Question

For a certain radioactive substance, it is observed that after\[4\,{\text{h}}\], only \[{\text{6}}{\text{.25% }}\] of the original sample is left undecayed. It follows that
This question has multiple correct options
A. The half-life of the sample is \[1\,{\text{h}}\]
B. The mean life of the sample is \[\dfrac{1}{{{\text{ln}}2}}{\text{h}}\]
C. The decay constant of the sample is \[\ln \left( 2 \right){{\text{h}}^{ - 1}}\]
D. After a further \[4\,{\text{h}}\], the amount of the substance left over would by only \[0.39\% \] of the original amount

Answer 1

Hint: We use the formula of radioactivity and we also use the given information for this solution and solve it accordingly. Tracers are a common radioisotope application.

Complete step by step answer:
We know that,

Law of radioactivity,
\[N = {N_{\text{0}}}{e^{ - \lambda t}}\] …… (1)

Where,
\[\,\lambda \,\] is decay constant.
Let us consider,
\[{N_0} = 100\]
So
\[N = 6.25\]
Here,
\[t = 4\,{\text{h}}\]
Now,
 $ 6.25 = 100{e^{ - \lambda \left( 4 \right)}} \\\implies {e^{4\lambda }} = 16 \\\implies 4\lambda = \ln 16 \\\implies \ln {2^4} = 4\ln 2 \\\implies \lambda = \lambda \nu \left( 2 \right){{\text{h}}^{ - 1}} \\ $
Mean life\[ = \dfrac{1}{\lambda } = \dfrac{1}{{\ln 2}}h\]
Half-life, \[t = {t_{1/2}}\],

So,
 $N = {N_0}/2 \\\implies {N_0}/2 = {N_0}{e^{ - \lambda {t_{1/2}}}} \\
  \implies {e^{\ln 2\left( {{t_{1/2}}} \right)}} = 2 \\ \implies \ln 2\left( {{t_{1/2}}} \right) = \ln 2 \\\implies {t_{1/2}} = 1\,h \\ $
For t = 4h , decay $N_0$ = 6.25
From equation (1)
 $ N = 6.25{e^{ - 4\ln 2}} \\$
$N= 0.39 \\$

So, the correct answer is “Option A,B,C&D”.

Additional Information:
Radioactive substance: Atoms that decay naturally are radioactive substances. It can produce alpha, beta and gamma-radiation particles. They cannot be switched off unlike X-ray sources, so it is harder to control them. Gamma radiation emitters are industrial x-rays sources such as iridium \[192\] and can be used for x-raying thick sections of the steel and other metals. These are also used inside protected enclosures but because the sources are not electrically switched off, they are housed in shielded containers. The source is projected from the container through a guiding tube and retracted to the point of use. With radioactive sources, living organisms, diseases, medical instruments and foods are diagnosed and treated, heat and electrical energy is produced and various steps are monitored in every type of industrial process. Tracers are a common radioisotope application.

Note:
Large amounts of exposure to radioactive substances can result in nausea, vibration, hair losing, diarrhea, blood flow, damage to the central nervous system and death. DNA damage is also caused and the risk of cancer is increased, especially in young children and fetuses.