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Half-Life Calculator

Computes first-order exponential decay: given two of half-life, elapsed time and remaining amount (with an initial amount) it solves the third and reports the remaining fraction, number of half-lives, decay constant and mean lifetime.

When to use

You need how much of a radioactive isotope, drug or other first-order decaying quantity remains after a time, or the half-life or elapsed time implied by a measured remaining amount.

Do not use when: The process is not first-order (e.g. zero-order elimination, growth, or compound interest – use compound-interest for money) or several isotopes decay in a chain.

Formula

N(t) = N0 × (1/2)^(t / t½) = N0 × e^(−λt); λ = ln 2 / t½; τ = 1 / λ; t½ = t / log2(N0 / N); t = t½ × log2(N0 / N)

First-order kinetics: the decay rate is proportional to the amount present, so the same fraction decays in every half-life regardless of the starting amount.

Inputs

Parameter	Type	Required	Description
`initial_amount`	number	default 100	Starting quantity in any unit (g, Bq, atoms, mg/L). Default 100 makes remaining_amount a percentage. Range: > 0
`half_life`	number	no	Half-life in any time unit (same unit as elapsed_time). Leave empty to solve for it. Range: > 0
`elapsed_time`	number	no	Time elapsed, same unit as half_life. Leave empty to solve for it. Range: ≥ 0
`remaining_amount`	number	no	Quantity left after elapsed_time, same unit as initial_amount. Leave empty to solve for it. Range: > 0

Outputs

Output	Type	Unit	Description
`remaining_amount`	number		N(t) = N0 × (1/2)^(t / half_life).
`remaining_fraction`	number		N(t) / N0.
`remaining_percent`	number	%	100 × N(t) / N0.
`decayed_amount`	number		N0 − N(t).
`number_of_half_lives`	number		elapsed_time / half_life.
`half_life`	number		Half-life, given or solved (time unit of the inputs).
`elapsed_time`	number		Elapsed time, given or solved (time unit of the inputs).
`decay_constant`	number		ln 2 / half_life, per time unit of the inputs.
`mean_lifetime`	number		1 / λ = half_life / ln 2.
`solved_for`	string		remaining_amount, half_life or elapsed_time.

Example

Carbon-14 after 11,460 years: {"initial_amount":100,"half_life":5730,"elapsed_time":11460} → {"remaining_amount":25,"remaining_fraction":0.25,"number_of_half_lives":2,"decay_constant":0.00012097,"mean_lifetime":8266.6,"solved_for":"remaining_amount"}

80 mg of iodine-131 falls to 10 mg in 24.06 days: {"initial_amount":80,"remaining_amount":10,"elapsed_time":24.06} → {"half_life":8.02,"number_of_half_lives":3,"remaining_percent":12.5,"decay_constant":0.086427,"solved_for":"half_life"}

GET https://tttkmbb.com/api/v1/calculate/half-life?initial_amount=100&half_life=5730&elapsed_time=11460

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/half-life (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/half-life · Markdown: https://tttkmbb.com/chemistry/half-life.md · JSON definition: https://tttkmbb.com/chemistry/half-life.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="half-life"
OpenAPI operationId: calculate_half_life_decay
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

OpenStax Chemistry 2e – 21.3 Radioactive Decay (textbook)
Wikipedia – Half-life (reference)

FAQ

Which time unit should I use?

Any, as long as half_life and elapsed_time share it; the decay constant and mean lifetime are then per that unit (e.g. per year for a half-life in years).

Does this work for drug elimination?

Yes for drugs with first-order (linear) elimination, using the plasma half-life; about 97% is eliminated after 5 half-lives. Zero-order elimination (e.g. ethanol at high doses) is not modelled.

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Moles to Grams Calculator — Convert the remaining mass of an isotope to a number of atoms.
Logarithm Calculator — Log2 and natural log arithmetic used in solving for time.