To find the half-life of Iodine-131 (I-131) given its activity, we can follow these steps:
### Step 1: Understand the relationship between activity, decay constant, and number of atoms.
The activity (A) of a radioactive substance is given by the formula:
\[ A = \lambda N \]
where:
- \( A \) is the activity in Bq (decays per second),
- \( \lambda \) is the decay constant (in s\(^{-1}\)),
- \( N \) is the number of radioactive atoms present.
### Step 2: Convert the mass of I-131 to moles and then to number of atoms.
Given that we have 1.0 mg of I-131, we first convert this mass to grams:
\[ 1.0 \, \text{mg} = 1.0 \times 10^{-3} \, \text{g} \]
Next, we need to convert grams to moles using the molar mass of I-131. The molar mass of I-131 is approximately 131 g/mol. Thus, the number of moles (n) is:
\[ n = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} = \frac{1.0 \times 10^{-3} \, \text{g}}{131 \, \text{g/mol}} \approx 7.63 \times 10^{-6} \, \text{mol} \]
Now, we convert moles to number of atoms using Avogadro's number (\( N_A \approx 6.022 \times 10^{23} \, \text{atoms/mol} \)):
\[ N = n \times N_A = 7.63 \times 10^{-6} \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 4.59 \times 10^{18} \, \text{atoms} \]
### Step 3: Calculate the decay constant (\( \lambda \)).
We know the activity \( A = 4.6 \times 10^{12} \, \text{Bq} \). Using the formula from Step 1:
\[ \lambda = \frac{A}{N} = \frac{4.6 \times 10^{12} \, \text{Bq}}{4.59 \times 10^{18} \, \text{atoms}} \approx 1.00 \times 10^{-6} \, \text{s}^{-1} \]
### Step 4: Calculate the half-life (\( t_{1/2} \)).
The half-life is related to the decay constant by the formula:
\[ t_{1/2} = \frac{0.693}{\lambda} \]
Substituting the value of \( \lambda \):
\[ t_{1/2} = \frac{0.693}{1.00 \times 10^{-6} \, \text{s}^{-1}} \approx 6.93 \times 10^{5} \, \text{s} \]
### Step 5: Convert the half-life from seconds to days.
To convert seconds to days, we use the conversion factor:
\[ 1 \, \text{day} = 86400 \, \text{s} \]
Thus,
\[ t_{1/2} \text{ (in days)} = \frac{6.93 \times 10^{5} \, \text{s}}{86400 \, \text{s/day}} \approx 8.02 \, \text{days} \]
### Final Answer:
The half-life of Iodine-131 is approximately **8 days**.
---
