In an experiment, a radioactive isotope is produced at a constant...

The rate of change of the number of active nuclei is given by the difference between the rate of production and the rate of decay.

$\dfrac{dN}{dt} = R - \lambda N$

where $R$ is the constant production rate and $\lambda$ is the decay constant.

After a long time ($t \gg t_{1/2}$), the number of active nuclei becomes constant, which means the system reaches a steady state.

At steady state, $\dfrac{dN}{dt} = 0$

$R - \lambda N = 0 \Rightarrow N = \dfrac{R}{\lambda}$

The decay constant $\lambda$ is related to the half-life $t_{1/2}$ by the equation:

$\lambda = \dfrac{\ln 2}{t_{1/2}}$

Substituting the value of $\lambda$ into the expression for $N$:

$N = \dfrac{R}{\left(\dfrac{\ln 2}{t_{1/2}}\right)} = \dfrac{R t_{1/2}}{\ln 2}$