The half life of `._38^90Sr` is 28 years. The disintegration rate of 15 mg of this isotope is of the order of

Options

A. `10^11` Bq
B. `10^10` Bq
C. `10^7` Bq
D. `10^9` Bq

Explanation

To solve the problem, we need to find the disintegration rate of 15 mg of the isotope \( _{38}^{90}Sr \) given that its half-life is 28 years. We will follow these steps:

### Step 1: 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 that the half-life is 28 years, we can substitute this value into the equation.

### Step 2: Convert half-life from years to seconds
To use the decay constant in calculations, we need to convert the half-life from years to seconds.
1 year = 365 days
1 day = 24 hours
1 hour = 3600 seconds
Thus,
\[ t_{1/2} = 28 \text{ years} \times 365 \text{ days/year} \times 24 \text{ hours/day} \times 3600 \text{ seconds/hour} \]

### Step 3: Calculate \( \lambda \)
Now we can calculate \( \lambda \):
\[ \lambda = \frac{0.693}{28 \times 365 \times 24 \times 3600} \]

### Step 4: Calculate the number of moles of \( _{38}^{90}Sr \)
Next, we need to find the number of moles of \( _{38}^{90}Sr \) in 15 mg:
1. Convert 15 mg to grams:
\[ 15 \text{ mg} = 15 \times 10^{-3} \text{ g} \]
2. The molar mass of \( _{38}^{90}Sr \) is approximately 90 g/mol.
3. Calculate the number of moles (\( n \)):
\[ n = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} = \frac{15 \times 10^{-3}}{90} \]

### Step 5: Calculate the number of atoms
Using Avogadro's number (\( N_A = 6.022 \times 10^{23} \) atoms/mol), we can find the total number of atoms:
\[ N = n \times N_A \]

### Step 6: Calculate the disintegration rate (activity)
The disintegration rate (activity, \( R \)) is given by:
\[ R = \lambda \times N \]

### Step 7: Final calculation
Substituting the values calculated in the previous steps will give us the disintegration rate in disintegrations per second (Becquerels).

### Step 8: Estimate the order of magnitude
Finally, we can express the disintegration rate in terms of its order of magnitude.