Question

At any given time a piece of radioactive material (${{t}_{1/2}}$=30 days) contains ${{10}^{12}}$ atoms. Calculate the activity of the sample in dps?
(A) $3.67\times {{10}^{5}}dps$
(B) $1.67\times {{10}^{5}}dps$
(C) $2.67\times {{10}^{5}}dps$
(D) $4.67\times {{10}^{5}}dps$

Answer 1

Hint: The half-life is the time taken for half the radionuclide's atoms to decay. The activity of a sample of a radioactive matter is defined by the number of disintegrations taking place at its core at any given moment.

Complete step by step solution:
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the most common types of decay are alpha decay, beta decay, and gamma decay, all of which involve emitting one or more particles or photons. The weak force is the mechanism that is responsible for beta decay.
For the case of one-decay nuclear reactions: This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer.
The activity also represents the number of radiations emitted. One call, therefore, alpha, beta, gamma activity the numbers of alpha, beta, gamma rays that are emitted and are in proportions of the number of disintegrations.
The piece of radioactive material (${{t}_{1/2}}$=30 days) contains ${{10}^{12}}$ atoms. Since half life is given , therefore, decay constant can be calculated using the formula:
\[\begin{align}
& \lambda =\dfrac{0.693}{{{t}_{1/2}}} \\
& \lambda =\dfrac{0.693}{30\times 24\times 3600}=2592000{{s}^{-1}} \\
\end{align}\]
Now the activity of the sample will be,
\[-\dfrac{dN}{dt}={{10}^{12}}\times 2592000=2.67\times {{10}^{5}}\]

Therefore, the correct answer is the (C) option.

Note: Half-lives of known radionuclides vary widely, from more than ${{10}^{24}}$ years for the very nearly stable nuclide $^{128}Te$ , to $2.3\text{ }\times \text{ }{{10}^{-23}}~$ seconds for highly unstable nuclides such as $^{7}H$. Activities are fundamental features of the radioactive sample and of the type of radiation emitted. They represent its ‘baseline radioactivity’. When the sample contains more than one element, the total activity is the sum of the individual activity values.