The maximum wavelength of radiation that can produce photoelectric effect in a certain metal is 200 nm . The maximum kinetic energy acquired by electron due to radiation of wavelength 100 nm will be

Options

A. 12.4 eV
B. 6.2 eV
C. 100 eV
D. 200 eV

Answer

To solve the problem, we need to find the maximum kinetic energy (KE_max) acquired by an electron due to radiation of wavelength 100 nm, given that the maximum wavelength for the photoelectric effect in a certain metal is 200 nm.

### Step-by-Step Solution:

1. **Understand the Given Data:**
- Maximum wavelength for the photoelectric effect (λ₀) = 200 nm
- Wavelength of the radiation (λ) = 100 nm

2. **Convert Wavelengths to Meters:**
- λ₀ = 200 nm = 200 x 10^-9 m
- λ = 100 nm = 100 x 10^-9 m

3. **Calculate the Work Function (Φ):**
The work function can be calculated using the formula:
\[ \Phi = \frac{hc}{\lambda_0} \]
where:
- \( h \) (Planck's constant) = \( 6.626 \times 10^{-34} \, \text{Js} \)
- \( c \) (speed of light) = \( 3 \times 10^8 \, \text{m/s} \)

Substituting the values:
\[ \Phi = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{200 \times 10^{-9}} \]

4. **Calculate the Energy of the Photon (E):**
The energy of the photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{100 \times 10^{-9}} \]

5. **Calculate the Maximum Kinetic Energy (KE_max):**
The maximum kinetic energy of the emitted electrons can be calculated using the formula:
\[ KE_{max} = E - \Phi \]
Substituting the expressions for E and Φ:
\[ KE_{max} = \frac{hc}{\lambda} - \frac{hc}{\lambda_0} \]
Factoring out \( hc \):
\[ KE_{max} = hc \left( \frac{1}{\lambda} - \frac{1}{\lambda_0} \right) \]

6. **Substitute the Values:**
Now substituting the values:
\[ KE_{max} = (6.626 \times 10^{-34})(3 \times 10^8) \left( \frac{1}{100 \times 10^{-9}} - \frac{1}{200 \times 10^{-9}} \right) \]

7. **Calculate the Result:**
- Calculate \( \frac{1}{100 \times 10^{-9}} = 10^{7} \)
- Calculate \( \frac{1}{200 \times 10^{-9}} = 5 \times 10^{6} \)
- Therefore, \( \frac{1}{100 \times 10^{-9}} - \frac{1}{200 \times 10^{-9}} = 10^{7} - 5 \times 10^{6} = 5 \times 10^{6} \)

Now substituting back:
\[ KE_{max} = (6.626 \times 10^{-34})(3 \times 10^8)(5 \times 10^{6}) \]

Calculate:
\[ KE_{max} = 9.939 \times 10^{-19} \, \text{J} \]

8. **Convert Joules to Electron Volts:**
To convert Joules to electron volts, use the conversion factor \( 1 eV = 1.6 \times 10^{-19} J \):
\[ KE_{max} = \frac{9.939 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 6.21 \, eV \]

Thus, the maximum kinetic energy acquired by the electron due to radiation of wavelength 100 nm is approximately **6.2 eV**.