14.33 Equation for the half-life of a 2nd order reaction

[Hannah Pablo 2H]

14.33 The half-life for the second-order reaction of a substance A is 50.5 s when [A]0 =0.84 mol/L. Calculate the time needed for the concentration of A to decrease to (a) one-sixteenth; (b) one- fourth; (c) one-fifth of its original value.

How do you obtain the equation to begin this problem?

[Niharika Reddy 1D]

Half life for second order

At the half life, t_1/2, half the initial concentration remains. Substituting this piece of information into the second order integrated rate law and solving for t gives the equation for the half life of a second order reaction.

[Neil DSilva 1L]

On Prof. Lavelle's Constants and Equations, the last line gives the two relevant equations for this problem.
For second order reactions, use and (the sheet uses R instead of A, but I used A because that's what the solutions manual uses).

Rearrange the half-life equation to solve for k.

Then rearrange the first equation to solve for t. Replace with where is the concentration after the initial concentration has decreased to one-nth its original value.