Question 42.1: CASE STUDY The Hydrogen Spectrum As shown in Figure 42.5, hy......
CASE STUDY The Hydrogen Spectrum
As shown in Figure 42.5, hydrogen produces four lines in the visible part of the spectrum: Hα (656 nm), Hβ (486 nm), Hγ (434 nm) and Hδ (410 nm). Find the energy associated with the photons that produce each of these lines.
Interpret and Anticipate
A photon’s energy is proportional to its frequency. The red line (Hα) has a lower frequency than the violet line (Hδ), so we expect the red line to be produced by photons that are less energetic than the ones that produce the violet line.
Solve
Use f = c/λ (Eq. 34.20) to find the frequency of each line.
Use E = hf (Eq. 40.8) to find the energy of each photon.
\begin{array}{lc}\hline \text { Line } & E \left(\times 1 0 ^{ -1 9 } J\right) \\ \hline H \alpha & 3.02 \\H \beta & 4.09 \\H \gamma & 4.58 \\H \delta & 4.85 \\\hline\end{array}The energies are very small. When the energy is so low, it is customary to work in electron volts (eV) instead of in SI units. The conversion factor is 1 eV = 1.6 \times 10^{-19} J.
\begin{array}{lc}\hline \text { Line } & E ( eV ) \\\hline H \alpha & 1.89 \\H \beta & 2.56 \\H \gamma & 2.86 \\H \delta & 3.03 \\\hline\end{array}Check and Think
As expected, the red line (Hα) is produced by photons having less energy than those producing the violet line (Hδ). In this chapter’s case study, we’ll show why these four unique lines are produced by hydrogen.