Difficulty: Intermediate | Prerequisites: Exponential functions, nuclear structure and decay types (see companion notes)
Big Picture
Once you know why nuclei decay (covered in the companion notes on structure and decay types), the next question is how fast. This section covers the mathematics of radioactive decay: the exponential decay law, half-life, the time constant, and activity. These concepts appear in everything from medical imaging dosimetry to archaeological dating. You should be comfortable with exponential functions and the natural logarithm before diving in.
Radioactive decay is random at the level of individual nuclei, but statistically predictable for large samples. The number of undecayed nuclei falls exponentially with time. The half-life is the time for half the sample to decay; the time constant is the time for the sample to drop to 37% of its original count. Activity (decays per second) decreases exponentially at the same rate.
Half-life (t₁/₂)
The time it takes for half the radioactive nuclei in a sample to decay. After one half-life, 50% remain; after two, 25%; after three, 12.5%, and so on. Think of it as the "halving time" of the sample.
Time constant (τ)
The time at which the number of remaining nuclei has dropped to e⁻¹ (about 37%) of the original number. Related to half-life by τ = t₁/₂ / ln 2 = t₁/₂ / 0.693, or equivalently t₁/₂ = 0.693τ. In simple terms, a slightly longer timescale than the half-life that simplifies the exponential maths.
Activity (R)
The number of decays per second. Measured in becquerels (Bq), where 1 Bq = 1 decay per second. An older unit, the curie (Ci), equals 3.7 × 10¹⁰ Bq. Think of it as how "busy" a radioactive source is right now.
Becquerel (Bq)
The SI unit of radioactive activity. 1 Bq = 1 decay per second.
Curie (Ci)
An older unit of activity. 1 Ci = 3.7 × 10¹⁰ Bq. Originally defined as the activity of 1 gram of radium-226.
Exponential decay
The mathematical pattern followed by radioactive samples: the rate of decay is proportional to the number of undecayed nuclei remaining. This produces the characteristic "ever-flattening" curve that never quite reaches zero.
N₀ (initial number of nuclei)
The number of radioactive nuclei present at time t = 0, before any decay has occurred.
Radioactive decay is a random process at the level of individual atoms. You cannot predict when a particular nucleus will decay, but you can predict how many out of a large sample will decay over a given time.
Nuclei do not vanish when they decay. A decayed nucleus becomes a different kind of nucleus (the daughter product).
The decay equation (half-life form):
N = N₀ × (1/2)^(t / t₁/₂)
N is the number of undecayed nuclei remaining after time t.
N₀ is the number at the start (t = 0).
t₁/₂ is the half-life.
The decay equation (exponential form):
N = N₀ × e^(−t/τ)
τ is the time constant.
These two forms are equivalent. Use whichever is more convenient for the problem at hand. The half-life form is often simpler when the elapsed time is a whole number of half-lives.
Half-life (t₁/₂): After each half-life, exactly half of the remaining undecayed nuclei have decayed.
After 1 half-life: 50% remain (N₀/2)
After 2 half-lives: 25% remain (N₀/4)
After 3 half-lives: 12.5% remain (N₀/8)
The curve never reaches zero. It asymptotically approaches it.
Time constant (τ): The time at which N = N₀ × e⁻¹ ≈ 0.37 N₀. So at t = τ, about 37% of the original nuclei remain.
Relationship between t₁/₂ and τ:
τ = t₁/₂ / ln 2 = t₁/₂ / 0.693
Or equivalently: t₁/₂ = 0.693 × τ
The time constant is always larger than the half-life by a factor of about 1.44.
Practical interpretation: the half-life tells you how quickly a sample becomes "safe" or "spent." A substance with a short half-life is intensely radioactive but decays quickly. A substance with a long half-life is less active at any given moment but persists for much longer.
Activity (R) is the number of decays per second. It tells you how radioactive a sample is right now.
Formula: R = N / τ = 0.693 N / t₁/₂
Activity is directly proportional to the number of undecayed nuclei and inversely proportional to the half-life.
A short half-life means high activity for a given number of nuclei.
Activity also decays exponentially:
R = R₀ × e^(−t/τ)
R₀ = N₀ / τ is the initial activity at t = 0.
Units:
1 Becquerel (Bq) = 1 decay per second (the SI unit).
1 Curie (Ci) = 3.7 × 10¹⁰ Bq. This is a very large unit; medical and laboratory sources are typically measured in millicuries (mCi) or microcuries (μCi).
Activity decreases over time at the same exponential rate as the number of undecayed nuclei.
Number remaining (half-life form): N = N₀ × (1/2)^(t / t₁/₂)
Number remaining (exponential form): N = N₀ × e^(−t/τ)
Time constant from half-life: τ = t₁/₂ / 0.693 = 1.44 × t₁/₂
Half-life from time constant: t₁/₂ = 0.693 × τ
Activity: R = N / τ = 0.693 N / t₁/₂
Activity over time: R = R₀ × e^(−t/τ)
Unit conversions: 1 Bq = 1 decay/s; 1 Ci = 3.7 × 10¹⁰ Bq
Carbon-14 dating uses the known half-life of ¹⁴C (about 5,730 years) to estimate the age of organic remains. In nuclear medicine, isotopes with short half-lives (e.g. ⁹⁹ᵐTc, t₁/₂ ≈ 6 hours) deliver diagnostic imaging with minimal long-term radiation exposure. Nuclear waste management depends entirely on half-life: spent fuel contains isotopes with half-lives from days to thousands of years, and storage must account for the slowest-decaying components.
Students often think that after two half-lives, the entire sample has decayed. After two half-lives, 25% of the original nuclei remain. The sample never fully reaches zero; it asymptotically approaches it.
Students sometimes confuse half-life and time constant. They are related but not equal. The time constant τ is always about 1.44 times the half-life.
Students often assume that a longer half-life means the substance is "more radioactive." The opposite is true at any given moment: for the same number of atoms, a shorter half-life produces a higher activity (more decays per second).
Students sometimes think decayed nuclei disappear. They do not. A decayed nucleus becomes a different nucleus (the daughter product).
⚠️ Be able to use both forms of the decay equation: the half-life form N = N₀(1/2)^(t/t₁/₂) and the exponential form N = N₀e^(−t/τ). Know when each is more convenient.
⚠️ Convert between half-life and time constant using t₁/₂ = 0.693τ.
⚠️ Calculate activity R = N/τ and understand that it decreases exponentially just like N.
⚠️ Convert between becquerels and curies. 1 Ci = 3.7 × 10¹⁰ Bq.
⚠️ Problems often ask: "After how many half-lives does the sample drop to X%?" Recognise that each half-life halves the count: 1 → 50%, 2 → 25%, 3 → 12.5%, 4 → 6.25%, etc.
True or false: After three half-lives, 1/8 of the original sample remains.
True. (1/2)³ = 1/8.
Fill in the blank: The SI unit of activity is the ______, equal to 1 decay per second.
Becquerel (Bq).
True or false: The time constant τ is shorter than the half-life t₁/₂.
False. τ = t₁/₂ / 0.693 ≈ 1.44 × t₁/₂, so τ is always longer.
Fill in the blank: At t = τ, the fraction of original nuclei remaining is approximately ______.
37% (or e⁻¹ ≈ 0.37).
True or false: A substance with a shorter half-life has a lower activity for the same number of atoms.
False. Shorter half-life means higher activity (more decays per second).
Q: A radioactive sample has a half-life of 4 hours. If you start with 1,000 nuclei, how many remain after 12 hours?
A: 12 hours = 3 half-lives. N = 1000 × (1/2)³ = 1000 × 1/8 = 125 nuclei.
Q: A sample's half-life is 10 minutes. What is its time constant?
A: τ = t₁/₂ / 0.693 = 10 / 0.693 ≈ 14.4 minutes.
Q: You measure an initial activity of 800 Bq. After one half-life, what is the activity?
A: 400 Bq. Activity halves with each half-life, just as the number of undecayed nuclei does.
Q: Convert 5 μCi to becquerels.
A: 5 × 10⁻⁶ × 3.7 × 10¹⁰ = 1.85 × 10⁵ Bq = 185,000 Bq.
Q: A sample has 2 × 10⁸ undecayed nuclei and a time constant of 500 seconds. What is its activity?
A: R = N / τ = 2 × 10⁸ / 500 = 4 × 10⁵ Bq (400,000 decays per second).
This material pairs with the companion notes on nuclear structure, binding energy, and decay types (same chapter). The exponential decay law is mathematically identical to the discharge of a capacitor in an RC circuit (Chapter 28), so if you are comfortable with one, the other follows the same logic. Carbon-14 dating connects to geology and archaeology courses. Activity and dosimetry appear again in any health physics or radiation safety module.
Radioactive decay, exponential decay, half-life, time constant, tau, decay constant, activity, becquerel, curie, Bq, Ci, N = N₀e^(−t/τ), decay curve, decay rate, decays per second, undecayed nuclei, daughter nucleus, daughter product, carbon-14 dating, radiocarbon, nuclear medicine, technetium-99m, radiation dosimetry, nuclear waste, spent fuel, RC circuit analogy, Chapter 30, General Physics, University of Washington