0.4 Quantum energy levels in atoms  (Page 5/9)

The Hydrogen atom spectrum also tells us what these energy levels are. Recall that the frequencies of radiation emittedby Hydrogen atoms are given by the Rydberg equation . Each choice of the positive integers $n$ and $m$ predicts a single observed frequency in the hydrogen atom spectrum.

Each emitted frequency must correspond to an energy $h\nu$ by Einstein’s equation . This photon energy must be the difference between two energy levels for a hydrogen electron, since that is the amount of energy released bythe electron moving from one level to the other. If the energies of the two levels are $E_m$ and $E_n$ , then we can write that

By comparing this to the Rydberg equation, each energy level must be given by the formula

We can draw two conclusions. First, the electron in a hydrogen atom can exist only with certain energies,corresponding to motion in what we now call a state or an orbital . Second, the energy of a state can be characterized by an integer quantum number , n = 1, 2, 3, ... which determines its energy.

These conclusions are reinforced by similar observations of spectra produced by passing a current through otherelements. Only specific frequencies are observed for each atom, although only the hydrogen frequencies obey the Rydbergformula.

We conclude that the energies of electrons in atoms are "quantized," that is, restricted to certainvalues. We now need to relate this quantization of energy to the existence of shells, as developed in a previous study .

Observation 3: photoelectron spectroscopy of multi-electron atoms

The ionization energy of an atom tells us the energy of the electron or electrons which are at highest energy inthe atom and are thus easiest to remove from the atom. To further analyze the energies of the electrons more tightly bound to thenucleus, we introduce a new experiment. The photoelectric effect can be applied to ionize atoms in a gas, in a process often called photoionization . We shine light on an atom and measure the minimum frequency of light, corresponding to a minimum energy,which will ionize an electron from an atom. When the frequency of light is too low,the photons in that light do not have enough energy to ionize electrons from an atom. As we increase thefrequency of the light, we find a threshold at which electrons begin to ionize. Above this threshold, the energy $h\nu$ of the light of frequency $\nu$ is greater than the energy required to ionize the atom, and the excess energy is retained by the ionizedelectron as kinetic energy.

In photoelectron spectroscopy, we measure the kinetic energy of the electrons which are ionized by light. Thisprovides a means of measuring the ionization energy of the electrons. By conservation of energy, the energy of the light isequal to the ionization energy $\mathrm{IE}$ plus the kinetic energy $\mathrm{KE}$ of the ionized electron:

Thus, if we use a known frequency $\nu$ and measure $\mathrm{KE}$ , we can determine $\mathrm{IE}$ . The more tightly bound an electron is to the atom, the higher the ionization energy and the smallerthe kinetic energy of the ionized electron. If an atom has more than one electron and these electrons have different energies, thenfor a given frequency of light, we can expect electrons to be ejected with different kinetic energies. The higher kineticenergies correspond to the weakly bound outer electrons, and the lower kinetic energies correspond to the tightly bound innerelectrons.