Half-Life Calculator - Remaining Amount, Time, or t½

A half-life calculator answers questions about a quantity that regularly halves: remaining amount after a wait, how long a wait must have been, what half-life fits two measured amounts, or what the starting amount was. The model is exponential decay, not a linear “lose the same number of grams every hour,” and not exponential growth. If remaining N is larger than initial N0, this page errors. Doubling times and bacterial growth belong somewhere else.

The default checksum is N0 = 100, t½ = 8, t = 24. That is exactly three half-lives (24 / 8 = 3). Remaining N = 100 × (1/2)³ = 100 × 1/8 = 12.5. If another widget prints 12.499999 or 13 on those inputs, check whether it used a rounded λ or a linear interpolation. This implementation uses N = N0 × (1/2)^(t / t½), which is exact for that ratio in binary floating point for these particular numbers.

Units are yours to keep consistent. If t½ is in years, t must be years. If N0 is milligrams, N is milligrams. The algebra never converts hours to seconds. Carbon-14 is mentioned below as a famous example of a half-life people type in by hand - not as a lookup table of isotopes or a calibrated radiocarbon age.

1. Pick the unknown. That field is disabled; enter the other three quantities.
2. Default demo: leave Solve for Remaining N, N0 = 100, t½ = 8, t = 24, and confirm N = 12.5 with t/t½ = 3.
3. Read λ = ln(2)/t½ and the number of half-lives t/t½ next to the solved value.
4. Copy the block or Clear back to the demo. Values stay in this browser for up to 30 days.

Rejected: N0 ≤ 0, t½ ≤ 0, t < 0, N ≤ 0 when that quantity is used, N > N0 (growth), and solving for t½ when N = N0 (undefined or contradictory). This is not a pharmacokinetics compartment model and not an isotope database.

After one half-life, t = t½, the exponent is 1, and N = N0/2. After two, N = N0/4. After three, N = N0/8. The default 100 → 12.5 is that third step. The exponential form with e is identical: λ is chosen so e^(−λ t½) = 1/2, which forces λ = ln(2)/t½. Powers of 1/2 and the exponent calculator are the same operation; logarithms when you solve for t are the inverse, which is why the log calculator is a sibling for checking log2(N0/N) by hand.

Discrete “count the halvings” works only when t is an integer multiple of t½. The formula still works for 1.5 half-lives: N = N0 / (2^1.5) = N0 / (2√2). This page does not round t/t½ to an integer. A remaining amount of 12.5 from 100 is three exact half-lives, not “about three.”

Start with 100. After 8 time units, 50. After 16, 25. After 24, 12.5. The table is not a special case of carbon dating; it is the definition of half-life. People sometimes average 100 and 0 and guess 50 after a long wait, or subtract 100/3 per half-life and land on a linear 0 at t = 24. Linear decay would hit zero in a finite time; exponential decay only approaches zero. At t = 24 with t½ = 8 you still have an eighth of the original, never zero.

If your lab reports 12.4 instead of 12.5, the mismatch is measurement or a different t½, not a different definition. Percent-style disagreement against a theoretical 12.5 is a job for the percent error calculator, after this page has told you the model value.

λ = ln(2)/t½ has units of 1/time. For t½ = 8, λ ≈ 0.0866434 per same time unit. Then N = N0 e^(−λt) is a calculus-friendly twin of the (1/2) power. This calculator always reports λ from the half-life that is known or just solved, plus t/t½ so you can see “how many halvings” without staring at λ.

Solving for time: t = t½ × log2(N0 / N). You need N0 > 0, 0 < N ≤ N0, t½ > 0. If N = N0, t = 0 (no time has elapsed). If N > N0, log2 of a number less than 1 would give negative time - growth - and is rejected. Rearrangement for half-life: t½ = t / log2(N0 / N), which needs t ≥ 0 and N < N0 (strictly less, otherwise the log vanishes).

Solving for N0: N0 = N × 2^(t/t½). That reconstructed start is always at least N for t ≥ 0. Solving for remaining N is the default path. A scientific calculator can reproduce the same powers if you want to audit a step; this page already does the rearrangement and the λ report.

Textbooks often quote a carbon-14 half-life around 5,730 years. You may type 5730 as t½ yourself if a homework problem uses that figure. This site does not fetch IntCal curves, δ¹³C corrections, reservoir ages, or an isotope menu. Radiocarbon dating in the field is a calibration problem; N = N0 (1/2)^(t/t½) is the idealized exponential. Mixing those two is how a class demo becomes a fake antiquity claim.

Other isotopes work the same way only as numbers you already know: if a problem says iodine-131 has t½ = 8.02 days, type 8.02 and keep t in days. There is no built-in table to go stale and no medical dose model. Pharmacy half-lives in a body are often effective half-lives (elimination plus decay) - still only valid here if you treat them as a single t½ in the exponential you were told to use.

Leave N0 = 100 and t½ = 8, switch Solve for to Time t, and type remaining N = 12.5. You should recover t = 24, because log2(100/12.5) = log2(8) = 3 and t = 8 × 3 = 24. Switch Solve for to Half-life t½, keep N = 12.5, N0 = 100, t = 24, and you should recover t½ = 8. Those two rearrangements are how a lab checks consistency: the same four numbers have to close under every unknown. If rounding in a report turned 12.5 into 12, solved t will not be exactly 24 - that is the data, not a second formula.

Solving for N0 from N = 12.5, t = 24, t½ = 8: N0 = 12.5 × 2³ = 12.5 × 8 = 100. The reconstructed start is a power-of-two scale-up of the remaining amount. If you measured remaining activity and know elapsed time and half-life, that is the estimate of what you began with, still under the single-exponential assumption.

A straight line from (0, 100) to (24, 0) would pass through 50 at t = 12 and 0 at t = 24. Half-life decay from the same start with t½ = 8 is at 50 when t = 8, 25 when t = 16, 12.5 when t = 24, and still positive afterward. Same endpoints in a sloppy sketch, completely different interiors. Homework that says “half the sample every 8 days” is the exponential, even if a student draws a line. This calculator will not average N0 and zero, and it will not hit zero at a finite t.

Plotting ln(N) against t is the diagnostic: exponential decay is a straight line with slope −λ. This page does not draw that graph, but it reports λ so you can check a slope from a log plot. If your points are not linear on a log scale, a single half-life is the wrong model (two-phase clearance, background counts, mixed isotopes). Do not force t½ out of N, N0, and t unless you believe the exponential.

Effective half-life in a body (physical decay plus biological clearance) can still be typed as t½ if a problem gives that combined number. Physical nuclear half-life alone is what carbon-14 examples usually mean. Mixing them without saying so is a units-and-model error, not something a solver can detect from four numbers.

Half-life is a duration. If t½ = 8 days and you type t = 24 thinking “24 hours,” you have compared days to hours and the exponent t/t½ becomes 3 when it should be 1. The calculator cannot see unit labels; 8 and 24 are dimensionless as far as IEEE-754 is concerned. Convert first. 24 hours with t½ = 8 days is t = 1 day, remaining N = 100 × (1/2)^(1/8) ≈ 91.7, not 12.5. The famous 12.5 checksum only holds when 24 and 8 share a unit.

λ inherits that unit. t½ = 8 days gives λ in 1/day. The same physical process with t½ = 8/365 years has a different numeric λ because the time axis was rescaled. Reporting λ without the unit is how two lab partners disagree by a factor of 24 or 365. This page prints a pure number next to λ; you supply “per hour” or “per year” in the sentence you copy into a report.

Amounts N and N0 likewise need one unit: grams with grams, becquerels with becquerels, percent of original with percent of original. Using N = 12.5 as a percent remaining when N0 = 100 is consistent (12.5%). Using N = 12.5 mg with N0 = 100 g is not; convert to 0.0125 g or 100000 mg first. The decay inequality N ≤ N0 is then meaningful. Growth rejection would misfire if you mixed milligrams remaining with grams initial and got a false N > N0.

Very large exponents (tiny remaining fractions after hundreds of half-lives) underflow to 0 in floating point. That is a computer limit, not “the sample is gone.” Chemically or radioactively you may still have atoms; IEEE-754 just cannot represent 100 × 2^(−1000) as a normal number. If a homework asks for an exact fraction, keep the (1/2)^(t/t½) form or a power of two, and do not trust a 0.000 display after extreme t.

In nuclear homework N is often activity (decays per second) or number of undecayed nuclei. In a chemistry demo it might be mass of a radioisotope, which is proportional to nucleus count if you stay in one species. In a toy “half the water each day” puzzle N is liters. The formula does not care which of those you meant as long as N and N0 are the same kind of quantity and the decay is actually exponential with a single t½. Mixing activity remaining with mass initial is a category error the solver cannot see.

Background subtraction belongs before you type N. If a counter reads 50 with 10 of background, the decaying signal is 40, not 50. Plugging 50 into a half-life solve against an N0 that was already background-corrected will invent a half-life. This page will happily compute that wrong t½ because 50 and N0 are just numbers. The carbon-14 warning is the same idea at larger scale: calibration and background are not inside the exponential button.

Pharmaceutical “half-life of a drug” in plasma is often fitted from a log-linear slope after distribution. Using this calculator for N = concentration, N0 = C0, t = hours, t½ = hours is valid only for that one-compartment log-linear piece. Multi-compartment curves need more than one exponential; a single t½ will not fit the whole time course. Honest limit: one λ, one t½, decay only.