Question

The work function of a substance is 4.0 eV. The longest wavelength of light that can cause photoelectron emission from this substance is approximately:

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

Hint:

The longest wavelength corresponds to the threshold frequency at which the kinetic energy of the ejected electron is zero. Use the equation \( \lambda = \frac{h \cdot c}{W} \) to calculate the wavelength.

Answer 1

The Correct Option is A

Step 1: Use of the Photoelectric Equation 
The photoelectric equation relates the energy of a photon, the work function of the material, and the maximum kinetic energy of the ejected photoelectrons: \[ E_{photon} = W + K.E. \] Where: 
\( E_{photon} = h \cdot f \) is the energy of the incoming photon. 
\( W \) is the work function of the material. 
\( K.E. \) is the maximum kinetic energy of the ejected electron (which is zero for the threshold frequency). At the threshold frequency, the kinetic energy is zero, so the energy of the photon is equal to the work function \( W \). 
Step 2: Calculation of the Longest Wavelength 
The energy of the photon can be expressed as \( E_{photon} = h \cdot c / \lambda \), where \( \lambda \) is the wavelength of light, \( h \) is Planck’s constant, and \( c \) is the speed of light. 
Rearranging for \( \lambda \): \[ \lambda = \frac{h \cdot c}{W} \] 
Substituting the known values: 
\(h = 6.626 \times 10^{34} \, \text{J.s}\)
\( c = 3.0 \times 10^8 \, \text{m/s} \) 
\( W = 4.0 \, \text{eV} = 4.0 \times 1.602 \times 10^{19} \, \text{J} \) \[ \lambda = \frac{6.626 \times 10^{34} \times 3.0 \times 10^8}{4.0 \times 1.602 \times 10^{19}} = 310 \, \text{nm} \] 

Final Answer: The longest wavelength that can cause photoelectron emission is approximately  310 nm .