How this calculation works
Track exponential parent depletion and daughter production minus daughter decay. Use the continuous equal-rate limit when half-lives coincide, and a stable-daughter convention when its half-life input is zero.
Parent = P₀ exp(−λp t); daughter = D₀ exp(−λd t) + b λp P₀ (exp(−λp t) − exp(−λd t)) / (λd − λp)A worked example
With 1,000 parent atoms, no initial daughter and equal 10-unit half-lives, time ten leaves 500 expected parent atoms and about 346.574 expected daughter atoms. Some produced daughters have already decayed onward.
Good to know
Isolated two-member expected-inventory model, not dose or handling guidance. All times share one unit; activity is per that unit, not automatically Bq. Other branches and later-chain products are excluded.
Reference: US DOE OSTI: decay chain equations (opens in a new tab) ↗
Daughter production and daughter loss occur together
A parent decay can produce a daughter while previously produced daughter atoms are already decaying. Subtracting the remaining parent inventory from the initial parent count only gives daughter inventory when the daughter is stable and the branch is complete.
The table retains parent and daughter separately. It does not label missing inventory as vanished matter; later decay products and other branches lie outside this two-member model.
Equal half-lives need a limit rather than division by zero
The usual two-exponential expression divides by the difference of decay constants. When those constants are equal, numerator and denominator both approach zero even though the physical solution remains finite.
The implementation uses the equal-rate limit directly and a stable exponential-difference form for nearby rates. Setting daughter half-life to zero explicitly selects a stable daughter, not instantaneous decay.
Activity inherits the entered time unit
Activity is decay constant multiplied by expected inventory. If the half-life and elapsed time are entered in hours, the activity is decays per hour; it is becquerels only when the common time unit is seconds.
Expected inventories can be fractional. The deterministic expectation does not simulate individual random events or provide radiation protection, material handling or dose calculations.
Use the result with the next measurement
The Arrhenius two temperature rate calculator describes a different exponential-rate model. Radioactive half-life is an entered chain constant here and is not inferred from a chemical activation-energy scenario.
Isolated two-member expected-inventory model, not dose or handling guidance. All times share one unit; activity is per that unit, not automatically Bq. Other branches and later-chain products are excluded.
A few questions, answered
Why is daughter not always initial parent minus final parent?
The daughter can also decay and branching may be incomplete.
What does daughter half-life zero mean?
Stable daughter in this tool.
Are activities always Bq?
Only if the common time unit is seconds.