What would the minimum work function for a metal have to be for visible light ($380$–$750$ nm) to eject photoelectrons?

Verified step by step guidance

1. Understand the problem: The work function (Φ) is the minimum energy required to eject an electron from the surface of a metal. For visible light to eject photoelectrons, the energy of the photons in the visible spectrum (380–750 nm) must be equal to or greater than the work function. Use the relationship between photon energy and wavelength.
2. Recall the formula for the energy of a photon: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J·s} \)), \( c \) is the speed of light (\( 3.00 \times 10^8 \; \text{m/s} \)), and \( \lambda \) is the wavelength of the photon in meters.
3. Determine the maximum photon energy in the visible spectrum: Use the shortest wavelength (\( \lambda = 380 \; \text{nm} \)) to calculate the maximum energy. Convert \( \lambda \) to meters by dividing by \( 10^9 \). Substitute \( h \), \( c \), and \( \lambda \) into the formula \( E = \frac{hc}{\lambda} \).
4. Interpret the result: The maximum photon energy corresponds to the minimum work function (Φ) required to eject photoelectrons. This is because the photon energy must be at least equal to the work function for the photoelectric effect to occur.
5. Conclude: The minimum work function for the metal is equal to the maximum photon energy calculated in the previous step. Ensure the units are consistent (e.g., convert the result to electronvolts if needed, using \( 1 \; \text{eV} = 1.602 \times 10^{-19} \; \text{J} \)).

Key Concepts

Work Function

The work function is the minimum energy required to remove an electron from the surface of a material. It is a critical parameter in photoelectric effect experiments, as it determines the threshold frequency of light needed to eject electrons. The work function is typically expressed in electron volts (eV) and varies among different materials.

Photoelectric Effect

The photoelectric effect is the phenomenon where electrons are emitted from a material when it absorbs light of sufficient energy. According to Einstein's photoelectric equation, the energy of the incoming photons must be greater than or equal to the work function of the material for electron ejection to occur. This effect demonstrates the particle-like behavior of light and is foundational in quantum physics.

Photon Energy and Wavelength

The energy of a photon is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. For visible light, which ranges from 380 to 750 nm, the corresponding photon energies can be calculated, and these energies must meet or exceed the work function for photoelectrons to be emitted from the metal.