Answer:
Carbon-14 will take 19,035 years to decay to 10 percent.
Step-by-step explanation:
Exponential Decay Function
A radioactive half-life refers to the amount of time it takes for half of the original isotope to decay.
An exponential decay can be described by the following formula:
[tex]N(t)=N_{0}e^{-\lambda t}[/tex]
Where:
No = The quantity of the substance that will decay.
N(t) = The quantity that still remains and has not yet decayed after a time t
[tex]\lambda[/tex] = The decay constant.
One important parameter related to radioactive decay is the half-life:
[tex]\displaystyle t_{1/2}=\frac {\ln(2)}{\lambda }[/tex]
If we know the value of the half-life, we can calculate the decay constant:
[tex]\displaystyle \lambda=\frac {\ln(2)}{ t_{1/2}}[/tex]
Carbon-14 has a half-life of 5,730 years, thus:
[tex]\displaystyle \lambda=\frac {\ln(2)}{ 5,730}[/tex]
[tex]\lambda=0.00012097[/tex]
The equation of the remaining quantity of Carbon-14 is:
[tex]N(t)=N_{0}e^{-0.00012097\cdot t}[/tex]
We need to calculate the time required for the original amout to reach 10%, thus N(t)=0.10No
[tex]0.10N_o=N_{0}e^{-0.00012097\cdot t}[/tex]
Simplifying:
[tex]0.10=e^{-0.00012097\cdot t}[/tex]
Taking logarithms:
[tex]ln 0.10=-0.00012097\cdot t[/tex]
Solving for t:
[tex]\displaystyle t=\frac{log 0.10}{-0.00012097}[/tex]
[tex]t\approx 19,035\ years[/tex]
Carbon-14 will take 19,035 years to decay to 10 percent.
