Hydrogen, atomic Balmer

In 1885 Balmer was able fo fif fhe discrete wavelengfhs X of part of fhe emission specfrum of fhe hydrogen atom, now called fhe Balmer series and illusfrafed in Figure 1.1, fo fhe empirical formula... 

Figure 1.1 Energy levels (vertical lines) and observed transitions (horizontal lines) of the hydrogen atom, including the Lyman, Balmer, Paschen, Brackett and Pfund series...

Question. Using Equations (1.11) and (1.12) calculate, to six significant figures, the wavenumbers, in cm of the first two (lowest n") members of the Balmer series of the hydrogen atom. Then convert these to wavelengths, in nm. 

Whereas the emission spectrum of the hydrogen atom shows only one series, the Balmer series (see Figure 1.1), in the visible region the alkali metals show at least three. The spectra can be excited in a discharge lamp containing a sample of the appropriate metal. One series was called the principal series because it could also be observed in absorption through a column of the vapour. The other two were called sharp and diffuse because of their general appearance. A part of a fourth series, called the fundamental series, can sometimes be observed. 

The study of the hydrogen atom also played an important role in the development of quantum theory. The Lyman, Balmer, and Paschen series of spectral lines observed in incandescent atomic hydrogen were found to obey the empirical equation... 

Calculate the frequency, wavelength, and wave number for the series limit of the Balmer series of the hydrogen-atom spectral lines. 

Derive an expression in terms of Rqo for the difference in wavelength, AA = Ah — Ad, between the first line of the Balmer series n = 2) for a hydrogen atom and the corresponding line for a deuterium atom Assume that the masses of the proton and the neutron are the same. 

There are in principle an infinite number of series beginning at higher quantum numbers with i = 6, 7, 8, 9... but they become increasingly difficult to observe. For the higher n series to be seen in the spectrum the levels have to be populated, so some hydrogen atoms must be in the n = 5 level to see the Pfund series. We shall see that the presence of the Balmer series in the spectrum of a star is indicative of the stellar temperature, which is a direct consequence of the population of the energy levels. More of this in Chapter 4. 

Determination of the temperature of the star from the intensity of the Balmer series and the population of n = 2 in the hydrogen atom... 

In gas clouds containing one or more hot stars (7 cn > 30 000 K), hydrogen atoms are ionized by the stellar UV radiation in the Lyman continuum and recombine to excited levels their decay gives rise to observable emission lines such as the Balmer series (see, for example, Fig. 3.22). Examples are planetary nebulae (PN), which are envelopes of evolved intermediate-mass stars in process of ejection and... 

The existence of outsized hydrogen atoms was inferred early on from the observation that 33 terms of the Balmer series could be observed in stellar spectra, compared to only 12 in the laboratory [58]. More recently [59] Rydberg atoms have been produced by exciting an atomic beam with laser light. When the outer electron of an atom is excited into a very high energy level, it enters a spatially extended orbital which is far outside the orbitals... 

Figure 2.1 Electronic orbitals and the resulting emission spectrum in the hydrogen atom, (a) Bohr orbitals of the hydrogen atom and the resulting spectral series, (b) emission spectrum of atomic hydrogen. The spectrum in (b) is calibrated in terms of wavenumber (P), which is reciprocal wavelength. The Balmer series, which consists of those transitions terminating on the second orbital, give rise to emission lines in the visible region of the spectrum. ( 1990 John Wiley Sons, Inc. Reprinted from Brady, 1990, by permission of the publisher.)...

It is noted without further comment that Moseley was able to characterize his observed x-ray spectra in such a way that the individual elements are identified purely by their respective atomic numbers without any use of an atomic model. Note again, however, that the denominators in the Qk and Ql expressions are exactly the frequencies of the hydrogenic Lymann and Balmer alpha lines (Moseley s nomenclature) and that therefore Qk and Ql exactly equal the nuclear charge of a one-electron hydrogenic atom which would be deduced from the frequency v of its observed Lymann and Balmer alpha lines2. 

Fig. 2-2.—Energy levels for the hydrogen atom. The arrows indicate the transitions that give the first four lines of the Balmer series in emission...

The wavelength of light at which the Balmer series converges (Problem 5.45) corresponds to the amount of energy required to completely remove an electron from the second shell of a hydrogen atom. Calculate this energy in kilojoules per mole. 

Bob, isn t it awesome that such a little formula captures a piece of reality You bet. And it was formulated by a secondary-school teacher in Basel, Switzerland, but Balmer never knew why the formula worked. We ll learn about the why in a moment. In 1913 Niels Bohr found that the Balmer formula supported his theory of discrete energy states within the hydrogen atom. ... 

Balmer Series This series is formed when excited electrons in a hydrogen atoms fall from a higher energy level to second energy level. This series lies in visible region. 

In 1913 Bohr amalgamated classical and quantum mechanics in explaining the observation of not only the Balmer series but also the Lyman, Paschen, Brackett, Pfund, etc., series in the hydrogen atom emission spectrum, illustrated in Figure 1.1. Bohr assumed empirically that the electron can move only in specific circular orbits around the nucleus and that the angular momentum pe for an angle of rotation 9 is given by... 

It was the analysis of the line spectrum of hydrogen observed by J. J. Balmer and others that led Neils Bohr to a treatment of the hydrogen atom that is now referred to as the Bohr model. In that model, there are supposedly allowed orbits in which the electron can move around the nucleus without radiating electromagnetic energy. The orbits are those for which the angular momentum, mvr, can have only certain values (they are referred to as quantized). This condition can be represented by the relationship... 

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