Half-Life

Half-life is the time it takes for the concentration of a reactant to be cut in half.

The longer the half-life, the slower the reaction. A reaction that takes 2 hours to reach its half-life, is slower than a reaction that only takes 1 hour to reach its half-life.

Zero-Order Reaction Integrated Rate Law: [A]t = -kt + [A]o For half-life, [A]t = 1/2[A]o and t = t1/2 Substitute: 1/2[A]o = -kt1/2 + [A]o Rearrange: kt1/2 = 1/2[A]o Solve for t1/2: t1/2 = [A]o 2k

Half-life for t1/2 = [A]o Zero-Order 2k Half-life and initial concentration are directly related. When initial concentration falls, half-life falls.

Let’s look at a graph of [A] v. time for a zero-order. Starting at the initial concentration of 0.200 M, let’s see how long it takes for the concentration to fall to 0.100 M. 0.200 M 0.100 M

1st Half-life: 260 Seconds

For the second half-life, the “initial concentration” is 0.100 M. Use the final concentration from the first half-life to be the “initial concentration” or [A]o for the second half-life. 0.200 M 1st 0.100 M 2nd 0.050 M

1st Half-life: 260 Seconds 2nd Half-Life: 130 Seconds 65 s

Zero-Order Reaction Integrated Rate Law: [A]t = -kt + [A]o Half-Life for t1/2 = [A]o Zero-Order 2k Each successive half-life is shorter than the previous half-life.

First-Order Reaction Integrated Rate Law: ln[A]t = -kt + ln[A]o For half-life, [A]t = 1/2[A]o and t = t1/2 Substitute: ln 1/2[A]o = -kt1/2 + ln[A]o Rearrange: ln 1/2[A]o - ln[A]o = -kt1/2 ln 1/2[A]o /ln[A]o = -kt1/2 ln 1/2 [A]o /[A]o = -kt1/2 ln 1/2= -kt1/2 -0.693 = -kt1/2 Solve for t1/2: 0.693 = t1/2 k

Half-life for 0.693 = t1/2 First-Order k Half-life and initial concentration are NOT related. When initial concentration falls, no effect on half-life is observed.

180 s 180 s 180 s

First-Order Reaction Integrated Rate Law: ln[A]t = -kt + ln[A]o Half-Life for t1/2 = 0.693 First-Order k Each successive half-life is the same length as the previous half-life.

Second-Order Reaction Integrated Rate Law: 1/[A]t = kt + 1/[A]o For half-life, [A]t = 1/2[A]o and t = t1/2 Substitute: 1/1/2[A]o = kt1/2 + 1/[A]o Solve for t1/2: 1 = t1/2 k[A]o

Half-life for: 1 = t1/2 Second-Order k[A]o Half-life and initial concentration are inversely related. When initial concentration falls, half-life gets longer.

130 s 260 s 520 s

Second-Order Reaction Integrated Rate Law: 1/[A]t = kt + 1/[A]o Half-Life for t1/2 = 1 Second-Order k[A]o Each successive half-life is longer than the previous half-life.

To determine half-life, you need to know: 1. the order of the reaction 2. the rate constant 3. the initial concentration