Question

If the half-life of a radioactive element is 4 days, how long will it take for three- fourths of a sample of the element to decay?

Answer 1

Answer:

$\boxed{\text{8 da}}$

Explanation:

The question will be easier to solve if we interpret it as, " How long will it take until one-fourth of a sample of the element remains,?"

The half-life of the element is the time it takes for half of it to decay.  

After one half-life, half (50 %) of the original amount will remain.  

After a second half-life, half of that amount (25 %) will remain, and so on.  

We can construct a table as follows:

$\begin{array}{cccl}\textbf{No. of} & & \textbf{Fraction} & \\\textbf{half-lives} & \textbf{t/da} & \textbf{remaining} & \\1 & 4 & \dfrac{1}{2} & \\\\2 & 8 & \dfrac{1}{4}& \\\\3 & 12 & \dfrac{1}{8}& \\\end{array}$

$\text{We see that 8 da is two half-lives, and the fraction of the element remaining is \frac{1}{4} .}\\\text{It takes \boxed{\textbf{8 da}} for three-fourths of the element to decay}$