38.5.1 Energy-Level Diagrams

An energy-level diagram, such as the one shown in FIGURE 38.15 , is a useful pictorial representation of the stationary-state energies. An energy-level diagram is less a graph than it is a picture. The vertical axis represents energy, but the horizontal axis is not a scale. Think of this as a picture of a ladder in which the energies are the rungs of the ladder. The lowest rung, with energy $E_{1}$ , is the ground state. Higher rungs are labeled by their quantum numbers, $n = 2, 3, 4, \dots$ .

The figure illustrates an energy level diagram.

FIGURE 38.15 Full Alternative Text

Energy-level diagrams are especially useful for showing transitions, or quantum jumps, in which a photon of light is emitted or absorbed. As examples, Figure 38.15 shows upward transitions in which a photon is absorbed by a ground-state atom $(n = 1)$ and downward transitions in which a photon is emitted from an $n = 4$ excited state.

EXAMPLE 38.10 Emission and absorption

An atom has stationary states $E_{1} = 0.00 eV, E_{2} = 3.00 eV$ , and $E_{3} = 5.00 eV$ . What wavelengths are observed in the absorption spectrum and in the emission spectrum of this atom?

MODEL Photons are emitted when an atom undergoes a transition from a higher energy level to a lower energy level. Photons are absorbed in a transition from a lower energy level to a higher energy level. But most of the atoms are in the $n = 1$ ground state, so the only transitions seen in the absorption spectrum start from the $n = 1$ state.

VISUALIZE FIGURE 38.16 shows an energy-level diagram for the atom.

The figure illustrates the energy level diagram of an atom at various states.

FIGURE 38.16 Full Alternative Text

SOLVE This atom will absorb photons on the $1 \to 2$ and $1 \to 3$ transitions, with $Δ E_{1 \to 2} = 3.00 eV$ and $Δ E_{1 \to 3} = 5.00 eV$ . From $f = Δ E_{atom} / h$ and $λ = c / f$ , we find that the wavelengths in the absorption spectrum are

\begin{array}{l} 1 \to 2 & f & = & 3.00 eV/ h = 7.25 \times 10^{14} Hz \\ λ & = & 414 nm (blue) \\ 1 \to 3 & f & = & 5.00 eV/ h = 1.21 \times 10^{15} Hz \\ λ & = & 248 nm (ultraviolet) \end{array}

The emission spectrum will also have the 414 nm and 248 nm wavelengths due to the $2 \to 1$ and $3 \to 1$ transitions from excited states 2 and 3 to the ground state. In addition, the emission spectrum will contain the $3 \to 2$ transition with $Δ E_{3 \to 2} = - 2.00 eV$ that is not seen in absorption because there are too few atoms in the $n = 2$ state to absorb. We found in Example 38.9 that a 2.00 eV transition corresponds to a wavelength of 621 nm. Thus the emission wavelengths are

\begin{array}{l} 2 \to 1 & λ & = & 414 nm (blue) \\ 3 \to 1 & λ & = & 248 nm (ultraviolet) \\ 3 \to 2 & λ & = & 621 nm (orange) \end{array}

38.5.1 Energy-Level Diagrams

FIGURE 38.15 An energy-level diagram.

EXAMPLE 38.10 Emission and absorption

FIGURE 38.16 The atom’s energy-level diagram.