Answer to Question #21959 in Atomic and Nuclear Physics for rahav

Question

75% of a sample was decayed in 24 hours , find the half life of the sample.

Answer 1

Let X0 be the initial amount of the sample.
Assume that the amount of sample at time t is given bythe formula
X(t) = X0 e^(-at)
for some constant a>0.
We should find T such that
X(T)/X0 = e^(-aT) = 0.5

For this we need to know value of a.
By assumption at time t=24 hours
0.75 = X(24)/X0= e^(-a*24),
whence
a = -ln(0.75) /24
= 0.011987

Thus
X(t) = X0e^(-0.011987 t).

Hence fior half life T we have that
e^(-0.011987 *T) = 0.5
and so
T =-ln(0.5)/0.011987
= 57.825hours

Thus the half life of the sample is 57.825 hours.