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 →Nuclear chemistry and kinetics share one small formula set that reliably returns marks once it is drilled. ABC Chemistry runs Class 11–12 chemistry coaching at the Gurugram centre and online classes across India, with home tuition available in Delhi-NCR — details at abcchemistry.in.