One mole of radium has an activity of 1/3.7 kilo curie. Its decay constant will be

Options

A. `1/6xx10^(-10) s^(-1)`
B. `10^(-10)s^(-1)`
C. `10^(-11)s^(-1)`
D. `10^(-8) s^(-1)`

Answer

To find the decay constant (\(\lambda\)) of one mole of radium given its activity, we can follow these steps:

### Step 1: Understand the given activity
The problem states that one mole of radium has an activity of \( \frac{1}{3.7} \) kilo curie.

### Step 2: Convert kilo curie to decay per second
1 kilo curie is equivalent to \( 3.7 \times 10^{10} \) decays per second. Therefore, we can convert the activity from kilo curie to decays per second:
\[ \text{Activity} = \frac{1}{3.7} \text{ kilo curie} = \frac{1}{3.7} \times 10^3 \text{ curie} \]
\[ = \frac{1}{3.7} \times 3.7 \times 10^{10} \text{ decays per second} = \frac{10^{10}}{3.7} \text{ decays per second} \]

### Step 3: Use Avogadro's number
Avogadro's number (\(N_A\)) is approximately \(6.023 \times 10^{23}\) particles per mole. The decay constant (\(\lambda\)) can be calculated using the formula:
\[ \lambda = \frac{\text{Activity}}{N_A} \]

### Step 4: Substitute the values
Substituting the activity and Avogadro's number into the formula:
\[ \lambda = \frac{\frac{10^{10}}{3.7}}{6.023 \times 10^{23}} \]

### Step 5: Simplify the expression
1. First, simplify the fraction:
\[ \lambda = \frac{10^{10}}{3.7 \times 6.023 \times 10^{23}} \]
2. Calculate \(3.7 \times 6.023\):
\[ 3.7 \times 6.023 \approx 22.29 \]
3. Therefore:
\[ \lambda \approx \frac{10^{10}}{22.29 \times 10^{23}} = \frac{10^{10}}{22.29} \times 10^{-23} \]
4. This simplifies to:
\[ \lambda \approx \frac{1}{22.29} \times 10^{-13} \text{ s}^{-1} \]

### Step 6: Final calculation
Calculating \( \frac{1}{22.29} \):
\[ \lambda \approx 0.0448 \times 10^{-13} \text{ s}^{-1} \approx 4.48 \times 10^{-15} \text{ s}^{-1} \]

### Conclusion
Thus, the decay constant (\(\lambda\)) for one mole of radium is approximately:
\[ \lambda \approx 4.48 \times 10^{-15} \text{ s}^{-1} \]