What is the difference between half-life and mean lifetime?

The half-life T½ is the time for exactly half of a sample to decay. The mean lifetime τ is the average time an individual atom survives before decaying, and it is longer: τ = T½ / ln2 ≈ 1.4427 · T½. The reason τ is bigger is that a few atoms hang around far longer than one half-life, dragging the average up past the median (which is what the half-life is). They are tied together through the decay constant: τ = 1/λ and T½ = ln2/λ. This tool reports all three for every calculation so you never have to convert between them by hand.

How does Carbon-14 dating work with this calculator?

Living things absorb Carbon-14 at a roughly constant rate, so a fresh sample has a known starting ratio. Once it dies, the C-14 decays with a half-life of 5,730 years and nothing replaces it. To date a sample, set Initial amount to 100, click the Carbon-14 preset to load 5,730 years, enter the measured remaining percent as the remaining amount (say 25), and solve for elapsed time — you get 11,460 years, which is two half-lives. The method is reliable out to roughly 50,000 years; past that, too little C-14 remains to measure accurately.

How is the decay constant λ calculated?

λ is the probability per unit time that any given atom decays, and it comes straight from the half-life: λ = ln2 / T½, where ln2 ≈ 0.6931. So a Carbon-14 sample with T½ = 5,730 years has λ ≈ 1.21 × 10⁻⁴ per year. λ is what appears in the e-based form of the law, N = N₀·e^(−λt). This calculator reports λ in whatever time unit you chose for the half-life, so if T½ is in days, λ is per day.

Why is radioactive decay exponential rather than linear?

Each atom decays independently with a fixed probability per unit time (λ) — it has no memory of how long it has already existed. When the rate of decay is proportional to how much is left (dN/dt = −λN), the solution is necessarily an exponential, N = N₀·e^(−λt). That is also why no sample ever reaches exactly zero: each half-life removes half of whatever remains, so you get 50%, 25%, 12.5%, and so on, approaching but never touching zero.

Can I solve for the original amount if I only know what is left?

Yes — set Solve for to Initial amount, then enter the remaining amount, the elapsed time, and the half-life. The tool computes N₀ = N · 2^(t/T½). This is the common forensic or archaeological case: you measure what survives today and work backwards to how much there was at the start.

Date an archaeological sample with Carbon-14

You measure that a charcoal fragment retains 18% of its original Carbon-14. Set Solve for to Elapsed time, click the Carbon-14 preset (loads 5,730 years), enter Initial amount 100 and Remaining amount 18, and read back roughly 14,170 years. Because the tool uses the true exponential law rather than rounding to whole half-lives, a measurement like 18% — which sits between two and three half-lives — gives an exact age instead of forcing you onto the nearest half-life mark.

Plan a medical isotope dose for a scan

A nuclear-medicine tech prepares a Technetium-99m dose (half-life 6.01 hours) for a scan three hours from now. Click the Tc-99m preset, set the initial activity, set elapsed time to 3 hours, and solve for the remaining amount to see how much activity is left at scan time — about 71% remains after half a half-life. That tells you how much to draw now so the delivered dose hits the target window.

Find the half-life from lab decay data

In a physics lab you start with 5,000 counts per minute and measure 1,250 cpm after 40 minutes. Set Solve for to Half-life, enter Initial amount 5000, Remaining amount 1250, and Elapsed time 40 (minutes). The tool returns 20 minutes — the activity dropped to a quarter, so exactly two half-lives passed. This is the standard way to identify an unknown isotope from its decay curve.

Track a drug's elimination half-life

A medication has an elimination half-life of 8 hours. You took 600 mg and want to know how much is still in your system after 24 hours. Set the half-life to 8 hours, Initial amount to 600, Elapsed time to 24, and solve for the remaining amount — about 75 mg, since 24 hours is three half-lives (600 → 300 → 150 → 75). The same exponential model that governs nuclei governs first-order drug clearance.

Check a homework answer on decay constants

A chemistry problem asks for the decay constant of Cobalt-60 (half-life 5.27 years). Click the Co-60 preset and read λ directly — about 0.1315 per year — along with the mean lifetime of about 7.6 years. Students often mix up λ and τ, or forget the ln2 factor; seeing all the derived quantities side by side makes the relationships concrete.

Half-Life Calculator — radioactive decay, decay constant & mean lifetime

Solve any radioactive-decay unknown — initial amount, remaining amount, elapsed time, or half-life — with exact exponential math, all in your browser

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Solve for

Time unit

Initial amount N₀

Remaining amount N

Elapsed time t (years)

Half-life T½ (years)

Common isotopesClick to fill the half-life

Remaining amount N50

Remaining fraction: 50%
Decay constant λ: 0.000120968 per years
Mean lifetime τ: 8266.64 years
Half-lives elapsed: 1

What this tool does

A radioactive-decay solver that handles the whole exponential model, not just the textbook "it halves every T½" shortcut. Pick which of the four quantities you want to solve for — initial amount N₀, remaining amount N, elapsed time t, or half-life T½ — type in the other three, and you get the answer plus everything that falls out of it: the remaining fraction (as a percent), the decay constant λ = ln2 / T½, the mean lifetime τ = T½ / ln2, and how many half-lives have passed. The math is the true continuous form N = N₀·(½)^(t/T½) = N₀·e^(−λt), so it is correct at any point in time, not only at whole half-lives — at t = T½/2 it reports 70.7% remaining, not a naive 75%. One click loads common isotopes (Carbon-14 at 5,730 years, Iodine-131 at 8.02 days, Technetium-99m at 6.01 hours, Uranium-238, and more) straight into the half-life field with the right time unit. Time can be entered in seconds, minutes, hours, days, or years, and t and T½ always share the same unit so there is no hidden conversion to second-guess. Every calculation is shareable as a URL and copyable in one click, and nothing ever leaves your browser tab.

Tool details

Input: Text + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 11 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Half-Life Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Date an archaeological sample with Carbon-14
You measure that a charcoal fragment retains 18% of its original Carbon-14. Set Solve for to Elapsed time, click the Carbon-14 preset (loads 5,730 years), enter Initial amount 100 and Remaining amount 18, and read back roughly 14,170 years. Because the tool uses the true exponential law rather than rounding to whole half-lives, a measurement like 18% — which sits between two and three half-lives — gives an exact age instead of forcing you onto the nearest half-life mark.
Plan a medical isotope dose for a scan
A nuclear-medicine tech prepares a Technetium-99m dose (half-life 6.01 hours) for a scan three hours from now. Click the Tc-99m preset, set the initial activity, set elapsed time to 3 hours, and solve for the remaining amount to see how much activity is left at scan time — about 71% remains after half a half-life. That tells you how much to draw now so the delivered dose hits the target window.
Find the half-life from lab decay data
In a physics lab you start with 5,000 counts per minute and measure 1,250 cpm after 40 minutes. Set Solve for to Half-life, enter Initial amount 5000, Remaining amount 1250, and Elapsed time 40 (minutes). The tool returns 20 minutes — the activity dropped to a quarter, so exactly two half-lives passed. This is the standard way to identify an unknown isotope from its decay curve.
Track a drug's elimination half-life
A medication has an elimination half-life of 8 hours. You took 600 mg and want to know how much is still in your system after 24 hours. Set the half-life to 8 hours, Initial amount to 600, Elapsed time to 24, and solve for the remaining amount — about 75 mg, since 24 hours is three half-lives (600 → 300 → 150 → 75). The same exponential model that governs nuclei governs first-order drug clearance.
Check a homework answer on decay constants
A chemistry problem asks for the decay constant of Cobalt-60 (half-life 5.27 years). Click the Co-60 preset and read λ directly — about 0.1315 per year — along with the mean lifetime of about 7.6 years. Students often mix up λ and τ, or forget the ln2 factor; seeing all the derived quantities side by side makes the relationships concrete.

Common pitfalls

Assuming decay is linear — that 50% remains after one half-life, so 0% remains after two. Wrong: each half-life removes half of what is left, so two half-lives give 25%, not 0%. The amount approaches zero but never reaches it.
Confusing the decay constant λ with the half-life T½. They are inversely related through λ = ln2 / T½, not equal. A longer half-life means a smaller λ (slower decay), and forgetting the ln2 factor is the most common arithmetic slip.
Mixing time units between t and the half-life. If the half-life is in years, elapsed time must be in years too. This tool keeps both on one selected unit so you cannot accidentally pair days with years.

Privacy

Every calculation here is plain JavaScript that runs inside your browser tab — the exponential formula, the decay constant, the mean lifetime, and the solver for whichever unknown you pick. Nothing you enter is sent anywhere, there is no logging, and there is no external API call. The one thing worth knowing: the shareable link encodes your inputs in the URL query string, so if you paste a share link somewhere, the destination server's log will record those numbers. For coursework that is harmless; if a value is sensitive, just copy the result instead of sharing the URL.

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Half-Life Calculator — radioactive decay, decay constant & mean lifetime

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Half-Life Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Date an archaeological sample with Carbon-14

Plan a medical isotope dose for a scan

Find the half-life from lab decay data

Track a drug's elimination half-life

Check a homework answer on decay constants

Common pitfalls

Privacy

FAQ

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