The kinetic energy of most energetic electrons emitted from a metallic surface is doubled when the wavelength `lamda` of the incident radiation is changed from 400 nm to 310 nm. The work function of the metal is

Options

A. 0.9 eV
B. 1.7 eV
C. 2.2 eV
D. 3.1 eV

Explanation

To find the work function of the metal given that the kinetic energy of the most energetic electrons emitted is doubled when the wavelength of the incident radiation changes from 400 nm to 310 nm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Photoelectric Effect Equation**:
The photoelectric effect can be described by the equation:
\[ E = W + KE_{max} \]
where \(E\) is the energy of the incident photons, \(W\) is the work function of the metal, and \(KE_{max}\) is the maximum kinetic energy of the emitted electrons.

2. **Calculate the Energy of Photons**:
The energy of the incident photons can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)) and \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)).

3. **Calculate Energy for Both Wavelengths**:
- For \(\lambda_1 = 400 \, \text{nm} = 400 \times 10^{-9} \, \text{m}\):
\[ E_1 = \frac{hc}{400 \times 10^{-9}} \]
- For \(\lambda_2 = 310 \, \text{nm} = 310 \times 10^{-9} \, \text{m}\):
\[ E_2 = \frac{hc}{310 \times 10^{-9}} \]

4. **Set Up the Equations**:
From the photoelectric effect equation, we have:
- For the first wavelength:
\[ E_1 = W + KE_{max} \]
- For the second wavelength, since the kinetic energy is doubled:
\[ E_2 = W + 2KE_{max} \]

5. **Substituting the Energies**:
Substitute the expressions for \(E_1\) and \(E_2\):
\[ \frac{hc}{400 \times 10^{-9}} = W + KE_{max} \quad \text{(1)} \]
\[ \frac{hc}{310 \times 10^{-9}} = W + 2KE_{max} \quad \text{(2)} \]

6. **Subtract Equation (1) from Equation (2)**:
\[ \frac{hc}{310 \times 10^{-9}} - \frac{hc}{400 \times 10^{-9}} = KE_{max} \]

7. **Finding a Common Denominator**:
The common denominator for the fractions is \(400 \times 310 \times 10^{-9} \times 10^{-9}\):
\[ KE_{max} = hc \left( \frac{400 - 310}{400 \times 310} \right) \times 10^{-9} \]
Simplifying gives:
\[ KE_{max} = \frac{hc \cdot 90}{400 \times 310} \times 10^{-9} \]

8. **Substituting Back to Find Work Function**:
Substitute \(KE_{max}\) back into either equation to find \(W\). Using equation (1):
\[ W = \frac{hc}{400 \times 10^{-9}} - KE_{max} \]

9. **Calculating Work Function**:
Substitute the values of \(h\) and \(c\) into the equation and solve for \(W\).

10. **Convert to Electron Volts**:
Since the work function is often expressed in electron volts (eV), convert the result from joules to eV using the conversion factor \(1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}\).

### Final Calculation:
After performing the calculations, you will find that the work function \(W\) is approximately:
\[ W \approx 2.21 \, \text{eV} \]