A certain radioactive substance is known to decay at a rate proportional to the

Question

A certain radioactive substance is known to decay at a rate proportional to the amount present. An experiment begins with 120g of the material. The differential equation of the decay substance is given by dN/dt - KN = 0 , where k is the proportionally decay constant. After 15 days, it is observed that only 90g remain. (a) what is the general expression for decay of substance as a function of time? (b) what are the value of decay constant k and the constant of integration? (c) What is the half life of the radioactive substance?

Explanation / Answer

a) Decay rate is first order reaction

hence

dN/dt = -kN

dN/N = -kdt

Integrating we get

ln N = -kt + C

when t = 0 we have N = 120 g

so ln 120 = 0 + c

c = ln120

and when t = 15 we have N = 90 g

so ln 90 = -k15 + ln120

Solving we get

k = 0.019

Hence the equation of decay becomes

ln N = -0.019t + 4.787

N = e^(-0.019t + 4.787)

b) k = 0.019 and constant = ln120 or 4.787

c) half life can be calculated as : T(1/2) = 0.693/k = 36.47 days.

Hence half life of the substance is 36.47 days

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