The activity of a freshly prepared radioactive sample is `10^(10)` disintegrations 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 sample is

Explanation

To solve the problem step by step, we will follow the outlined approach based on the information provided in the question.

### Step 1: Understand the relationship between activity, mean life, and number of atoms
The activity \( A \) of a radioactive sample is given by the formula:
\[ A = \alpha N \]
where \( \alpha \) is the decay constant and \( N \) is the number of radioactive atoms. The decay constant \( \alpha \) is related to the mean life \( \tau \) by:
\[ \alpha = \frac{1}{\tau} \]
Thus, we can express the activity in terms of the mean life:
\[ A = \frac{N}{\tau} \]

### Step 2: Rearranging the equation to find the number of atoms
From the activity equation, we can rearrange to find \( N \):
\[ N = A \cdot \tau \]
Substituting the given values:
- Activity \( A = 10^{10} \) disintegrations per second
- Mean life \( \tau = 10^{9} \) seconds

Calculating \( N \):
\[ N = 10^{10} \cdot 10^{9} = 10^{19} \]

### Step 3: Calculate the mass of the radioactive sample
The mass \( m \) of the radioactive sample can be calculated using the formula:
\[ m = N \cdot m_{\text{atom}} \]
where \( m_{\text{atom}} \) is the mass of a single atom of the radioisotope. Given:
- Mass of an atom \( m_{\text{atom}} = 10^{-25} \) kg

Substituting the values:
\[ m = 10^{19} \cdot 10^{-25} = 10^{-6} \text{ kg} \]

### Step 4: Convert the mass from kg to mg
To convert kilograms to milligrams, we use the conversion factor \( 1 \text{ kg} = 1000 \text{ mg} \):
\[ m = 10^{-6} \text{ kg} \cdot 10^{3} \text{ mg/kg} = 10^{-3} \text{ mg} \]

### Step 5: Final answer
Thus, the mass of the radioactive sample is:
\[ m = 1 \text{ mg} \]

### Summary
The mass of the radioactive sample is **1 mg**.

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