`.^(60 m)Co rarr .^(60)Co` emits `gamma`-radiations of wavelength `3 xx 10^(-10)`. Assuming each nuclei emits one wavelength, with what mass per mole of two nuclei differ?
a. `4.43 xx 10^(-9) g` b. `4.43 xx 10^(-4) g`
c. `4.43 xx 10^(-3) g` d. `4.43 g`

Options

A. a. `4.43 xx 10^(-9) g`
B. b. `4.43 xx 10^(-6) g`
C. c. `4.43 xx 10^(-3) g`
D. d. `4.43 g`

Explanation

To solve the problem, we need to find the difference in mass per mole of two nuclei of cobalt-60 that emit gamma radiation of a given wavelength. We will use the formula relating energy, wavelength, and mass.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Wavelength (λ) = \(3 \times 10^{-10}\) m
- Avogadro's number (N) = \(6.022 \times 10^{23}\) mol\(^{-1}\)
- Planck's constant (h) = \(6.626 \times 10^{-34}\) J·s
- Speed of light (c) = \(3 \times 10^{8}\) m/s

2. **Calculate Energy (E) of One Photon:**
The energy of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34} \text{ J·s})(3 \times 10^{8} \text{ m/s})}{3 \times 10^{-10} \text{ m}} \]
\[ E = \frac{1.9878 \times 10^{-25}}{3 \times 10^{-10}} = 6.626 \times 10^{-16} \text{ J} \]

3. **Calculate the Mass Equivalent of the Energy:**
Using Einstein's equation \(E = mc^2\), we can find the mass (m) equivalent of the energy:
\[ m = \frac{E}{c^2} \]
Substituting the values:
\[ m = \frac{6.626 \times 10^{-16} \text{ J}}{(3 \times 10^{8} \text{ m/s})^2} \]
\[ m = \frac{6.626 \times 10^{-16}}{9 \times 10^{16}} = 7.362 \times 10^{-33} \text{ kg} \]

4. **Convert Mass to Grams:**
Since \(1 \text{ kg} = 1000 \text{ g}\):
\[ m = 7.362 \times 10^{-33} \text{ kg} \times 1000 = 7.362 \times 10^{-30} \text{ g} \]

5. **Calculate Mass per Mole:**
The mass per mole of photons (for one nucleus) is:
\[ m_{\text{mole}} = N \times m = (6.022 \times 10^{23}) \times (7.362 \times 10^{-30} \text{ g}) \]
\[ m_{\text{mole}} \approx 4.43 \times 10^{-6} \text{ g} \]

6. **Final Calculation for Two Nuclei:**
Since we are interested in the difference in mass for two nuclei:
\[ m_{\text{difference}} = 2 \times m_{\text{mole}} = 2 \times 4.43 \times 10^{-6} \text{ g} \approx 8.86 \times 10^{-6} \text{ g} \]

7. **Select the Correct Option:**
The closest option to our calculated value is:
- b. \(4.43 \times 10^{-4} \text{ g}\)