By β^ + -emission, ^ 57 Co decays into ^ 57 Fe . The produced ^ 57...

The decay sequence is ^{57}\text{Co} \xrightarrow{\beta^+} {}^{57}\text{Fe}^{*} \xrightarrow{\gamma} {}^{57}\text{Fe}$.<br><br>Since the half-life of ^{57}\text{Co}$ ($270 \text{ days}$) is much greater than the half-life of the excited ^{57}\text{Fe}^{*}$state ($10^{-8} \text{ s}$), secular equilibrium is established almost instantaneously.<br><br>In secular equilibrium, the rate of formation of the daughter nucleus is equal to its rate of decay. Thus, the activity of ^{57}\text{Fe}^{*}$ ($\gamma$-emission rate) is equal to the activity of ^{57}\text{Co}$($\beta^{+}$-emission rate).<br><br>The$\gamma$-emission rate will therefore decay with the half-life of the parent ^{57}\text{Co}$ nucleus.

The $\gamma$-emission rate drops from $5.0 \times 10^{9} \text{ s}^{-1}$ to $2.5 \times 10^{9} \text{ s}^{-1}$, which is exactly half of its initial value.

The time required for the activity to become half is one half-life of ^{57}\text{Co}$, which is$270 \text{ days}.