Half-Life Calculator: Formula and Worked Examples

A half-life calculator works out how much of a substance remains after a given time, or how long it takes to reach a specified amount. Enter the starting amount, elapsed time, and half-life; the result shows the remaining amount and the number of half-lives elapsed.

Formula

The remaining amount after repeated half-life intervals is R = A × (1/2)^t/T. Here, R is the amount remaining in the same units as the starting amount, A; t is elapsed time; and T is the half-life. The elapsed time and half-life must use the same time unit. The calculator also reports n = t/T, the number of half-lives elapsed.

This equation assumes exponential decay: a fixed fraction, rather than a fixed amount, is lost per unit of time. After one half-life, half remains; after two, one quarter remains. The calculator treats the inputs as nonnegative and the half-life as positive.

Worked Example

Using the default inputs, the starting amount is 100 units, elapsed time is 12 time units, and the half-life is 4 time units.

1. Find the number of half-lives: n = 12 / 4 = 3.
2. Apply the remaining-amount formula: R = 100 × 0.5³.
3. Evaluate the fraction: 0.5³ = 0.125, so R = 12.50 units.

The displayed results are 12.50 units remaining and 3.00 half-lives elapsed. The units chosen for time do not affect the answer as long as both time inputs match.

How Half-Life Works

A half-life is the time needed for an amount undergoing exponential decay to fall to half its initial value. The process repeats: each interval halves what is present at that point, not the amount originally present. With a starting amount of 80 grams, for example, the idealized amounts after successive half-lives are 40, 20, 10, and 5 grams.

That distinction explains why a substance does not generally disappear after one or two half-lives. In the mathematical model, the amount approaches zero but does not become exactly zero at any finite time. In real measurements, however, a tiny remaining amount can be below the detection limit or practically negligible.

Half-life is used in several fields, including radioactive decay, chemical processes, and models of how quantities diminish. The meaning and measurement of a “remaining amount” depend on the situation. For radioactive substances, calculations describe statistical behavior across many atoms; they do not predict the exact decay time of one particular atom.

When to Use the Calculator

Use the calculator when the starting quantity, elapsed duration, and half-life are known and you want an estimate of the amount that remains under the standard exponential model. It can also make comparisons easier: a duration equal to four half-lives leaves 1/16 of the starting amount, or 6.25%.

• Choose consistent units: if elapsed time is in days, enter the half-life in days too. Convert one value before calculating if they are given in different units.
• Keep amount units consistent: the answer uses the same unit as the starting amount, whether that is grams, milligrams, or another quantity.
• Interpret the output as a model result: it is not a direct measurement and can differ from observations because of uncertainty, measurement limits, or conditions outside the model.

For a given half-life, the starting amount changes the size of the answer but not the fraction remaining. For example, after three half-lives, 12.5% remains whether the starting amount was 8 units or 800 units.

Solving for Time or Starting Amount

The same relationship can be rearranged if a different quantity is unknown. If the starting and remaining amounts are known, solve for elapsed time with t = T × log(A/R) / log(2), where log can be any logarithm base as long as the same base is used in numerator and denominator. For instance, reaching one-eighth of the starting amount takes three half-lives because 1/8 = (1/2)³.

If the remaining amount and time are known, the starting amount is A = R × 2^t/T. These rearrangements require meaningful positive amounts. A measured value of zero cannot be used in the logarithm to calculate elapsed time; “zero” may instead mean the amount is below the instrument’s detection limit.

Common Mistakes and Limits

• Mixing time units: entering 30 days for elapsed time and 2 months for half-life without conversion gives a misleading result.
• Subtracting half the original amount each time: the loss per interval gets smaller because each half-life removes half of the amount then present.
• Confusing half-life with total lifetime: a half-life is not a deadline when all material vanishes.
• Rounding too early: retain unrounded values through intermediate steps, then round the final display to the precision justified by the inputs.

The simple formula may not fit a process whose decay rate changes, whose quantity is replenished, or whose behavior is not exponential. Some real systems also involve multiple components with different half-lives. A single half-life cannot fully describe those cases; use a model that accounts for the separate components and the available measurements.

Frequently Asked Questions

How do I calculate half-life from a graph?
Identify the starting amount and the time when the measured amount reaches half of it. That time difference is one half-life, provided the data follow an exponential-decay pattern. Repeating the comparison at lower amounts can help check whether the half-life is consistent.

How much remains after three half-lives?
One-eighth, or 12.5%, remains because (1/2)³ = 1/8. Multiply the starting amount by 0.125 to get the remaining amount.

Can the remaining amount be less than one whole atom?
A mathematical model can produce a fractional expected amount. Individual atoms are not divided by the calculation; in small samples, actual outcomes are discrete and can differ from the statistical average.

Does half-life depend on how much material I start with?
For a process described by a constant half-life, no: the half-life sets the time for the fraction to halve, while the starting quantity sets the scale of the amount remaining. Whether that model applies depends on the substance or process.

Why does a measured amount not match the calculator exactly?
Measurements have uncertainty and finite precision, and real conditions may differ from the assumptions of the model. Small predicted amounts can also be below a measuring instrument’s detection limit.

Related

• Calculate Your Age Online: Exact Age in Years, Months and Days
• Lower and Upper Limits: How to Calculate Both
• Calculate Impedance: Formula, Examples, and AC Circuits

in a | Starting amount | units | 100 | 0
in t | Elapsed time | time units | 12 | 0
in h | Half-life | time units | 4 | 0
out r | Remaining amount | units | a * 0.5 ^ (t / h) | 2
out n | Half-lives elapsed | half-lives | t / h | 2