The Line Spectrum of Hydrogen

In 1913, not long after Planck s and Einstein s discoveries, a theoretical explanation of the emission spectrum of the hydrogen atom was presented by the Danish physicist Niels Bohr. Bohr s treatment is very complex and is no longer considraed to be correct in all its details. We will concentrate only on his important assumptions and final results, which account for the observed spectral lines and which provide an important step toward the understanding of quantum theory. 

Johann Jakob Balmer (1825-1898). Swiss matheinatician. From 1859 until his death in 1898 Balmer taught math at a secondary school for girls in Basel, Switzerland. Although physicists did not understand why his equation woilced until long after his death, the visible series of lines in the spectrum of hydrogen is named for him. 

Johannes Robert Rydberg (1854-1919). Swedish mathematician and physicist. Rydberg analyzed many atomic spectra in an effort to understand the periodic properties of elements. Although he was nominated twice for the Nobel Prize in Physics, he never received it. 

Niels Henrik David Bohr (1885-1962). Danish physicist. One of the founders of modem physics, he received the Nobel Prize in Physics in 1922 for his theory explaining the line spectrum of hydrogen. 

The radius of each circular orbit in Bohr s model depends on rr. Thus, as n increases from 1 to 2 to 3, the orbit radius increases very rapidly. The higher the excited state, the ferther away the electron is from the nucleus (and the less tightly held it is by the nucleus). 

Student Annotation The Rydberg equation is a mathematical relationship that was derived from experimental data. Although it predates quantum theory by decades, it agrees remarkably well with it for one-electron systems such as the hydrogen atom. 

---

This accounts for the fact that the line spectrum of hydrogen shows only lines having certain wavelengths. In order for the electron to move in a stable orbit, the electrostatic attraction between it and the proton must be balanced by the centrifugal force that results from its circular motion. As shown in Figure 1.7, the forces are actually in opposite directions, so we equate only the magnitudes of the... 

In 1913 Niels Bohr proposed his atomic theory with the help of the line spectrum of hydrogen atoms and Planck s quantum theory. His postulates can be summarized as follows ... 

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... 

What is the significance of the line spectrum of hydrogen It indicates that only certain energies are allowed for the electron in the hydrogen atom. In other words, the energy of the electron in the hydrogen atom is quantized. 

How was Bohr s atomic model able to explain the line spectrum of hydrogen ... 

Bohr s theory explained the line spectrum of hydrogen exactly. It did not explain measurements made for atoms other than hydrogen. 

The line spectrum of hydrogen in the visible region. [Reproduced from http //en.wikipedia.org/wiki /Hydrogen spectral series (accessed November 30, 2013).]... 

Example 3-4. Calculate the wavelengths (nm) of the five lines shown in Figure 3.16 for the line spectrum of hydrogen if = 2 for the Balmer series. 

The Paschen series of lines in the line spectrum of hydrogen occur in the near-IR. (a) Calculate the wavelength (in nm) of the series limit for the Paschen series of lines in the line spectrum of hydrogen, (b) The frequency of one line in the Paschen series of hydrogen is 2.34 x 10 Hz. Using the Bohr model of the atom with its circular orbits, sketch this specific electronic transition. 

Use the Rydberg equation to calculate the wavelengths of the first three lines in the Brackett series of the line spectrum of hydrogen. 

This equation, with n" and n given various integral values, accounts for the line spectrum of hydrogen. 

The discovery that waves could have matterlike properties and that matter could have wavelike properties was revolutionary. Although scientists had long believed that energy and matter were distinct entities, the distinction between them, at lea.st at the atomic level, was no longer clear. Bohr s theory was tremendously successful in explaining the line spectrum of hydrogen, but it failed to explain the spectra of atoms with more than one electron. The electron appeared to behave as a particle in some circumstances and as a wave in others. Neither description could completely... 

In spite of its initial snccess in explaining the line spectrum of hydrogen, the Bohr model left many nnanswered qnestions. It did, however, serve as an intermediate model between a classical view of the electron and a fnUy qnantum-mechanical view, and therefore has great historical and conceptnal importance. Nonetheless, it was ultimately replaced by a more complete quantum-mechanical theory that fully incorporated the wave nature of the electron. 

---