The half life of a radioactive substance is 13 years. The decay constant is

Options

A. `1.69xx10^(-10) S^(-1)`
B. `1.96xx10^(-9) S^(-1)`
C. `1.69xx10^(-9) S^(-1)`
D. `1.29xx10^(-7) S^(-1)`

Answer

To find the decay constant (\( \lambda \)) of a radioactive substance given its half-life, we can use the following relationship:

### Step 1: Understand the relationship between half-life and decay constant
The decay constant is related to the half-life (\( t_{1/2} \)) by the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
where \( \ln(2) \) is approximately 0.693.

### Step 2: Substitute the half-life value
Given that the half-life of the substance is 13 years, we can substitute this value into the formula:
\[ \lambda = \frac{0.693}{13 \text{ years}} \]

### Step 3: Convert years to seconds
To express the decay constant in seconds inverse, we need to convert years into seconds. The conversion is as follows:
- 1 year = 365 days
- 1 day = 24 hours
- 1 hour = 3600 seconds

Calculating the total number of seconds in one year:
\[ \text{Seconds in one year} = 365 \times 24 \times 3600 = 31,536,000 \text{ seconds} \]

### Step 4: Substitute the converted time into the decay constant formula
Now we can substitute this value back into the decay constant formula:
\[ \lambda = \frac{0.693}{13 \times 31,536,000} \]

### Step 5: Calculate the decay constant
Now, we perform the calculation:
\[ \lambda = \frac{0.693}{409,968,000} \approx 1.69 \times 10^{-9} \text{ s}^{-1} \]

### Final Result
Thus, the decay constant (\( \lambda \)) is approximately:
\[ \lambda \approx 1.69 \times 10^{-9} \text{ s}^{-1} \]