Question

Element X decays radioactively with a half life of 10 minutes. If there are 340 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 13 grams?

Answer 1

It will take 47.1 minutes

Explanation:

Half life refers to the time taken for exactly half of the original mass to be reduced into half

Firstly, we write the equation that describes how a radioactive isotope decays;

m(t) = I * e^-&t

where m(t) is the mass at a particular time t

I is the initial mass

& is the decay constant

t is the time taken

Mathematically, the decay constant & is related to the half life by the equation;

& = ln2/half life

here, half life is 10 minutes

& = ln2/10 = 0.0693 min^-1

also for the element in question

m = 13g

I = 340 g

Plugging these into the equation alongside the decay constant, we have

13 = 340 * e^(-0.693 * t)

we divide both sides by 340

0.0382 = e^(-0.693t)

taking the ln of both sides, we have

ln 0.0382 = ln e^-0.0693t

-3.264 = -0.0693t

t = -3.264/-0.0693

t = 47.089

t = 47.1 minutes