In his explanation of the threshold frequency in the...

In his explanation of the threshold frequency in the photoelectric effect, Einstein reasoned that the absorbed photon must have a minimum energy to dislodge an electron from the metal surface. This energy is called the work function (\phi) of the metal. What is the longest wavelength of radiation (in $\mathrm{nm}$ ) that could cause the photoelectric effect in each of these metals: (a) calcium, $\phi=4.60 \times 10^{-19}$ $\mathrm{J} ;$ (b) titanium, $\phi=6.94 \times 10^{-19} \mathrm{~J} ;$ (c) sodium, $\phi=4.41 \times 10^{-19} \mathrm{~J} ?$

Key Concepts

Photoelectric Effect

The photoelectric effect is the process by which electrons are emitted from the surface of a material, typically a metal, when it is exposed to electromagnetic radiation of sufficient energy. This phenomenon supports the concept that light can be understood as discrete packets of energy called photons, and it was pivotal in the development of quantum mechanics.

Work Function

The work function is the minimum amount of energy required to liberate an electron from the surface of a material. It is a material-specific property that determines the threshold energy necessary for the photoelectric effect to occur, ensuring that only photons with energy equal to or greater than this value can eject electrons.

Photon Energy

Photon energy is defined as the energy carried by a single photon and is calculated using the equation E = hf, where h is Planck's constant and f is the frequency of the photon. This concept is crucial in the photoelectric effect as it directly relates the energy of light to its ability to overcome the work function of the metal.

Wavelength and Energy Relationship

The relationship between the energy of a photon and its wavelength is described by the equation E = hc/?, where c is the speed of light and ? is the wavelength. This inverse relationship allows for the determination of the longest wavelength (or lowest energy) that can still cause the photoelectric effect when equated to the work function.

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