Question

A sample of radioactive material has mass $m$, decay constant $\lambda$ , and molecular mass $M$. If NA​ is Avogadro's number, the initial activity of the sample is:

Answer 1

Step 1: Given data

1. The mass of the material is $m$.
2. the molecular mass of the material is $M$
3. Avogadro's number $N_A$.

Step 2: The activity of Radioactive decay

1. The number of decays per unit time is called the activity of a radioactive element.
2. The activity of a radioactive element is defined by the form, $A=\lambda N$, which, $N$ is the number of radioactive nuclei present at a time and $\lambda$ is the radioactive constant.

Step 3: Formulae and concepts

1. In radioactivity, the number of radioactive nuclei present at a time t is defined by the form, $N=N_o{e}^{-\lambda t}$, where, $N_o$ is the number of un-decayed nuclei at a time, $t=0$.
2. We know, the total number of nuclei (molecules) in a material is $N=\frac{m}{M}\times N_A$, where, $N_A$ is the Avogadro's number, M is the molecular mass and m is the initial mass of the material.

Step 4: Finding the activity

As we know, at time $t=0$ (initial) activity is $A=\lambda N_o$.

So, activity,

$$\begin{gathered} A=\lambda N_o=\lambda \frac{m}{M}\times N_A \\ orA=\lambda \frac{m{N}_A}{M} \end{gathered}$$

Therefore, the initial activity of the sample is $A=\lambda \frac{m{N}_A}{M}$.