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Chapter Four: Single electron atom (H-atom)

الكلية كلية التربية للعلوم الصرفة     القسم قسم الفيزياء     المرحلة 3
أستاذ المادة فؤاد عطية مجيد       13/11/2013 18:24:07
It is found that hydrogen always gives a set of line spectra in the save position, sodium another set, iron still another and so on the line structure of the spectrum extends into both the ultraviolet and infrared regions. It is impossible to explain such a line spectrum phenomenon without using quantum theory. For many years, unsuccessful attempts were made to correlate the observed frequencies with those of a fundamental and its overtones (denoting other lines here). Finally, in 1885, Balmer found a simple formula which gave the frequencies of a group lines emitted by atomic hydrogen. Since the spectrum of this element is relatively simple, and fairly typical of a number of others, we shall consider it in more detail.
Under the proper conditions of excitation, atomic hydrogen may be made to emit the sequence of lines illustrated in Fig. 4.1. This sequence is called series.

There is evidently a certain order in this spectrum, the lines becoming crowded more and more closely together as the limit of the series is approached. The line of longest wavelength or lowest frequency, in the red, is known as Ha, the next, in the blue-green, as Hb, the third as Hg, and so on.
Balmer found that the wavelength of these lines were given accurately by the simple formula

where l is the wavelength, R? is a constant called the Rydberg constant, and n may have the integral values 3, 4, 5, etc., if l is in meters,

Substituting R and n = 3 into the above formula, one obtains the wavelength of the Ha-line:

For n = 4, one obtains the wavelength of the Hb-line, etc. for n=¥, one obtains the limit of the series, at l = 364.6nm –shortest wavelength in the series.
Other series spectra for hydrogen have since been discovered. These are known, after their discoveries, as Lymann, Paschen, Brackett and Pfund series. The formulas for these are

The Lymann series is in the ultraviolet, and the Paschen, Brackett, and Pfund series are in the infrared. All these formulas can be generalized into one formula which is called the general Balmer series.

All the spectra of atomic hydrogen can be described by this simple formula. As no one can explain this formula, it was ever called Balmer formula puzzle.
4.1 Bohr model for H-atom
¬¬¬Let’s suppose an atom consists of nucleus with charge (Ze) and mass M and electron of charge (e) and mass m moving in circular orbit around the nucleus.
The electron moving around the nucleus under the influence of the Coulomb force keeps the electron in it’s orbit therefore;
…(4.1)
and by using the second postulate of Bohr
(4.2)







And by substituting the value of v from eq. (4.2) into eq. (4.1) we get;
…(4.3)
…(4.4) where rn is nth Bohr radius
…(4.5) vn is the velocity of the electron in the
nth orbit.
For hydrogen atom Z=1 and we take n=1 (ground state) and substitute into eq. (4.4) we will find first Bohr radius (ao)

Therefore eq. 4.4 can be written in terms of first Bohr radius as

to calculate the total energy for the H-atom we use rn and vn from equations (4.4) and (4.5) respectively.
… (4.6)
To calculate the electron energy in the ground state we put n=1 and Z=1 in eq. (4.6) we get;

(-13.6 eV) is called the binding energy of the electron in the first orbit in H-atom.
The other energy levels can be calculated from the relation

It is easy to see that all the energy in atoms should be discrete not continuous. When the electron transits from nth orbit to kth orbit, the

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