Half-Life Formula: Easy Calc & Examples

This calculator computes the remaining quantity of a substance after a given time using the standard exponential decay formula, and it also shows the elapsed time as a fraction of the half-life. You enter the initial amount, the half-life, and the elapsed time, then get the remaining amount and the decay constant. The formula is N(t) = N₀ × (1/2)^(t / t½), where N₀ is the starting amount, t½ is the half-life, and t is the time that has passed.

Formula

The half-life formula describes exponential decay. In words: the amount remaining equals the initial amount multiplied by one-half raised to the power of the number of half-lives that have passed.

Symbolically:

N(t) = N₀ × (1/2)^(t / t½)

• N(t) – amount remaining after time t (same unit as Nâ‚€, e.g., grams, milligrams, moles, becquerels)
• Nâ‚€ – initial amount at t = 0
• t – elapsed time (any unit, but must match the half-life unit)
• t½ – half-life (time for the amount to reduce to half)

An equivalent form uses the decay constant λ (lambda):

N(t) = N₀ × e^(−λt), where λ = ln(2) / t½ ≈ 0.693 / t½

Both forms give the same result. The first is often easier because it avoids logarithms.

Worked Example

Using the calculator defaults: initial amount N₀ = 100 g, half-life t½ = 8 hours, elapsed time t = 16 hours.

1. Calculate the number of half-lives: t / t½ = 16 / 8 = 2.
2. Raise 0.5 to that power: 0.5² = 0.25.
3. Multiply by the initial amount: 100 g × 0.25 = 25 g.

So after 16 hours, 25 g remains. The decay constant is λ = ln(2) / 8 ≈ 0.08664 per hour. The fraction remaining is 0.25, or 25%.

When to Use the Half-Life Formula

The half-life formula applies to any process that follows first-order kinetics – meaning the rate of decay is proportional to the amount present. Common examples include:

• Radioactive decay (carbon-14, iodine-131, etc.)
• Drug elimination from the body (pharmacokinetics)
• Chemical reactions that are first-order
• Discharge of capacitors in electronics (though often expressed as time constant Ï„)

It is not appropriate for zero-order reactions (where decay rate is constant) or second-order reactions (where rate depends on concentration squared). For those, different equations are needed.

How to Read the Result

The remaining amount tells you exactly how much of the original substance is left after the elapsed time. The fraction remaining is a dimensionless number between 0 and 1 (or 0% to 100%). For example, a fraction of 0.125 means 12.5% remains – that is three half-lives (0.5³ = 0.125).

The decay constant λ is the probability per unit time that a single atom decays. A larger λ means faster decay. The half-life and λ are inversely related: shorter half-life → larger λ.

In practical terms, after one half-life, 50% remains; after two, 25%; after three, 12.5%; after ten, about 0.1%. The amount never reaches zero mathematically, but it becomes negligible.

Common Mistakes and Pitfalls

• Unit mismatch: The elapsed time and half-life must use the same unit. If half-life is in hours and time is in days, convert first (1 day = 24 hours).
• Using the wrong base: The formula uses 0.5, not 0.5 times something else. Some people mistakenly use (1/2)^(t) without dividing by the half-life.
• Confusing half-life with mean lifetime: The mean lifetime (Ï„) is longer: Ï„ = t½ / ln(2) ≈ 1.443 × t½. They are not the same.
• Rounding too early: In multi-step problems, keep extra digits until the final answer.
• Applying to non-first-order processes: For example, bacterial growth (doubling) uses a similar but different formula with base 2 and a positive exponent.

Limits of the Formula

The half-life formula assumes a constant half-life, which is true for most radioactive isotopes and many chemical reactions under fixed conditions. However, some processes have half-lives that change with conditions:

• Biological half-life can vary with age, health, and organ function.
• Chemical reaction half-lives can depend on temperature and concentration.
• In nuclear physics, the half-life is essentially constant for a given isotope, but it can be altered by extreme conditions (e.g., electron capture in highly ionized atoms).

Also, the formula gives the expected remaining amount. In radioactive decay, the actual number of atoms that decay in a short time is random, but for large numbers of atoms (e.g., >10¹�), the statistical fluctuation is negligible.

Frequently Asked Questions

What is the half-life formula in simple terms?

It’s N(t) = N₀ × (1/2)^(t/t½). You take the initial amount, halve it for each half-life that passes. For example, if the half-life is 10 days and 30 days pass, you halve three times: N₀ → N₀/2 → N₀/4 → N₀/8.

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

Rearrange the formula: t½ = t / (log₂(N₀/N(t))). In practice, use logarithms: t½ = t × ln(2) / ln(N₀/N(t)). For example, if 100 g decays to 25 g in 16 hours, then N₀/N = 4, log₂(4)=2, so t½ = 16/2 = 8 hours.

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

The half-life is the time for half the substance to decay. The decay constant λ is the probability of decay per unit time. They are related by λ = ln(2)/t½. A short half-life gives a large λ, meaning fast decay.

Can the half-life formula be used for drug elimination?

Yes, but with caution. The biological half-life of a drug is often estimated using this formula, but it assumes first-order elimination, which is true for many drugs at therapeutic doses. However, some drugs follow zero-order kinetics (e.g., alcohol at high concentrations), where the formula does not apply.

How do I calculate the time for a substance to decay to a specific percentage?

Use the formula t = t½ × log₂(N₀/N(t)). For example, to find when 10% remains, N₀/N = 10, log₂(10) ≈ 3.32. So t = 3.32 × t½. If the half-life is 8 hours, that’s about 26.6 hours.

Does the half-life ever change?

For a given radioactive isotope under normal conditions, the half-life is constant. But for biological or chemical processes, it can change with temperature, pressure, concentration, or the organism’s metabolic state. Always check if the conditions match the assumptions.

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in N0 | Initial amount | g | 100 | 0
in th | Half-life | h | 8 | 0
in t | Elapsed time | h | 16 | 0
out remaining | Remaining amount | g | N0 * 0.5^(t / th) | 2
out decay_const | Decay constant λ | per h | ln(2) / th | 4
out fraction | Fraction remaining | - | 0.5^(t / th) | 3