# How do you calculate half life decay?

##### 1 Answer

It depends on the order of the reaction with respect to the reactant undergoing half-life decay.

The common half-life expressions for one reactant

#t_"1/2" = ([A]_0)/(2k)# (zero order)

#t_"1/2" = (ln2)/k# (first order)

#t_"1/2" = 1/(k[A]_0)# (second order)

#t_"1/2" = 3/(2k[A]_0^2)# (third order)where

#[A]_0# is the initial concentration in#"M"# and#k# is the rate constant in the appropriate units to obtain time units of#"s"# .

If you wish to derive them, that requires some calculus. I can do first order as an example, as that is the hardest one. For the general rate law

#r(t) = k[A] = -(d[A])/(dt)# ,of the first-order reaction

#A -> B# ,where the negative sign indicates disappearance of reactant,

separation of variables gives:

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

Now, integrate from

#-int_(0)^(t) kdt = int_([A]_0)^([A]) 1/([A])d[A]#

The integral of

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

For a half-life,

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

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

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