Answer:
The current amount of the Potassium-40 sample is approximately 37.521 grams.
Explanation:
The amount of the sample of the radioactive isotope decays exponentially in time, the amount of mass of the sample as a function of time ([tex]m (t)[/tex]), in grams, is described below:
[tex]m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }[/tex] (1)
Where:
[tex]m_{o}[/tex] - Initial mass, in grams.
[tex]t[/tex] - Time, in years.
[tex]\tau[/tex] - Time constant, in years.
The time constant can be found from half life ([tex]t_{1/2}[/tex]), in years, described in statement:
[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex] (2)
If we know that [tex]m_{o} = 300\,g[/tex], [tex]t = 3.9\times 10^{9}\,yr[/tex] and [tex]t_{1/2} = 1.3\times 10^{9}\,yr[/tex], then the current amount of the sample is:
[tex]\tau = \frac{1.3\times 10^{9}\,yr}{\ln 2}[/tex]
[tex]\tau \approx 1.876\times 10^{9}\,yr[/tex]
[tex]m = (300\,g)\cdot e^{-\frac{3.9\times 10^{9}\,yr}{1.876\times 10^{9}\,yr} }[/tex]
[tex]m\approx 37.521\,g[/tex]
The current amount of the Potassium-40 sample is approximately 37.521 grams.
