Half-life is the time required for a quantity to reduce to half its initial value under first-order exponential decay.

Can I use this for drug elimination or radioactivity?

Yes. The calculator models first-order exponential decay common in pharmacokinetics and radioactive decay. Always validate assumptions for your use case.

Do I need half-life or decay constant?

You can supply either. They are equivalent via λ = ln(2)/t1/2. Provide one and the tool computes the other.

What units are supported?

Seconds, minutes, hours, days, and years for time; the decay constant can be expressed per any of those units.

How accurate are the results?

Results are computed with double precision and standard formulas. Accuracy depends on the precision of your inputs and whether decay is truly first-order.

Can it compute time to reach a target amount?

Yes. Choose "Time t" as the unknown and provide the other inputs.

Half-Life Calculator

Half life calculator for exponential decay. Compute remaining quantity, initial quantity, time, half-life, or decay constant with unit conversions, clear steps, and WCAG 2.1 AA accessibility.

Half-Life Calculator

This professional half life calculator solves exponential decay problems in seconds. It’s built for scientists, engineers, educators, and students to compute remaining quantity, initial quantity, time, half-life, or decay constant with precise unit handling and fully accessible design.

Choose what to solve for

You may provide either half-life or decay constant; the tool converts between them automatically.

Initial quantity N0 *

Remaining quantity N(t) *

Time elapsed t *

Half-life t½ *

Decay constant λ *

Results

Remaining N(t) —

Initial N0 —

Time t —

Half-life t½ —

Decay constant λ —

Fraction remaining (N/N0) —

Percent decayed —

Number of half-lives (t / t½) —

Authoritative Data Source and Methodology

Primary reference: IUPAC Compendium of Chemical Terminology (the “Gold Book”), entry “half-life; t_1/2”, 3rd edition (2014). DOI: 10.1351/goldbook. Link: https://goldbook.iupac.org/terms/view/H02723. All formulas employ first-order exponential decay, consistent with the IUPAC definition and standard kinetics.

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

The Formula Explained

Exponential decay (first order):

$$N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$$

Relationship between half-life and decay constant:

$$\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}$$

Solving for time to reach a target:

$$t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)$$

Glossary of Variables

Symbol	Name	Units	Notes
N0	Initial quantity	Any (e.g., mg, Bq)	Starting amount at t = 0
N(t)	Remaining quantity	Same as N0	Amount after time t
t	Time elapsed	s, min, h, d, y	Must be ≥ 0
t½	Half-life	s, min, h, d, y	Positive
λ	Decay constant	1/s, 1/min, 1/h, 1/d, 1/y	Positive
n	Number of half-lives	—	n = t / t½

Worked Example

How It Works: A Step-by-Step Example

Suppose a drug has a half-life of 8 hours. If the initial dose is N0 = 120 mg, how much remains after t = 24 hours?

Number of half-lives: n = t / t½ = 24 / 8 = 3.
Apply decay: N = N0 × (1/2)^n = 120 × (1/2)^3 = 120 × 0.125 = 15 mg.
Cross-check using λ: λ = ln 2 / t½ = 0.693147/8 h = 0.086643 h⁻1; N = 120 × e^(−0.086643×24) ≈ 15 mg.

Therefore, 15 mg remain after 24 hours. Fraction remaining is 0.125; 87.5% has decayed.

Frequently Asked Questions (FAQ)

Is this calculator valid for any decay process?

It models first-order exponential decay, which covers radioactive decay and many pharmacokinetic cases. Processes that are zero-order or mixed-order require different models.

Which should I enter: half-life or decay constant?

Either one is sufficient. The tool converts them via λ = ln(2)/t½ and uses whichever you provide.

Can I solve for time to reach a target amount?

Yes. Select “Time t” as the unknown, then provide N0, N(t), and either t½ or λ.

Do the units have to match?

Time units are selectable and internally converted. N0 and N(t) must use the same unit (e.g., both mg or both Bq).

What precision do you use?

Internally we use double precision. Results are rounded for readability, but the underlying calculation retains higher precision.

Why do I get an error when N(t) ≥ N0?

For a decaying process, N(t) must be less than or equal to N0. If greater, check your inputs or whether your process involves growth instead of decay.

How can I cite this tool?

You can cite the IUPAC Gold Book entry for half-life (2014) as the methodological reference and include the URL of this calculator.

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

\[N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}\]

N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}

Formula (extracted LaTeX)

\[\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}\]

\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}

Formula (extracted LaTeX)

\[t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)\]

t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)

Formula (extracted text)

Exponential decay (first order): $N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$ Relationship between half-life and decay constant: $\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}$ Solving for time to reach a target: $t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)$

Variables and units

No variables provided in audit spec.

Sources (authoritative):

Science — calcdomain.com · Accessed 2026-01-19
https://calcdomain.com/science
https://goldbook.iupac.org/terms/view/H02723 — goldbook.iupac.org · Accessed 2026-01-19
https://goldbook.iupac.org/terms/view/H02723

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

Full original guide (expanded)

Half-Life Calculator

Choose what to solve for

You may provide either half-life or decay constant; the tool converts between them automatically.

Initial quantity N0 *

Remaining quantity N(t) *

Time elapsed t *

Half-life t½ *

Decay constant λ *

Results

Remaining N(t) —

Initial N0 —

Time t —

Half-life t½ —

Decay constant λ —

Fraction remaining (N/N0) —

Percent decayed —

Number of half-lives (t / t½) —

Authoritative Data Source and Methodology

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

The Formula Explained

Exponential decay (first order):

$$N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$$

Relationship between half-life and decay constant:

$$\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}$$

Solving for time to reach a target:

$$t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)$$

Glossary of Variables

Symbol	Name	Units	Notes
N0	Initial quantity	Any (e.g., mg, Bq)	Starting amount at t = 0
N(t)	Remaining quantity	Same as N0	Amount after time t
t	Time elapsed	s, min, h, d, y	Must be ≥ 0
t½	Half-life	s, min, h, d, y	Positive
λ	Decay constant	1/s, 1/min, 1/h, 1/d, 1/y	Positive
n	Number of half-lives	—	n = t / t½

Worked Example

How It Works: A Step-by-Step Example

Suppose a drug has a half-life of 8 hours. If the initial dose is N0 = 120 mg, how much remains after t = 24 hours?

Number of half-lives: n = t / t½ = 24 / 8 = 3.
Apply decay: N = N0 × (1/2)^n = 120 × (1/2)^3 = 120 × 0.125 = 15 mg.
Cross-check using λ: λ = ln 2 / t½ = 0.693147/8 h = 0.086643 h⁻1; N = 120 × e^(−0.086643×24) ≈ 15 mg.

Therefore, 15 mg remain after 24 hours. Fraction remaining is 0.125; 87.5% has decayed.

Frequently Asked Questions (FAQ)

Is this calculator valid for any decay process?

It models first-order exponential decay, which covers radioactive decay and many pharmacokinetic cases. Processes that are zero-order or mixed-order require different models.

Which should I enter: half-life or decay constant?

Either one is sufficient. The tool converts them via λ = ln(2)/t½ and uses whichever you provide.

Can I solve for time to reach a target amount?

Yes. Select “Time t” as the unknown, then provide N0, N(t), and either t½ or λ.

Do the units have to match?

Time units are selectable and internally converted. N0 and N(t) must use the same unit (e.g., both mg or both Bq).

What precision do you use?

Internally we use double precision. Results are rounded for readability, but the underlying calculation retains higher precision.

Why do I get an error when N(t) ≥ N0?

For a decaying process, N(t) must be less than or equal to N0. If greater, check your inputs or whether your process involves growth instead of decay.

How can I cite this tool?

You can cite the IUPAC Gold Book entry for half-life (2014) as the methodological reference and include the URL of this calculator.

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

\[N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}\]

N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}

Formula (extracted LaTeX)

\[\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}\]

\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}

Formula (extracted LaTeX)

\[t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)\]

t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)

Formula (extracted text)

Exponential decay (first order): $N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$ Relationship between half-life and decay constant: $\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}$ Solving for time to reach a target: $t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)$

Variables and units

No variables provided in audit spec.

Sources (authoritative):

Science — calcdomain.com · Accessed 2026-01-19
https://calcdomain.com/science
https://goldbook.iupac.org/terms/view/H02723 — goldbook.iupac.org · Accessed 2026-01-19
https://goldbook.iupac.org/terms/view/H02723

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

Half-Life Calculator

Choose what to solve for

You may provide either half-life or decay constant; the tool converts between them automatically.

Initial quantity N0 *

Remaining quantity N(t) *

Time elapsed t *

Half-life t½ *

Decay constant λ *

Results

Remaining N(t) —

Initial N0 —

Time t —

Half-life t½ —

Decay constant λ —

Fraction remaining (N/N0) —

Percent decayed —

Number of half-lives (t / t½) —

Authoritative Data Source and Methodology

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

The Formula Explained

Exponential decay (first order):

$$N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$$

Relationship between half-life and decay constant:

$$\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}$$

Solving for time to reach a target:

$$t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)$$

Glossary of Variables

Symbol	Name	Units	Notes
N0	Initial quantity	Any (e.g., mg, Bq)	Starting amount at t = 0
N(t)	Remaining quantity	Same as N0	Amount after time t
t	Time elapsed	s, min, h, d, y	Must be ≥ 0
t½	Half-life	s, min, h, d, y	Positive
λ	Decay constant	1/s, 1/min, 1/h, 1/d, 1/y	Positive
n	Number of half-lives	—	n = t / t½

Worked Example

How It Works: A Step-by-Step Example

Suppose a drug has a half-life of 8 hours. If the initial dose is N0 = 120 mg, how much remains after t = 24 hours?

Number of half-lives: n = t / t½ = 24 / 8 = 3.
Apply decay: N = N0 × (1/2)^n = 120 × (1/2)^3 = 120 × 0.125 = 15 mg.
Cross-check using λ: λ = ln 2 / t½ = 0.693147/8 h = 0.086643 h⁻1; N = 120 × e^(−0.086643×24) ≈ 15 mg.

Therefore, 15 mg remain after 24 hours. Fraction remaining is 0.125; 87.5% has decayed.

Frequently Asked Questions (FAQ)

Is this calculator valid for any decay process?

It models first-order exponential decay, which covers radioactive decay and many pharmacokinetic cases. Processes that are zero-order or mixed-order require different models.

Which should I enter: half-life or decay constant?

Either one is sufficient. The tool converts them via λ = ln(2)/t½ and uses whichever you provide.

Can I solve for time to reach a target amount?

Yes. Select “Time t” as the unknown, then provide N0, N(t), and either t½ or λ.

Do the units have to match?

Time units are selectable and internally converted. N0 and N(t) must use the same unit (e.g., both mg or both Bq).

What precision do you use?

Internally we use double precision. Results are rounded for readability, but the underlying calculation retains higher precision.

Why do I get an error when N(t) ≥ N0?

For a decaying process, N(t) must be less than or equal to N0. If greater, check your inputs or whether your process involves growth instead of decay.

How can I cite this tool?

You can cite the IUPAC Gold Book entry for half-life (2014) as the methodological reference and include the URL of this calculator.

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

\[N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}\]

N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}

Formula (extracted LaTeX)

\[\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}\]

\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}

Formula (extracted LaTeX)

\[t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)\]

t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)

Formula (extracted text)

Exponential decay (first order): $N(t) = N_0 \, e^{-\lambda t} \quad\text{and}\quad N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$ Relationship between half-life and decay constant: $\lambda = \frac{\ln 2}{t_{1/2}} \quad\Longleftrightarrow\quad t_{1/2} = \frac{\ln 2}{\lambda}$ Solving for time to reach a target: $t = \frac{\ln\!\left(\frac{N_0}{N(t)}\right)}{\lambda} = t_{1/2}\,\log_2\!\left(\frac{N_0}{N(t)}\right)$

Variables and units

No variables provided in audit spec.

Sources (authoritative):

Science — calcdomain.com · Accessed 2026-01-19
https://calcdomain.com/science
https://goldbook.iupac.org/terms/view/H02723 — goldbook.iupac.org · Accessed 2026-01-19
https://goldbook.iupac.org/terms/view/H02723

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

Formulas

(Formulas preserved from original page content, if present.)

Version 0.1.0-draft

Citations

Add authoritative sources relevant to this calculator (standards bodies, manuals, official docs).

Changelog

0.1.0-draft — 2026-01-19: Initial draft (review required).