Half-Life Calculator

Formula

N = N₀ × (1/2)^(t / t½)

N is the remaining quantity of the substance; N₀ is the initial quantity; t is the total elapsed time; t½ is the half-life of the substance (measured in the exact same time unit as t). Additionally, the decay constant λ is calculated as λ = ln(2) / t½, and the number of half-lives elapsed is t / t½.

Worked example

Initial amount

100

Half-life

5730

Time elapsed (same unit)

10000

• Remaining amount: 29.829244
• Percent remaining: 29.8292%
• Half-lives passed: 1.7452
• Decay constant λ: 0.00012097

What the Half-Life Calculator Computes

The Half-Life Calculator determines how much of a decaying substance remains after a specific period of time. Exponential decay is a fundamental concept across nuclear physics, radiocarbon dating, pharmacokinetics, environmental science, and continuous rate finance.

Whenever a quantity degrades at a rate directly proportional to its current amount, it follows first-order kinetics. Rather than losing a fixed amount per day or year, the substance loses a fixed percentage over fixed intervals. This calculator processes three user inputs:

• Initial amount ($N_0$): The starting mass, concentration, activity, or percentage.
• Half-life ($t_{1/2}$): The time required for the initial quantity to diminish by 50%.
• Time elapsed ($t$): The total duration over which decay takes place, expressed in the same time units as the half-life.

From these inputs, the calculator outputs the Remaining amount ($N$), the Percent remaining, the exact number of Half-lives passed, and the continuous Decay constant ($\lambda$).

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Step-by-Step Worked Example

To understand how the formula works in practice, consider the radioactive decay of Carbon-14 across an extended archaeological timeframe.

Given Parameters

• Initial amount ($N_0$): 100 (e.g., 100 grams or 100% baseline activity)
• Half-life ($t_{1/2}$): 5,730 years
• Time elapsed ($t$): 10,000 years

Step 1: Calculate the Number of Half-Lives Elapsed

Divide the elapsed time by the half-life period: $$\text{Half-lives passed} = \frac{t}{t_{1/2}} = \frac{10000}{5730} \approx 1.7452007$$

Step 2: Compute the Decay Constant ($\lambda$)

The continuous decay constant uses the natural logarithm of 2 (approximately 0.69314718): $$\lambda = \frac{\ln(2)}{t_{1/2}} = \frac{0.69314718}{5730} \approx 0.00012097 \text{ per year}$$

Step 3: Compute the Remaining Amount ($N$)

Apply the standard half-life decay formula: $$N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}$$ $$N = 100 \times (0.5)^{1.7452007} \approx 100 \times 0.29829244 = 29.829244$$

Step 4: Express as a Percentage

$$\text{Percent remaining} = \left(\frac{N}{N_0}\right) \times 100 = \left(\frac{29.829244}{100}\right) \times 100 = 29.8292%$$

After 10,000 years, roughly 29.83 units of the original 100 units remain.

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Interpreting the Results and Decay Benchmarks

Exponential decay produces predictable benchmark intervals regardless of the substance involved. Comparing your calculated half-lives passed against standard whole-number decay milestones helps validate your findings:

• 1 Half-Life ($t = t_{1/2}$): Exactly 50.0% of the original substance remains.
• 2 Half-Lives ($t = 2 \times t_{1/2}$): 25.0% remains ($0.5 \times 0.5$).
• 3 Half-Lives ($t = 3 \times t_{1/2}$): 12.5% remains.
• 4 Half-Lives ($t = 4 \times t_{1/2}$): 6.25% remains.
• 5 Half-Lives ($t = 5 \times t_{1/2}$): 3.125% remains.
• 7 Half-Lives ($t = 7 \times t_{1/2}$): Under 1.0% (0.78125%) remains.
• 10 Half-Lives ($t = 10 \times t_{1/2}$): Approximately 0.0977% remains.

In our worked example, 1.7452 half-lives passed. Because this value falls between 1 and 2 half-lives, the remaining amount (29.83%) correctly falls between 50% and 25%.

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Practical Tips for Accurate Calculations

1. Strict Unit Alignment: The formula does not convert time units internally. If your isotope half-life is listed in hours (e.g., Technetium-99m at 6 hours) and elapsed time is 2 days, convert the elapsed time to 48 hours before calculating.
2. Consistent Quantity Units: The units for $N_0$ and $N$ are interchangeable—grams, milligrams, Becquerels, microcuries, or parts per million—as long as the input and output share the exact same measurement scale.
3. Use Percentages for Unknown Masses: If the starting mass is unknown but you need to know how much activity remains, enter 100 for initial amount to get the exact percentage remaining.

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When Not to Rely on This Model

While the half-life equation is exact for pure nuclear physics, certain real-world systems deviate from standard first-order decay:

• Zero-Order Metabolic Elimination: High doses of substances like ethanol or aspirin saturate liver enzymes, causing clearance at a constant absolute rate per hour (linear decline) rather than a constant percentage.
• Multi-Compartment Pharmacokinetics: Many medications distribute into fat, tissue, and plasma at different rates, requiring multi-exponential models ($t_{1/2\alpha}$ distribution half-life vs. $t_{1/2\beta}$ elimination half-life) rather than a single half-life figure.
• Decay Chains with Radioactive Daughters: If an isotope decays into another radioactive daughter isotope (such as Uranium-238 decaying into Thorium-234), overall radioactivity cannot be modeled with a single half-life equation; differential equations such as the Bateman equations are required.

Comparison

Scenario	Input	Result
Lower input	Initial amount: 80	Remaining amount: 23.863395
Base example	Initial amount: 100	Remaining amount: 29.829244
Higher input	Initial amount: 120	Remaining amount: 35.795092

Common mistakes

• Mixing incompatible time units, such as entering half-life in days and elapsed time in years without conversion.
• Assuming decay is linear by subtracting 50% for the first half-life and reaching zero after two half-lives.
• Applying radioactive half-life formulas to non-first-order processes, such as zero-order chemical reactions where a fixed amount degrades per hour.
• Confusing biological half-life (metabolism and excretion) with physical nuclear half-life in medical or pharmacology contexts.
• Rounding intermediate values like elapsed half-lives or the decay constant too early, which compounds error across large timeframes.

Checklist

• Verify that elapsed time (t) and half-life (t½) use the exact same time unit (e.g., both in years, hours, or seconds).
• Enter the initial quantity (N₀) as a positive number representing mass, percentage, count, or concentration.
• Confirm that the process follows first-order exponential decay rather than linear or zero-order kinetics.
• Check whether environmental conditions (such as biological clearance vs. nuclear decay) affect the applicable half-life.
• Use the decay constant λ if your model requires the continuous exponential form N = N₀ × e^(−λt).

Calculators Percentage Editorial Team

Calculator guides from the Calculators Percentage team. These introductory examples are educational, not professional advice.

Last reviewed: November 12, 2026

This calculator provides general mathematical and scientific estimates based on ideal first-order exponential decay models. It should not be used for medical dosing, safety protocols, or nuclear engineering without professional verification.