The work function for a metal is 4eV. To emit a photoelectron of zero velocity from the surface of the metal, the wavelength of incident light should be :

Options

A. 2700 A
B. 1700 A
C. 5900 A
D. 3100 A

Answer

To solve the problem of finding the wavelength of incident light required to emit a photoelectron of zero velocity from a metal with a work function of 4 eV, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Work Function**: The work function (Φ) is the minimum energy required to remove an electron from the surface of a metal. In this case, Φ = 4 eV.

2. **Kinetic Energy of the Photoelectron**: The problem states that the photoelectron is emitted with zero velocity. The kinetic energy (KE) of an electron is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
Since the velocity (v) is zero, the kinetic energy is also zero:
\[ KE = 0 \text{ eV} \]

3. **Energy Conservation Equation**: According to the photoelectric effect, the energy of the incident photon (E) is equal to the work function plus the kinetic energy of the emitted electron:
\[ E = \Phi + KE \]
Substituting the known values:
\[ E = 4 \text{ eV} + 0 \text{ eV} = 4 \text{ eV} \]

4. **Relate Energy to Wavelength**: The energy of a photon can also be expressed in terms of its wavelength (λ) using the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant (approximately \( 4.1357 \times 10^{-15} \) eV·s),
- \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s).

5. **Using the Value of hc in eV·Å**: For convenience, when using energy in electron volts and wanting the wavelength in angstroms, we can use the value:
\[ hc \approx 12375 \text{ eV·Å} \]

6. **Calculate the Wavelength**: Rearranging the energy-wavelength equation gives:
\[ \lambda = \frac{hc}{E} \]
Substituting the values:
\[ \lambda = \frac{12375 \text{ eV·Å}}{4 \text{ eV}} = 3093.75 \text{ Å} \]

7. **Final Result**: Rounding to a reasonable number of significant figures, we find:
\[ \lambda \approx 3100 \text{ Å} \]

### Conclusion:
The wavelength of incident light required to emit a photoelectron of zero velocity from the surface of the metal is approximately **3100 Å**.