Half-Life Reaction Formula and Calculator

The half-life reaction calculator uses a substance’s initial amount, elapsed time, and half-life to calculate how much remains in a first-order decay process. Enter those three values in consistent time units; the result is the remaining amount and the fraction or percentage left.

Formula

The standard half-life equation for first-order decay is N(t) = N₀(1/2)^t/t_1/2. An equivalent exponential form is N(t) = N₀e^−λt, where λ = ln(2)/t_1/2. The calculator uses the exponential form: N(t) = N₀ × exp(−ln(2) × t/t_1/2).

• N(t) is the amount remaining after elapsed time t, in the same amount unit used for N₀.
• N₀ is the initial amount, such as a mass, number of particles, or concentration.
• t is elapsed time.
• t_1/2 is the half-life: the time required for the amount to fall to half its value in this model.
• λ is the first-order rate constant, with units reciprocal to time, such as s⁻¹ or day⁻¹.

The time units for t and t_1/2 must match. For example, use hours for both or convert both to seconds. The ratio t/t_1/2 is dimensionless, so the exponential function receives a unitless value. The fraction remaining is (1/2)^t/t_1/2; multiplying that fraction by 100 gives the percentage remaining.

Worked Example

Use the calculator’s defaults: initial amount N₀ = 100 amount units, elapsed time t = 10 time units, and half-life t_1/2 = 5 time units.

1. Calculate the number of half-lives elapsed: t/t_1/2 = 10/5 = 2.
2. Calculate the fraction remaining: (1/2)² = 0.25. Equivalently, exp(−ln(2) × 2) = exp(−1.38629436) ≈ 0.25.
3. Calculate the remaining amount: N(t) = 100 × 0.25 = 25.00 amount units.
4. Convert the fraction to a percentage: 100 × 0.25 = 25.00%.

The outputs are 25.00 amount units remaining, a fraction of 0.2500, and 25.00% remaining. They describe the same result in different forms.

Assumptions

• The process follows first-order decay with a constant rate and a constant half-life over the elapsed period.
• The measured quantity is proportional to the amount of the decaying substance. The equation does not account for material entering or leaving the system, or for a changing reaction environment.
• Initial amount and remaining amount use the same units; elapsed time and half-life use the same time units.

How Half-Life Works

Half-life describes a repeated proportional decrease, not a fixed quantity lost during each interval. After one half-life, half remains; after two, one quarter remains; after three, one eighth remains. Each interval halves the amount that exists at the start of that interval. Thus a sample starting at 100 units loses 50 in the first half-life, 25 in the next, and 12.5 in the third.

For a first-order process, the fraction remaining after n half-lives is (1/2)ⁿ, where n = t/t_1/2. This also works for a non-integer number of half-lives. If 1.5 half-lives pass, the fraction left is (1/2)^1.5 ≈ 0.3536, or about 35.36%. The formula is continuous, so elapsed time need not be a whole number of half-lives.

In chemistry, first-order kinetics means the reaction rate is proportional to the concentration of the reacting substance. For A → products, a common integrated rate law is ln([A]_t/[A]₀) = −kt, where k is the rate constant. In that setting, the half-life is t_1/2 = ln(2)/k. Here concentration can stand in for amount if the reaction volume stays constant. Radioactive decay follows the same mathematical form, though its underlying process is nuclear rather than a chemical reaction.

When to Use This Formula

Use this equation when a process is reasonably modeled by first-order decay and its half-life is known or can be determined from a rate constant. It can estimate remaining reactant in a first-order reaction, remaining radioactive material under the usual decay model, or another quantity that decreases exponentially with a constant proportional rate. The equation calculates a mathematical expectation; it does not identify the correct half-life for a particular substance or reaction.

If the rate constant k is known instead of the half-life, first calculate t_1/2 = ln(2)/k, keeping reciprocal time units consistent. Conversely, if a half-life is measured, calculate k = 0.693147…/t_1/2. A value of k in min⁻¹ gives a half-life in minutes; a value in s⁻¹ gives one in seconds.

For bench work, this calculator supports study and planning arithmetic only; follow your lab method, the safety data sheet (SDS) and your supervisor.

Interpreting Results and Common Errors

• Mixing time units: If the elapsed time is in minutes but the half-life is in hours, convert one before calculating. A factor-of-60 mismatch can radically alter the answer.
• Confusing half-life with reaction time: A half-life is not the time for the substance to disappear. In the ideal exponential model, the amount approaches zero but does not reach exactly zero in a finite time.
• Subtracting the same amount each interval: That would describe a linear decrease, not first-order decay. The amount lost per half-life becomes smaller as the remaining amount shrinks.
• Using the wrong starting quantity: N₀ is the amount at the start of the interval being modeled. If timing begins after some decay has already occurred, use the amount at that new starting point.
• Rounding too early: Keep full precision in intermediate calculations and round the final result to a sensible number of significant figures. The calculator’s displayed decimals are not a claim that measurements are that precise.

A calculated fraction above 1 would require negative elapsed time under this model, meaning a point before the chosen starting time. This calculator accepts nonnegative elapsed times and a positive half-life. A zero half-life would make the formula undefined.

Limits and Alternative Kinetic Models

Not every reaction has a constant half-life. For a zero-order reaction, the rate is independent of reactant concentration and the amount decreases by a fixed quantity per unit time; its half-life depends on the starting concentration. For a second-order reaction, the half-life also depends on the starting concentration. Applying the first-order equation to either case can give a misleading estimate.

Real systems can depart from an ideal model because conditions change, multiple reactions occur, products react further, or measurements include background signal. A reported half-life may also be an estimate with experimental uncertainty. Check that the source’s value applies to the relevant substance, reaction conditions, and units. When the observed data do not follow a straight line in a plot of ln(concentration) against time, a single first-order rate constant may not describe the full process.

Frequently Asked Questions

What is the half-life reaction formula?
The first-order formula is N(t) = N₀(1/2)^t/t_1/2, or equivalently N(t) = N₀e^−λt, with λ = ln(2)/t_1/2.

How do I calculate the amount left after a half-life?
Divide elapsed time by the half-life to find how many half-lives have passed, raise 1/2 to that value, and multiply by the initial amount. After exactly one half-life, 50% remains.

How many half-lives until 10% remains?
Set (1/2)ⁿ = 0.10. Then n = ln(0.10)/ln(0.5) ≈ 3.322 half-lives. Multiply 3.322 by the half-life to get the elapsed time.

Does half-life depend on the starting amount?
For an ideal first-order reaction or radioactive decay, the half-life does not depend on the initial amount. For zero-order and second-order reactions, it generally does.

Can a half-life reaction reach zero?
The ideal exponential equation predicts that the amount gets progressively smaller and approaches zero, but it does not become exactly zero at a finite time. In practice, measurements eventually fall below a detection limit.

Related

• Theoretical Yield Formula and Worked Example
• Percent Composition Calculator and Formula
• Empirical to Molecular Formula Calculator

in n0 | Initial amount | amount units | 100 | 0
in t | Elapsed time | time units | 10 | 0
in th | Half-life | time units | 5 | 0
out nr | Amount remaining | amount units | n0 * exp(-ln(2) * t / th) | 2
out f | Fraction remaining | - | exp(-ln(2) * t / th) | 4
out p | Percent remaining | % | 100 * f | 2