# How do you find first order half life?

##### 1 Answer

Well, consider a general **first-order rate law** for a one-reactant reaction:

#A -> B#

#r(t) = k[A]# where

#k# is the rate constant and#[A]# is the concentration of#A# in#"M"# .

This is numerically equal to:

#= -(d[A])/(dt)# where

#(d[A])/(dt)# denotes an instantaneous rate of change in concentration of#A# over time.

By separation of variables:

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

Integrate the left-hand side from time zero to time

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

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

Thus, we obtain the **first-order integrated rate law**:

#ul(ln[A] = -kt + ln[A]_0)#

For a half-life, we have a current concentration of

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

#t_"1/2"# is the half-life.

Using the properties of logarithms:

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

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

#=> ln2 = kt_"1/2"#

#=> color(blue)barul(|stackrel(" ")(" "t_"1/2" = (ln2)/k" ")|)#