1 mg redium has `2.68xx10^18` atoms. Its half life is 1620 years. How many radium atoms will disintegrate from 1 mg of pure radium in 3240 years ?

Options

A. `2.01xx10^9`
B. `2.01xx10^18`
C. `1.01xx10^9`
D. `1.01xx10^18`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We are given:
- The number of radium atoms in 1 mg of radium: \( N_0 = 2.68 \times 10^{18} \) atoms.
- The half-life of radium: \( t_{1/2} = 1620 \) years.
- The time period for which we want to calculate the disintegration: \( t = 3240 \) years.

### Step 2: Calculate the number of half-lives
To find out how many half-lives fit into the time period of 3240 years, we can use the formula:
\[ \text{Number of half-lives} = \frac{t}{t_{1/2}} \]
Substituting the values:
\[ \text{Number of half-lives} = \frac{3240 \text{ years}}{1620 \text{ years}} = 2 \]

### Step 3: Determine the remaining number of atoms
After each half-life, the number of remaining active nuclei is halved. Therefore, after 2 half-lives, the remaining number of atoms can be calculated as:
\[ N = \frac{N_0}{2^n} \]
where \( n \) is the number of half-lives. Substituting the values:
\[ N = \frac{2.68 \times 10^{18}}{2^2} = \frac{2.68 \times 10^{18}}{4} = 0.67 \times 10^{18} \text{ atoms} \]

### Step 4: Calculate the number of disintegrated atoms
The number of atoms that have disintegrated can be calculated by subtracting the remaining atoms from the initial number of atoms:
\[ \text{Disintegrated atoms} = N_0 - N \]
Substituting the values:
\[ \text{Disintegrated atoms} = 2.68 \times 10^{18} - 0.67 \times 10^{18} = 2.01 \times 10^{18} \text{ atoms} \]

### Final Answer
Thus, the number of radium atoms that will disintegrate from 1 mg of pure radium in 3240 years is:
\[ \boxed{2.01 \times 10^{18} \text{ atoms}} \]

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