The half-life of pononium is `140` days. After how many days. `16 gm` polonium will be reduced to `1 gm` (or `15 gm` will decay) ?

Options

A. 700 days
B. 280 days
C. 560 days
D. 420 days

Explanation

To solve the problem, we need to determine how many days it will take for 16 grams of polonium to decay to 1 gram, given that the half-life of polonium is 140 days.

### Step-by-Step Solution:

1. **Understand the Half-Life Concept**:
The half-life of a substance is the time required for half of the substance to decay. For polonium, the half-life is given as 140 days.

2. **Identify Initial and Final Amounts**:
- Initial amount of polonium (N₀) = 16 grams
- Final amount of polonium (N) = 1 gram

3. **Calculate the Decay**:
The amount of polonium that has decayed is:
\[ \text{Decay} = N₀ - N = 16 \text{ grams} - 1 \text{ gram} = 15 \text{ grams} \]

4. **Use the Decay Formula**:
The relationship between the remaining amount and the initial amount can be expressed as:
\[ \frac{N}{N₀} = \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]
where \(T_{1/2}\) is the half-life, and \(t\) is the time elapsed.

5. **Substitute Known Values**:
We know:
- \(N = 1\) gram
- \(N₀ = 16\) grams
- \(T_{1/2} = 140\) days

Therefore, we can write:
\[ \frac{1}{16} = \left(\frac{1}{2}\right)^{\frac{t}{140}} \]

6. **Express \(\frac{1}{16}\) in Terms of Powers of 2**:
We can express \(\frac{1}{16}\) as:
\[ \frac{1}{16} = \frac{1}{2^4} \]

7. **Set the Exponents Equal**:
Since the bases are the same, we can equate the exponents:
\[ \frac{t}{140} = 4 \]

8. **Solve for \(t\)**:
Multiply both sides by 140 to find \(t\):
\[ t = 4 \times 140 = 560 \text{ days} \]

### Final Answer:
It will take **560 days** for 16 grams of polonium to decay to 1 gram.

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