8.5 Atomic spectra and x-rays  (Page 2/12)

The energy-level diagram for hydrogen is similar to sodium, because both atoms have one electron in the outer shell. The valence electron of sodium moves in the electric field of a nucleus shielded by electrons in the inner shells, so it does not experience a simple 1/ r Coulomb potential and its total energy depends on both n and l . Interestingly, mercury has two separate energy-level diagrams; these diagrams correspond to two net spin states of its 6 s (valence) electrons.

The sodium doublet

Strategy

We must first determine the quantum number of the initial state that satisfies the selection rule. Then, we can use this number to determine the magnitude of orbital angular momentum of the initial state.

Solution

1. Allowed transitions must obey the selection rule. If the quantum number of the initial state is $l=0$ , the transition is forbidden because $\text{Δ}l=0$ . If the quantum number of the initial state is $l=2,3,4$ ,…the transition is forbidden because $\text{Δ}l>1.$ Therefore, the quantum of the initial state must be $l=1$ . The orbital angular momentum of the initial state is
   
   $L=\sqrt{l(l+1)}\hslash =1.41\hslash .$
2. Because the final state for both transitions is the same (3 s ), the difference in energies of the photons is equal to the difference in energies of the two excited states. Using the equation
   
   $\text{Δ}E=hf=h\left(\frac{c}{\lambda }\right),$
   
   we have
   
   $\begin{array}{cc}\hfill \text{Δ}E& =hc\left(\frac{1}{{\lambda }_1}-\frac{1}{{\lambda }_2}\right)\hfill \\ & =\left(4.14\times {10}^{\mathrm{-15}}\text{eVs}\right)\left(3.00\times {10}^8\text{m/s}\right)\times \left(\frac{1}{589.00\times {10}^{\mathrm{-9}}\text{m}}-\frac{1}{589.59\times {10}^{\mathrm{-9}}\text{m}}\right)\hfill \\ & =2.11\times {10}^{\mathrm{-3}}\text{eV}\text{.}\hfill \end{array}$

Significance

The energy difference $\text{Δ}E$ is about 0.1% (1 part in 1000) of this average energy. However, a sensitive spectrometer can measure the difference.

Atomic fluorescence

Fluorescence occurs when an electron in an atom is excited several steps above the ground state by the absorption of a high-energy ultraviolet (UV) photon. Once excited, the electron “de-excites” in two ways. The electron can drop back to the ground state, emitting a photon of the same energy that excited it, or it can drop in a series of smaller steps, emitting several low-energy photons. Some of these photons may be in the visible range. Fluorescent dye in clothes can make colors seem brighter in sunlight by converting UV radiation into visible light. Fluorescent lights are more efficient in converting electrical energy into visible light than incandescent filaments (about four times as efficient). [link] shows a scorpion illuminated by a UV lamp. Proteins near the surface of the skin emit a characteristic blue light.