Half Life Calculator: Remaining Amount

This half-life calculator works out how much of a 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 the fraction left. The calculation uses the standard exponential decay formula, so it works for any radioactive isotope or any process that follows first-order kinetics.

Formula

The remaining amount N(t) after time t is given by:

N(t) = N₀ × (1/2)^{t / T_1/2}

• N_{0 = initial amount (grams, moles, becquerels, or any unit)}
• t = elapsed time (same time unit as half-life)
• T_1/2 = half-life (time for half the substance to decay)

The exponent t / T_1/2 is the number of half-lives that have passed. Each half-life reduces the amount by a factor of 0.5. The fraction remaining is simply 0.5 raised to that power.

Worked Example

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

1. Number of half-lives = 10 / 5 = 2
2. Fraction remaining = 0.5² = 0.25
3. Remaining amount = 100 × 0.25 = 25 g

The calculator shows 25 g remaining and a fraction of 0.25, matching the manual calculation.

Understanding Half-Life

Half-life is the time required for half of the atoms in a radioactive sample to decay. It is a constant for a given isotope, unaffected by temperature, pressure, or chemical state. For example, carbon-14 has a half-life of about 5,730 years, while iodine-131 has a half-life of about 8 days. The concept also applies to drug elimination from the body, chemical reactions, and even capacitor discharge, as long as the process follows first-order kinetics.

Because decay is exponential, the amount never reaches zero mathematically. After 10 half-lives, only about 0.1% of the original remains; after 20 half-lives, it is about one part per million.

How to Use the Calculator

Enter the initial amount in any unit (grams, milligrams, becquerels, moles). Then enter the elapsed time and the half-life in the same time unit (hours, days, years). The calculator returns the remaining amount in the same unit as the initial amount, and the fraction remaining as a decimal (0.25 = 25%).

• If you know the fraction remaining and want the half-life, rearrange the formula: T_1/2 = t / (log_0.5(fraction))
• If you want the time to reach a certain amount, use t = T_1/2 × log_0.5(N/N₀)

This calculator handles the direct case: given initial amount, time, and half-life, find the remaining amount.

Common Mistakes and Limits

The most frequent error is mixing time units. If the half-life is in years and the elapsed time is in days, convert one to the other first. Another mistake is using the decay constant λ instead of the half-life; the relationship is T_1/2 = ln(2)/λ, so the formula can also be written N(t) = N₀ e^−λt.

This formula assumes a constant half-life and a single decay process. It does not apply to multi-step decay chains, where daughter products also decay, or to processes that are not first-order. For very short half-lives (milliseconds) or very long ones (billions of years), the math still works, but measurement uncertainties become significant.

Rounding: the calculator shows 4 decimal places, but in practice, the precision is limited by the input values. If you measure an initial amount to 2 significant figures, the result should not be quoted to 6.

Real-World Applications

Half-life calculations are used in:

• Radiometric dating (carbon-14, potassium-argon)
• Nuclear medicine (dosing and decay of radiopharmaceuticals)
• Radiation safety (decay of waste or contamination)
• Pharmacokinetics (drug clearance from the body)
• Archaeology and geology (age estimation)

For example, if a sample of iodine-131 (half-life 8 days) starts with 40 MBq of activity, after 24 days (3 half-lives) the activity is 40 × 0.5³ = 5 MBq. This helps hospital physicists plan safe handling and disposal.

Frequently Asked Questions

What is the difference between half-life and decay constant? The decay constant λ is the probability per unit time that a nucleus decays. The half-life is the time for half the sample to decay. They are related by λ = ln(2)/T_1/2. The decay formula can use either: N(t) = N₀ e^(−λt) is equivalent to N(t) = N₀ (1/2)^(t/T_1/2).

Can half-life be used for non-radioactive substances? Yes, any process that follows first-order kinetics has a half-life. This includes many chemical reactions, drug elimination, and even the discharge of a capacitor through a resistor. The same formula applies, but the “half-life� may depend on conditions like temperature or concentration.

Why does the amount never reach zero? Because exponential decay approaches zero asymptotically. After each half-life, half of the remaining atoms decay, so there is always a tiny fraction left. In practice, the amount becomes negligible after many half-lives, but mathematically it never reaches exactly zero.

What if the elapsed time is not a multiple of the half-life? The formula works for any time, not just whole multiples. For example, after 1.5 half-lives, the fraction remaining is 0.5^1.5 ≈ 0.3536, or about 35.4%. The calculator handles fractional exponents automatically.

How do I convert between different time units? Use the same unit for both time and half-life. If the half-life is in years and the time is in days, divide the days by 365.25 to get years, or multiply the half-life by 365.25 to get days. The calculator does not convert units, so you must do this yourself.

Does the initial amount have to be in grams? No, you can use any unit of quantity: grams, milligrams, moles, becquerels, counts per minute, or even number of atoms. The remaining amount will be in the same unit. The fraction remaining is dimensionless and independent of the unit.

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in a0 | Initial amount | g | 100 | 0
in t | Elapsed time | h | 10 | 0
in h | Half-life | h | 5 | 0
out rem | Remaining amount | g | a0 * 0.5^(t/h) | 4
out frac | Fraction remaining | - | 0.5^(t/h) | 4