you are given the initial amount of 10 milligrams of the ele

you are given the initial amount of 10 milligrams of the element phosphorus 32 which has a half-life of 14 days. assuming the element decays according to the function: A(t)=A0e^kt where A0 is the inital amount present and k is the decay constant, find a expression for the decay constant, k.

Solution

A(t) = A0 e^kt

A(t) /A0 = e^kt

apply ln on both sides

ln (A(t) /A0) = ln(e^kt)

ln(A(t) /A0) = kt ln(e) [ by fromula ln a^m = m lna ]

ln(A(t) /A0) = kt *1 [ ln(e) =1]

kt = ln(A(t)/A0)

k = [ ln(A(t) /A0 ] /t is the expression for finding decay constant k

given half life is 14 days

so A(t) /A0 = 1/2

we have to convert 14 days into years 14/365 = 0.0384

k = ln(1/2)/0.0384

k= -0.693/0.0384

k = -18.051