Question

The half-life of a radioactive element is 8 hours. A given number of nuclei of that element is reduced to 1/4 of that number after two hours.
a.) True
b.) False

Answer 1

In order to solve the given problem first we will define the half life of a radioactive element. Further we will understand the meaning of half life in general terms relating it to the number of nuclei and the disintegration of the nuclei. We will also try to find out some formula relating the two. Further we will check for the truth of the given statement by taking the number of nuclei as variable and finding the nuclei left after the given time. We will also find the correct number of nuclei left if the statement will be false.

Complete step by step answer:
First let us define the half life of radioactive elements.
Half-life is the amount of time taken to minimise a quantity to half of its original value. In nuclear physics, the term is widely used to explain how easily unstable atoms undergo radioactive decay or how long stable atoms last. In order to describe some form of exponential or non-exponential decay, the term is often used more broadly.
In general terms the half-life cycle is characterised as the time during which half of the volume of radioactive matter in a given sample disintegrates.
So if there were $N$ number of nuclei in the radioactive element initially then after one half life of the element the number of nuclei left will be $\dfrac{N}{2}$ . And after one more half life or after two half live the number of nuclei of the element left will be $\dfrac{N}{4}$ .
This is the reason behind the exponential decay of the nuclei.
Given in the problem that:
The half-life of a radioactive element is 8 hours.

Statement:
A given number of nuclei of that element is reduced to 1/4 of that number after two hours.
As we have the half life of the element is 8 hours. So after every 8 hours the number of nuclei left will be half the initial.
Let the initial number of nuclei for the given case is $X$ . So after 8 hours the number of nuclei left will be $\dfrac{X}{2}$ . Which is $\dfrac{1}{2}$ of the initial. And after another 8 hours or after 16 hours the number of nuclei left will be $\dfrac{X}{4}$ or $\dfrac{1}{4}$ of the initial.
As the time taken for nuclei to reduce to $\dfrac{1}{4}$ is 16 hours.
Hence, the given statement is false.
So, the correct answer is “Option B”.

Note: In order to solve such types of problems students must not use the formula as that method will be more difficult and tiresome. The students must proceed with the basic definition and basic relation. The half-life of carbon-14 can be used by scientists to assess the estimated age of organic objects. They assess how much has been converted by carbon-14. Radioactive isotopes are used by physicians as diagnostic tracers. In radiation regulation, the radiological half-life is significant because long-lived radionuclides are around for longer periods than shorter-lived organisms once released. For longer duration than short-lived nuclides, long-lived radionuclides released to the atmosphere may be around.