How do you derive first order half life?

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Answer 1 · 2017-02-05T21:14:21

You can always start from the first-order rate law:

#r(t) = k[A] = -(d[A])/(dt)#

for the reaction:

#A -> B#

Separate variables:

#-kdt = 1/([A])d[A]#

Integrate from the initial to the final state. It is convenient to set #t_0 = 0#.

#-int_(0)^(t)kdt = int_([A]_0)^([A])d[A]#

#-(kt - k*0) = ln[A] - ln[A]_0#

#=> ln[A] - ln[A]_0 = -kt#

#=> ln\frac([A])([A]_0) = -kt#

For the half-life, at time #t = t_"1/2"#, the concentration of #A# dropped to #1/2[A]_0#. Therefore:

#ln\frac(1/2cancel([A]_0))(cancel([A]_0)) = -kt_"1/2"#

#ln (1/2) = -kt_"1/2"#

#-ln (1/2) = kt_"1/2"#

#ln (1/2)^(-1) = kt_"1/2"#

#ln 2 = kt_"1/2"#

Therefore:

#color(blue)(t_"1/2" = ln2/k)#