Question 42.3: CASE STUDY Other Hydrogen Lines Use the information in Figur......
CASE STUDY Other Hydrogen Lines
Use the information in Figure 42.7 to find the wavelength of the first (lowest energy) line of the Lyman, Paschen, and Brackett series. In what part of the electromagnetic spectrum are these lines found?
INTERPRET and ANTICIPATE
Find the wavelength of the emitted photons from the energy differences shown in Figure 42.7. The greater the energy difference, the greater the energy of the photon. High-energy photons have short wavelengths. Of the three series, the Lyman transition has the greatest energy difference, so we expect its photon will have the shortest wavelength. Check the results by using the empirical fit (Eq. 42.3).
SOLVE
Find the energy difference for each of the lines. The subscripts L, P, and B refer to the first line of the Lyman, Paschen and Brackett series, respectively.
In each case, the energy lost by the atom equals the photon energy. Use E = hc/λ (Eq. 40.17) to find the photon’s wavelength.
Use Figure 34.11 and Table 34.2 (page 1101) to identify the part of the electromagnetic spectrum in which each line is found.
\begin{aligned}& \lambda_{ L }=1.22 \times 10^{-7} m \text { in the UV } \\ & \lambda_{ P }=1.88 \times 10^{-6} m \text { in the IR } \\& \lambda_{ B }=4.05 \times 10^{-6} m \text { in the IR }\end{aligned}CHECK AND THINK
As expected, the Lyman line has the shortest wavelength. We can check our results using Equation 42.3.
| TABLE 34.2 The electromagnetic spectrum broken into convenient bands. Values are approximate. | ||
| Name of band | Wavelength λ (m) |
Frequency f (Hz) |
| Radio | >10^{-2} | <10^{11} |
| Microwave | 10^{-4}-1 | 10^9-10^{13} |
| Infrared (IR) | 10^{-6}-10^{-4} | 10^{12}-10^{14} |
| Visible light | 10^{-7}-10^{-6} | 10^{14}-10^{15} |
| Ultraviolet (UV) | 10^{-9}-10^{-7} | 10^{15}-10^{18} |
| X-rays | 10^{-12}-10^{-9} | 10^{17}-10^{20} |
| Gamma rays | <10^{-10} | >10^{19} |
