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The half-life for rhodium -101 is 3.3 years. After 10 years, about what fraction of some original amount of rhodium -101 would remain?A. One-thirdB. One-eighthC. One-tenthD. One-halfE. One-fourth

Last updated date: 08th Sep 2024
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Hint Instability of an atom's nucleus may result from an excess of either neutrons or protons. A radioactive atom will attempt to reach stability by ejecting nucleons (protons or neutrons), as well as other particles, or by releasing energy in other forms. The definition of elimination half-life is the length of time required for the concentration of a particular substance (typically a drug) to decrease to half of its starting dose in the body.

Complete step by step answer
We know that radioactivity refers to the particles which are emitted from nuclei as a result of nuclear instability. Because the nucleus experiences the intense conflict between the two strongest forces in nature, it should not be surprising that there are many nuclear isotopes which are unstable and emit some kind of radiation.
We know that the definition of elimination half-life is the length of time required for the concentration of a particular substance (typically a drug) to decrease to half of its starting dose in the body.
$\mathrm{t}_{1 / 2}=3.8$ day
It can be said that the half-life usually describes the decay of discrete entities, such as radioactive atoms. In that case, it does not work to use the definition that states "half-life is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom, and its half-life is one second, there will not be "half of an atom" left after one second.
$\therefore \lambda =\dfrac{0.693}{{{t}_{1/2}}}=\dfrac{0.693}{3.8}=0.182$
If the initial number of atoms is a $=\mathrm{A}_{0}$ then after time $t$ the number
of atoms is $a/20=\text{A}$.
Now we can add that the term "half-life" is almost exclusively used for decay processes that are exponential (such as radioactive decay or the other examples above), or approximately exponential (such as biological half-life discussed below). In a decay process that is not even close to exponential, the half-life will change dramatically while the decay is happening. In this situation it is generally uncommon to talk about half-life in the first place, but sometimes people will describe the decay in terms of its "first half-life", "second half-life", etc.,
$\mathrm{t}=\dfrac{2.303}{\lambda} \log \dfrac{\mathrm{A}_{0}}{\mathrm{A}}=\dfrac{2.303}{0.182} \log \dfrac{\mathrm{a}}{\mathrm{a} / 20}$
After the evaluation we get that:
$=\dfrac{2.303}{0.182} \log 20=16.46$

So the correct answer is option B.

Note Knowing about half-lives is important because it enables you to determine when a sample of radioactive material is safe to handle. They need to be active long enough to treat the condition, but they should also have a short enough half-life so that they don't injure healthy cells and organs. The rate at which a radioactive isotope decays is measured in half-life. The term half-life is defined as the time it takes for one-half of the atoms of a radioactive material to disintegrate. Half-lives for various radioisotopes can range from a few microseconds to billions of years.