Calculating Photon Energy and the Photoelectric Effect: A Violet Light

To solve part (a), note that the energy of a photon is given by $E=\text{hf}$. For part (b), once the energy of the photon is calculated, it is a straightforward application of ${\text{KE}}_e=\text{hf}\text{–BE}$ to find the ejected electron’s maximum kinetic energy, since BE is given.

Photon energy is given by

Since we are given the wavelength rather than the frequency, we solve the familiar relationship $c=\mathrm{f\lambda }$ for the frequency, yielding

Combining these two equations gives the useful relationship

Now substituting known values yields

Converting to eV, the energy of the photon is

Finding the kinetic energy of the ejected electron is now a simple application of the equation ${\text{KE}}_e=\text{hf}\text{–BE}$. Substituting the photon energy and binding energy yields

The energy of this 420-nm photon of violet light is a tiny fraction of a joule, and so it is no wonder that a single photon would be difficult for us to sense directly—humans are more attuned to energies on the order of joules. But looking at the energy in electron volts, we can see that this photon has enough energy to affect atoms and molecules. A DNA molecule can be broken with about 1 eV of energy, for example, and typical atomic and molecular energies are on the order of eV, so that the UV photon in this example could have biological effects. The ejected electron (called a photoelectron) has a rather low energy, and it would not travel far, except in a vacuum. The electron would be stopped by a retarding potential of but 0.26 eV. In fact, if the photon wavelength were longer and its energy less than 2.71 eV, then the formula would give a negative kinetic energy, an impossibility. This simply means that the 420-nm photons with their 2.96-eV energy are not much above the frequency threshold. You can show for yourself that the threshold wavelength is 459 nm (blue light). This means that if calcium metal is used in a light meter, the meter will be insensitive to wavelengths longer than those of blue light. Such a light meter would be completely insensitive to red light, for example.