Half-Life Calculator: Solve Remaining Amount, Half-Life, and Elapsed Time
Use the Half-Life Calculator for radioactive and exponential decay. Solve for remaining quantity, initial amount, half-life, or elapsed time with isotope presets and a decay timeline.
Table of Contents
Half-life problems look simple until you rearrange the equation. You might know a sample started at 100 units, the half-life of Carbon-14 is 5730 years, and 11,460 years have passed — remaining amount is easy. Flip any one of those values into the unknown and the algebra gets messy: logarithms, decay constants, and the risk of mixing years with days. The Half-Life Calculator solves for remaining amount (N), initial amount (N₀), half-life (t½), or elapsed time (t) from the other three, using the same exponential decay law that nuclear physics and pharmacokinetics share.
Everything runs in the browser. There is no account, no upload, and no round-trip to a server, so classroom homework, a lab notebook, or a nuclear-medicine check stays on the device.
This guide covers the four solve modes, the built-in isotope presets, how the decay timeline is built, and when to keep half-life and elapsed time in the same unit so the answer is physically meaningful.
Why Use the Half-Life Calculator?
- Four-way solver. Switch among remaining amount, initial amount, half-life, and elapsed time. The form hides the field you are solving for, so you only enter what you already know.
- Isotope presets. One click fills t½ for Carbon-14 (5730 y), Iodine-131 (8.02 d), Cobalt-60 (5.27 y), Uranium-238 (4.47 Gy), or Radon-222 (3.82 d).
- Worked formula. Each result shows the substituted equation, not only a number, so you can check the rearrangement against a textbook.
- Decay timeline. A table lists remaining quantity and percent of original after 0 through 6 half-lives, which is the chart most lab reports already expect.
- Wide numeric range. Astronomical half-lives and tiny remnants use scientific notation instead of overflowing a pocket calculator display.
- Private and offline. After the page loads, inputs never leave the device. Useful on a locked-down school network or a clinic workstation with no extra accounts.
Key Features
| Feature | What it does |
|---|---|
| Solve for N | N₀, t½, and t → remaining amount and % of original |
| Solve for N₀ | N, t½, and t → starting quantity |
| Solve for t½ | N₀, N, and t → half-life |
| Solve for t | N₀, N, and t½ → elapsed time |
| Isotope presets | C-14, I-131, Co-60, U-238, Rn-222 |
| Decay timeline | Remaining quantity and % after 0–6 half-lives |
| Copy | Copies the numeric result plus the worked formula |
The tool uses first-order exponential decay: each half-life halves the remaining quantity, regardless of the starting size. Units are yours to choose. The calculator does not convert years to days, so half-life and elapsed time must share one unit — years with years, days with days.
Remaining amount must be greater than zero and strictly less than the initial amount when you solve for half-life or elapsed time. That is the physical constraint for decay (not growth).
How to Use
- Open the Half-Life Calculator and pick Solve for: Remaining amount (N), Initial amount (N₀), Half-life (t½), or Elapsed time (t).
- Enter the three known values. Leave units consistent. For radiocarbon, that usually means years; for I-131, days.
- Optionally tap an isotope chip to fill t½. Carbon-14 is the default half-life (5730) with sample values N₀ = 100 and t = 11460 — two half-lives.
- Read the primary result, the substituted formula, and the percent of original when you solved for N. Scroll to Decay timeline to see 100%, 50%, 25%, 12.5%, and so on.
- Copy the result if you need it in a report, or Clear and try a second scenario without reloading.
Example: N₀ = 100, t½ = 5730 y, t = 11460 y. Remaining N = 25 (25% of original). The timeline row 2 × t½ matches that 25, which is the standard “two half-lives left a quarter” check.
Understanding Half-Life and Exponential Decay
Half-life (t½) is the time for a quantity to fall to half of its current value. For radioactive nuclei, that quantity is the number of undecayed atoms (or activity, which tracks the same exponential). For a first-order drug elimination model, it is the plasma concentration, as long as clearance stays proportional to concentration.
The calculator uses the half-life form of the decay law, not a separate decay-constant input:
- Remaining: N = N₀ × (1/2)^(t / t½)
- Initial: N₀ = N / (1/2)^(t / t½)
- Half-life: t½ = t × ln(2) / ln(N₀ / N)
- Elapsed time: t = t½ × ln(N₀ / N) / ln(2)
The exponent t / t½ is the number of half-lives that have elapsed. After 1 half-life the remainder is 1/2, after 2 it is 1/4, after 3 it is 1/8. The timeline table is that sequence applied to your N₀, for integer counts 0 through 6. If elapsed time is not an integer number of half-lives — 1.5 half-lives, for example — the main solver still returns the exact fractional power; the table is a discrete reference, not a substitute for the formula.
The decay constant λ = ln(2) / t½ is implicit in the last two rearrangements. You do not need to type λ. If a problem gives λ instead of t½, convert first: t½ = ln(2) / λ ≈ 0.693 / λ, then paste t½ into the tool.
Carbon-14 dating uses the same equation with a known atmospheric (or calibration) starting ratio. If a sample has 25% of the modern C-14 activity, two half-lives have passed: t ≈ 2 × 5730 = 11,460 years. Real radiocarbon ages apply calibration curves; the calculator gives the uncalibrated exponential age, which is the number most homework problems want.
Activity (becquerels) decays with the same half-life as the nuclide. You can treat N and N₀ as activities, masses, or “percent remaining” — the ratio is what matters. Do not mix activity with mass unless you know they are proportional for that sample.
Very large t½ values such as Uranium-238 (4.47 × 10⁹ years) and very small remainders print in exponential notation so a 4.47e9 half-life does not look like a rounding error.
Practical Use Cases
Estimate a radiocarbon age from remaining C-14
A charcoal fragment measures 40% of the modern Carbon-14 activity. Choose Elapsed time, set N₀ = 100, N = 40, tap C-14 so t½ = 5730, and read t. That is the uncalibrated age in years. If the lab reported 12.5% remaining instead, you should land near three half-lives (~17,190 y), which the timeline’s 3 × t½ row already shows as 12.5% — a useful sanity check before you write the number on a worksheet.
Check remaining I-131 activity after a delay
Iodine-131 is used in nuclear medicine and has t½ = 8.02 days. A vial is calibrated at 100 MBq on Monday. You need the activity on Friday (4 days later). Solve for Remaining amount with N₀ = 100, t½ from the I-131 preset, t = 4. The answer is a bit above 50 MBq because 4 days is just under one half-life. Keeping t in days matches the preset; switching t to hours without converting t½ would be wrong by a factor of 24.
Recover an unknown half-life from two measurements
A lab notes 80 units at t = 0 and 20 units at t = 12 hours. Solve for Half-life with N₀ = 80, N = 20, t = 12. Two halvings in 12 hours means t½ = 6 hours. The same mode works for a drug concentration that fell from 64 mg/L to 8 mg/L over a known interval — still first-order decay, still the same formula.
Plan a Co-60 or U-238 homework check
Cobalt-60 (5.27 y) and Uranium-238 (4.47 Gy) are common textbook nuclides. For Co-60, “what fraction remains after 10.54 years?” is two half-lives → 25%. Enter N₀ = 1 (or 100), tap Co-60, t = 10.54, solve for N. For U-238, elapsed times of millions of years still sit on the same curve; scientific notation on the result is expected, not a bug. Use the timeline when the question asks “after n half-lives,” and the solver when t is an awkward fraction of t½.
Best Practices
- Keep t and t½ in the same unit. The presets are labeled in years (y), days (d), or gigayears (Gy). Convert elapsed time to that unit before you calculate.
- Treat N and N₀ as the same kind of quantity. Both masses, both activities, or both percentages. Mixing grams with becquerels without a conversion factor is not a half-life problem.
- Use remaining < initial when solving for t or t½. Growth would need a doubling-time tool; this calculator models decay.
- Trust the formula line. If the substituted equation does not match the problem statement, an input field is wrong — usually t and t½ swapped.
- Use the timeline for integer half-lives, the solver for fractions. 1.37 half-lives will not appear as a table row; the main result is the one to report.
- Copy the worked formula into the lab notebook. Graders want N = 100 × (1/2)^(11460/5730), not only “25.”
Ready to stop rearranging logarithms by hand? Open the Half-Life Calculator, pick the unknown, tap an isotope if you have one, and read remaining quantity, half-life, or elapsed time with the decay timeline beside it.
Related Tools You Might Like:
- Scientific Calculator — logs, powers, and ln(2) when a problem still needs a manual check
- Molar Mass Calculator — molecular weight for the same chemistry set as a decay lab
- Sig Fig Calculator — count and round significant figures on N, t½, and reported ages
Happy decaying — may your units match and your remainders stay honest.