Question

Help with test review to study for final next week.

Answer 1

The exponential decay is given by:

$A=A_0e^{rt}$

where A0 is the initial amount of the element and r is the decay rate.

To find the decay rate we use the fact that the half life is 1590 years; this means that it takes 1590 years for the amount of substance to be half the original amount, that is:

$\frac{1}{2}A_0=A^{}_0e^{1590r}$

Solving for r we have:

$\begin{gathered} \frac{1}{2}A_0=A^{}_0e^{1590r} \\ \frac{1}{2}=e^{1590r} \\ \ln \frac{1}{2}=\ln (e^{1590r}) \\ \ln \frac{1}{2}=1590r \\ r=\frac{1}{1590}\ln \frac{1}{2} \end{gathered}$

Hence the decay rate is:

$r=\frac{1}{1590}\ln \frac{1}{2}$

Now that we have the decay rate we have that the function describing the amount of radium for our example is:

$A=100e^{(\frac{1}{1590}\ln \frac{1}{2})t}$

To determine how much radium we have after 1000 years we plug t=1000 in the function above:

$\begin{gathered} A=100e^{(\frac{1}{1590}\ln \frac{1}{2})(1000)} \\ A=64.67 \end{gathered}$

Therefore after 1000 years we have 64.67 mg of radium-226