How the Half-Life Calculator Works (Radioactive Decay)

What Is Half-Life?

Half-life (symbol: t½) is the time required for exactly half of a quantity of a radioactive substance to undergo nuclear decay. It is a characteristic property of each radioactive isotope — constant and independent of temperature, pressure, chemical form, or how much of the substance is present.

After one half-life, 50% remains. After two half-lives, 25% remains. After three, 12.5% — and so on. This is exponential decay: the quantity never reaches zero mathematically, but becomes negligibly small after many half-lives (typically considered safe after 10 half-lives, when less than 0.1% remains).

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The Formulas

Primary Exponential Decay Formula

$$ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} $$

Where:

• N(t) = quantity remaining at time t
• N₀ = initial quantity
• t = elapsed time
• t½ = half-life of the isotope

Alternative Form Using the Decay Constant

The decay constant λ (lambda) represents the probability per unit time that a given atom will decay.

$$ \lambda = \frac{\ln(2)}{t_{1/2}} \approx \frac{0.6931}{t_{1/2}} $$

The formula then becomes:

$$ N(t) = N_0 \times e^{-\lambda t} $$

Both forms are mathematically equivalent and will yield identical results.

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Solving for Each Variable

Finding Remaining Quantity N(t)

Given N₀, t, and t½, plug directly into the primary formula:

$$ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} $$

Finding Elapsed Time t

Rearrange using logarithms:

$$ t = t_{1/2} \times \frac{\ln\left(\frac{N(t)}{N_0}\right)}{\ln\left(\frac{1}{2}\right)} = t_{1/2} \times \log_2\left(\frac{N_0}{N(t)}\right) $$

Finding Initial Quantity N₀

$$ N_0 = \frac{N(t)}{\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}} $$

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Step-by-Step Guide

Step 1 — Identify the Isotope and Its Half-Life

Look up the half-life of your isotope from literature, a reference table, or the calculator's built-in presets. Ensure time units are consistent throughout your calculation.

Step 2 — Determine What You Know

Identify which three of the four variables (N₀, N(t), t, t½) you have. The fourth is what you are solving for.

Step 3 — Select the Solve-For Mode

In the Nexus Half-Life Calculator, select the tab for what you want to find: Remaining Amount, Elapsed Time, or Initial Amount.

Step 4 — Enter Your Values

Input the known values. The calculator accepts any time unit (seconds, minutes, hours, days, years) — just be consistent between the elapsed time and half-life inputs.

Step 5 — Read and Interpret the Result

The result is displayed in both absolute quantity and as a percentage of the original. The decay curve graph shows the complete decay trajectory.

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Reference Table — Common Isotopes

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Real-World Example 1 — Radiocarbon Dating

A sample of ancient wood charcoal contains 340 grams of Carbon-14. A living tree today would contain 1,200 grams of C-14 in an equivalent sample. How old is the charcoal?

Known values:

• N₀ = 1,200 g (C-14 in a living equivalent)
• N(t) = 340 g (C-14 remaining)
• t½ = 5,730 years

Apply the elapsed-time formula:

$$ t = 5730 \times \log_2\left(\frac{1200}{340}\right) = 5730 \times \log_2(3.529) = 5730 \times 1.819 \approx 10{,}423 \text{ years} $$

The charcoal is approximately 10,400 years old — placing it in the early Holocene period.

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Real-World Example 2 — Medical Imaging (Technetium-99m)

A patient is administered 400 MBq of Technetium-99m for a bone scan. How much radioactivity remains after 18 hours?

Known values:

• N₀ = 400 MBq
• t = 18 hours
• t½ = 6 hours

$$ N(18) = 400 \times \left(\frac{1}{2}\right)^{\frac{18}{6}} = 400 \times \left(\frac{1}{2}\right)^3 = 400 \times 0.125 = 50 \text{ MBq} $$

After 18 hours (3 half-lives), only 50 MBq remains — just 12.5% of the administered dose. This short half-life is why Tc-99m is ideal for medical imaging: it delivers the diagnostic scan with minimal long-term radiation exposure to the patient.

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Frequently Asked Questions

Q: Does half-life change with temperature or chemical environment? A: No. Nuclear decay is a property of the nucleus and is unaffected by chemical bonding, temperature, or pressure (with extremely rare exceptions in specific electron-capture isotopes under extraordinary conditions).

Q: Why is "half-life" used instead of "full decay time"? A: Because radioactive substances never fully reach zero — the decay is asymptotic. Half-life is a constant, measurable property that makes calculations predictable. Each successive half-life reduces the quantity by half, regardless of the current amount.

Q: What is the relationship between half-life and the decay constant? A: They are inversely related: λ = ln(2) / t½ ≈ 0.6931 / t½. A shorter half-life means a larger decay constant, meaning atoms decay faster.

Q: How many half-lives until a substance is considered safe? A: A general rule of thumb in radiation safety is 10 half-lives, after which less than 0.1% of the original activity remains. For Iodine-131 (t½ = 8 days), this is about 80 days.

Q: Can half-life be used for things other than radioactive decay? A: Yes. The half-life concept applies to any exponential decay process, including drug elimination from the bloodstream (pharmacokinetics), population decline, and capacitor discharge in electronics.