The work function of a metal surface is 4.2 eV. The maximum wavelength which can eject electrons from this metal surface is

Options

A. 3076 Å
B. 4116 Å
C. 2946 Å
D. 5088 Å

Answer

To find the maximum wavelength that can eject electrons from a metal surface with a given work function, we can use the relationship between the work function and the wavelength of light. The work function (φ) is the minimum energy required to remove an electron from the surface of the metal.

### Step-by-Step Solution:

1. **Understand the Work Function**: The work function (φ) is given as 4.2 eV. This is the energy required to eject an electron from the metal surface.

2. **Use the Energy-Wavelength Relation**: The energy (E) of a photon can be expressed in terms of its wavelength (λ) using the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \(E\) is the energy of the photon,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \(c\) is the speed of light (\(3.0 \times 10^8 \, \text{m/s}\)),
- \(\lambda\) is the wavelength in meters.

3. **Set Up the Equation**: For the maximum wavelength that can eject electrons, we set the energy of the photon equal to the work function:
\[ \phi = \frac{hc}{\lambda} \]
Rearranging this gives:
\[ \lambda = \frac{hc}{\phi} \]

4. **Convert Work Function to Joules**: Since we will use SI units, we need to convert the work function from electron volts to joules. The conversion factor is:
\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \]
Therefore, the work function in joules is:
\[ \phi = 4.2 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 6.72 \times 10^{-19} \, \text{J} \]

5. **Calculate the Wavelength**:
Substitute the values of \(h\), \(c\), and \(\phi\) into the equation for \(\lambda\):
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{Js})(3.0 \times 10^8 \, \text{m/s})}{6.72 \times 10^{-19} \, \text{J}} \]
\[ \lambda = \frac{1.9878 \times 10^{-25}}{6.72 \times 10^{-19}} \approx 2.95 \times 10^{-7} \, \text{m} \]

6. **Convert to Angstroms**: To express the wavelength in angstroms (1 Å = \(10^{-10}\) m):
\[ \lambda = 2.95 \times 10^{-7} \, \text{m} = 2950 \, \text{Å} \]

### Final Answer:
The maximum wavelength that can eject electrons from the metal surface is approximately **2950 Å**.