What is the binding energy (in J/mol or kJ/mol) of an electron in a metal, whose threshold frequency for photoelectron is `3.5 xx 10^(13) s^(-1)` ?

Options

A. `66.38 kJ//mol`
B. `6.95 kJ//mol`
C. `13.91 kJ//mol`
D. `27.82 kJ//mol`

Answer

To find the binding energy of an electron in a metal given the threshold frequency, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Binding Energy**:
- Binding energy is the energy required to remove an electron from the metal. It is also known as the work function (φ) of the metal.

2. **Use the Formula for Work Function**:
- The work function can be calculated using the formula:
\[ \phi = h \nu \]
where:
- \( \phi \) is the work function (binding energy),
- \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{J s}\)),
- \( \nu \) is the threshold frequency.

3. **Substitute the Given Values**:
- Given:
\[ \nu = 3.5 \times 10^{13} \, \text{s}^{-1} \]
- Now, substitute the values into the formula:
\[ \phi = (6.626 \times 10^{-34} \, \text{J s}) \times (3.5 \times 10^{13} \, \text{s}^{-1}) \]

4. **Calculate the Binding Energy**:
- Performing the multiplication:
\[ \phi = 6.626 \times 3.5 \times 10^{-34 + 13} = 23.155 \times 10^{-21} \, \text{J} \]
- Thus, the binding energy in joules is:
\[ \phi \approx 2.3155 \times 10^{-20} \, \text{J} \]

5. **Convert to Joules per Mole**:
- To convert this energy to Joules per mole, multiply by Avogadro's number (\(N_A = 6.022 \times 10^{23} \, \text{mol}^{-1}\)):
\[ \phi_{\text{per mole}} = 2.3155 \times 10^{-20} \, \text{J} \times 6.022 \times 10^{23} \, \text{mol}^{-1} \]
- Performing the multiplication:
\[ \phi_{\text{per mole}} \approx 13913.6 \, \text{J/mol} \]

6. **Convert to Kilojoules per Mole**:
- To convert Joules to kilojoules, divide by 1000:
\[ \phi_{\text{kJ/mol}} = \frac{13913.6 \, \text{J/mol}}{1000} \approx 13.9136 \, \text{kJ/mol} \]

### Final Answer:
The binding energy of the electron in the metal is approximately **13.91 kJ/mol**.