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" ")|)#