Half-Life Calculator: Decay & Carbon-14

Half-life calculator: exponential decay and decay constants

This calculator solves N(t) = N₀(1/2)^(t/t₁/₂) for any one variable and converts half-life to mean lifetime (τ) or decay constant (λ). Single-isotope model; radiocarbon calibration not included.

What this calculator does

The widget computes any one of remaining quantity (N_t), initial quantity (N₀), elapsed time (t), or half-life (t₁/₂) from the other three using exponential decay. A constant converter links half-life, mean lifetime (τ), and decay constant (λ). The dark results card shows the formula and substituted steps for each solve. It does not apply radiocarbon calibration curves, isotope production corrections, or multi-step decay chains.

Half-life form:

$N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$
Time from remaining fraction:

$t = t_{1/2} \cdot \frac{\ln(N_0/N_t)}{\ln 2}$
Constant relations:

$t_{1/2} = \tau \ln 2 = \frac{\ln 2}{\lambda}$

How the math works

Start with the default decay solve: Time (t) selected, N₀ = 100, N_t = 50, and t₁/₂ = 5730 years (carbon-14). Because half the sample remains, elapsed time equals one half-life: t = 5730 × ln(100/50) / ln 2 = 5730 years. The results card prints that value plus the ln ratio step.

Switch the dropdown to Quantity remaining (N_t) and enter t = 5730 with the same N₀ and t₁/₂. The widget evaluates N_t = N₀ × (1/2)^(t/t₁/₂) = 100 × (1/2)¹ = 50. For an unknown half-life instead, enter N₀, N_t, and t; the solve uses t₁/₂ = t × ln 2 / ln(N₀/N_t).

In Constant converter, entering half-life 5730 yields mean lifetime τ = t₁/₂ / ln 2 ≈ 8266.6 years and decay constant λ = 1/τ ≈ 0.000121 per year in the Converted constants list. Textbook carbon-14 problems stop at this algebra; field and museum work layers calibration tables and contamination checks on top.

Limits of the model

This page assumes single-step exponential decay with no new production after t = 0. Real radiocarbon dating compares ratios to modern standards and applies calibration curves; nuclear chains and instrument background are not modeled. Time and half-life must share units; the widget does not convert seconds to years or normalize activity units for you.

Half-Life Calculator FAQ

What is the difference between half-life and mean life?

Half-life (t₁/₂) is the time for a quantity to fall to half its starting value. Mean lifetime (τ) is the average time until one decay event in the exponential model. They relate by

t_{1/2} = \tau \ln 2

. Enter any one value in the Constant converter; the widget fills the other two in the Converted constants panel.

How does carbon-14 dating work in this tool?

Carbon-14 has a half-life of about 5,730 years. With defaults (N₀ = 100, N_t = 50, t₁/₂ = 5730, solve for Time (t)), elapsed time is 5730 years—one half-life. Real radiocarbon work also applies calibration curves and production-ratio corrections; this page is the textbook exponential model only.

What is the decay constant?

The decay constant λ is the rate in

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

. It links to half-life by

\lambda = \ln 2 / t_{1/2}

. In Constant converter, pick Decay constant (λ) under Given or read λ from the converted list after entering t₁/₂.

How do I solve for time given half-life and remaining quantity?

Choose Solve for: Time (t) in Decay solver, then enter N₀, N_t, and t₁/₂ in matching time units. The widget uses

t = t_{1/2} \times \ln(N_0/N_t) / \ln 2

. N_t must be less than N₀.

Can I mix units for time and quantity?

Time (t) and half-life (t₁/₂) must use the same units (seconds, years, etc.). N₀ and N_t only need to share a ratio scale (grams, moles, counts, or normalized activity). The widget does not convert between time units or quantity units for you.

Why is the result blank when I solve for time?

For time or half-life solves, remaining quantity must be strictly less than initial (N_t < N₀), and all three entered values must be positive. Growth or stable ratios (N_t ≥ N₀) are outside single-step exponential decay; the dark results card shows Enter three values to solve instead.

How do I solve for remaining quantity N_t?

Select Quantity remaining (N_t) under Solve for, enter N₀, t, and t₁/₂. Example: N₀ = 100, t = 5730, t₁/₂ = 5730 → N_t = 100 × (1/2)^(5730/5730) = 50. The results card shows the half-life formula and substituted arithmetic.

What does the Constant converter panel show?

Enter one positive value and choose whether it is half-life, mean lifetime (τ), or decay constant (λ). The white Converted constants card lists all three: at default half-life 5730, τ ≈ 8266.6 and λ ≈ 0.000121 per year (same time units as your entry).

Half-Life Calculator

Decay solver

Constant converter

Converted constants

How to use this calculator

Reading the decay solver and constants

Example: carbon-14 age from 50% remaining (default)

Example: quantity left after 5730 years

Converted constants panel at t₁/₂ = 5730

Scope limits