The luminous dials of watches are usually made by mixing a zinc sulphide phosphor with an `alpha`-particles emitter. The mass of radium (mass number `226,half-life 1620 years`)that is needed to produce an average of `10` alpha`-particles per second for this purpose is

Options

A. `2.77 mg`
B. `2.77 g`
C. `2.77 xx10^(-23)g`
D. `2.77 xx10^(-13)kg`

Explanation

To solve the problem, we need to determine the mass of radium-226 required to produce an average of 10 alpha particles per second. We will use the concepts of radioactive decay, half-life, and the relationship between activity, decay constant, and the number of radioactive nuclei.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Mass number of radium-226 (Ra-226) = 226
- Half-life (T₁/₂) = 1620 years
- Desired activity (A) = 10 alpha particles per second

2. **Convert Half-life to Seconds:**
\[ T_{1/2} = 1620 \text{ years} \times (3.16 \times 10^7 \text{ seconds/year}) = 5.12 \times 10^{10} \text{ seconds} \]

3. **Calculate the Decay Constant (λ):**
The decay constant (λ) is given by the formula:
\[ \lambda = \frac{0.693}{T_{1/2}} \]
Substituting the half-life in seconds:
\[ \lambda = \frac{0.693}{5.12 \times 10^{10}} \approx 1.354 \times 10^{-11} \text{ s}^{-1} \]

4. **Relate Activity to Number of Nuclei:**
The activity (A) is related to the decay constant and the number of radioactive nuclei (N) by the formula:
\[ A = \lambda N \]
Rearranging gives:
\[ N = \frac{A}{\lambda} \]
Substituting the values:
\[ N = \frac{10}{1.354 \times 10^{-11}} \approx 7.39 \times 10^{11} \text{ nuclei} \]

5. **Calculate the Mass of Radium-226:**
The number of moles (n) of radium can be calculated using Avogadro's number (N_A = \(6.022 \times 10^{23}\)):
\[ n = \frac{N}{N_A} = \frac{7.39 \times 10^{11}}{6.022 \times 10^{23}} \approx 1.23 \times 10^{-12} \text{ moles} \]
The mass (m) of radium can be calculated using the molar mass (M = 226 g/mol):
\[ m = n \times M = 1.23 \times 10^{-12} \text{ moles} \times 226 \text{ g/mol} \approx 2.78 \times 10^{-10} \text{ grams} \]

6. **Convert Mass to Kilograms:**
\[ m \approx 2.78 \times 10^{-10} \text{ grams} = 2.78 \times 10^{-13} \text{ kg} \]

### Final Answer:
The mass of radium-226 needed is approximately \(2.78 \times 10^{-10} \text{ grams}\) or \(2.78 \times 10^{-13} \text{ kg}\).