A radioactive substance decays according to the formula Q ( t ) = Q 0 e ? kt whe
ID: 2869292 • Letter: A
Question
A radioactive substance decays according to the formula
Q(t) = Q0e?kt
where Q(t) denotes the amount of the substance present at time t (measured in years), Q0 denotes the amount of the substance present initially, and k (a positive constant) is the decay constant.
(a) Find the half-life of the substance in terms of k.
(b) Suppose a radioactive substance decays according to the formula
Q(t) = 15e?0.0001438t
How long will it take for the substance to decay to half the original amount? (Round your answer to the nearest whole number.)
yr
Explanation / Answer
A) Q(t) = Q0e-kt
At t = 0 , Q(t) = Q(0) = Q0
Thus,
At half life Q(t) = Q0/2
Hence, Q0/2 = Q0*e-kt
or, (1/2) = e-kt
Taking log to the base 'e' both sides we get
ln(1/2) = -k*t
or, ln2 = k*t
or, t = ln2/k = half life
B) Using the equation in (A) ; we get that Q0 = 15 & k = 1.438*10-4 yr-1
Thus, time taken to decay half of its original = half life of the decay process = ln2/k = 0.693/(1.438*10-4) 4.82*103 years