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:
[tex]m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }[/tex] (1)
Where:
- [tex]m_{o}[/tex] - 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 ([tex]t_{1/2}[/tex]), in minutes, through the following expression:
[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex] (2)
If we know that [tex]t_{1/2} = 8\,min[/tex], [tex]m_{o} = 170\,g[/tex] and [tex]m(t) = 5\,g[/tex], then the time needed for the decay is:
τ ≈ 11.541 min
[tex]t = -\tau \cdot \ln \frac{m(t)}{m_{o}}[/tex]
t ≈ 40.698 min
A time of 40.7 minutes is taken for 170 grams of element X to decay to 5 grams. [tex]\blacksquare[/tex]
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