{U}$ )	Alpha ( $\alpha$ )	703.8 million years	Fission reactor enrichment
Uranium-238 ( $^{238}\text{U}$ )	Alpha ( $\alpha$ )	4.468 billion years	Geochronological age of Earth

Questions

What does “half-life” mean?

The half-life ( $t_{1/2}$ ) of a substance is the time required for exactly one-half of the unstable atomic nuclei in a radioactive sample to undergo spontaneous nuclear transformation. After one half-life, 50% of the sample remains. After two half-lives, 25% remains. After three half-lives, 12.5% remains.

Can temperature or pressure change an isotope’s half-life?

Under terrestrial physical conditions, no. Radioactive decay is an intrinsic property of the atomic nucleus governed by the weak and strong fundamental nuclear interactions. Chemical bonding, temperature, pressure, magnetic fields, and physical state do not alter the decay rate of radionuclides.

How does radiocarbon dating work?

Cosmic rays continuously interact with atmospheric nitrogen-14 to create Carbon-14 ( $^{14}\text{C}$ ). Living organisms ingest Carbon-14 through carbon dioxide exchange or the food chain, maintaining a constant ratio of $^{14}\text{C}$ to stable $^{12}\text{C}$ . When an organism dies, metabolic intake halts, and the trapped $^{14}\text{C}$ decays with a half-life of 5,730 years. By measuring the residual $^{14}\text{C}$ activity relative to $^{12}\text{C}$ , scientists calculate how many years have passed since death.

Half-Lives (n)	Fraction Left	Percentage	Remaining (grams (g))
0 × t½	1	100.00%	100
1 × t½	1/2	50.00%	50
2 × t½	1/4	25.00%	25
3 × t½	1/8	12.50%	12.5
4 × t½	1/16	6.25%	6.25
5 × t½	1/32	3.13%	3.125
6 × t½	1/64	1.56%	1.5625

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Half-Life Calculator

Decay kinetics summary

Half-life decay schedule (first 6 cycles)

About this tool

When to use it

How to use it

Mathematical formulas & decay kinetics

Solving for each variable

1. Remaining quantity ( $N(t)$ )

2. Initial quantity ( $N_0$ )

3. Half-life ( $t_{1/2}$ )

4. Elapsed time / Age ( $t$ )

Decay constant and mean lifetime

Reference table: common radioactive isotopes

Questions

What does “half-life” mean?

Can temperature or pressure change an isotope’s half-life?

How does radiocarbon dating work?

Half-Life Calculator

Decay kinetics summary

Half-life decay schedule (first 6 cycles)

About this tool

When to use it

How to use it

Mathematical formulas & decay kinetics

Solving for each variable

1. Remaining quantity (N(t)N(t)N(t))

2. Initial quantity (N0N_0N0​)

3. Half-life (t1/2t_{1/2}t1/2​)

4. Elapsed time / Age (ttt)

Decay constant and mean lifetime

Reference table: common radioactive isotopes

Questions

What does “half-life” mean?

Can temperature or pressure change an isotope’s half-life?

How does radiocarbon dating work?

1. Remaining quantity ( $N(t)$ )

2. Initial quantity ( $N_0$ )

3. Half-life ( $t_{1/2}$ )

4. Elapsed time / Age ( $t$ )