Overview of Radioactive Equilibrium
In a parent-daughter decay chain, after a certain period of time, the activities of the parent and daughter nuclei can reach a constant ratio. This state is known as radioactive equilibrium. The ratio between the activities of the parent P and daughter D is time-dependent but will stabilize at a constant value under certain conditions.
The activity of the parent and daughter nuclei evolves with time, and their relationship is described by the equation:
\[ \frac{A_D(t)}{A_P(t)} = \frac{\lambda_D A_P(0)}{\lambda_P - \lambda_D} \left( e^{-\lambda_D t} - e^{-\lambda_P t} \right) \]
Conditions for Radioactive Equilibrium
The relationship between the parent and daughter activities varies depending on the relative decay constants or half-lives of the parent and daughter. There are three main cases to consider:
Case 1: The Half-Life of the Daughter Exceeds that of the Parent
If the half-life of the daughter nucleus (T1/2)D is longer than that of the parent (T1/2)P, this implies λD < λP. In this case, the activity ratio increases exponentially with time and equilibrium is never reached. The equation for this scenario is:
\[ \frac{A_D(t)}{A_P(t)} = e^{-\lambda_P t} - e^{-\lambda_D t} \]
This shows that the activity ratio increases with time and never stabilizes.
Case 2: The Half-Life of the Daughter is Shorter than the Parent
If the daughter’s half-life (T1/2)D is shorter than the parent’s (T1/2)P, implying λD > λP, the activity ratio stabilizes at a constant value as time increases:
\[ \frac{A_D(t)}{A_P(t)} \to \frac{\lambda_D}{\lambda_D - \lambda_P} \]
This indicates that the activity ratio becomes constant and independent of time, representing transient equilibrium.
Case 3: The Half-Life of the Daughter is Much Shorter Than the Parent
If the daughter’s half-life (T1/2)D is much shorter than the parent’s (T1/2)P, meaning λD ≫ λP, the activity of the daughter approximates that of the parent and the ratio approaches 1:
\[ \frac{A_D(t)}{A_P(t)} \approx 1 \]
This condition represents secular equilibrium, where the activities of the parent and daughter are essentially the same and decay together.
Example Calculation of Activity Ratio
Suppose we have the following values for the decay constants:
- λP = 0.0001 s-1
- λD = 0.0002 s-1
We can calculate the activity ratio AD(t) / AP(t) at any time t using the formula from above. For large time t, the ratio becomes:
\[ \frac{A_D(t)}{A_P(t)} = \frac{\lambda_D}{\lambda_D - \lambda_P} \]
Substituting the values:
Activity ratio at large time: AD(t) / AP(t) = 2