Two different radioactive isotopes decay to 10% of their respective original amounts. Isotope A does this is 33 days, while isotope B does this in 43 days. What is the approximate difference in the half-lives of the isotopes?
3 days
10 days
13 days
33 days

Question

Two different radioactive isotopes decay to 10% of their respective original amounts. Isotope A does this is 33 days, while isotope B does this in 43 days. What is the approximate difference in the half-lives of the isotopes?
3 days
10 days
13 days
33 days

anonymous · Answer

For either isotope, you can compute the half life using the exponential decay formula,
$$y=Ce^{kt}~~\implies~~\frac{y}{C}=e^{kt}$$
where $\dfrac{y}{C}$ represents the proportion of the original amount remaining, $k$ is the relative decay factor, and $t$ is time (in days, in this case).

For isotope A, you have
$$0.10=e^{33k_A}~~\implies~~k_A=\cdots$$
Solve for $k_A$, then you can use it to find the half life. The half life is the time it takes for half of the original amount of the substance to decay, so $\dfrac{y}{C}=0.5$.
$$0.5=e^{k_At_A}~~\implies~~t_A=\cdots$$
where $t_A$ is the half life of isotope A.

Use the same process to find the half life of isotope B, $t_B$. Your final answer is the absolute difference, $|t_A-t_B|$.