Half Life Calculation Formula Explained

This calculator computes the remaining quantity of a substance after a given time using the half-life formula, and it also works in reverse to find the half-life itself from initial and remaining amounts. You enter the initial quantity, the elapsed time, and either the half-life or the remaining amount, and you get the missing value instantly. The formula is N(t) = N₀ × (1/2)^(t / t½), where N₀ is the initial amount, t is elapsed time, and t½ is the half-life.

Formula

The half-life formula describes exponential decay: after each half-life period, exactly half of the substance remains. Written mathematically:

N(t) = N₀ × (1/2)^(t / t½)

• N(t) – amount remaining after time t (same unit as Nâ‚€, e.g., grams, moles, becquerels)
• Nâ‚€ – initial amount at t = 0
• 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. For example, if t = 10 hours and t½ = 5 hours, then t / t½ = 2, meaning two half-lives have elapsed, so the fraction remaining is (1/2)² = 0.25.

Worked Example

Using the default values from the calculator above:

• Initial amount Nâ‚€ = 100 g
• Elapsed time t = 10 hours
• Half-life t½ = 5 hours

Step 1: Compute the number of half-lives: t / t½ = 10 / 5 = 2

Step 2: Raise 0.5 to that power: (0.5)² = 0.25

Step 3: Multiply by the initial amount: 100 × 0.25 = 25 g

The calculator shows 25.0000 g remaining, which matches. The fraction remaining is 0.25, meaning 25% of the original substance is left.

When to Use the Half-Life Formula

Half-life calculations appear in several fields, and the formula is the same everywhere:

• Nuclear physics – decay of radioactive isotopes (e.g., Carbon-14 has a half-life of about 5,730 years).
• Medicine – how long a drug stays in the body; a drug with a 4-hour half-life drops to half its concentration every 4 hours.
• Chemistry – first-order reaction rates, where the half-life is constant regardless of concentration.
• Environmental science – persistence of pollutants or pesticides in soil.

In all cases, the formula assumes exponential decay, which means the rate of decay is proportional to the amount present. This holds for radioactive decay and most first-order chemical reactions, but not for zero-order or second-order reactions where the half-life changes over time.

How to Read the Result

The remaining amount tells you how much of the original substance is left after the specified time. The fraction remaining is a dimensionless number between 0 and 1 (or 0% to 100%). For example:

• After 1 half-life: 50% remains
• After 2 half-lives: 25% remains
• After 3 half-lives: 12.5% remains
• After 10 half-lives: about 0.1% remains (0.5¹â�° ≈ 0.00098)

In practical terms, a substance is often considered “gone� after about 10 half-lives, though mathematically it never fully reaches zero. For radioactive waste, regulators use multiples of half-lives to define safe storage times, but the formula itself only gives the fraction.

Common Mistakes and Unit Mix-Ups

The most frequent error is using inconsistent time units. If the half-life is given in years, the elapsed time must also be in years. Convert first if needed:

• 1 day = 24 hours
• 1 year ≈ 365.25 days (for precise radiometric dating)
• 1 hour = 60 minutes

Another common mistake is confusing half-life with the decay constant (λ). The relationship is t½ = ln(2) / λ, where ln(2) ≈ 0.693. If you accidentally use the decay constant directly in the exponent, you’ll get the wrong answer. The formula above uses half-life directly, so no conversion is needed.

Rounding too early can also cause errors. For instance, if t / t½ is 2.7, don’t round to 3. Use the exact exponent: (0.5)^2.7 ≈ 0.154. The calculator handles this automatically.

Limitations of the Formula

The half-life formula assumes a constant half-life, which is true for radioactive isotopes and first-order reactions. However, it does not apply to:

• Second-order reactions – half-life depends on initial concentration and increases as the reaction proceeds.
• Zero-order reactions – half-life decreases as concentration drops.
• Biological processes – drug clearance often follows more complex kinetics (e.g., Michaelis-Menten), so a single half-life is an approximation.
• Mixtures – if you have multiple isotopes or compounds with different half-lives, the overall decay is a sum of exponentials, not a single half-life.

Also, the formula gives the theoretical remaining amount. In real life, measurement errors, impurities, or environmental factors can cause deviations, especially at very small amounts.

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 / (log₂(N₀ / N(t))). In words: divide the elapsed time by the base-2 logarithm of the ratio of initial to remaining amount. For example, if 100 g decays to 25 g in 10 hours, then N₀/N(t) = 4, log₂(4) = 2, so t½ = 10/2 = 5 hours.

What does half-life mean in simple terms?

Half-life is the time it takes for half of the substance to decay or be eliminated. After one half-life, 50% remains; after two, 25%; after three, 12.5%. It does not mean that the substance disappears after two half-lives – it just keeps halving.

Is half-life the same as the decay constant?

No, but they are related. The decay constant (λ) is the probability of decay per unit time, and the half-life is t½ = ln(2)/λ ≈ 0.693/λ. The formula N(t) = N₀ × e^(-λt) is equivalent to the half-life formula, but uses λ instead of t½.

Can half-life be used for non-radioactive substances?

Yes, any process that follows exponential decay has a half-life. This includes drug metabolism, chemical reactions of first order, and even the discharge of a capacitor in electronics. The formula is the same, but the underlying mechanism differs.

Why is the half-life constant for radioactive elements?

Radioactive decay is a random quantum process, and the probability of decay per nucleus per unit time is constant. This makes the decay exponential, so the half-life is independent of how much material you have, temperature, pressure, or chemical state (with rare exceptions like electron capture, which can be slightly affected by chemical environment).

How do I convert half-life to a percentage remaining?

Multiply the fraction remaining by 100. For example, a fraction of 0.25 corresponds to 25%. The calculator’s “Fraction remaining� output gives you this directly; just multiply by 100 for a percentage.

Related

• Change Calculator Icon: How to Do It
• Change Calculator: How to Count Money Fast
• Change Calculator Python: Code & Examples

in N0 | Initial amount | g | 100 | 0
in t | Elapsed time | h | 10 | 0
in th | Half-life | h | 5 | 0
out remaining | Remaining amount | g | N0 * (0.5)^(t / th) | 4
out fraction | Fraction remaining | - | (0.5)^(t / th) | 4