[kellycox]

I have struggled with a problem for over 1 1/2 days. Here's the question.
   "The reaction A->B+C. is 2nd order in A. When [A]0=0.100M, the reaction is 20% complete in 40.0 minutes. Calculate the value of the rate constant(in L/min *mol)."
    The formula that I have been working with is Half-life
                                  t1/2 = 1/k[A]0
    I am solving for "k", but I keep getting an answer of 1.00 x 10-1. I don't think that I am using the right formula for the information that I have. Is there another formula that I should be using to solve this question? Thanks for any direction/help that you can give me.

[pantone159]

Half-life is for first order reactions, not second.
There would be a different formula for second order.

[Donaldson Tan]

first order reactions exhibit constant half-life.

second order reactions exhibit half-life which is a function of the concentration. The derrivation is shown below:

let A be the concentration of component A.
-dA/dt = k.A² => -dA/A² = k.dt
integrate from Ao to 0.5Ao ( -dA/A² )  = integrate from 0 to t_1/2 ( k.dt )
=> 1/Ao = k.t_1/2 => t_1/2 = 1/k.Ao

Kelly's equation is correct. But there is no point using half-life to calculate k. That is because the reaction is 20% complete and not 50% complete.

Since we are given the time taken for reaction to complete 20%, then
concentration of A at 20% complete = (1 - 0.2)Ao = 0.8 Ao
t_0.2 = 40min = 40x60 s = 2400s
let A be the concentration of component A.
-dA/dt = k.A² => -dA/A² = k.dt
integrate from Ao to 0.8Ao ( -dA/A² )  = integrate from 0 to t_0.2 ( k.dt )
=> 1/0.8Ao - 1/Ao = k.t_0.2 => (1/Ao)(0.8^-1-1) = k.t_0.2

The rate constant k is this given as k = 1/Ao.t_0.2)(0.8^-1-1)