What is Radioactive Decay?
Radioactive decay is the spontaneous disintegration of an unstable atomic nucleus, emitting alpha (α), beta (β), or gamma (γ) radiation. The half-life (T½) is the time required for exactly half of the radioactive atoms in a sample to decay. It follows a first-order exponential law and is independent of temperature, pressure, or chemical state.
Core Radioactive Decay Formulas
| Symbol | Quantity | Unit | Notes |
|---|
| N₀ | Initial number of atoms | atoms (or any unit) | At time t = 0 |
| N(t) | Remaining atoms at time t | atoms (or any unit) | Always < N₀ |
| λ | Decay constant | s−1 (or per unit time) | Probability of decay per atom per second |
| T½ | Half-life | s, min, h, days, years | Fixed for each nuclide; independent of N |
| A | Activity | Becquerel (Bq) or Curie (Ci) | 1 Ci = 3.7 × 1010 Bq |
| t | Elapsed time | Same units as T½ | Must use consistent units |
Solved Examples (NCERT / JEE Level)
Example 01 — Remaining Quantity (Carbon Dating)
A sample of wood contains 1/8th of the original C-14 activity. Given that the half-life of C-14 is 5730 years, calculate the age of the sample.
Given: N = N₀/8 → N/N₀ = 1/8, T½ = 5730 years
Using: N = N₀ × (1/2)t/T½
→ 1/8 = (1/2)t/5730
→ (1/2)3 = (1/2)t/5730
→ 3 = t / 5730
→ t = 3 × 5730
Age = 17,190 years | 3 half-lives have elapsed
Example 02 — Decay Constant and Activity
A radioactive sample of Iodine-131 (T½ = 8.02 days) contains 3.0 × 1015 atoms. Find (a) the decay constant in s−1 and (b) the initial activity in Becquerel.
Given: T½ = 8.02 days = 8.02 × 24 × 3600 = 692,928 s
N = 3.0 × 1015 atoms
(a) λ = ln(2) / T½ = 0.6931 / 692928
λ = 1.0003 × 10−6 s−1
(b) A = λ × N
A = 1.0003 × 10−6 × 3.0 × 1015
A = 3.001 × 109 Bq = 3.001 GBq
λ = 1.00 × 10−6 s−1 | A = 3.00 × 109 Bq (3.00 GBq)
Example 03 — Finding Half-Life from Experimental Data
A radioactive element has its activity reduced from 6400 counts/s to 100 counts/s in 30 hours. Calculate its half-life.
Given: A₀ = 6400, A = 100, t = 30 h
Since activity is proportional to N: A/A₀ = N/N₀
Using: t = T½ × log₂(N₀/N)
30 = T½ × log₂(6400/100)
30 = T½ × log₂(64)
30 = T½ × 6 [since 26 = 64]
T½ = 30/6
T½ = 5 hours | 6 half-lives elapsed in 30 hours
Frequently Asked Questions (FAQs)
What is the difference between decay constant and half-life?
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The decay constant (λ) is the probability that a single nucleus will decay per unit time. The half-life (T½) is the time for half the nuclei to decay. They are inversely related: λ = ln(2) / T½ = 0.6931 / T½. A large decay constant means rapid decay and a short half-life. Both are fixed properties of a given radioisotope and do not depend on temperature, pressure, or the amount present.
Why does radioactive decay follow an exponential law?
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Radioactive decay is a stochastic (random) quantum process. Each nucleus has an equal, constant probability (λ) of decaying per unit time, regardless of the nucleus's age. When this property is applied to a large number of atoms, the rate of decay at any moment is proportional to the number of undecayed atoms present (dN/dt = −λN). Solving this first-order differential equation gives the exponential decay law N(t) = N₀ e−λt.
What is the unit of radioactivity and how is Becquerel different from Curie?
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The SI unit of radioactivity is the Becquerel (Bq), equal to 1 nuclear disintegration per second. The older unit, the Curie (Ci), was defined as the activity of 1 gram of Radium-226 and equals 3.7 × 1010 Bq. The Curie is still widely used in nuclear medicine and radiation safety. For comparison, a typical smoke detector contains about 1 microcurie (37,000 Bq) of Am-241.
How is radioactive decay used in carbon dating?
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Radiocarbon dating (C-14 dating) uses the known half-life of Carbon-14 (5,730 years) to estimate the age of organic materials up to about 50,000 years old. Living organisms maintain a constant ratio of C-14 to C-12 by exchanging carbon with the atmosphere. When an organism dies, it stops exchanging carbon and its C-14 begins to decay. By measuring the remaining C-14 / C-12 ratio and comparing it to the known atmospheric ratio, scientists can calculate when the organism died using the formula t = T½ × log₂(N₀/N).
Does a radioactive substance ever fully decay to zero?
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Theoretically, no — the exponential decay curve approaches zero asymptotically, meaning there will always be some fraction of atoms remaining, no matter how many half-lives pass. However, in practice, when the expected number of atoms drops below 1 (for a finite sample), the sample is effectively considered fully decayed. For example, after 10 half-lives, only 0.098% of the original atoms remain, which for most practical samples is negligible. This is why nuclear waste with long half-lives (like Pu-239 at 24,110 years) remains hazardous for extraordinarily long periods.