Question

Element X decays radioactively with a half life of 8 minutes. If there are 980 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 43 grams?
y=a(.5)^t/h

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Answer 1

Answer:

$t \approx 36.1\,min$

Step-by-step explanation:

The time constant for the isotope decay is:

$\tau = \frac{8\min}{\ln 2}$

$\tau \approx 11.542\,min$

Now, the decay of the isotope is modelled after the following expression:

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

The time is now cleared with some algebraic handling:

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

$t = -\tau \cdot \ln \frac{m(t)}{m_{o}}$

Finally, the time need for the element X to decay to 43 grams is:

$t = - (11.542\,min)\cdot \ln\left(\frac{43\,g}{980\,g} \right)$

$t \approx 36.1\,min$