Answer:
19.47 years
Step-by-step explanation:
In order to find the answer we need to use the following formula:
[tex]m(t)=m(0)*e^{kt}[/tex] where:
m(t)=amount after 't' years
m(0)=initial amount
k=decay constant
t= time in years
If the initial amount of substance is 'x' then the problem establishes:
m(0)=x*100
m(t)=x*96.5
t=1 year
Using the equation we have:
[tex]x*96.5=x*100*e^{k*1}[/tex]
[tex]ln(96.5/100)=k[/tex]
[tex]-0.0356=k[/tex]
Now, the half-life of a substance is the time 't' which allows half of the substance to be decayed. So:
m(0)=x*100
m(t)=x*50
k=-0.0356
Then:
[tex]x*50=x*100*e^{-0.0356t}[/tex]
[tex]ln(50/100)/(-0.0356)=t[/tex]
[tex]19.47=t[/tex]
In conclusion, the half-life of the substance is 19.47 years.
Answer:
The half-life of the substance is 19.47 years.
Step-by-step explanation:
The equation for the amount of substance remaining is given by the following equation:
[tex]Q(t) = Q(0)e^{-rt][/tex]
In which Q(t) is the amount remaining after t years, Q(0) is the initial amount and r is the rate that this amount decreases.
A sample of a radioactive substance decayed to 96.5% of its original amount after a year.
This means that [tex]Q(1) = 0.965Q(0)[/tex]
We use this to find r. So
[tex]Q(t) = Q(0)e^{-rt][/tex]
[tex]0.965Q(0) = Q(0)e^{-r][/tex]
[tex]e^{-r} = 0.965[/tex]
[tex]\ln{e^{-r}} = \ln{0.965}[/tex]
[tex]-r = =0.0356[/tex]
[tex]r = 0.0356[/tex]
So
[tex]Q(t) = Q(0)e^{-0.0356t}[/tex]
(a) What is the half-life of the substance?
This is t when Q(t) = 0.5Q(0). So
[tex]Q(t) = Q(0)e^{-0.0356t}[/tex]
[tex]0.5Q(0) = Q(0)e^{-0.0356t]}[/tex]
[tex]e^{-0.0356t} = 0.5[/tex]
[tex]\ln{e^{-0.0356t}} = \ln{0.5}[/tex]
[tex]-0.0356t = \ln{0.5}[/tex]
[tex]t = -\frac{\ln{0.05}}{0.0356}[/tex]
[tex]t = 19.47[/tex]
The half-life of the substance is 19.47 years.
