Download Bohr Model of the Hydrogen Atom: Stable Electron Orbits and Energy Levels and more Thesis Physics in PDF only on Docsity!
Electronic Physics Lecture 2(Control & Systems engineering /University Of Technology first year ) Dr.Ekhlas Energy Levels and Atomic Structure ( continued ) : 1.2.3Bohr Atomic Model Although the electromagnetic theory agrees with a lot of experimental results, it does not agree with the presence of the atom in the stable state. The reason beyond the failure of the classical physics’ laws in explaining the atomic structure is that it deals with things as either waves or particles without any duality. In order to understand the atomic structure, this dual character of wave and particle must be taking into account and this was done by Bohr when he proposed his atomic
. structure that combines between classical and modern physics : In 1913, Bohr postulated two main assumptions : The first assumption The electron revolves around the nucleus continuously and without radiating energy if its orbit contains full number of the electron De Broglie wavelengths. This assumption is based on the wave- particle duality of the electron because the electron wavelength is calculated in terms of the classical velocity of the electron as the following equation shows
.Where: λ: De Broglie wavelength (m) h: Planck's Constant =6.625× 10 -34^ (J .sec)
. p : The electron momentum (.m kg/sec ) . m: The electron mass . v: The electron velocity Substituting v Gives the following equation Since the orbit of the electron is a circular circumference with a radius ( r (^) n) and equals (2 π r ), therefore the condition of a stable orbit : i . n: The principal quantum number . rn : The orbit radius that contains ( n) wavelengths : Substituting (λ) in the above equation gives
This equation shows that there is a definite energy for each orbit and that the energy of the electron is determined by the radius of the orbit (rn ) or by the principal quantum number ( n). These energies (E (^) Tn ) represent the energy levels of the atom The lowest energy state (E (^) T at n= 1 ) is called the normal, or ground , level, and the other stationary states of the atom are called excited radiating, critical, or resonance levels As (n) increases, the energy gradually approaches to zero and at ( n =∞) the energy becomes zero and the electron is detached from the nucleus. The idea of the separate energy levels of the atom interprets the origin of the atomic spectra which will be explained by the second assumption of Bohr The electron of the Hydrogen atom occupies the lowest energy level ( n= 1 ) which is the closest to the nucleus. From the equation of (E (^) Tn ) the energy of this level is (-13.6 eV), this energy is the required energy to separate the electron from the nucleus of the Hydrogen atom and is called the ionization potential. The second energy level ( 2 = n ) in the Hydrogen atom is (10.2 eV) from the ground state so in order to raise the electron to this level, the required energy must be supplied by an external source : The second assumption It is possible for the electron under the right conditions to transfer from one orbit to another and since these orbits have definite energies; therefore the transition of the electron requires either absorption of energy (if the transition is from a low energy level to a high energy level) or radiation of energy (if the transition is from a high energy level to a low energy level) according to the conservation of energy. As mentioned in the first assumption, the energies of the atomic levels are definite or quantized; therefore the absorption or radiation of energy will be quantized too, hence, the change in the energy will equal one quantum :as the following equation shows (hf )
Ei –Ef = hf ……. Eq ( 1.14) : Where
. Ei : The energy of the level from which the transition initiates . Ef : The energy of the level at which the transition terminates . h: Planck's Constant f : The frequency of the radiation emitted from or absorbed by the atom . (Hz) . If ( Ei –Ef = + ), then there is a radiation of energy If( Ei –Ef = - ) , then there is an absorption of energy According to the second assumption of Bohr, the energy is radiated (emitted) from the atom only when the electron moves from a high level to a lower level, this energy equals the difference between the two levels' energies and it will be released once in the form of photons (discrete energy packets (i.e. not in a gradual form)). The frequency of the emitted : photon equals F= Ei –Ef / h Hydrogen spectral series 1.2.3. A line spectrum contains only certain discreet wavelengths of light. Each element gives a characteristic line spectrum, the lines arising from electron transitions within the atom. With one electron and one proton, hydrogen is the simplest element and gives the simplest line spectrum. The spectral lines of hydrogen correspond to particular jumps of the electron between energy levels. When an electron jumps from a higher energy level to a lower, a photon of a specific wavelength is
:The result is Rydberg formula : Where
. f : Frequency of emitted photon (Hz) c : Speed of light. (c= 2.9979×10^8 m/sec) . ƛ : The wavelength (m) ƛ : The wave number (m -1 , cm -1 )/ 1 R : Rydberg constant. ( R= 1.0974 × 7 m
) nf : The number of the level at which the transition terminates . nf : The number of the level at which the transition terminates ni : The number of the level from which the transition initiates The transitions of the electron in the hydrogen atom result in the : following spectral series The Lyman series: The wavelengths in the ultra violet (UV) band, - 1 resulting from electrons dropping from higher energy levels into the (n = 1) orbit. ( nf =1, ni =2,3,4,….,∞) The Balmer series : The wavelengths in the visible spectrum, - 2 resulting from electrons falling from higher energy levels into the (n = 2) orbit. ( nf =1, ni =3,4,5,….,∞)
The Paschen series : The wavelengths in the infrared spectrum, - 3 resulting from electrons falling from higher energy levels into the (n = 3) orbit. ( nf =1, ni =4,5,6,….,∞) The Brackett series: The wavelengths in the infrared spectrum, - 4 resulting from electrons falling from higher energy levels into the (n = 4) orbit.. ( nf =1, ni =5,6,7,….,∞) The Pfund series : The wavelengths in the infrared spectrum, - 5 - 5 resulting from electrons falling from higher energy levels into the (n = 5) orbit.. ( nf =1, ni =6,7,8,….,∞) :The Wave-Mechanics Model 1.2. Although Bohr theory was consistent well with the atoms containing one electron such as hydrogen as well as ions of hydrogen, it was found that this theory can not be applied to the spectrum of complex atoms consisting of two electrons or more, it was also found that some spectral lines decompose into close multiple lines known as (fine structure). This fine structure can not be explained by Bohr theory which assumed the existence of only one orbit for each quantum number while the fine structure refers to the existence of several levels (n ) with slightly different energy for each quantum number (n) Schrodinger has been able to develop a more comprehensive theory during the period 1925-1926 under the title (The Wave-Mechanics ). Schrodinger has been able to formulate a differential wave equation to describe the behavior of the electron when it falls under the effect of nucleus attraction. When solving this equation for the electron of the hydrogen atom, the quantum number (n) appears automatically as one of the results of solving Schrodinger equation. It has been found that the electrons that have the same principal quantum number (n) clustered around the nucleus in an atomic shell
The Hydrogen atom electron moves from the second energy level (n=2) to the- 2 first energy level (n=1). Calculate the released energy and the wavelength caused by this movement For the Hydrogen atom in the ground state, determine: 1) The kinetic energy- 3 of the electron. 2) The potential energy of the electron. 3) The total energy of the electron. 4) The required energy to release (remove) the electron from the atom. 5) The required wavelength to release (remove) the electron from the atom Calculate the wavelength of an electron which starts its motion from rest- 4 through a potential difference of (500 v), then calculate the wavelength of a proton under the same conditions. Note: mass of proton= 1.67310 -27 Kg A photon with a wavelength (600 nm), find: 1) The photon frequency. 2) - 5 The photon momentum. 3) The photon energy in (e.v) Determine the centripetal force on the Hydrogen atom electron with a - 6 wavelength (3.3 10 -10 m Calculate the frequency and the wavelength of the electron in Balmer series- 7 from the fourth level The electron of the Hydrogen atom moves from a certain level to the first- 8 level with a frequency of (2.9310 15 Hz). Determine the number of the initial level Determine the maximum frequency and the minimum frequency in: 1) - 9 ... Lyman series. 2) Balmer series. 3) Brackett series*
