The half-life of radium is `1500 years`. In how many years will `1 g` of pure radium be reduced to one centigram?

Options

A. `3.927 xx 10^(2)years`
B. `9.927 xx 10^(2)years`
C. `99.927 xx 10^(2)years`
D. `0.927 xx 10^(2)years`

Explanation

To solve the problem of how long it takes for 1 gram of pure radium to be reduced to 1 centigram, we can follow these steps:

### Step 1: Understand the given information
- The half-life of radium (\( t_{1/2} \)) is given as 1500 years.
- The initial amount of radium (\( N_0 \)) is 1 gram.
- The final amount of radium (\( N \)) we want to find is 1 centigram, which is 0.01 grams.

### Step 2: Use the half-life formula
The relationship between the initial amount, final amount, and half-life can be expressed using the formula:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
Where:
- \( N \) = final amount of substance
- \( N_0 \) = initial amount of substance
- \( t \) = time elapsed
- \( t_{1/2} \) = half-life of the substance

### Step 3: Plug in the known values
Substituting the known values into the formula:
\[ 0.01 = 1 \left( \frac{1}{2} \right)^{\frac{t}{1500}} \]

### Step 4: Simplify the equation
This simplifies to:
\[ 0.01 = \left( \frac{1}{2} \right)^{\frac{t}{1500}} \]

### Step 5: Convert 0.01 to a power of 1/2
We know that:
\[ 0.01 = \frac{1}{100} = \left( \frac{1}{2} \right)^6.644 \]
(Here, \( 100 = 2^{6.644} \) because \( 2^{6.644} \approx 100 \))

### Step 6: Set the exponents equal
Now we can set the exponents equal:
\[ \frac{t}{1500} = 6.644 \]

### Step 7: Solve for \( t \)
To find \( t \), multiply both sides by 1500:
\[ t = 1500 \times 6.644 \approx 9966 \text{ years} \]

### Final Answer
Thus, it will take approximately **9966 years** for 1 gram of pure radium to be reduced to 1 centigram.

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