Question

Radium has a half-life of 1500 years. How long will it take for 250kg of Radium to decay down to less than 10kg

Answer 1

Answer:

6967 years

Explanation:

The radioactive substance left after a periodical t year can be expressed by using the formula:

$Q(t) = Q_oe^{-kt}$

here;

$Q_o$ = the radioactive initial value.

We need to understand that provided that the radioactive substance will get reduced to half of the provided initial amount  after a periodic time, Then:

the half-life of the radioactive substance left is:

$Q(h) = \dfrac{Q_o}{2}$

Given that:

the half-life = 1500 years

$\dfrac{Q_o}{2}= Q_o e^{-k\times 1500} \\ \\ \dfrac{Q_o}{2}= Q_o e^{ -1500k}$

Divide both sides by $Q_o$

$\dfrac{1}{2} =e^{-1500k}$

Then, find the natural logarithm of both side;

$\mathtt{In \dfrac{1}{2} = -1500 k}$

$k = \dfrac{1}{-1500}\mathtt{In}\dfrac{1}{2}$

k = 0.000462

So, after a particular (t) time, a 250 kg radium sample was reduced to 10 kg;

Then:

$10 = 250 e^{-0.000462t}$

$0.04 = e^{-0.000462t}$

From both sides, finding the natural logarithm, we have:

In(0.04) = -0.000462t

$t = \dfrac{In(0.04)}{-0.000462}$

t = 6967.26

Thus, it will take approximately 6967 years for a 250 kg radium sample to get reduced to 10 kg.