Mass of Freshly Prepared Radioactive Sample Given Mean Life and Activity

To find the mass of the radioactive sample, we use the fundamental laws of radioactive decay:

1. Relation between Activity and Number of Nuclei:
   The activity $A$ of a radioactive sample is related to the number of undecayed radioactive nuclei $N$ and the decay constant $\lambda$ by:
   $A = \lambda N$

2. Relation between Decay Constant and Mean Life:
   The mean life $\tau$ of the radioactive substance is the reciprocal of the decay constant:
   $\tau = \frac{1}{\lambda} \implies \lambda = \frac{1}{\tau}$

   Substituting this into the activity expression gives:
   $A = \frac{N}{\tau} \implies N = A \cdot \tau$

3. Calculating the Number of Atoms ($N$):
   Given:

   • Activity, $A = 10^{10} \text{ disintegrations per second (dps)}$
   • Mean life, $\tau = 10^9 \text{ s}$

   $N = (10^{10} \text{ s}^{-1}) \times (10^9 \text{ s}) = 10^{19} \text{ atoms}$

4. Calculating the Total Mass of the Sample:
   Given the mass of a single atom, $m = 10^{-25} \text{ kg}$:
   $M = N \times m = 10^{19} \times 10^{-25} \text{ kg} = 10^{-6} \text{ kg}$

5. Converting the Mass to Milligrams ($\text{mg}$):
   Since $1 \text{ kg} = 10^6 \text{ mg}$:
   $M = 10^{-6} \times 10^6 \text{ mg} = 1 \text{ mg}$

Thus, the mass of the radioactive sample is $1$.