The work function of a substance is 12.4 eV. The longest wavelength of light that can cause photoelectron emission from this substance is approximately

Options

A. 540 nm
B. 400 nm
C. 310 nm
D. 100 nm

Answer

To find the longest wavelength of light that can cause photoelectron emission from a substance with a work function of 12.4 eV, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between energy and wavelength**: The energy of a photon is given by the equation:
\[ E = h \nu = \frac{hc}{\lambda} \]
where \(E\) is the energy of the photon, \(h\) is Planck's constant, \(\nu\) is the frequency, \(c\) is the speed of light, and \(\lambda\) is the wavelength.

2. **Identify the condition for photoelectron emission**: For photoelectron emission to occur, the energy of the incoming photon must be equal to or greater than the work function (\(\phi\)) of the substance. In this case, the work function is given as 12.4 eV.

3. **Set up the equation for the longest wavelength**: At the threshold of emission (the longest wavelength), the energy of the photon is equal to the work function:
\[ E = \phi = 12.4 \text{ eV} \]

4. **Use the relationship between energy and wavelength**: Substitute the energy into the equation for energy:
\[ 12.4 \text{ eV} = \frac{hc}{\lambda} \]

5. **Rearrange the equation to solve for \(\lambda\)**:
\[ \lambda = \frac{hc}{12.4 \text{ eV}} \]

6. **Substitute known values**: Use the known values for \(h\) and \(c\):
- Planck's constant \(h = 4.135667696 \times 10^{-15} \text{ eV s}\)
- Speed of light \(c = 3 \times 10^8 \text{ m/s}\)
- Convert \(hc\) into electron volts and nanometers:
\[ hc \approx 1240 \text{ eV nm} \]

7. **Calculate the wavelength**:
\[ \lambda = \frac{1240 \text{ eV nm}}{12.4 \text{ eV}} = 100 \text{ nm} \]

8. **Conclusion**: The longest wavelength of light that can cause photoelectron emission from this substance is approximately 100 nm.

### Final Answer:
The longest wavelength of light that can cause photoelectron emission from the substance is **100 nm**.

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