Half Life Decay Calculator: Formula & How to Use

A half-life decay calculator computes how much of a radioactive substance remains after a given time, based on its half-life. You enter the initial amount, the elapsed time, and the half-life, and it returns the remaining quantity and often the fraction decayed. The calculation uses the exponential decay formula N(t) = N₀ × (1/2)^(t / t½).

Formula

The half-life formula is:

N(t) = N₀ × (1/2)^(t / t½)

• N(t) – remaining quantity after time t (same unit as Nâ‚€)
• Nâ‚€ – initial quantity (grams, moles, becquerels, etc.)
• t – elapsed time (any unit, as long as it matches t½)
• t½ – half-life (same time unit as t)

The exponent t / t½ tells you how many half-lives have passed. Each half-life multiplies the amount by 0.5, so two half-lives leave 0.5² = 0.25, three leave 0.125, and so on.

Worked Example

Using the default values: initial amount = 100 g, elapsed time = 10 days, half-life = 5 days.

1. Number of half-lives: t / t½ = 10 / 5 = 2
2. Fraction remaining: (1/2)^2 = 0.25
3. Remaining amount: 100 g × 0.25 = 25 g
4. Amount decayed: 100 g – 25 g = 75 g

The calculator shows 25 g remaining, which matches exactly.

What Is Half-Life?

Half-life is the time required for half of the atoms in a radioactive sample to decay. It is a statistical property: you cannot predict when a single atom will decay, but for a large number of atoms, the decay rate is remarkably consistent. Half-lives range from fractions of a second (e.g., polonium-214 at 164 microseconds) to billions of years (e.g., uranium-238 at 4.5 billion years).

The concept applies beyond radioactivity. It is used in pharmacokinetics to describe how quickly a drug is eliminated from the body, in nuclear medicine for tracer doses, and in carbon dating for archaeological samples. In every case, the same exponential decay law applies, so the calculator works for any process that follows first-order kinetics.

How to Use the Calculator

Enter three values:

• Initial amount – the starting quantity in any unit (grams, milligrams, becquerels, curies, moles).
• Elapsed time – how long the decay has been happening.
• Half-life – the known half-life of the substance.

The calculator returns the remaining amount, the fraction remaining, and the amount decayed. You can use any consistent time unit (seconds, hours, years) as long as elapsed time and half-life use the same unit. The amount unit does not matter because the formula is linear – if you double the initial amount, the remaining amount doubles too.

Common Mistakes and Misconceptions

• Mixing time units – entering half-life in years but elapsed time in days gives a wrong exponent. Always convert to the same unit first.
• Confusing half-life with mean life – mean life (Ï„) is the average time an atom survives, equal to t½ / ln(2) ≈ 1.443 × t½. The decay formula using mean life is N(t) = Nâ‚€ × e^(−t/Ï„), which gives the same result.
• Assuming linear decay – after two half-lives, 25% remains, not 0%. The decay is exponential, not linear.
• Rounding too early – for small remaining amounts, rounding the fraction to 2 or 3 decimals can cause large errors. Keep extra digits until the final step.

Limits and Accuracy

The formula assumes a constant half-life and a large number of atoms. For extremely small samples (a few atoms), statistical fluctuations become significant, and the exact remaining amount is not deterministic. Also, the half-life value itself is an experimental measurement with uncertainty; for long-lived isotopes like uranium-238, the half-life is known to about 0.1% precision, but for some isotopes the uncertainty is larger.

The calculator does not account for decay chains. If a radioactive parent decays into a radioactive daughter, the total activity of the sample may initially increase or decrease in a more complex way. For a single isotope, however, the simple exponential formula is accurate.

Real-World Applications

• Nuclear medicine – determining how much of a radioactive tracer remains in a patient after a scan.
• Carbon dating – estimating the age of organic materials by comparing the remaining carbon-14 to the initial amount.
• Waste management – calculating how long radioactive waste must be stored before it decays to safe levels.
• Pharmacology – predicting drug concentration in the body over time, using the biological half-life.

Frequently Asked Questions

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

The decay constant (λ) is the probability per unit time that an atom decays. It relates to half-life by λ = ln(2) / t½. The decay formula can also be written as N(t) = N₀ × e^(−λt). Both forms are equivalent; the calculator uses the half-life form because it is more intuitive.

Can I use this calculator for non-radioactive decay?

Yes, if the process follows first-order kinetics. For example, the elimination of many drugs from the body is described by a half-life. However, some processes (like zero-order reactions) do not follow this law, so check the underlying kinetics before applying the formula.

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

Rearrange the formula: t½ = t × ln(2) / ln(N₀ / N(t)). You need the elapsed time, the initial amount, and the remaining amount. This is useful for experimental data or when the half-life is not listed in reference tables.

What units should I use for the initial amount?

Any unit works because the formula is linear. Common choices are grams, milligrams, becquerels (for activity), or moles. Just be consistent – if you enter grams, the remaining amount will also be in grams.

Why does the calculator show a fraction remaining when I only asked for the amount?

The fraction remaining is a useful intermediate value. It tells you what proportion of the original sample is left, independent of the amount unit. For example, a fraction of 0.125 means 12.5% remains, which is often easier to interpret than a raw mass.

Is there a way to calculate the time needed to reach a specific remaining amount?

Yes, rearrange the formula to solve for t: t = t½ × ln(N₀ / N(t)) / ln(2). For instance, to reduce a sample to 1% of its original, you need about 6.64 half-lives (since 0.5^6.64 ≈ 0.01). This is a common calculation in waste management and nuclear safety.

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in n0 | Initial amount | g | 100 | 0
in t  | Elapsed time | days | 10 | 0
in hl | Half-life | days | 5 | 0
out rem | Remaining amount | g | n0 * (0.5)^(t / hl) | 4
out frac | Fraction remaining | - | (0.5)^(t / hl) | 4
out decayed | Amount decayed | g | n0 - n0 * (0.5)^(t / hl) | 4