Half Life Formula Geology: How to Calculate Age

This calculator works out the age of a rock or mineral using the half-life formula from radiometric dating. You enter the half-life of the isotope, the amount of parent isotope remaining (as a fraction or percentage), and the calculator returns the elapsed time in the same units as the half-life. It is a direct application of the exponential decay equation and is the same math used in carbon-14, potassium-argon, and uranium-lead dating.

Formula

The half-life formula in geology is:

t = (t�/₂ / ln 2) × ln(N₀ / N)

Where:

• t = age of the sample (time elapsed since the mineral closed)
• tâ‚�/â‚‚ = half-life of the radioactive parent isotope (same time unit as t)
• Nâ‚€ = original amount of parent isotope (often set to 1, i.e., 100%)
• N = current amount of parent isotope remaining (as a fraction of Nâ‚€)
• ln = natural logarithm

An equivalent form uses the decay constant λ = ln 2 / t�/₂, giving t = (1/λ) × ln(N₀ / N). Both are the same equation; the half-life version is more intuitive for most people.

Worked Example

Using the default values from the calculator above: half-life = 5730 years (the half-life of carbon-14), parent remaining = 0.5 (50%).

1. Calculate the decay constant: λ = ln 2 / 5730 = 0.693147 / 5730 = 0.000120968 per year.
2. Plug into the age equation: t = (1/0.000120968) × ln(1 / 0.5).
3. ln(1 / 0.5) = ln(2) = 0.693147.
4. So t = (1/0.000120968) × 0.693147 = 8267 × 0.693147 ≈ 5730 years.

Unsurprisingly, if half of the parent remains, the age equals one half-life. The calculator gives 5730.00, matching exactly.

How Radiometric Dating Works

Radioactive isotopes decay at a constant, probabilistic rate. The half-life is the time it takes for half of the atoms in a sample to decay. Geologists measure the ratio of parent isotope to daughter product (or to a stable reference isotope) and use the half-life formula to calculate how many half-lives have passed.

For the calculation to be valid, the system must have been “closed� – no parent or daughter atoms added or removed except by radioactive decay. In practice, geologists choose minerals that lock in the parent isotope at formation, such as zircon for uranium-lead dating or feldspar for potassium-argon dating.

Choosing the Right Isotope

The half-life must match the expected age of the rock. If the half-life is too short, all the parent will be gone; if too long, too little daughter will have formed to measure accurately. Common systems include:

• Carbon-14 (tâ‚�/â‚‚ = 5,730 years) – for organic material up to ~50,000 years old.
• Potassium-40 to Argon-40 (tâ‚�/â‚‚ = 1.25 billion years) – for rocks older than ~100,000 years.
• Uranium-238 to Lead-206 (tâ‚�/â‚‚ = 4.47 billion years) – for the oldest rocks and meteorites.
• Rubidium-87 to Strontium-87 (tâ‚�/â‚‚ = 48.8 billion years) – for very old rocks, often used as a cross-check.

No single isotope works for all ages; the half-life must be comparable to the sample’s age.

Common Mistakes and Misconceptions

• Using the wrong half-life – mixing up carbon-14 (5730 years) with carbon-13 or carbon-12, which are stable.
• Assuming the initial amount is known – in many methods, Nâ‚€ is inferred from the daughter/parent ratio, not measured directly. The formula above assumes you know Nâ‚€, which is often not the case in real samples.
• Ignoring daughter product loss – if the daughter is a gas (like argon), it can escape when the rock is heated, making the age appear too young.
• Rounding too early – using 0.693 instead of ln 2 introduces small errors that matter in old rocks.

A common misconception is that half-life is affected by temperature, pressure, or chemical environment. It is not – radioactive decay rates are essentially constant under Earth’s surface conditions. This constancy is what makes radiometric dating reliable.

Limitations of the Half-Life Formula

The formula assumes a closed system and a known initial amount of parent. In reality, rocks can lose or gain parent or daughter atoms through weathering, metamorphism, or fluid flow. Geologists use several techniques to detect such disturbances:

• Isochron dating – plots multiple samples to find the initial ratio without assuming Nâ‚€.
• Concordia diagrams – used in uranium-lead dating to identify lead loss.
• Cross-checking with multiple isotope systems – if two different methods agree, the age is likely correct.

Also, the formula only gives the time since the mineral closed (crystallized or last cooled below a certain temperature), not the age of the Earth or the rock’s components. For example, a zircon crystal may be older than the sedimentary rock that contains it.

Frequently Asked Questions

What is the half-life formula in geology?

The basic formula is t = (t�/₂ / ln 2) × ln(N₀ / N), where t is the age, t�/₂ is the half-life, N₀ is the initial amount of parent isotope, and N is the remaining amount. If you know the fraction remaining (N/N₀), you can compute age directly.

How do you calculate age from half-life?

Divide the natural log of the remaining fraction by the natural log of 0.5, then multiply by the half-life. For example, if 25% of the parent remains, ln(0.25)/ln(0.5) = 2, so the age is two half-lives.

Why is carbon-14 not used for rocks?

Carbon-14 has a half-life of only 5,730 years. After about 10 half-lives (57,000 years), the remaining carbon-14 is too small to measure accurately. Rocks are usually millions to billions of years old, so longer-lived isotopes like potassium-40 or uranium-238 are needed.

What does a half-life of 1.25 billion years mean?

It means that after 1.25 billion years, half of the original potassium-40 atoms in a mineral will have decayed to argon-40. After another 1.25 billion years, half of the remaining potassium-40 will decay, leaving one-quarter of the original.

Can the half-life formula be used for carbon dating?

Yes, but with a twist. In carbon dating, you measure the ratio of carbon-14 to carbon-12 in a sample and compare it to the atmospheric ratio. The formula still applies, but Nâ‚€ is the initial carbon-14 ratio, which is assumed to be constant over time.

What is the difference between half-life and decay constant?

The half-life (t�/₂) is the time for half the atoms to decay. The decay constant (λ) is the probability of decay per unit time. They are related by λ = ln 2 / t�/₂. The age formula can use either: t = (1/λ) × ln(N₀/N).

Related

• Annual Percent Change Calculator: Formula & Examples
• Greenhouse Surface Area Calculator: How It Works
• Greatest Common Factor Calculator & How It Works

in t_half | Half-life | Ma | 5730 | 0
in parent_frac | Parent remaining (0-1) | - | 0.5 | 0
out age | Age | Ma | t_half * (log(parent_frac) / log(0.5)) | 2
out decay_const | Decay constant (λ) | per Ma | log(2) / t_half | 6