# 9.5: Atomic Energy States and Line Spectra

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The energies of the hydrogenlike orbitals of various atoms were mentioned in Chapter 6 and, in particular, we showed a diagram of the most stable state $$\left( 1s \right)^2 \left( 2s \right)^2 \left( 2p \right)^2$$ of a carbon atom (Figure 6-4). Transfer of one of the $$2p$$ electrons to the $$3s$$ orbital requires excitation of the atom to a higher energy state and this can be achieved by absorption of electromagnetic radiation of the proper wavelength. The usual way that such excitation occurs is by absorption of a single quantum of radiant energy, and we can say that the absorption of this amount of energy $$\Delta{E_{12}}$$, corresponds to excitation of the atom from the ground state with energy $$E_1$$ to an excited state of configuration $$\left( 1s \right)^2 \left( 2s \right)^2 \left( 2p \right)^1 \left( 3s \right)^1$$ and energy $$E_2$$: The difference in energy, $$\Delta{E_{12}}$$, is related directly to the frequency ($$\nu$$, $$\text{sec}^{-1}$$) or wavelength ($$\lambda$$, $$\text{nm}$$)$$^4$$ of the absorbed quantum of radiation by the equation

$\Delta{E_{12}} = h \nu = \dfrac{hc}{\lambda} \tag{9.1}$

in which $$h$$ is Planck's constant and $$c$$ is the velocity of light. The relationship $$\Delta{E} = h\nu$$ often is called the Bohr frequency condition.

For chemical reactions, we usually express energy changes in $$\text{kcal mol}^{-1}$$. For absorption of one quantum of radiation by each atom (or each molecule) in one mole, the energy change is related to $$\lambda$$ by

$\Delta{E_{12}} = \dfrac{28,600}{\lambda(nm)} \; kcal/mol \tag{9.2}$

As defined, $$\Delta{E_{12}}$$ corresponds to one einstein of radiation.

What we have developed here is the idea of a spectroscopic change being related to a change in energy associated with the absorption of a quantum of energy. Spectra are the result of searches for such absorptions over a range of wavelengths (or frequencies). If one determines and plots the degree of absorption by a monoatomic gas such as sodium vapor as a function of wavelength, a series of very sharp absorption bands or lines are observed, hence the name line spectra. The lines are sharp because they correspond to specific changes in electronic configuration without complication from other possible energy changes.

$$^4$$See Section 9-3 for discussion of the units of frequency and wavelength.

## References

John D. Robert and Marjorie C. Caserio (1977) Basic Principles of Organic Chemistry, second edition. W. A. Benjamin, Inc. , Menlo Park, CA. ISBN 0-8053-8329-8. This content is copyrighted under the following conditions, "You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format."