Half-Life Calculator

Math

Calculate radioactive decay, remaining quantity, half-life duration, and elapsed time with decay constant, mean lifetime, and interactive graph.

Isotopes:

Remaining Quantity (N_t)

Percent Remaining 25.00%

Half-Lives Elapsed 2.00 cycles

Mean Lifetime (τ) 8,266.6

Radioactive Decay Curve (N vs Time)

Decay Kinetics & Mathematical Steps

1. Radioactive Decay Law Formula

N(t) = N₀ × (1/2)^(t / t_½)

Quantity halves after every elapsed half-life period.

2. Decay Constant & Mean Lifetime

λ = ln(2) / t_½ = 1.2097e-4, τ = 1 / λ = 8,266.64

The decay constant represents the probability of decay per unit time.

3. Number of Half-Lives Elapsed

n = t / t_½ = 11,460 / 5,730 = 2.0000 half-lives

After 2.00 cycles, 25.00% of the original sample remains.

How This Calculator Works

Half-life is the characteristic time duration required for a decaying quantity to reduce to exactly half of its initial value. It governs nuclear radioactive decay, pharmacokinetics (how fast medications clear from the human bloodstream), and chemical first-order reaction kinetics.

📐 Exponential Decay & Half-Life Formulas

N(t) = N₀ · (1/2)^(t / t_½) = N₀ · e^(-λt) | λ = ln(2) / t_½ ≈ 0.69315 / t_½

N(t): Remaining quantity at elapsed time t

N₀: Initial quantity of the substance at t = 0

t_½: Half-life period of the specific isotope or drug

λ (lambda): Decay constant rate parameter

τ (tau): Mean lifetime: τ = 1 / λ = t_½ / ln(2)

Key Mathematical Rules & Properties

Constant Rate Property

Radioactive half-life is strictly constant and independent of the initial substance mass, temperature, chemical state, or ambient pressure.

Elapsed Halflives (n)

After n half-lives, the remaining fraction of initial material is (1/2)ⁿ: after 1 half-life = 50%, after 2 = 25%, after 3 = 12.5%, after 4 = 6.25%, and after 10 = ~0.1%.

Mean Lifetime Relationship

The average lifetime τ of an unstable nucleus is related to its half-life by τ = t_½ / ln(2) ≈ 1.4427 × t_½.

Radiocarbon Dating Application

Carbon-14 (half-life 5,730 years) is used to date organic archaeological artifacts by comparing remaining ¹⁴C activity to modern atmospheric equilibrium.

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Worked Example: Carbon-14 Decay Over 11,460 Years

A 100g sample of Carbon-14 (half-life = 5,730 years) decays over 11,460 years. Calculate remaining mass.

Identify parameters: N₀ = 100g, t_½ = 5,730 years, t = 11,460 years.
Calculate number of half-lives: n = 11,460 / 5,730 = 2.
Apply half-life formula: N(t) = 100 × (0.5)² = 100 × 0.25 = 25g.
Calculate decay constant: λ = ln(2) / 5730 ≈ 0.00012097 per year.

Final Answer: 25g remaining (75g decayed)

Frequently Asked Questions

Does half-life ever change with temperature or pressure?

No. Nuclear decay is a quantum process governed by the weak and strong nuclear forces inside the atomic nucleus, completely impervious to external chemical bonds, heat, or pressure.

What is biological half-life vs physical half-life?

Physical half-life is the radioactive decay rate of an isotope. Biological half-life is the time it takes for an organism to metabolize and excrete half of an administered drug or toxin.

Will a radioactive sample ever reach exactly zero?

Mathematically, exponential decay approaches zero asymptotically. Physically, when the number of remaining unstable atoms reaches zero, the radioactivity ceases completely.

How many half-lives are required for a medical isotope to be considered safe?

In clinical medicine and health physics, a radioactive substance is generally considered completely decayed to negligible background levels after 7 to 10 half-lives (< 0.1% remaining).

Understanding your calculation

Method, example and limitations

How the calculation works

For a constant half-life H, the remaining quantity after time t is the initial quantity multiplied by (1/2)^(t/H).

Example to check

Starting with 80 units and a half-life of 10 years leaves 20 units after 20 years, or two half-lives.

Before using the result

The model assumes a constant decay rate and consistent time units. Biological clearance, replenishment and multiple decay processes can require a different model.

Calculation methodology · Report a result that needs review