The half life of radium is 1600 years. After how much time will 1 g radium be reduced to 125 mg ?

Options

A. 4800 years
B. 4500 years
C. 5000 years
D. 4750 years

Answer

To solve the problem of how long it will take for 1 g of radium to reduce to 125 mg, we can follow these steps:

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

### Step 2: Convert the Final Amount to Grams
We start with 1 g of radium and want to find out how long it will take to reduce to 125 mg. We need to convert 125 mg to grams:
\[ 125 \text{ mg} = 0.125 \text{ g} \]

### Step 3: Use the Radioactive Decay Formula
The formula for radioactive decay is:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
Where:
- \(N\) is the remaining quantity of the substance (0.125 g),
- \(N_0\) is the initial quantity of the substance (1 g),
- \(n\) is the number of half-lives.

### Step 4: Set Up the Equation
Substituting the known values into the equation:
\[ 0.125 = 1 \left(\frac{1}{2}\right)^n \]
This simplifies to:
\[ \left(\frac{1}{2}\right)^n = 0.125 \]

### Step 5: Convert 0.125 to a Power of 2
Recognizing that \(0.125\) can be expressed as a power of 2:
\[ 0.125 = \frac{1}{8} = \frac{1}{2^3} \]
Thus, we can rewrite the equation:
\[ \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3 \]

### Step 6: Solve for \(n\)
By comparing the exponents, we find:
\[ n = 3 \]

### Step 7: Calculate the Total Time
Since each half-life is 1600 years, the total time taken for 3 half-lives is:
\[ \text{Total Time} = n \times \text{Half-life} = 3 \times 1600 \text{ years} = 4800 \text{ years} \]

### Final Answer
The time required for 1 g of radium to reduce to 125 mg is **4800 years**.

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