Calculate radioactive decay activities with the Bateman equation for parent-daughter pairs, single isotopes, or linear chains over time.

Predict daughter activity including growth from its parent. Half-lives and branch yields must describe your actual decay chain.

One year = 365.25 days.

Initial daughter and branch settings

Optional: blank means zero initial activity.

Optional: blank assumes 100% of this memberโ€™s decays feed the next member.

Optional: blank means 99%. Time estimate assumes zero initial daughter and a finite limiting activity ratio.

Bateman Equation Formula

Bateman equations describe decay and growth in radioactive chains. The single-isotope special case is shown below; this calculator uses activity A rather than atom count N, with ฮป = ln(2) / half-life.

A(t) = A0 ร— e(- ฮป ร— t)

Variables:

  • A(t) is the activity after elapsed time
  • A0 is the initial activity
  • ฮป (lambda) is the decay constant, which is the probability per unit time that a particle will decay
  • t is the time elapsed

For a parent-daughter pair: AD(t) = AD0 exp(โˆ’ฮปD t) + b AP0 ฮปD [exp(โˆ’ฮปP t) โˆ’ exp(โˆ’ฮปD t)] / (ฮปD โˆ’ ฮปP). Here b is the branching fraction. If rates are equal, AD(t) = [AD0 + b AP0 ฮปt] exp(โˆ’ฮปt). Linear-chain modes include up to four radioactive members with constant rates, no external source and no physical removal.

What is the Bateman Equation?

The Bateman equation is a fundamental equation in nuclear physics and chemistry that describes the behavior of radioactive decay chains. It is also applicable to any process that can be described by first-order kinetics, such as certain chemical reactions or biological processes. The equation provides a way to predict the quantity of a substance that remains after a period of time, assuming that the decay process is the only significant factor affecting the quantity.

How to Calculate Using the Bateman Equation?

The following steps describe single-isotope forward decay. Select the parentโ€“daughter or linear-chain calculation for daughter ingrowth; supply every included half-life and any nonzero initial daughter activities.

  1. First, select Single isotope: forward decay and enter initial activity with its unit.
  2. Next, enter the half-life; the calculator obtains ฮป = ln(2) / half-life.
  3. Then, determine the time elapsed (t).
  4. Use the single-isotope relation A(t) = A0 ร— exp(โˆ’ฮปt).
  5. Select Calculate to obtain the activity at the end of the interval.
  6. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem :

Use the following variables as an example problem to test your knowledge.

Initial activity A0 = 100 MBq

Decay constant ฮป = 0.1 per minute, equivalent to a half-life of ln(2)/0.1 โ‰ˆ 6.931471806 minutes

Time elapsed t = 5 minutes. Activity = 100 ร— exp(โˆ’0.5) โ‰ˆ 60.653066 MBq.