The energy required to remove and electron from the surface of sodium metal is ` 2. 3 eV`. What is the longest wavelength of radiation with which it can shown photoelectric effect ?

Options

A. `5.4 xx 10^(-17) m`
B. `5. 4 xx 10^(-8) m`
C. `5.4 xx 10^(-7) m`
D. `5.4 xx 10^(-9) m`

Answer

To solve the problem, we need to find the longest wavelength of radiation that can cause the photoelectric effect for sodium metal, given that the energy required to remove an electron is 2.3 eV.

### Step-by-Step Solution:

1. **Understand the energy requirement**: The energy required to remove an electron from sodium is given as 2.3 eV. This is the work function (ϕ) of sodium.

2. **Convert electron volts to joules**:
- We know that 1 eV = \(1.6 \times 10^{-19}\) J.
- Therefore, to convert 2.3 eV to joules:
\[ E = 2.3 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 3.68 \times 10^{-19} \, \text{J} \]

3. **Use the photoelectric effect formula**: The energy of a photon is related to its wavelength (λ) by the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \(E\) is the energy of the photon,
- \(h\) is Planck's constant (\(6.63 \times 10^{-34} \, \text{Js}\)),
- \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)),
- \(\lambda\) is the wavelength in meters.

4. **Rearrange the formula to find wavelength**:
\[ \lambda = \frac{hc}{E} \]

5. **Substitute the values**:
- Substitute \(h\), \(c\), and \(E\) into the equation:
\[ \lambda = \frac{(6.63 \times 10^{-34} \, \text{Js}) \times (3 \times 10^8 \, \text{m/s})}{3.68 \times 10^{-19} \, \text{J}} \]

6. **Calculate λ**:
\[ \lambda = \frac{1.989 \times 10^{-25}}{3.68 \times 10^{-19}} \approx 5.4 \times 10^{-7} \, \text{m} \]

7. **Convert to nanometers** (optional):
- To convert meters to nanometers, multiply by \(10^9\):
\[ \lambda \approx 5.4 \times 10^{-7} \, \text{m} = 540 \, \text{nm} \]

8. **Final answer**: The longest wavelength of radiation that can show the photoelectric effect on sodium is approximately \(5.4 \times 10^{-7} \, \text{m}\) or \(540 \, \text{nm}\).