A radioactive element has half-life period `800 yr`. After `6400 yr`, what amount will remain?

Options

A. `1/2`
B. `1/16`
C. `1/8`
D. `1/256`

Answer

To solve the problem of how much of a radioactive element remains after 6400 years given its half-life of 800 years, we can follow these steps:

### Step 1: Determine the number of half-lives
The first step is to find out how many half-lives fit into the total time period of 6400 years. The formula to calculate the number of half-lives (n) is:

\[ n = \frac{T}{t_{1/2}} \]

where \( T \) is the total time elapsed (6400 years) and \( t_{1/2} \) is the half-life (800 years).

Substituting the values:

\[ n = \frac{6400 \text{ years}}{800 \text{ years}} = 8 \]

### Step 2: Apply the radioactive decay formula
Next, we can use the radioactive decay formula to find the remaining amount of the substance after the elapsed time. The formula is:

\[ N = N_0 \left(\frac{1}{2}\right)^n \]

where:
- \( N \) is the remaining amount of the substance,
- \( N_0 \) is the initial amount,
- \( n \) is the number of half-lives.

From Step 1, we found that \( n = 8 \). Thus, we can express the remaining amount as:

\[ N = N_0 \left(\frac{1}{2}\right)^8 \]

### Step 3: Calculate the remaining amount
Calculating \( \left(\frac{1}{2}\right)^8 \):

\[ \left(\frac{1}{2}\right)^8 = \frac{1}{256} \]

So, the remaining amount of the radioactive element after 6400 years is:

\[ N = N_0 \cdot \frac{1}{256} \]

### Conclusion
After 6400 years, the amount of the radioactive element that remains is \( \frac{1}{256} \) of the initial amount.

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