BOHR ATOMIC MODEL

 

                                            Bohr Atomic Model

In Bohr model, Niles Bohr atom with a positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus-similar in structure to the solar system, but with attraction provided by electrostatic forces.

He suggested that electrons could only have certain classical motions:

 1. Electrons in atoms orbit the nucleus.

 2. The electrons can only orbit stably, without radiating, in certain orbits (called by Bohr the "stationary orbits"): at a certain discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron's acceleration does not result in radiation and energy loss as required by classical electromagnetic.

 3. Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ν determined by the energy difference of the levels according to the Planck relation:

                                          DE= E₂ -E₁ =hn                where h is Planck’s constant .

The frequency of the radiation emitted at an orbit of period T is as it would be in classical mechanics; it is the reciprocal of the classical orbit period:  n = 1/T

 Bohr Quantization Rule

The significance of the Bohr model is that the laws of classical mechanics apply to the motion of the electron about the nucleus only when restricted by a quantum rule. The angular momentum L is restricted to be an integer multiple of a fixed unit:

                                       L = nℏ        where n = 1, 2,3...

                                     ℏ =h/2p

 n is called the principle quantum number,  . The lowest value of n is 1       

this gives a smallest possible orbital radius of 0 0.592 A⁰  known as the Bohr radius.

   

                                           Hydrogen Atom

      A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. According to Bohr electron revolve about the nucleus in different quantized circular orbits whose angular momentum is given by L n = h where n = 1, 2,3...      .The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.

                                        m_ev²/r  =(1/4pe₀ ) e²/r²

 where m_e is the electron's mass            e is the charge of the electron, (1/4pe₀ )  is Coulomb's constant and v is velocity of electrons in orbit.

 This equation determines the electron's speed at any radius:  v = √(ke²/ m_er)

                                          K=1/4pe₀

    It also determines the electron's total energy at any radius:

                                              E= m_ev²/2 - ke²/r

Putting the value of v one will get  ;    E=- ke²/2r

 The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a  motionless electron infinitely far from the proton. The total energy is half the potential energy,

From the quantization the angular momentum

               L= n ℏ =m_evr =nℏ

 

Substituting the expression for the velocity gives an equation for r in terms of n

                                              √(ke²/ m_er) =ℏn

             So that the allowed orbit radius at any n is

                                                              r_n=n²ℏ²/ ke²m_e

                                                                                                  r₁=0.53*10^-10m (For n =1)

      The smallest possible value of r in the hydrogen atom is called the Bohr radius r₁.

    The energy of the n th level for any atom is determined by the radius and quantum number:

                         E_n = ke²/ r_n =-( ke² )²m_e /2n²ℏ²     =(-13.6/n²)ev

     The combination of natural constants in the energy formula is called the Rydberg energy R which is given by                     R= (ke² )²m_e /2ℏ²

        This expression is clarified by interpreting it in combinations which form more natural units: m_ec²    is the rest mass energy of the electron (511 ) keV .

                              (ke²) /ℏc =α= 1/137    is the fine structure constant .

                             R=1/2(m_ec² ) α² = ( 1.097 ´10⁷ m^-1)

   For nuclei with Z protons, the energy levels are (to a rough approximation):

                       E_n= -Z²R/n²

     

 

                          Atomic Spectra

 The spectrum of atomic hydrogen arises from transitions between its permitted states.

· Each element has a characteristic line spectrum

 · When an atomic gas is excited by passing electric current, it emits radiation. The radiation has a spectrum which contains certain specific wavelength, called Emission line spectrum.

 · When while light is passed through a gas, gas absorb light of certain wavelength present in its emission spectrum. Resulting spectrum is called Absorption line spectrum.

· The number, intensity and exact wavelength of the lines in the spectrum depend on Temperature, Pressure, Presence of Electric field, Magnetic field, and the motion of the source.

Spectral series

When an electric discharge is passed through gaseous hydrogen, the H2 molecules dissociate and the energetically excited H atoms that are produced emit light of discrete frequencies, producing a spectrum of a series of lines

(i) Lyman Series:    1/λ=R(1/1²  -1/n²)   ;   n =  2,3,4,5,…(In U.V. region)

                         Where, R is Rydberg constant = ( 1.097 ´10⁷ m^-1)

 (ii) Balmer Series:     1/λ=R(1/2²  -1/n²)   ;   n =  3,4,5,6,…(In Visible region)

n= 3 for  Ha line  n=4 for  Hb Line,  n =5 for Hg  

(iii) Paschen series: 1/λ=R(1/3²  -1/n²)   ;   n=4, 5, 6 …(Near Infra Red)

(iv) Bracket Series:        1/λ =R(1/4²  -1/n²)   ;   n=5, 6,7,8,….(Infra Red)

(v) Pfund Series:  1/λ=R(1/5²  -1/n²)   ;   n=6,7,8,9….(Far Infra Red)

 

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