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 is

Explanation

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

### Step 1: Understand the given data
- Activity \( A = 10^{10} \) disintegrations per second
- Mean life \( \tau = 10^{9} \) seconds
- Mass of one atom \( m = 10^{-25} \) kg

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

### Step 3: Relate activity to the number of nuclei
The activity \( A \) is also given by:
\[ A = \lambda N \]
where \( N \) is the number of radioactive nuclei. We can rearrange this to find \( N \):
\[ N = \frac{A}{\lambda} = \frac{10^{10}}{10^{-9}} = 10^{10} \times 10^{9} = 10^{19} \]

### Step 4: Calculate the total mass of the sample
The total mass \( M \) of the radioactive sample can be calculated using the number of nuclei and the mass of one atom:
\[ M = N \times m = 10^{19} \times 10^{-25} \text{ kg} \]
Calculating this gives:
\[ M = 10^{19 - 25} = 10^{-6} \text{ kg} \]

### Step 5: Convert the mass to milligrams
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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