Question

Element X decays radioactively with a half life of 7 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 40 grams?
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Answer 1

The element X will decay radioactively to 40 grams in a time of approximately 32.3 minutes.

How to model the decay of a radioactive isotope

In this question we must model the decay of a isotope. The element X decays exponentially and is described by the following model:

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

Where:

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

$t$ - Time, in minutes

$\tau$ - Time constant, in minutes

$m(t)$ - Current mass, in grams

The time constant ($\tau$), in minutes, is calculated by the following expression:

$\tau = \frac{t_{1/2}}{\ln 2}$   (2)

Where $t_{1/2}$ is the halflife, in minutes.

If we know that $t_{1/2} = 7\,min$, $m_{o} = 980\,g$ and $m = 40\,g$, then the decay time is:

$\tau = \frac{7\,min}{\ln 2}$

$\tau \approx 10.099\,min$

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

$t = -10.099\cdot \ln \frac{40}{980}$

$t \approx 32.303\,min$

The element X will decay radioactively to 40 grams in a time of approximately 32.3 minutes. $\blacksquare$

To learn more on radioactive decay, we kindly invite to check this verified question: https://brainly.com/question/26847087