Engineering & Physics CalcyZone Calculator

Half-Life Calculator

Calculate how much of a radioactive or decaying substance remains after a given time.

Amount remaining

25

Fraction remaining

25%

Half-lives elapsed

2

Decay constant λ

0.000121

Step-by-step working

N = N₀ × (½)^(t / t½) = 100 × 0.5^(11460 / 5730) = 25

Mathematical Formula

N(t) = N₀ × (½)^(t / t½); λ = ln 2 / t½.

Overview & Explanation

In first-order decay, a fixed fraction decays per unit time; the half-life is the time for half to remain.

How It Works

Divide elapsed time by half-life.
Raise ½ to that power.
Multiply by the initial amount.

Practical Applications

Radiocarbon dating.
Drug elimination (pharmacokinetics).

The decay formula

Remaining amount N = N0 x (1/2)^(t / T), where N0 is the starting amount, t is the elapsed time and T is the half-life, in the same time units. The exponent t / T counts how many half-lives have passed.

Example: carbon-14 has a half-life of about 5,730 years. After 11,460 years (two half-lives), 100 g becomes 25 g. For a substance with a 6-hour half-life, 80 mg after 15 hours is 80 x 0.5^2.5 = 14.14 mg.

Decay constant

The same process can be written as N = N0 x e^(-lambda t), with lambda = ln 2 / T. For T = 6 hours, lambda = 0.1155 per hour. To solve for time, use t = T x log2(N0 / N).

Example 1: C-14 after 11,460 years

Two half-lives → 25%.

1

Inputs

Initial amount N₀: 100 · Half-life t½: 5730 · Elapsed time t (same unit): 11460

= entered

2

Step 1

N = N₀ × (½)^(t / t½) = 100 × 0.5^(11460 / 5730) = 25

3

Amount remaining

Result

= 25

Example 2: I-131 (8.02 days) after 30 days

Medical isotope.

1

Inputs

Initial amount N₀: 50 · Half-life t½: 8.02 · Elapsed time t (same unit): 30

= entered

2

Step 1

N = N₀ × (½)^(t / t½) = 50 × 0.5^(30 / 8.02) = 3.7404

3

Amount remaining

Result

= 3.7404

How it's calculated & sources

N(t) = N₀ × (½)^(t / t½); λ = ln 2 / t½.

U.S. NRC – Half-life (radiological)

Frequently Asked Questions

Is decay linear?

No, it is exponential: each half-life halves what remains.

Units?

Any time unit, as long as half-life and elapsed time match.

What is the decay constant?

λ = ln 2 / t½, the fraction decaying per unit time.

Does the substance ever reach zero?

Mathematically it only approaches zero. In practice, after about 10 half-lives less than 0.1% remains.

Does half-life change with amount?

No. For first-order decay, each half-life takes the same time whatever the starting amount.

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