The activity of a freshly prepared radioactive sample is `10^(10)` disintergrations per second, whose mean life is `10^(9)s`. The mass of an atom of this radioisotope is `10^(-25) kg`. The mass ( in `mg`) of the radioactive samples is

Explanation

To find the mass of the radioactive sample, we can follow these steps:

### Step 1: Understand the relationship between activity, decay constant, and number of atoms
The activity \( A \) of a radioactive sample is given by the formula:
\[ A = \lambda N \]
where:
- \( A \) is the activity (in disintegrations per second),
- \( \lambda \) is the decay constant,
- \( N \) is the number of radioactive atoms.

### Step 2: Calculate the decay constant \( \lambda \)
The decay constant \( \lambda \) is related to the mean life \( \tau \) by the equation:
\[ \lambda = \frac{1}{\tau} \]
Given that the mean life \( \tau = 10^9 \) seconds, we can calculate \( \lambda \):
\[ \lambda = \frac{1}{10^9} \, \text{s}^{-1} \]

### Step 3: Rearrange the activity formula to find \( N \)
We can rearrange the activity formula to solve for \( N \):
\[ N = \frac{A}{\lambda} \]
Substituting the values of \( A \) and \( \lambda \):
\[ A = 10^{10} \, \text{disintegrations/s} \]
\[ N = \frac{10^{10}}{\frac{1}{10^9}} = 10^{10} \times 10^9 = 10^{19} \]

### Step 4: Calculate the mass of the radioactive sample
The mass \( m \) of the radioactive sample can be calculated using:
\[ m = N \times m_{\text{atom}} \]
where \( m_{\text{atom}} \) is the mass of a single atom of the radioisotope, given as \( 10^{-25} \, \text{kg} \):
\[ m = 10^{19} \times 10^{-25} \, \text{kg} \]
\[ m = 10^{-6} \, \text{kg} \]

### Step 5: Convert the mass from kg to mg
To convert kilograms to milligrams, we use the conversion factor \( 1 \, \text{kg} = 10^6 \, \text{mg} \):
\[ m = 10^{-6} \, \text{kg} \times 10^6 \, \text{mg/kg} = 1 \, \text{mg} \]

### Final Answer
The mass of the radioactive sample is \( 1 \, \text{mg} \).

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