Half-Life Calculator — Free Online Radioactive Decay Tool

How to Use the Half-Life Calculator

1. Select what to solve for: The calculator offers three modes. "Remaining Amount" calculates how much substance is left after a specified time. "Time Elapsed" determines how long it takes to reach a certain remaining amount. "Half-Life Period" calculates the half-life from observed decay data.
2. Enter the initial amount: Input the starting quantity of your radioactive substance. This can be expressed in any consistent unit — grams, milligrams, number of atoms, activity in becquerels, or any proportional measure. The decay ratio is independent of the unit chosen.
3. Select or enter the half-life: Use the isotope preset dropdown to automatically fill in the half-life for common isotopes like Carbon-14, Iodine-131, or Cesium-137. Alternatively, select "Custom Value" and enter any half-life period manually.
4. Enter additional values: Depending on your selected mode, enter the time elapsed or the remaining amount. The calculator needs two of the three values (besides the initial amount) to solve for the third.
5. Review the results: The results panel displays the calculated answer along with supporting information: the number of halvings that have occurred, the percentage of material remaining, the decay formula used, and the decay constant (lambda) for your isotope.

Make sure your time unit selection matches the units of your half-life value. For example, if using Carbon-14 with a half-life of 5,730 years, set the time unit to "years" and enter the elapsed time in years as well.

Half-Life Formula

Variables Explained

• N(t): The amount of radioactive substance remaining at time t. This decreases exponentially as atoms undergo radioactive decay, transforming into daughter products.
• N₀ (N-zero): The initial amount of substance at time t = 0. This is the starting quantity before any decay has occurred.
• t: The elapsed time since the initial measurement. Must be in the same units as the half-life period.
• t½ (t-half): The half-life period — the time required for exactly half of the remaining substance to decay. This is a constant specific to each radioactive isotope.
• λ (lambda): The decay constant, representing the probability of decay per unit time. A larger lambda means faster decay and a shorter half-life.

Step-by-Step Example

Calculate the remaining amount of Carbon-14 in a sample after 17,190 years, starting with 800 grams:

1. Identify values: N_0 = 800 g, t_half = 5,730 years, t = 17,190 years
2. Calculate the number of halvings: 17,190 / 5,730 = 3.0 halvings
3. Apply the formula: N(t) = 800 x (1/2)^3
4. Calculate: N(t) = 800 x 0.125 = 100 grams
5. Verify: 800 to 400 (1st), 400 to 200 (2nd), 200 to 100 (3rd)

After exactly three half-lives (17,190 years), only 100 grams of the original 800 grams of Carbon-14 remain. The other 700 grams have decayed into Nitrogen-14 through beta-minus decay, emitting electrons and antineutrinos in the process.

Practical Examples

Example 1: Dr. Chen's Archaeological Carbon Dating

Dr. Chen discovers a wooden artifact at an excavation site. Laboratory analysis reveals that the sample contains 35% of the Carbon-14 expected in a living organism. She uses the half-life calculator to determine the artifact's age:

• Initial amount (N_0): 100 (using percentage as a convenient unit)
• Remaining amount: 35
• Half-life of C-14: 5,730 years
• Number of halvings: ln(100/35) / ln(2) = 1.0496 / 0.6931 = 1.515
• Time elapsed: 1.515 x 5,730 = 8,681 years

The artifact is approximately 8,681 years old, dating it to around 6,655 BCE, during the early Neolithic period. This dating helps Dr. Chen place the artifact in the context of early agricultural settlements in the region.

Example 2: Maria's Thyroid Treatment with Iodine-131

Maria receives a 150 millicurie dose of Iodine-131 for thyroid cancer treatment. Her physician needs to know when the radioactivity will decrease to a safe level (below 5 millicuries) for unrestricted contact with others:

• Initial activity: 150 mCi
• Target activity: 5 mCi
• Half-life of I-131: 8.02 days
• Halvings needed: ln(150/5) / ln(2) = 3.4012 / 0.6931 = 4.907
• Time required: 4.907 x 8.02 = 39.4 days

Maria will need approximately 39 days of radiation precautions before her body's radioactivity drops below the threshold. Her physician advises her to maintain distance from children and pregnant women during this period, with progressively fewer restrictions as the Iodine-131 decays.

Example 3: Professor Okafor's Strontium-90 Contamination Assessment

Professor Okafor is evaluating soil contamination from a nuclear facility. The initial Strontium-90 deposition was estimated at 4,000 becquerels per kilogram of soil in 1986. He needs to calculate the remaining contamination in 2026:

• Initial activity: 4,000 Bq/kg
• Elapsed time: 40 years (1986 to 2026)
• Half-life of Sr-90: 28.8 years
• Halvings: 40 / 28.8 = 1.389
• Remaining: 4,000 x (0.5)^1.389 = 4,000 x 0.382 = 1,527 Bq/kg

After 40 years, approximately 38.2% of the original Strontium-90 remains. The contamination level of 1,527 Bq/kg may still exceed regulatory limits depending on the jurisdiction. Professor Okafor calculates it will take another 34 years (74 years total) for the contamination to drop below 500 Bq/kg, a common remediation target.

Example 4: Jason's Geology Lab — Uranium-238 Dating

Jason is analyzing a rock sample in his geology lab. Mass spectrometry shows a uranium-238 to lead-206 ratio indicating that 15% of the original uranium has decayed. He needs to determine the rock's age:

• Remaining fraction: 85% (100% - 15% decayed)
• Half-life of U-238: 4.5 billion years
• Halvings: ln(100/85) / ln(2) = 0.1625 / 0.6931 = 0.2345
• Age: 0.2345 x 4,500,000,000 = 1.055 billion years

The rock formed approximately 1.055 billion years ago during the Mesoproterozoic era. Uranium-lead dating is one of the most reliable methods for dating ancient rocks because it provides two independent decay chains (U-238 to Pb-206 and U-235 to Pb-207) that can cross-check each other.

Common Isotope Half-Life Reference Table

Tips and Complete Guide

Understanding Radioactive Decay Chains

Many radioactive isotopes do not decay directly into stable elements but instead produce other radioactive isotopes in a chain. Uranium-238, for instance, undergoes 14 successive decay steps before finally becoming stable Lead-206. Each step has its own half-life, from microseconds (Polonium-214) to hundreds of thousands of years (Uranium-234). The half-life calculator addresses single-step decay, which is sufficient for most practical applications because each step's decay rate depends only on the amount of that specific isotope present. For complex decay chains in nuclear engineering, specialized simulation software tracks all daughter products simultaneously.

Half-Life in Pharmacology and Medicine

The concept of half-life extends beyond radioactive decay to pharmacology, where it describes how quickly drugs are eliminated from the body. While drug half-life follows similar exponential mathematics, the biological processes involved (metabolism, excretion) are more complex than nuclear decay. A drug with a 4-hour half-life will have 50% remaining after 4 hours, 25% after 8 hours, and so on. Physicians use this information to determine dosing intervals — most drugs are administered at intervals of about one half-life to maintain therapeutic levels. The same exponential decay formula applies, making our calculator useful for pharmacokinetic estimates as well.

Radiation Safety and the 10 Half-Life Rule

A practical rule in radiation safety is that after 10 half-lives, the radioactivity of a sample has decreased to less than 0.1% of its original value (specifically, 0.0977%). This means the radiation level is roughly one-thousandth of the starting level. For short-lived isotopes used in medical imaging, like Technetium-99m (6-hour half-life), 10 half-lives equals 60 hours or 2.5 days, after which radioactive waste can often be disposed of as regular waste. For longer-lived isotopes in nuclear waste, such as Plutonium-239 (24,100-year half-life), 10 half-lives means 241,000 years of storage, which is why nuclear waste disposal is such a challenging problem.

Radiocarbon Calibration

Raw Carbon-14 dates do not always correspond directly to calendar years because the atmospheric C-14 concentration has varied over time due to changes in solar activity, Earth's magnetic field, and human activities like nuclear testing. Scientists address this through calibration curves that convert raw radiocarbon years to calendar years. The IntCal20 calibration curve (2020) extends back 55,000 years and is based on tree ring data, coral records, and lake sediments. When using our calculator for C-14 dating estimates, remember that the results represent raw radiocarbon years that may need calibration for precise dating. For educational purposes and approximate dating, the uncalibrated results are sufficiently accurate.

Common Mistakes to Avoid

• Mismatching time units: Ensure the time elapsed and half-life are in the same units. Entering 30 days as elapsed time with a half-life given in years will produce meaningless results.
• Assuming linear decay: Radioactive decay is exponential, not linear. After 3 half-lives, 12.5% remains — not 0% (which linear extrapolation might suggest at 2 half-lives if you assumed 50% loss per half-life cumulatively).
• Confusing activity with amount: Activity (measured in becquerels or curies) is proportional to the number of atoms but also depends on the decay constant. Two samples with the same number of atoms but different half-lives will have different activities.
• Ignoring branching ratios: Some isotopes can decay through multiple pathways, each with a different probability. The effective half-life accounts for all pathways combined, but the product distribution requires knowing the branching ratios.
• Applying C-14 dating beyond its range: Carbon-14 dating is reliable up to about 50,000 years (roughly 9 half-lives). Beyond this, so little C-14 remains that measurement uncertainties dominate. For older samples, use uranium-lead or potassium-argon dating methods.

Frequently Asked Questions