Carbon-14 will take 19,035 years to decay to 10 per cent.
What is the time of decay?
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_oe^{-\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]t_{1/2}=\dfrac{Ln(2)}{\lambda}[/tex]
If we know the value of the half-life, we can calculate the decay constant:
[tex]\lambda =\dfrac{ln(2)}{t_{1/2}}[/tex]
Carbon-14 has a half-life of 5,730 years, thus:
[tex]\lambda =\dfrac{ln(2)}{5730}[/tex]
[tex]\lambda[/tex] = 0.00012097
The equation of the remaining quantity of Carbon-14 is:
[tex]N(t)=N_oe^{-0.00012097t}[/tex]
We need to calculate the time required for the original amount to reach 10%, thus N(t)=0.10No
[tex]0.10N_o=-N_oe^{-0.00012097t}[/tex]
Simplifying:
[tex]0.10=e^{0.00012097t}[/tex]
Taking logarithms:
ln(0.10) = -0.00012097t
Solving for t:
[tex]t=\dfrac{Log0.10}{-0.00012097}[/tex]
t = 19035 years
Carbon-14 will take 19,035 years to decay to 10 per cent.
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