To solve the problem, we need to determine which wavelength of light cannot be used for the photoelectric effect given the work function range of the metal (2 eV to 5 eV). We will use the formula for energy in terms of wavelength:
### Step 1: Understand the relationship between energy and wavelength
The energy \( E \) of a photon is given by the equation:
\[
E = \frac{hc}{\lambda}
\]
where:
- \( h \) is Planck's constant,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength.
### Step 2: Determine the minimum and maximum wavelengths
The work function \( \phi \) is the minimum energy required to eject an electron from the metal surface. For the photoelectric effect to occur, the energy of the incoming photon must be greater than or equal to the work function.
1. **Maximum Wavelength (\( \lambda_{\text{max}} \))**:
- This corresponds to the minimum energy (work function = 5 eV):
\[
\lambda_{\text{max}} = \frac{hc}{E_{\text{min}}}
\]
\[
\lambda_{\text{max}} = \frac{(4 \times 10^{-15} \, \text{eV s}) \times (3 \times 10^{8} \, \text{m/s})}{5 \, \text{eV}}
\]
\[
\lambda_{\text{max}} = \frac{12 \times 10^{-7} \, \text{eV m}}{5 \, \text{eV}} = 240 \, \text{nm}
\]
2. **Minimum Wavelength (\( \lambda_{\text{min}} \))**:
- This corresponds to the maximum energy (work function = 2 eV):
\[
\lambda_{\text{min}} = \frac{hc}{E_{\text{max}}}
\]
\[
\lambda_{\text{min}} = \frac{(4 \times 10^{-15} \, \text{eV s}) \times (3 \times 10^{8} \, \text{m/s})}{2 \, \text{eV}}
\]
\[
\lambda_{\text{min}} = \frac{12 \times 10^{-7} \, \text{eV m}}{2 \, \text{eV}} = 600 \, \text{nm}
\]
### Step 3: Determine the valid range of wavelengths
The valid range of wavelengths for the photoelectric effect is:
\[
240 \, \text{nm} < \lambda < 600 \, \text{nm}
\]
### Step 4: Identify the wavelength that cannot be used
Now, we need to check which of the given wavelengths falls outside this range. If a wavelength is less than 240 nm or greater than 600 nm, it cannot be used for the photoelectric effect.
- If we have options such as:
- 200 nm
- 300 nm
- 650 nm
- 500 nm
**Analysis of Options**:
- 200 nm: Less than 240 nm (not valid)
- 300 nm: Within the range (valid)
- 650 nm: Greater than 600 nm (not valid)
- 500 nm: Within the range (valid)
### Conclusion
The wavelengths that cannot be used for the photoelectric effect are 200 nm and 650 nm. However, since the question asks for one specific wavelength, we can conclude that **650 nm** is the answer.
### Final Answer
**650 nm cannot be used for the photoelectric effect.**
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