# Transient equilibrium

In nuclear physics, transient equilibrium is a situation in which equilibrium is reached by a parent-daughter radioactive isotope pair where the half-life of the daughter is shorter than the half-life of the parent. Contrary to secular equilibrium, the half-life of the daughter is not negligible compared to parent's half-life. An example of this is a molybdenum-99 generator producing technetium-99 for nuclear medicine diagnostic procedures. Such a generator is sometimes called a cow because the daughter product, in this case technetium-99, is milked at regular intervals.[1] Transient equilibrium occurs after four half-lives, on average.

## Activity in transient equilibrium

The activity of the daughter is given by the Bateman equation:

$A_d = ([A_P(0)\frac{\lambda_d}{\lambda_d-\lambda_P} \times (e^{-\lambda_Pt}-e^{-\lambda_dt})] \times BR ) + A_d(0)e^{-\lambda_dt}$,

where $A_P$ and $A_d$ are the activity of the parent and daughter, respectively. $T_P$ and $T_d$ are the half-lives of the parent and daughter, respectively, and BR is the branching ratio.

In transient equilibrium, the Bateman equation cannot be simplified by assuming the daughter's half-life is negligible compared to the parent's half-life. The ratio of daughter-to-parent activity is given by:

$\frac{A_d}{A_P} = \frac{T_P}{T_P-T_d} \times BR$.

## Time of maximum daughter activity

In transient equilibrium, the daughter activity increases and eventually reaches a maximum value that can exceed the parent activity. The time of maximum activity is given by:

$t_{max} = \frac{1.44 \times T_P T_d}{T_P-T_d} \times ln(T_P/T_d)$,

where $T_P$ and $T_d$ are the half-lives of the parent and daughter, respectively. In the case of $^{99m}Tc$-$^{99}Mo$ generator, the time of maximum activity ($t_{max}$) is approximately 24 hours which makes it convenient for medical use. [2]