Chapter 8, Problem 115FP

The Rydberg-Ritz combination principle is an empirical relationship proposed by Walter Ritz in 1908 to explain the relationship among spectral lines of the hydrogen atom. The principle states that the spectral lines of the atom include frequencies that are either the sum or the difference of the frequencies of two other lines. This principle is obvious to us, because we now know that spectra arise from transitions between energy levels, and the energy of a transition is proportional to the frequency.
The frequencies of the first ten lines of an emission spectrum of hydrogen are given in the table at the bottom of this page. In this problem, use ideas from this chapter to identify the transitions involved, and apply the Rydberg-Ritz combination principle to calculate the frequencies of other the spectrum of hydrogen.
a. Use Balmers original equation, $\lambda =B{m}^2\left(m^2-n^2\right)$ , with $B=346.6$ nm, to develop an expression for the frequency $v_{m,n}$ of a line involving a transition from level m to level n, where m>n.
b. Use the expression you derived in (a) to calculate the expected ratio of the frequencies of the first two lines in each of the Lyman, Balmer, and Paschen series: (for the Lyman series); (for the Balmer series); and (for the Paschen series). Compare your calculated ratios to the observed ratio 2.465263/2.921793 = 0.843750 to identity the series as the Lyman, Balmer, or Paschen series. For each line in the series, specify the transition (quantum numbers) involved. Use a diagram, such as that given in Figure 8-13 to summarize your results.
c. Without performing any calculations, and starting from the Rydberg formula, equation (8.4), show that $v_{21}+v_{22}=v_{21}$ and thus, $v_{21}-v_{21}=v_{22}$ , This is an illustration of the Rydberg-Ritz combination principle: the frequency of a spectral line is equal to the sum or difference of frequencies of other lines.
d. Use the Rydberg-Ritz combination principle to determine, if possible, the frequencies for the other two series named in (b). (Hint The diagram you drew in part (b) might help you density the appropriate combinations of frequencies.)
e. Identify the transition associated with a of frequency $2.422405\times {10}^{12}{s}^{-1},$ one a series of discovered in 1953 by C. J. Humphreys.