Radium `Ra^236` has a half-life of 1590 years. How much of the original amount of `Ra^236` would remain after 6360 year ?

Options

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

Explanation

To solve the problem of how much of the original amount of Radium \( Ra^{236} \) remains after 6360 years, given its half-life of 1590 years, we can follow these steps:

### Step 1: Determine the number of half-lives
First, we need to find out how many half-lives fit into the total time period of 6360 years.

\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{6360 \text{ years}}{1590 \text{ years}} = 4 \]

### Step 2: Understand the decay process
Each half-life reduces the remaining amount of the substance by half. If we start with an initial amount \( N_0 \), after each half-life, the remaining amount can be calculated as follows:

- After 1st half-life: \( N_0 \times \frac{1}{2} \)
- After 2nd half-life: \( N_0 \times \frac{1}{2^2} = N_0 \times \frac{1}{4} \)
- After 3rd half-life: \( N_0 \times \frac{1}{2^3} = N_0 \times \frac{1}{8} \)
- After 4th half-life: \( N_0 \times \frac{1}{2^4} = N_0 \times \frac{1}{16} \)

### Step 3: Calculate the remaining amount
After 4 half-lives, the remaining amount of \( Ra^{236} \) is:

\[ \text{Remaining amount} = N_0 \times \frac{1}{16} \]

### Step 4: Express the remaining amount as a fraction of the original amount
To express how much of the original amount remains, we can write:

\[ \frac{\text{Remaining amount}}{N_0} = \frac{N_0 \times \frac{1}{16}}{N_0} = \frac{1}{16} \]

### Conclusion
Thus, after 6360 years, the remaining amount of \( Ra^{236} \) is \( \frac{1}{16} \) of the original amount.

### Final Answer
The correct answer is \( \frac{1}{16} \).

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