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Hint - To solve this question, first we have to understand the term half-life and then we will learn about the decaying process and its terms. Lastly, we will be calculating the answer with the help of the half-life formula.

Half-life is defined as the time required for a quantity of substance to reduce to half of its initial value.Commonly used to describe how spontaneously unstable atoms undergo, or how long stable atoms survive, radioactive decay.

> Radioactive decay – Radioactive decay (also known as nuclear decay) is defined as the spontaneous breakdown of an atomic nucleus resulting in the release of energy and matter from the nucleus. A material that contains an unstable nuclei is said to be radioactive.

> It has three types which are alpha, decay, gamma decay and beta decay. Each one involves the emission of one or more photons (or particles).

Now let us Calculate our answer with the half-life formula –

\[N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{\dfrac{1}{2}}}}}}}\]

$N\left( t \right) = $ quantity of the substance remaining

${N_0} = $ initial quantity of the substance

$t = $ time elapsed

${t_{\dfrac{1}{2}}} = $ half-life of the substance

$n = \dfrac{{total\,time}}{{half - life\,period}} = \dfrac{{90}}{{30}} = 3$

$N = {N_0} \times {\left( {\dfrac{1}{2}} \right)^n}$

$ = 600 \times {\left( {\dfrac{1}{2}} \right)^3}$

$ = \dfrac{{600}}{8}$

$ = 75$ atoms

So, the correct answer is option C. 75 atoms

Note – Radioactive decay is random (i.e. stochastic) at the single atom level. According to the quantum theory, it is impossible to predict the decay time of a particular atom, regardless of its existence. However, when we talk about a significant number of identical atoms, the overall decay rate can be expressed by decay constant or as half-life.

Half-life is defined as the time required for a quantity of substance to reduce to half of its initial value.Commonly used to describe how spontaneously unstable atoms undergo, or how long stable atoms survive, radioactive decay.

__Complete answer:__> Radioactive decay – Radioactive decay (also known as nuclear decay) is defined as the spontaneous breakdown of an atomic nucleus resulting in the release of energy and matter from the nucleus. A material that contains an unstable nuclei is said to be radioactive.

> It has three types which are alpha, decay, gamma decay and beta decay. Each one involves the emission of one or more photons (or particles).

Now let us Calculate our answer with the half-life formula –

\[N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{\dfrac{1}{2}}}}}}}\]

$N\left( t \right) = $ quantity of the substance remaining

${N_0} = $ initial quantity of the substance

$t = $ time elapsed

${t_{\dfrac{1}{2}}} = $ half-life of the substance

$n = \dfrac{{total\,time}}{{half - life\,period}} = \dfrac{{90}}{{30}} = 3$

$N = {N_0} \times {\left( {\dfrac{1}{2}} \right)^n}$

$ = 600 \times {\left( {\dfrac{1}{2}} \right)^3}$

$ = \dfrac{{600}}{8}$

$ = 75$ atoms

So, the correct answer is option C. 75 atoms

Note – Radioactive decay is random (i.e. stochastic) at the single atom level. According to the quantum theory, it is impossible to predict the decay time of a particular atom, regardless of its existence. However, when we talk about a significant number of identical atoms, the overall decay rate can be expressed by decay constant or as half-life.

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