Question

The activity of a freshly prepared radioactive sample of a radioisotope is \(10^{10}\) disintegrations per second. The average life (\(\tau\)) of this isotope is \(10^{9}\ \text{s}\), and the mass of a single atom of this isotope is \(10^{-25}\ \text{kg}\). The total mass of the radioactive sample is

• A. \(0.1\ \text{mg}\)
• B. \(1\ \text{mg}\)
• C. \(10\ \text{mg}\)
• D. \(100\ \text{mg}\)

Hint:

Activity equals the decay constant times the number of atoms, and the decay constant is the reciprocal of the average life.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Activity is \(A = \lambda N\), and the decay constant is \(\lambda = \dfrac{1}{\tau}\).

Step 2: Find the number of atoms
\(N = A\tau = 10^{10}\times 10^{9} = 10^{19}\) atoms.

Step 3: Calculation
Mass \(= N\times\) mass of one atom \(= 10^{19}\times 10^{-25}\ \text{kg} = 10^{-6}\ \text{kg}\).

Step 4: Convert units
\(10^{-6}\ \text{kg} = 10^{-3}\ \text{g} = 1\ \text{mg}\).

Final Answer:
The total mass of the sample is 1 mg. \[ \boxed{\text{(B) }1\ \text{mg}} \]