The half-life period of radium is `1600` years. The fraction of a sample of radium that would remain after `6400` years is.

Options

A. `1/4`
B. `1/2`
C. `1/8`
D. `1/16`

Answer

To solve the problem, we need to determine the fraction of a sample of radium that remains after a given time, using its half-life. Here’s a step-by-step breakdown:

### Step 1: Understand the half-life concept
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. For radium, the half-life is given as 1600 years.

### Step 2: Calculate the number of half-lives
To find out how many half-lives fit into the total time of 6400 years, we use the formula:

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

Where:
- \( T \) = total time (6400 years)
- \( t_{1/2} \) = half-life (1600 years)

Substituting the values:

\[ n = \frac{6400}{1600} = 4 \]

### Step 3: Determine the remaining fraction
After \( n \) half-lives, the remaining fraction of the substance can be calculated using the formula:

\[ \text{Remaining fraction} = \frac{1}{2^n} \]

Substituting \( n = 4 \):

\[ \text{Remaining fraction} = \frac{1}{2^4} = \frac{1}{16} \]

### Step 4: Conclusion
The fraction of the sample of radium that would remain after 6400 years is \( \frac{1}{16} \).

### Final Answer
The fraction of the sample of radium that would remain after 6400 years is \( \frac{1}{16} \).

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