Radon `(Ra)` decays into Polonium `(P_0)` by emitting an `alpha-`particle with half-life of `4` days. A sample contains `6.4 xx 10^10` atoms of `R_a`. After `12` days, the number of atoms of `R_a` left in the sample will be

Options

A. `3.2xx10^(10)`
B. `0.53xx10^(10)`
C. `2.1xx10^(10)`
D. `0.8xx10^(10)`

Answer

To solve the problem of how many atoms of Radon (Ra) are left after 12 days, we will use the concept of half-lives. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Half-life of Radon (Ra), \( t_{1/2} = 4 \) days
- Total time elapsed, \( t = 12 \) days
- Initial number of atoms of Radon, \( N_0 = 6.4 \times 10^{10} \) atoms

### Step 2: Calculate the number of half-lives
To find out how many half-lives have passed in 12 days, we can use the formula:
\[ n = \frac{t}{t_{1/2}} \]
Substituting the values:
\[ n = \frac{12 \, \text{days}}{4 \, \text{days}} = 3 \]
So, 3 half-lives have passed.

### Step 3: Use the half-life formula to find remaining atoms
The formula to calculate the remaining number of atoms after \( n \) half-lives is:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
Substituting the values we have:
\[ N = 6.4 \times 10^{10} \left(\frac{1}{2}\right)^3 \]

### Step 4: Calculate \( \left(\frac{1}{2}\right)^3 \)
Calculating \( \left(\frac{1}{2}\right)^3 \):
\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

### Step 5: Substitute back to find \( N \)
Now substituting back into the equation:
\[ N = 6.4 \times 10^{10} \times \frac{1}{8} \]
Calculating \( \frac{6.4}{8} \):
\[ \frac{6.4}{8} = 0.8 \]
Thus:
\[ N = 0.8 \times 10^{10} \]

### Final Answer
The number of atoms of Radon left in the sample after 12 days is:
\[ N = 0.8 \times 10^{10} \text{ atoms} \]