Question

Calculate the amount, in grams, of an original 300-gram sample of radioactive isotope Potassium-40 remaining after

Answer 1

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 ($m (t)$), in grams, is described below:

$m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }$ (1)

Where:

$m_{o}$ - Initial mass, in grams.

$t$ - Time, in years.

$\tau$ - Time constant, in years.

The time constant can be found from half life ($t_{1/2}$), in years, described in statement:

$\tau = \frac{t_{1/2}}{\ln 2}$ (2)

If we know that $m_{o} = 300\,g$, $t = 3.9\times 10^{9}\,yr$ and $t_{1/2} = 1.3\times 10^{9}\,yr$, then the current amount of the sample is:

$\tau = \frac{1.3\times 10^{9}\,yr}{\ln 2}$

$\tau \approx 1.876\times 10^{9}\,yr$

$m = (300\,g)\cdot e^{-\frac{3.9\times 10^{9}\,yr}{1.876\times 10^{9}\,yr} }$

$m\approx 37.521\,g$

The current amount of the Potassium-40 sample is approximately 37.521 grams.