The reaction of certain chemical compounds can be modeled using a differential e

Question

The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -ky^n(t). where y(t) is the concentration of the compound for t greaterthanorequalto 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume that the initial concentration of the compound is y(0) = y_0 > 0. (a) Consider a first-order reaction (n = 1) and show that the solution of the initial value problem y(t) = y_0e^-kt. (b) Consider a second-order reaction (n = 2) and show that the solution of the initial value problem is y(t) = y_0/y_0 kt + 1.

Explanation / Answer

From the given question,

(a)dy/dt= -ky

dy/y=-k dt

integrating both sides

ln y =-kt +c

when t=0, y=y0

ln y0 = -k(0) +c

c=ln y0

ln y =-kt +ln y0

ln(y/y0 )=-kt

y/y0 = e-kt

y=y0 e-kt

(b)dy/dt= -ky2

dy/y2=-kdt

-1/y=-kt+c

when t=0, y=y0

-1/y0 = c

-1/y=-kt-1/y0

-1/y= (-kty0-1)/y0

y=y0/(1+kty0)

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