Can I solve for different variables?

Yes! This calculator supports reverse solving. You can solve for: Decay constant (λ), Half-life time (T), Initial quantity (N(0)), Mean lifetime (τ), Remaining quantity (N(t)), Total time (t).

Half life

Solve half life relationships quickly using core chemistry formulas.

Formula shown5 inputs definedUpdated Nov 2025By Automated Chemistry GeneratorFree, no sign-up

Quick Summary

Use this when you need:

Predict how much radioisotope remains after storage
Convert reported decay constants into intuitive half-lives
Estimate elapsed time from a measured remaining fraction

You’ll need:Half-life time (T), Initial quantity (N(0)), Mean lifetime (τ), Remaining quantity (N(t)), and 1 more

You’ll get:Decay constant (λ)

How this is calculated

Formula:

N(t) = N0 * (1/2)^(t / t1/2) and t1/2 = ln(2) / λ

In plain language:

First-order decay leaves half the nuclei every half-life t1/2, so raising 1/2 to the ratio t/t1/2 scales the initial amount, and the relation λ = ln 2 / t1/2 links decay constant and half-life.

The formula is always visible so you can verify the math and explain your numbers to anyone who asks.

What these terms mean

Decay constant (λ)

Decay constant (λ) used in the calculation.

Half-life time (T)

Half-life time (T) used in the calculation.

Initial quantity (N(0))

Initial quantity (N(0)) used in the calculation.

Mean lifetime (τ)

Mean lifetime (τ) used in the calculation.

Remaining quantity (N(t))

Remaining quantity (N(t)) used in the calculation.

Total time (t)

Total time (t) used in the calculation.

When to use this calculator

Predict how much radioisotope remains after storage
Convert reported decay constants into intuitive half-lives
Estimate elapsed time from a measured remaining fraction

Last updated: November 19, 2025

Half-life calculator explained

Radioactive decay and first-order chemical decomposition both follow exponential kinetics. This calculator links the decay constant {decay_constant}, half-life {half_life_time}, mean lifetime {mean_lifetime}, elapsed time {total_time}, and quantities ({initial_quantity}, {remaining_quantity}) so you can answer any half-life question without re-deriving the math.

Use it to plan isotope shipments, estimate shelf life for unstable reagents, or convert between decay constants and the more intuitive half-life.

How the conversion works

The fundamental relationships are:

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

and

t_{1/2} = \\frac{\\ln 2}{\\lambda}, \\qquad \\tau = \\frac{1}{\\lambda}, \\qquad t_{1/2} = (\\ln 2) \\tau

The calculator keeps these expressions synchronized: change any variable and it updates the rest instantly.

Units and conversions

Variable	Units	Notes
$N_0$ , $N(t)$	activity (Bq, Ci) or mass	Keep input and output units consistent.
$t$	s, min, h, days, years	Match the unit used for $\\lambda$ .
$t_{1/2}$	same as $t$	Half the time needed to lose half the material.
$\\lambda$	reciprocal time	e.g., day $^{-1}$ .
$\\tau$	same unit as $t$	Mean lifetime $= 1/\\lambda$ .

Worked examples

Storage loss for Co-60 sources
$t_{1/2} = 5.27$ years, $N_0 = 10\\ \\text{mCi}$ , $t = 2.0$ years.

N(t) = 10 \\times \\left(\\frac{1}{2}\\right)^{2.0/5.27} = 7.69\\ \\text{mCi}

About 77% of the original activity remains after two years.

Converting decay constant to half-life
Measured decay constant $\\lambda = 3.0 \\times 10^{-3}\\ \\text{day}^{-1}$ .

t_{1/2} = \\frac{\\ln 2}{3.0 \\times 10^{-3}} = 2.31 \\times 10^{2}\\ \\text{days}

Mean lifetime $\\tau = 1/\\lambda = 333\\ \\text{days}$ ; the calculator shows both values.

Tips and pitfalls

Keep units consistent; if $\\lambda$ is per hour but time is entered in days, convert before calculating.
For nuclides that decay through chains, apply the single half-life model to each independent step or use Bateman equations for more accuracy.
When dealing with chemical half-lives, confirm the reaction is first-order before applying these formulas.
Use logarithmic plots ($\ln N$ vs. $t$ ) to verify the decay constant from experimental data before plugging it into the calculator.

References and further reading

Frequently Asked Questions

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Calculator info

Version:: v1.0.0
Last reviewed:: Nov 15, 2025
Sensitivity:: low
References:: 1

Last updated: November 19, 2025

Half-life calculator explained

Use it to plan isotope shipments, estimate shelf life for unstable reagents, or convert between decay constants and the more intuitive half-life.

How the conversion works

The fundamental relationships are:

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

and

t_{1/2} = \\frac{\\ln 2}{\\lambda}, \\qquad \\tau = \\frac{1}{\\lambda}, \\qquad t_{1/2} = (\\ln 2) \\tau

The calculator keeps these expressions synchronized: change any variable and it updates the rest instantly.

Units and conversions

Variable	Units	Notes
$N_0$ , $N(t)$	activity (Bq, Ci) or mass	Keep input and output units consistent.
$t$	s, min, h, days, years	Match the unit used for $\\lambda$ .
$t_{1/2}$	same as $t$	Half the time needed to lose half the material.
$\\lambda$	reciprocal time	e.g., day $^{-1}$ .
$\\tau$	same unit as $t$	Mean lifetime $= 1/\\lambda$ .

Worked examples

Storage loss for Co-60 sources
$t_{1/2} = 5.27$ years, $N_0 = 10\\ \\text{mCi}$ , $t = 2.0$ years.

N(t) = 10 \\times \\left(\\frac{1}{2}\\right)^{2.0/5.27} = 7.69\\ \\text{mCi}

About 77% of the original activity remains after two years.

Converting decay constant to half-life
Measured decay constant $\\lambda = 3.0 \\times 10^{-3}\\ \\text{day}^{-1}$ .

t_{1/2} = \\frac{\\ln 2}{3.0 \\times 10^{-3}} = 2.31 \\times 10^{2}\\ \\text{days}

Mean lifetime $\\tau = 1/\\lambda = 333\\ \\text{days}$ ; the calculator shows both values.

Tips and pitfalls

Keep units consistent; if $\\lambda$ is per hour but time is entered in days, convert before calculating.
For nuclides that decay through chains, apply the single half-life model to each independent step or use Bateman equations for more accuracy.
When dealing with chemical half-lives, confirm the reaction is first-order before applying these formulas.
Use logarithmic plots ($\ln N$ vs. $t$ ) to verify the decay constant from experimental data before plugging it into the calculator.

Quick Summary

Calculator Tool

How this is calculated

What these terms mean

Decay constant (λ)

Half-life time (T)

Initial quantity (N(0))

Mean lifetime (τ)

Remaining quantity (N(t))

Total time (t)

When to use this calculator

Half-life calculator explained

How the conversion works

Units and conversions

Worked examples

Tips and pitfalls

References and further reading

Frequently Asked Questions

Related Calculators

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Beer-Lambert law

Molarity

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Was this calculator helpful?

Found an error?

Calculator info

Half-life calculator explained

How the conversion works

Units and conversions

Worked examples

Tips and pitfalls

References and further reading

Frequently Asked Questions

Related Calculators

pKa

Beer-Lambert law

Molarity

Mole

Neutralization

Normality

Was this calculator helpful?

Found an error?

Calculator info