#### Answer

Zero order: $t_{\frac{1}{2}}=\frac{[A]_0}{2k}$
First order: $t_{\frac{1}{2}}=\frac{ln2}{k}$
Second order: $t_{\frac{1}{2}}=\frac{1}{k[A]_0}$
The half life of a zero order reaction is directly proportional to the concentration of the reactant remaining. The half life of a first order reaction is independent of the concentration of the reactant remaining. The half life of a second order reaction is inversely proportional to the concentration of the reactant reaming.
If the half life of a zero order reaction is 20 seconds, then the second half life will be 10 seconds.
If the half life of a first order reaction is 20 seconds, then the second half life will also be 20 seconds.
If the half life of a second order reaction is 20 seconds, then the second half life will be 40 seconds.

#### Work Step by Step

To find the half life, we use the fact that the concentration once the half life is completed will be 1/2 its initial concentration or $[A]=\frac{1}{2}[A]_0$. Plugging this into the respective integrated rate laws and solving for $t$ will yield the half life equation.
The half life of a zero order reaction is directly proportional to the concentration of the reactant remaining because doubling the concentration of the reactant will double the half life. The half life of a first order reaction is independent of the concentration of the reactant remaining because doubling the concentration of the reactant will not change the half life. The half life of a second order reaction is inversely proportional to the concentration of the reactant reaming because doubling the concentration of the reactants will half the half life.
The second half life of a zero order reaction will be half its first half life because its half life is directly proportional to the concentration of the reactant and there will be half as much reactant once the first half life is over. The second half life of a first order reaction will be the same as its first half life because it is independent of the concentration of the reactant. The second half life of a second order reaction will be double its first half life because its half life is inversely proportional to the concentration of the reactant and there will be half as much reactant when the first half life is over.