# 2.2.3 Radioactivity in Equilibrium

References: Sorenson and Phelps; Hendee.
Created October 6, 1995

Often in radioactive decay the parent decays into a daughter nucleus which itself is radioactive. This chain is depicted as:

This section will cover radioactive "equilibrium." Equilibrium is the situation in which the ratio between the activities of the successive members of the decay series remains constant.

### 2.2.3.1 The Bateman Equations

The second term in the activity equation is just the residual daughter product activity remaining from any that was present at t=0. This equation is simpler if we set this value to zero and just examine the first term. The equation for activity of the daughter is known as the Bateman equation.
### 2.2.3.2 Secular Equilibrium

When the half-life of the parent is very long compared to that of the daughter, the decrease of parent activity is negligible over the course of the observation period. An example parent-daughter pair is Ra-226 (half-life =1620 yr) which decays to Rn-226 (half-life 4.8 days). The observation period is so small relative to 1620 years that 1 gram of Ra-226 was used to define the unit of radioactivity - the Curie.
When the half-life of the parent is much greater than the daughter, the Bateman Equation reduces to:

The plot of parent and daughter activity in secular equilibrium is:

### 2.2.3.3 Transient Equilibrium

"Transient Equilibrium" is the situation when the parent half-life is of the order of the observation time and the daughter half-life is considerably shorter (but not negligibly shorter). Examples include Te-132 (78 hours) decaying to I-132 (2.3 hours) and Sn-113 decaying to In-113m (1.7 hours). However, the best example for us is the Mo-99 parent - Tc-99m daughter relationship.

The upper curve for Tc-99m is the straight application of the Bateman equation. The lower curve is the actual curve for Tc-99m which takes into account the fact that about 10% of the Mo-99 decays go promptly to Tc-99 and therefore do not contribute to the activity of the daughter, Tc-99m.
The ratio of the daughter activity to the parent activity is given by:

where we assume all parent decays result in daughters. This ratio for Tc-99m is 95%, taking into account the alternative branch of Mo-99 decay, and 110% if we assume all Mo-99 decays result in Tc-99m formation.

If we use calculus to find the time at which the daughter reaches a maximum, we find the following equation:

For Tc-99m, it turns out that t-max = 22.8 hours.
Interesting aside: When I took the ABR exam, I did not remember this equation but rather simply remembered that Tc-99m reaches a maximum a little less than 24 hours. The multiple choice answers where, of course, 22.4 hours, 22.8 hours, 23.2 hours, 23.6 hours, and 24 hours!

### 2.2.3.4 No Equilibrium

When the daughter half-life is longer than the parent half-life, there is **no** equilibrium established between them. As the short-lived parent dies off, the activity of the daughter starts from zero, grows to a maximum, then falls slowly at its own decay rate (the parent having since died off and not able to influence daughter rate any further).
### 2.2.3.5 Mixtures of Unrelated Nuclides

The total activity of a sample of mixed nuclides, all of which decay to stable nuclides, is simply given by:

The mixture will always eventually assume the slope of the longest lived nuclide as the shorter lived nuclides die off. If one observes the mixture for a very long time, the longest lived half-life can be determined by the behavior at long times. Subtracting the longest lived curve from the mixture's decay curve leaves a remaining decay curve, who's behavior at moderately long times is that of the second longest lived nuclide. This in turn can be subtracted from the reamining decay curve, and so one.

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Douglas J. Wagenaar, Ph.D., wagenaar@nucmed.bih.harvard.edu