Half-Life Formula — Radioactive Decay Calculations
Half-life is the time a sample takes to fall to half of what it was. The idea is simple; the marks are lost on the arithmetic — mixing up the decay constant with the half-life, using the wrong exponent, or assuming every reaction has a constant half-life. This guide fixes all three, with four worked problems.
The formulas
Half-life: t½ = ln 2 / λ = 0.6931 / λ
Half-life form (easiest when t is a whole number of half-lives):
N = N₀ × (1/2)n, where n = t / t½
Solving for time: t = (1/λ) × ln(N₀/N)
All four are the same statement. Radioactive decay is a first-order process, so the fraction lost per unit time is constant and does not depend on how much you started with.
What each symbol means
| Symbol | Meaning | Unit |
|---|---|---|
| N₀ | Amount at the start — atoms, moles, grams or activity | any, if used consistently |
| N | Amount remaining after time t | same as N₀ |
| λ | Decay constant — probability of decay per unit time | s⁻¹, h⁻¹, day⁻¹, yr⁻¹ |
| t½ | Half-life | reciprocal of λ's unit |
| A | Activity = λN — decays per second | becquerel (Bq); 1 Ci = 3.7 × 10¹⁰ Bq |
Because A = λN and λ is fixed, activity falls in exactly the same proportion as the number of atoms. That is why a question may give you counts per minute instead of grams and the working does not change at all.
The fraction-remaining table worth memorising
| Half-lives elapsed (n) | Fraction left | Percent left | Percent decayed |
|---|---|---|---|
| 1 | 1/2 | 50 % | 50 % |
| 2 | 1/4 | 25 % | 75 % |
| 3 | 1/8 | 12.5 % | 87.5 % |
| 4 | 1/16 | 6.25 % | 93.75 % |
| 10 | 1/1024 | ≈ 0.098 % | ≈ 99.9 % |
The rule of thumb that ten half-lives makes a source "essentially gone" comes straight from that last row.
Worked example 1 — whole number of half-lives
A 100 g sample of a nuclide has t½ = 5.0 days. How much remains after 20 days?
n = t ÷ t½ = 20 ÷ 5.0 = 4 half-lives
N = 100 × (1/2)⁴ = 100 ÷ 16 = 6.25 g
and 100 − 6.25 = 93.75 g has decayed. Whenever n is a whole number, never touch the exponential — the halving form is faster and safer.
Worked example 2 — a non-integer number of half-lives
Technetium-99m, used in medical imaging, has t½ = 6.01 hours. A 12.0 mCi dose is prepared. What is its activity 24.0 hours later?
n = 24.0 ÷ 6.01 = 3.9933 half-lives
23.9933 = e3.9933 × 0.6931 = e2.7678 = 15.92
A = 12.0 ÷ 15.92 = 0.754 mCi
Just under 4 half-lives, so just over 1/16 of the original — 12.0 ÷ 16 = 0.75 mCi is the mental check, and the exact answer sits right beside it. That short half-life is deliberate: the tracer does its job and then leaves the patient quickly.
Worked example 3 — solving for time
Iodine-131 has t½ = 8.02 days. How long until 90 % of a sample has decayed?
First get λ: λ = 0.6931 ÷ 8.02 = 0.08643 day⁻¹
90 % decayed means 10 % remains, so N₀/N = 10
t = (1/λ) × ln(N₀/N) = ln 10 ÷ 0.08643 = 2.3026 ÷ 0.08643
t = 26.6 days
Cross-check with the halving picture: 3 half-lives (24.06 days) leaves 12.5 %, and a little more time takes it to 10 %. 26.6 days fits.
Worked example 4 — the same maths in chemical kinetics
A first-order decomposition has k = 2.5 × 10⁻³ s⁻¹. Find its half-life and the time for the reaction to be 75 % complete.
t½ = 0.6931 ÷ k = 0.6931 ÷ 0.0025 = 277 s
75 % complete means 25 % remains, which is exactly 2 half-lives:
t = 2 × 277 = 554 s
For a first-order reaction the decay constant λ and the rate constant k are the same quantity with different names. That is why the nuclear-chemistry and kinetics chapters share one formula sheet.
Half-life is constant only for first order
| Order | Half-life expression | Behaviour as reaction proceeds |
|---|---|---|
| Zero | t½ = [A]₀ / 2k | Gets shorter |
| First | t½ = 0.6931 / k | Constant — independent of [A]₀ |
| Second | t½ = 1 / (k[A]₀) | Gets longer |
A useful exam shortcut runs backwards: if successive half-lives measured from one experiment are equal, the reaction is first order.
Common mistakes
- Using λ where t½ belongs. They are reciprocally related through ln 2, not equal. λ = 0.6931/t½.
- Mismatched time units. If λ is in day⁻¹, t must be in days. Convert before substituting.
- Confusing "percent decayed" with "percent remaining". 90 % decayed means N/N₀ = 0.10. Half of all wrong answers in this topic come from this one line.
- Assuming every reaction has a fixed half-life. Only first-order does.
- Halving the half-life instead of the amount. t½ never changes; N does.
- Rounding ln 2 to 0.69 in long calculations. Use 0.6931; the error compounds through several half-lives.
Where it appears in exams
| Exam | Typical question |
|---|---|
| CBSE/ICSE Class 12 | t½ from k, amount left after n half-lives, first-order proofs |
| JEE / NEET | Activity ratios, mixed-nuclide samples, average life τ = 1/λ |
| IIT-JAM / CUET-PG | Order determination from half-life data, radiometric dating arithmetic |
| GATE / CSIR-NET | Parent–daughter decay chains, secular equilibrium, tracer applications |
Let the calculator handle the exponent. Give it any three of N₀, N, t, t½ and λ, and it returns the missing quantity together with the number of half-lives elapsed — the value that makes every answer easy to sanity-check.
Open the Half-Life Calculator →