Write the integrated rate law for a first-order reaction.

For a first-order reaction, the integrated rate law is given by:

where $[A]_0$ is the initial concentration of the reactant, $[A]_t$ is the concentration at time $t$, and $K$ is the rate constant.
This can be rearranged as:

The half-life ($t_{1/2}$) is the time required for the concentration of a reactant to decrease to half of its initial value.
So, at $t = t_{1/2}$, the concentration $[A]_t$ becomes $\frac{[A]_0}{2}$.

Substitute $t = t_{1/2}$ and $[A]_t = \frac{[A]_0}{2}$ into the integrated rate law:

Since $\ln(2) \approx 0.693$, the expression can also be written as:

The expression for the half-life period of a first-order reaction is: