Half-Life Formula: Easy Calculation Guide

This calculator computes the remaining quantity of a substance after a given time using the standard half-life formula. You enter the initial amount, the half-life, and the elapsed time, and it returns the remaining amount and the fraction left. The formula is N(t) = N₀ × (1/2)^(t / t½), where N₀ is the initial amount, t is the elapsed time, and t½ is the half-life.

Formula

The half-life formula describes how an exponentially decaying quantity decreases over time. In words: the remaining amount equals the initial amount multiplied by one-half raised to the power of the number of half-lives that have passed. The number of half-lives is simply the elapsed time divided by the half-life duration.

Symbolically: N(t) = N₀ × (1/2)^(t / t½)

• N(t) – the remaining quantity after time t (same unit as Nâ‚€)
• Nâ‚€ – the initial quantity (grams, milligrams, moles, counts, etc.)
• t – the elapsed time (any time unit: seconds, hours, years)
• t½ – the half-life (same time unit as t)

The formula works for any unit of quantity and any unit of time, as long as t and t½ use the same time unit. It applies to radioactive decay, drug elimination, chemical reactions, and capacitor discharge—any process where the rate of decay is proportional to the amount present.

Worked Example

Using the calculator defaults: initial amount N₀ = 100 g, half-life t½ = 10 hours, elapsed time t = 30 hours.

1. Calculate the number of half-lives: t / t½ = 30 / 10 = 3
2. Compute the decay factor: (1/2)^3 = 1/8 = 0.125
3. Multiply by the initial amount: 100 × 0.125 = 12.5 g

The calculator shows 12.5 g remaining, which matches. The fraction remaining is 0.125, so 12.5% of the original substance is left after 30 hours.

When to Use the Half-Life Formula

The half-life formula is used in any field that deals with exponential decay. Common applications include:

• Nuclear physics: determining how much of a radioactive isotope remains after a given period, or how long until a sample decays to a safe level.
• Medicine: calculating how long a drug stays in the body—important for dosing intervals and withdrawal times.
• Archaeology: radiocarbon dating uses the half-life of carbon-14 (about 5,730 years) to estimate the age of organic materials.
• Environmental science: estimating how long pollutants or pesticides persist in soil or water.
• Electronics: predicting how quickly a capacitor discharges through a resistor (though that uses the time constant Ï„, not half-life, the math is similar).

Use the formula whenever you know the half-life and want the remaining amount at a specific time, or when you know the remaining amount and want to find the elapsed time (by solving for t).

How to Read the Result

The result tells you exactly how much of the original substance is still present. A fraction remaining of 1.0 means none has decayed; 0.5 means one half-life has passed; 0.25 means two half-lives; and so on. The result never reaches zero mathematically—it just gets infinitely small. In practice, after about 10 half-lives, less than 0.1% remains, which is often considered negligible.

When reading the result, keep the units in mind. If you entered the initial amount in grams, the remaining amount is in grams. If you entered the half-life in hours, the elapsed time must also be in hours. The fraction remaining is unitless and always between 0 and 1.

Common Mistakes and Pitfalls

The most frequent error is mixing time units. If the half-life is given in days but the elapsed time is in hours, you must convert one to the other before dividing. For example, a half-life of 2 days and an elapsed time of 12 hours: convert 2 days to 48 hours, then t/t½ = 12/48 = 0.25 half-lives.

Another common mistake is confusing half-life with the decay constant (λ). The relationship is t½ = ln(2)/λ, which is about 0.693/λ. If you have the decay constant, use that to find the half-life first, then apply the formula.

Rounding too early can also cause errors. Keep at least three significant figures in intermediate steps, especially when the number of half-lives is not an integer. For instance, if t/t½ = 2.7, the decay factor is 0.5^2.7 ≈ 0.154, not 0.125 (which would be 3 half-lives).

Finally, remember that the formula assumes a constant half-life. For some processes (like certain chemical reactions of order other than first), the “half-life� changes over time, and this simple formula does not apply.

Limits of the Formula

The half-life formula is exact only for first-order exponential decay, where the rate of decay is proportional to the amount present. This holds for most radioactive isotopes and for many drugs that follow first-order kinetics. However, it does not apply to:

• Zero-order decay: where a constant amount decays per unit time (e.g., alcohol elimination at high concentrations).
• Second-order reactions: where the half-life depends on the initial concentration.
• Systems with multiple decay paths: where the effective half-life is a combination of several half-lives.

Also, the formula gives the theoretical remaining amount, but real measurements can deviate due to measurement error, environmental factors, or if the substance is not pure. In medicine, individual metabolism varies, so the actual half-life can differ from the population average.

Frequently Asked Questions

How do I find the half-life if I know the initial and remaining amounts?

Rearrange the formula to solve for t½: t½ = t × ln(2) / ln(N₀ / N(t)). For example, if 100 g decays to 25 g in 20 hours, then t½ = 20 × ln(2) / ln(100/25) = 20 × 0.693 / ln(4) ≈ 20 × 0.693 / 1.386 = 10 hours.

What is the difference between half-life and decay constant?

The decay constant (λ) is the probability per unit time that a single atom will decay. The half-life is the time for half the sample to decay. They are related by t½ = ln(2)/λ. If you have λ, you can compute the half-life and then use the half-life formula.

Can the half-life formula be used for doubling (growth) as well?

Yes, the same exponential formula works for growth if you replace “half-life� with “doubling time.� The formula becomes N(t) = N₀ × 2^(t / t_double). The calculator here is for decay, but the math is identical—just replace 0.5 with 2.

Why does the half-life never reach zero?

Because exponential decay is asymptotic: the amount approaches zero but never actually reaches it in finite time. In practice, after 10 half-lives, less than 0.1% remains, and after 20 half-lives, less than one millionth. For most purposes, that is effectively zero.

Does the half-life change with temperature or pressure?

For radioactive decay, the half-life is essentially constant regardless of temperature, pressure, or chemical state—it is a property of the nucleus. For chemical reactions, however, the half-life can depend on temperature and concentration, so the simple formula may not apply.

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in N0 | Initial amount | g | 100 | 0
in t_half | Half-life | hours | 10 | 0
in t | Elapsed time | hours | 30 | 0
out remaining | Remaining amount | g | N0 * (0.5)^(t / t_half) | 2
out fraction | Fraction remaining | - | (0.5)^(t / t_half) | 3