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igor_vitrenko [27]
3 years ago
6

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

Geography
1 answer:
Aleksandr [31]3 years ago
4 0

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.

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