Question

Determine the integrated form of the third-order rate law of the form:

-d[A]/dt = k[A]^3.

Starting from a 0.5 M initial concentration of A, and a rate constant of 0.1 M^-2 * s^-1, how much time would be required before reaching a 0.3M concentration of A?

Answer Options

		78.2 s
		35.6 s
		22.3 s
		13.7 s

Answer 1

Given rate law is :   -d[A]/dt = k[A]³

                             d[A]/[A]³ = -kdt

                     apply integration we get

                           d[A]/[A]³ = - kdt

                              -(1/(2[A]² )) = -kt + I                           I = integration constant

Obtaining the value of I :

If [A_o] be the initial concentration.

When t= 0 s ---> [A] = [A_o] then above equation becomes     -(1/(2[A_o]² )) = -k(0) + I  

                                                                                                        I = -(1/(2[A_o]² ))

Therefore the integration equation is     -(1/(2[A]² )) = -kt + -(1/(2[A_o]² ))

       kt = (1/(2[A]² )) -(1/(2[A_o]² ))  

kt = (1/2) [(1/[A]²) - (1/[A_o]²)]

Given [A_o] = 0.5 M

[A] = 0.3M

k = rate constant = 0.1 M^-2 s^-1

Plug the values we get

t = (1/2k) [(1/[A]²) - (1/[A_o]²)]

t = (1/(2x0.1)) [(1/0.3²) - (1/0.5²)]

= 35.6 s

Therefore the required time is 35.6 s