Respuesta :
Answer:- 9.4 minutes.
Solution:- Radioactive decay obeys first order reaction kinetics and the equation used to solve this type of problems is:
[tex]lnN=-kt+lnN_0[/tex]
where, k is decay constant and t is the time. [tex]N_0[/tex] is the initial amount of the radioactive substance and N is the remaining amount.
Since the value of decay constant is not given, so we need to calculate it first from given half life by using the formula:
[tex]k=\frac{0.693}{t_1_/_2}[/tex]
where [tex]t_1_/_2[/tex] stands for half life.
Given half life is 3.0 minutes.
So, [tex]k=\frac{0.693}{3.0min}[/tex]
[tex]k=0.231min^-^1[/tex]
Let's plug in the values in the first order reaction equation and solve it for t.
[tex]ln4.5g=-0.231min^-^1(t)+ln40.0g[/tex]
It could also be written as:
[tex]ln(\frac{4.5g}{40.0g})=-0.231min^-^1}[/tex]
[tex]-2.18=-0.231min^-^1}[/tex]
[tex]t=\frac{-2.18}{-0.231min^-^1}[/tex]
k = 9.4 min
So, the radioactive substance would take 9.4 minutes to decay from 40.0 grams to 4.5 grams.
