Activity Calculator

Activity equals decay constant times quantity

Solution

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How It Works

Activity (A) is the rate of radioactive disintegrations per second in a sample. It is the product of the decay constant and the number of radioactive atoms present: A = λN. The SI unit is the becquerel (Bq), defined as one disintegration per second. The older curie (Ci) equals 3.7 × 10¹⁰ Bq — the activity of one gram of radium-226. Activity decreases exponentially over time at the same rate as the number of atoms.

Example Problem

A sample contains 1.0 × 10²⁰ atoms of a radioactive isotope with decay constant λ = 0.001 s⁻¹. Calculate the activity in becquerels and curies.

  1. Use A = λN. Substitute the given values.
  2. A = 0.001 s⁻¹ × 1.0 × 10²⁰ atoms.
  3. A = 1.0 × 10¹⁷ disintegrations per second = 1.0 × 10¹⁷ Bq.
  4. Convert to curies: 1.0 × 10¹⁷ Bq ÷ 3.7 × 10¹⁰ Bq/Ci ≈ 2.7 × 10⁶ Ci.
  5. Interpretation: this sample emits 10¹⁷ particles per second — millions of curies, an extremely active source.

Key Concepts

Activity is what radiation detectors measure directly; mass-to-mass comparisons of radioactive samples are deceptive because one gram of a short-lived isotope can be enormously more active than one gram of a long-lived isotope with the same atomic structure. Activity scales linearly with the number of atoms (N) and inversely with half-life (small t½ → large λ → large A). Because A = λN and N decays exponentially, activity itself follows A(t) = A₀e^(-λt). Specific activity (Bq per gram) is a useful normalized quantity for comparing isotopes.

Applications

  • Nuclear medicine — radiopharmaceutical dose is specified in MBq or mCi at calibration time and back-calculated to administration time using the half-life.
  • Radiation safety — exposure limits are set in terms of activity per unit area or volume, not mass.
  • Smoke detector design — the 0.9 µCi americium-241 button in residential detectors emits alpha particles to ionize the air gap.
  • Industrial radiography — Ir-192 sources (typically 1–100 Ci) image welds in pipelines and pressure vessels.
  • Environmental monitoring — bequerels per kg of soil or per liter of water benchmark contamination levels after nuclear releases.

Common Mistakes

  • Confusing activity with dose — activity counts disintegrations; dose (Gy, Sv) accounts for energy absorbed and biological weighting.
  • Forgetting that activity decreases with time — a medical isotope's activity at administration is lower than at calibration; correct using A(t) = A₀ e^(-λt).
  • Using mixed units — if N is in atoms and λ is in s⁻¹, A comes out in Bq; if λ is in min⁻¹, multiply by 1/60 to get Bq.
  • Equating curies and becquerels — 1 Ci = 3.7 × 10¹⁰ Bq; orders of magnitude apart.

Frequently Asked Questions

How do you calculate radioactive activity?

Multiply the decay constant by the number of radioactive atoms: A = λN. The result is in becquerels (decays per second) when λ is in s⁻¹.

What is the formula for activity?

A = λN. Equivalently, since λ = ln(2)/t½, you can write A = N × ln(2) / t½ — useful when the half-life is given instead of the decay constant.

What units is activity measured in?

The SI unit is the becquerel (Bq), one decay per second. The older curie (Ci) equals 3.7 × 10¹⁰ Bq and is still widespread in medical and nuclear industry contexts. Submultiples kBq, MBq, GBq and mCi, µCi are common.

How is activity different from radiation dose?

Activity tells you how many disintegrations are happening per second; dose tells you how much energy is being absorbed per unit mass of tissue (gray, Gy) and how much biological harm that energy does (sievert, Sv). A high-activity source far away may deliver less dose than a lower-activity source held in your hand.

What is specific activity?

Specific activity is activity divided by mass — typically expressed in Bq/g or Ci/g. It's a property of the isotope (and any carrier isotopes), useful for comparing samples of different sizes or estimating how much mass a given activity represents.

Does activity stay constant over time?

No — activity decreases exponentially at the same rate as the number of atoms, since A = λN. A freshly produced medical isotope is far more active than the same physical sample after a few half-lives.

Reference:

Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Worked Examples

Nuclear Medicine

What is the activity of a 1 × 10¹⁰ atom Tc-99m dose for a SPECT scan?

Technetium-99m is the most common radiotracer in nuclear medicine; its 6-hour half-life corresponds to λ ≈ 3.21 × 10⁻⁵ s⁻¹. A patient dose contains roughly 10¹⁰ Tc-99m atoms. Compute the activity that drives the gamma camera image.

  • Knowns: λ = 3.21 × 10⁻⁵ s⁻¹, N = 1 × 10¹⁰ atoms
  • Formula: A = λ × N
  • A = (3.21 × 10⁻⁵) × (1 × 10¹⁰)

A ≈ 3.21 × 10⁵ Bq (321 kBq)

Real clinical Tc-99m doses are in the MBq–GBq range; this scaled-down example highlights the linear A = λN relationship rather than a literal injected dose.

Smoke Detector

How many americium-241 atoms sit behind a 37 kBq smoke-detector ionization chamber?

A typical residential ionization smoke detector contains about 37 kBq of americium-241. With Am-241's very long half-life of 432.6 years (λ ≈ 5.078 × 10⁻¹¹ s⁻¹), solve A = λN backward to find the number of source atoms.

  • Knowns: A = 37,000 Bq, λ = 5.078 × 10⁻¹¹ s⁻¹
  • Formula: N = A / λ
  • N = 37,000 / (5.078 × 10⁻¹¹)

N ≈ 7.29 × 10¹⁴ atoms

That's a fraction of a microgram of Am-241 — small enough to be a closed, low-risk source while still emitting a steady alpha flux that ionizes the air gap in the detector.

Nuclear Forensics

What decay constant does a 50,000 Bq sample with 1.5 × 10¹⁵ atoms imply?

A health-physics technician measures a sealed reference sample at 50,000 Bq and, by mass spectrometry, counts 1.5 × 10¹⁵ atoms of the radionuclide. Invert A = λN to extract the decay constant — and from there, the half-life t½ = ln(2)/λ.

  • Knowns: A = 50,000 Bq, N = 1.5 × 10¹⁵ atoms
  • Formula: λ = A / N
  • λ = 50,000 / (1.5 × 10¹⁵)

λ ≈ 3.33 × 10⁻¹¹ s⁻¹

Once λ is in hand, the half-life follows from t½ = ln(2)/λ ≈ 0.693 / (3.33 × 10⁻¹¹) ≈ 2.08 × 10¹⁰ s ≈ 660 years — useful for identifying an unknown isotope by its decay constant.

Radioactive Activity Formula

Activity is the rate at which atoms in a radioactive sample disintegrate. It's the product of the decay constant and the number of radioactive atoms present:

A = λ × N

Where:

  • A — activity, in becquerels (Bq = decays per second) or curies (1 Ci = 3.7 × 10¹⁰ Bq)
  • λ (lambda) — decay constant, units of inverse time (s⁻¹, hr⁻¹, yr⁻¹). Equal to ln(2) / t½
  • N — number of radioactive atoms currently in the sample

Activity itself decays exponentially over time because N decays exponentially: A(t) = A₀ × e^(−λt). A freshly produced isotope is far more active than the same physical sample after a few half-lives — a fact medical-imaging departments rely on when calibrating doses for delayed administration.

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