Question

Element X decays radioactively with a half life of 13 minutes. If there are 260 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 15 grams?
y=a(.5)^t/h

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Answer 1

Answer:

It would take 54 minutes to the element X to decay to 15 grams.

Step-by-step explanation:

A radioactive half-life refers to the amount of time it takes for half of the original isotope to decay and its given by

                                                $N(t)=N_0(\frac{1}{2})^\frac{t}{t_{1/2}}$

where,

$N(t)$ = quantity of the substance remaining

$N_0$ = initial quantity of the substance

$t$ = time elapsed

$t_{1/2}$ = half life of the substance

We know that the element X decays radioactively with a half life of 13 minutes ($t_{1/2}$), there are 260 grams of it ($N_0$) and we want to find how long ($t$) would it take the element to decay to 15 grams ($N(t)$).

Using the above formula and solving for $t$, we get that

$15=260(\frac{1}{2})^\frac{t}{13}\\\\260\left(\frac{1}{2}\right)^{\frac{t}{13}}=15\\\\\left(\frac{1}{2}\right)^{\frac{t}{13}}=\frac{3}{52}\\\\\ln \left(\left(\frac{1}{2}\right)^{\frac{t}{13}}\right)=\ln \left(\frac{3}{52}\right)\\\\\frac{t}{13}\ln \left(\frac{1}{2}\right)=\ln \left(\frac{3}{52}\right)\\\\t=-\frac{13\ln \left(\frac{3}{52}\right)}{\ln \left(2\right)} \approx 54 \:min$