For second-order reactions, how does each subsequent half-life relate to the previous one?

For second-order reactions, the relationship between concentration and time is such that as the concentration of the reactant decreases, the time required for the concentration to decrease by half (the half-life) increases. The formula for the half-life of a second-order reaction is given by:

[ t_{1/2} = \frac{1}{k [A]_0} ]

where ( t_{1/2} ) is the half-life, ( k ) is the rate constant, and ( [A]_0 ) is the initial concentration of the reactant.

In a second-order reaction, each time a half-life occurs, the concentration of the reactant reduces from ( [A]_0 ) to ( \frac{[A]_0}{2} ) after the first half-life. During the second half-life, the concentration will drop from ( \frac{[A]_0}{2} ) to ( \frac{[A]_0}{4} ). Because the half-life is inversely related to the initial concentration, the half-life for the second instance is:

[ t_{1/2,2} = \frac{1}{k \left(\frac{[A]_