A radioactive substance has a half-life of 1,700 years. Calculate the time taken for substance to reduce from one gram to one milligram.

Answer

To solve the problem of how long it takes for a radioactive substance to reduce from one gram to one milligram given its half-life of 1,700 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial and Final Masses**:
- Initial mass \( m_0 = 1 \, \text{gram} \)
- Final mass \( m_t = 1 \, \text{milligram} = 10^{-3} \, \text{grams} \)

2. **Use the Formula for Radioactive Decay**:
The mass of a radioactive substance at time \( t \) can be expressed as:
\[ m_t = m_0 \cdot e^{-\lambda t} \]
where \( \lambda \) is the decay constant.

3. **Express the Ratio of Masses**:
We can express the ratio of the final mass to the initial mass:
\[ \frac{m_t}{m_0} = e^{-\lambda t} \]
Substituting the values:
\[ \frac{10^{-3}}{1} = e^{-\lambda t} \]
This simplifies to:
\[ e^{-\lambda t} = 10^{-3} \]

4. **Take the Natural Logarithm of Both Sides**:
Taking the natural logarithm gives:
\[ -\lambda t = \ln(10^{-3}) \]
Using the property of logarithms:
\[ \ln(10^{-3}) = -3 \ln(10) \]
Thus, we have:
\[ -\lambda t = -3 \ln(10) \]
or
\[ \lambda t = 3 \ln(10) \]

5. **Calculate the Decay Constant \( \lambda \)**:
The decay constant \( \lambda \) is related to the half-life \( t_{1/2} \) by the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
Given \( t_{1/2} = 1700 \, \text{years} \):
\[ \lambda = \frac{\ln(2)}{1700} \]

6. **Substitute \( \lambda \) Back into the Equation**:
Substitute \( \lambda \) into the equation:
\[ \frac{\ln(2)}{1700} t = 3 \ln(10) \]
Rearranging gives:
\[ t = \frac{3 \ln(10) \cdot 1700}{\ln(2)} \]

7. **Calculate the Value of \( t \)**:
Using \( \ln(10) \approx 2.303 \) and \( \ln(2) \approx 0.693 \):
\[ t = \frac{3 \cdot 2.303 \cdot 1700}{0.693} \]
Performing the calculation:
\[ t \approx \frac{11751.3}{0.693} \approx 16948.48 \, \text{years} \]

### Final Answer:
The time taken for the substance to reduce from one gram to one milligram is approximately **16,948.48 years**.