Question

The half-life of a specific radionuclide is 8 days. How much of a 80 mg sample will be left after 24 days?

Answer 1

Hint : The half-life of a radioactive nuclide refers to the amount of time taken by one half of the radioactive sample to decay. Using the concept of half-life, we can obtain the amount of sample will be left. We need to start by calculating the number of half-lives that the sample can undergo in the given time.

Complete Step-By-Step Solution:
We know as given in the question, since $8days$ is the half-life of the element, therefore, in the given time which is $24days$, the number of half-lives the element will have$ = \dfrac{{24}}{8} = 3$
Therefore, the element will have $3$ half-lives.
Now, the amount of sample given is $80mg$
Hence, we use the formula to calculate the amount of the sample left after $n$ half-lives.
Now, we know the formula is:
$N = {N_o}{\left( {\dfrac{1}{2}} \right)^n}$
Where,
$N$ is the amount of the sample left after $n$ half-lives
 ${N_o}$ is the initial amount of the sample with which we started.
$\dfrac{1}{2}$ Corresponds to the half-life of the particle.
$n$ Denotes the number of half-live the particle can undergo in the given time.
Putting the values, we obtain:
$N = 80{\left( {\dfrac{1}{2}} \right)^3}$
On solving the above equation, we get:
$N = 80 \times \dfrac{1}{8} = 10mg$
Therefore, we obtain the amount of the element that will be left after $24days$ is $10mg$. This is our required solution.

Note:
The calculation for half-life of a substance is necessary as it is used to calculate the rate at which a radioactive element will decay to give away radiation, thus we can also infer the amount of the substance that will be left after a certain time. Moreover, the half-life of a substance is also necessary as it ensures safe handling, elements having short half-lives are not injurious to health.