Chemistry question - using the integrated 1st order rate law...please help!?

Consider two reaction vessels, one containing A and the other containing B. At t = 0, [A]o = [B]o. A and B decompose by first order kinetics with rate constants of
kA = 4.50 × 10-4 s-1 and kB = 3.70 × 10-3 s-1, respectively. Calculate the time that must pass to reach the condition such that [A] = (4.18)[B].

Please work out the problem and explain to me how you find the concentrations of A and B to start with. Thank you!

joeyeehu...

Best Answer - Chosen by Voters

For A: ln [A]o/[A] = k' t
ln[A]o - ln[A] = k' t
ln[A]o = k' t + ln[A]....................(1)

For B:
ln [B]o = k" t + ln[B]

Since [A]o = [B]o

ln [A]o = k" t + ln [B] ...............(2)

(2) = (1)

k" t + ln [B] = k' t + ln [A]

Given that [A] = (4.18) [B]

(k" - k' ) t = ln (4.18)[B] - ln [B]

(k" - k') t = ln 4.18 .......................{ln X - ln Y = ln X/Y}

t = ln (4.18) / (3.70 x 10^-3 - 4.5 x 10^-4)

t= 440.1 s