Sure, let's go through the problem step-by-step to find the decay constant and the activity rate.
### Step 1: Understand the given values
- Number of atoms ([tex]\( N \)[/tex]) = [tex]\( 10^{12} \)[/tex]
- Half-life ([tex]\( t_{1/2} \)[/tex]) = 15 days
### Step 2: Calculate the decay constant
The decay constant ([tex]\( \lambda \)[/tex]) is related to the half-life by the formula:
[tex]\[ \lambda = \frac{\ln(2)}{t_{1/2}} \][/tex]
Where:
- [tex]\( \ln(2) \)[/tex] is the natural logarithm of 2 (approximately 0.693).
- [tex]\( t_{1/2} \)[/tex] is the half-life.
So,
[tex]\[ \lambda = \frac{0.693}{15} \approx 0.046209812037329684 \, \text{days}^{-1} \][/tex]
### Step 3: Calculate the activity rate
The activity rate ([tex]\( A \)[/tex]) is given by the formula:
[tex]\[ A = \lambda \cdot N \][/tex]
Where:
- [tex]\( \lambda \)[/tex] is the decay constant.
- [tex]\( N \)[/tex] is the number of atoms.
Thus,
[tex]\[ A = 0.046209812037329684 \times 10^{12} \approx 46209812037.32968 \, \text{decays per day} \][/tex]
### Conclusion
- The decay constant ([tex]\( \lambda \)[/tex]) is approximately [tex]\( 0.046209812037329684 \, \text{days}^{-1} \)[/tex].
- The activity rate ([tex]\( A \)[/tex]) is approximately [tex]\( 46209812037.32968 \, \text{decays per day} \)[/tex].
So, the material with [tex]\( 10^{12} \)[/tex] atoms and a half-life of 15 days has an activity rate of around [tex]\( 46209812037.32968 \)[/tex] decays per day.
