The work function of a substance is `4.0 eV`. The longest wavelength of light that can cause photo electron emission from this substance is approximately. (a) `540 nm` (b ) `400nm` (c ) `310 nm` (d) `220 nm`

Options

A. 540 nm
B. 400 nm
C. 310 nm
D. 220 nm

Explanation

To find the longest wavelength of light that can cause photoelectron emission from a substance with a given work function, we can use the formula that relates energy and wavelength:

1. **Understand the Work Function**: The work function (ϕ) is the minimum energy required to remove an electron from the surface of a material. In this case, the work function is given as 4.0 eV.

2. **Convert Work Function to Joules**: Since we will be using SI units, we need to convert the work function from electron volts to joules. The conversion factor is:
\[ 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \]
Therefore, the work function in joules is:
\[ \phi = 4.0 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} = 6.4 \times 10^{-19} \text{ J} \]

3. **Use the Energy-Wavelength Relation**: The energy of a photon can be related to its wavelength (λ) using the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \(E\) is the energy of the photon,
- \(h\) is Planck's constant (\(6.63 \times 10^{-34} \text{ J s}\)),
- \(c\) is the speed of light (\(3 \times 10^8 \text{ m/s}\)),
- \(\lambda\) is the wavelength in meters.

4. **Rearranging the Formula**: To find the wavelength, we rearrange the equation:
\[ \lambda = \frac{hc}{E} \]

5. **Substituting Values**: Now we can substitute the values into the equation:
\[ \lambda = \frac{(6.63 \times 10^{-34} \text{ J s}) \times (3 \times 10^8 \text{ m/s})}{6.4 \times 10^{-19} \text{ J}} \]

6. **Calculating λ**:
\[ \lambda = \frac{1.989 \times 10^{-25} \text{ J m}}{6.4 \times 10^{-19} \text{ J}} \approx 3.11 \times 10^{-7} \text{ m} \]
Converting this to nanometers:
\[ \lambda \approx 310 \text{ nm} \]

7. **Conclusion**: The longest wavelength of light that can cause photoelectron emission from this substance is approximately **310 nm**.