One gram of a radiactive substance takes `50` to lose `1` centigram. Find its half-life period.

Explanation

To solve the problem of finding the half-life period of a radioactive substance, we will follow these steps:

### Step 1: Understand the problem
We have 1 gram of a radioactive substance that loses 1 centigram in 50 seconds. We need to find its half-life period.

### Step 2: Define the initial and remaining quantities
- Initial mass, \( n_0 = 1 \) gram
- Mass lost = 1 centigram = 0.01 gram
- Remaining mass after 50 seconds, \( n = n_0 - \text{mass lost} = 1 - 0.01 = 0.99 \) grams

### Step 3: Use the radioactive decay formula
The formula for radioactive decay is given by:
\[ n = n_0 e^{-\alpha t} \]
Where:
- \( n \) is the remaining quantity
- \( n_0 \) is the initial quantity
- \( \alpha \) is the decay constant
- \( t \) is the time elapsed

### Step 4: Substitute the known values into the equation
Substituting \( n = 0.99 \) grams, \( n_0 = 1 \) gram, and \( t = 50 \) seconds into the equation:
\[ 0.99 = 1 \cdot e^{-50\alpha} \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ e^{-50\alpha} = 0.99 \]

### Step 6: Take the natural logarithm of both sides
Taking the natural logarithm:
\[ -50\alpha = \ln(0.99) \]
This can be rewritten as:
\[ 50\alpha = -\ln(0.99) \]

### Step 7: Calculate \(-\ln(0.99)\)
Using a calculator:
\[ -\ln(0.99) \approx 0.01005 \]
Thus:
\[ 50\alpha = 0.01005 \]

### Step 8: Solve for \(\alpha\)
Now, solving for \(\alpha\):
\[ \alpha = \frac{0.01005}{50} \approx 0.000201 \text{ seconds}^{-1} \]

### Step 9: Use the half-life formula
The half-life \( T_{1/2} \) is given by:
\[ T_{1/2} = \frac{0.693}{\alpha} \]

### Step 10: Substitute \(\alpha\) into the half-life formula
Substituting the value of \(\alpha\):
\[ T_{1/2} = \frac{0.693}{0.000201} \approx 3447.76 \text{ seconds} \]

### Step 11: Convert seconds to minutes
To convert seconds to minutes:
\[ T_{1/2} \approx \frac{3447.76}{60} \approx 57.46 \text{ minutes} \]

### Final Answer
The half-life period of the radioactive substance is approximately **57 minutes**.

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