(a) Estimate the wavelength of the photon emitted as an...

(a) Estimate the wavelength of the photon emitted as an electron falls from the $n=2$ shell to the $n=1$ shell in the gold atom $(Z=79) .$ (b) About how much energy must bombarding electrons have to excite gold to radiate this emission line? (a) As noted in Problem 44.1, to a first approximation the energies of the innermost electrons of a large-Z atom are given by $\mathrm{E}_{n}=-13.6 \mathrm{Z}^{2} / \mathrm{n}^{2} \mathrm{eV}$. Thus, $$ \Delta E_{2,1}=13.6(79)^{2}\left(\frac{1}{1}-\frac{1}{4}\right)=63700 \mathrm{eV} $$ This corresponds to a photon with $$ \lambda=(1240 \mathrm{~nm})\left(\frac{1 \mathrm{eV}}{63700 \mathrm{eV}}\right)=0.0195 \mathrm{~nm} $$

(a) Estimate the wavelength of the photon emitted as an electron falls from the $n=2$ shell to the $n=1$ shell in the gold atom $(Z=79) .$ (b) About how much energy must bombarding electrons have to excite gold to radiate this emission line?
(a) As noted in Problem 44.1, to a first approximation the energies of the innermost electrons of a large-Z atom are given by $\mathrm{E}_{n}=-13.6 \mathrm{Z}^{2} / \mathrm{n}^{2} \mathrm{eV}$. Thus,
$$
\Delta E_{2,1}=13.6(79)^{2}\left(\frac{1}{1}-\frac{1}{4}\right)=63700 \mathrm{eV}
$$
This corresponds to a photon with
$$
\lambda=(1240 \mathrm{~nm})\left(\frac{1 \mathrm{eV}}{63700 \mathrm{eV}}\right)=0.0195 \mathrm{~nm}
$$

Key Concepts

Atomic Energy Levels and the Bohr Model

This concept refers to the quantization of electron energies in atoms. For hydrogen-like atoms, the energy levels can be approximated by a formula (often originating from the Bohr model) where the energy of a level is proportional to -Z²/n². This quantization framework is foundational in atomic physics and explains why atoms emit or absorb photons only at specific energies corresponding to transitions between these discrete levels.

Electronic Transitions and Photon Emission

When an electron moves from a higher energy level to a lower one, the atom releases energy in the form of a photon whose energy is equal to the difference between the two levels. This process is pivotal in spectroscopy, as analyzing the emitted photon energies (or wavelengths) reveals details about the electronic structure of the atom.

Relationship Between Photon Energy and Wavelength

The energy of a photon is inversely proportional to its wavelength, as expressed by the Planck-Einstein relation (E = hc/?). This relationship is key to converting a known energy difference between atomic levels into the corresponding wavelength of the emitted photon, allowing predictions about the spectral line positions.

Threshold Energy for Electron Bombardment and Excitation

In processes where atoms are excited by electron impact, the bombarding electrons must have kinetic energy at least equal to the energy gap between the relevant electron levels. This excitation threshold ensures that the electrons can impart sufficient energy to an atom to promote an electron to a higher energy state, thereby eventually leading to photon emission upon its decay.

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