Question

Element X decays radioactively with a half life of 8 minutes. If there are 170
a
grams of Element X, how long, to the nearest tenth of a minute, would it take
the element to decay to 5 grams?
y = a(.5)

Answer 1

A time of 40.7 minutes is taken for 170 grams of element X to decay to 5 grams.

How to analyze a radioactive decay case

Let suppose that element X experiments a simple radioactive decay, that is, that the element X becomes gradually into another less radioactive stable element in time.

We know that decay behaves exponentially and follows this model:

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

Where:

• $m_{o}$ - Initial mass, in grams
• t - Time, in minutes
• τ - Time constant, in minutes
• m(t) - Current mass, in grams

The time constant can be described in terms of half-life ($t_{1/2}$), in minutes, through the following expression:

$\tau = \frac{t_{1/2}}{\ln 2}$   (2)

If we know that $t_{1/2} = 8\,min$, $m_{o} = 170\,g$ and $m(t) = 5\,g$, then the time needed for the decay is:

τ ≈ 11.541 min

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

t ≈ 40.698 min

A time of 40.7 minutes is taken for 170 grams of element X to decay to 5 grams. $\blacksquare$

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