# What is the first-order half-life for #"0.50 M"# of a substance whose radioactive decay has a rate constant of #0.69 xx 10^(-2) "min"^(-1)#?

##### 1 Answer

May 20, 2017

You don't need its concentration. (Why?)

#t_("1/2") = "100.5 min"#

The **Integrated Rate Law** for the first-order process is:

#ln[A] = -kt + ln[A]_0# where:

#[A]# is the concentration of the reactant.#[A]_0# is its initial concentration.#k# is the rate constant in#"time"^(-1)# .#t# is of course the time in#"time"# units.

For a half-life, we take

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

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

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

But since

#-ln2 = -kt_"1/2"# ,

and:

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

See, the concentration doesn't matter for first-order...

#color(blue)(t_"1/2") = (ln2)/(0.69 xx 10^(-2) "min"^(-1))#

#=# #color(blue)("100.5 min")#