Half-Life Calculator

Half-life: The time it takes for a quantity to drop to half its starting value.

Initial amount: The amount present at the start of the decay process.

Remaining amount: The amount left after some time has passed.

Elapsed time: How long the sample has been decaying.

Decay constant: A number that tells how fast exponential decay happens. A larger value means faster decay.

Mean lifetime: The average lifetime in the exponential model. It is longer than the half-life.

Percent remaining: The share of the original sample still present, written as a percent.

Percent decayed: The share of the original sample that has disappeared, written as a percent.

Mistake: Entering Initial amount in grams and Remaining amount in milligrams.
Fix: Convert one amount first so Initial amount and Remaining amount use the same unit.

Mistake: Mixing up Elapsed time unit and Half-life unit and then reading the answer as if both were the same.
Fix: Check both unit fields carefully. The calculator converts units, so read Calculated value in the unit tied to the mode you chose.

Mistake: Typing a Remaining amount larger than Initial amount for a decay problem.
Fix: For this model, Remaining amount must be less than or equal to Initial amount . Recheck the numbers or the problem statement.

Mistake: Entering 0 for Half-life , Elapsed time , or an amount that must be positive.
Fix: Use positive values for any known decay quantity. If time is truly zero, then Remaining amount must equal Initial amount .

Mistake: Expecting Percent decayed and Percent remaining to be independent outputs.
Fix: They always add to 100%. If one looks wrong, check the amount inputs and time values.

Mistake: Reading Decay constant as a percent instead of a rate.
Fix: Decay constant is an inverse-time rate, such as per year or per day, not a percent.

• This calculator uses the standard exponential decay model, so it only fits processes that decay at a constant proportional rate.
• Initial amount and Remaining amount must be in the same amount unit.
• Elapsed time and Half-life may use different units, but both are converted internally before solving.
• For a decay interpretation, Remaining amount cannot be greater than Initial amount.
• If Elapsed time is 0, then Remaining amount must equal Initial amount or the inputs are inconsistent.
• The model approaches zero over time but does not reach exactly zero in finite time, so a remaining amount of 0 cannot be used in logarithm-based solve modes.
• If the amounts show no decrease over a positive time interval, the implied half-life is infinite rather than a normal finite value.

Core model

This calculator uses the standard half-life form of exponential decay.

N(t) = N0 * (1/2)^(t / t_half)

Here, N(t) is the remaining amount, N0 is the initial amount, t is elapsed time, and t_half is half-life.

Equivalent conversion formulas

The calculator also converts between half-life, decay constant, and mean lifetime.

N(t) = N0 * e^(-lambda * t)

t_half = ln(2) / lambda

tau = 1 / lambda

tau = t_half / ln(2)

lambda means decay constant, and tau means mean lifetime. The calculator uses ln(2) = 0.6931471805599453.

Mode-specific solving

Depending on Solve for, the calculator rearranges the same model.

t = t_half * ln(N0 / N) / ln(2)

t_half = t * ln(2) / ln(N0 / N)

N0 = N / (1/2)^(t / t_half)

N = N0 * (1/2)^(t / t_half)

After the main value is found, it also computes percent outputs.

percent remaining = 100 * N / N0

percent decayed = 100 - percent remaining

Time-unit conversion

To prevent unit mismatch errors, Elapsed time and Half-life are first converted to a common base unit of seconds using 60 seconds per minute, 3,600 seconds per hour, 86,400 seconds per day, and 31,557,600 seconds per Julian year. The final answer is then reported in the unit that matches the selected mode.

Worked mini-example

Suppose Initial amount is 100, Remaining amount is 25, and Elapsed time is 10 years. Then the sample went from 100 to 25, which means it fell to one-fourth of the original amount. One-fourth equals two half-lives, so the half-life must be 5 years.

t_half = 10 * ln(2) / ln(100 / 25) = 10 * ln(2) / ln(4) = 5

From that, the decay constant is about 0.1386 per year, and the mean lifetime is about 7.2135 years.

Assumptions behind the result

The result is only valid when the process follows exponential decay with a constant half-life. Real measurements may differ because of rounding, measurement error, or because the physical system does not follow one simple decay rate over the whole time period.