✤ Thomson's model: (1898 J.J. Thomson) => According to this model, atom is a positively charged sphere in which -ve charge electrons are

                                       embedded so atom is electrical neutral.

 ❈ Failure:      1.     It could not explain the origin sphere.

                               (Limitation) of several spectral series in hydrogen atom.

                       2. It failed to explain the large angle scattering of α particle.

    Thomson model work b/w (1898-1911)

✤ Rutherford's model: On the basis of α Particle scattering experiment  (geiger-marsden)

           1. An atom consists a small and massive central core in which entire positive charge and mass of atom are concentrated which is called nucleus.

           2. Size of nucleus is very small as compare to atom (10-10m)

           3. Electrons revolve around the nucleus in circular orbits.

                    Limitations: 1. It could not explain the stability of atom.

                                         2. According to electromagnetic theory revolving e- should radiate energy continuously, so eventually it jump in the nucleus,                                              which is not possible, Rutherford could not explain it.

✤ BOHR'S atomic model.: (Postulates)

         1. Electrons revolve around the nucleus in circular orbits. in these orbits centripetal force required for their rotation. is provided by electrostatic              attraction b/w electron and nucleus.

       i.e mv2/r = kze2/r2

         2. Electron revolve in those orbit in which angular momentum become equal to the integral multiple of nh / 2π 

     So mvr = nh/2π

         3. In a particular orbit electron does not radiate energy but when electron jumps from one orbit to another then it radiate energy.

                               i.e hv = E₂-E₁ E₁ and E₂ are energies in two orbits.

                       Total energy of electron in its orbit: It is equal to the Sum of K.E and P.E.

      Here K.E = Kze²/2r

      and P.E = -Kze²/r

     Total energy = K.E+P.E

            = -Kze²/2r

En = (2π²mk²e⁴/h2)*z²/n²

     En = [ -13.6/n² ] ev for hydrogen atom

  { En -- energy of electron in nth orbit }

  Note :      1. P.E = -2K.E

                      P.E = 2T.E

                 2. T.E = -K.E

                 3. r ∝ n²

                 4. V ∝ 1/n

                 5. E ∝ 1/n²

                 6. E ∝ z²/n²

  Note : 1. Energy of atom is least (largest negative values) when n=1(ground state) (-13.6ev)

             2. Energy of atom increase of n↑. These energies are called energy of excited state.

Q)      The energy level diagram of an element is given here. Which transition corresponds to the emission of spectral line of wavelength 102.7 nm.

Sol.   Here λ = 102.7nm.

                        = 102.7 x 10⁻⁹ m

             E = hc/λ = 6.6 x 10⁻³⁴ x 3 x 10⁸/102.7 x 10⁻⁹ J

       = 1.935 x 10-18

         / 1.6 x 10⁻¹⁹ ev.

       = 12.1 ev

  corresponding to transition D

     E = -1.5 – (-13.6) we can use shortcut as

         = 12.1 eV hc/λ = 12.375 x 10⁻⁷ ev

          :. E = 12.375 x 10⁻⁷/λ ev.

❊ Hydrogen spectrum: When electricity is passing through hydrogen (monoatomic) gas at low pressure, spectrum of different colour and wave length are                                         emitted. These spectrum are called hydrogen spectrum.

                                   Wave No. of radiation emitted (no. of complete wave in unit length)

                            

                                                    1/ λ = R (1/n12 – 1/n22)

                                                                   Here R = Rydberg constant.

                                                           = 1.097x10⁷ m⁻¹

                                        n₁, n₂ are orbital Numbers.

❈ Distance of closest approach ⇒ It is the minimum distance of α-particle from nucleus at which it get retrace its path by 180° i.e. K.E of α is converted                                     into electrostatic potential energy.

                                                

                                                                Ro = 2kZe² / ½ mv²

                                                                    Here k = 9x10⁹

                                                                 Z = atomic No.

✤ Impact parameter ⇒ It is destined as the ⊥ distance of velocity vector & 2He4(α) from the centre of the nucleus.

    b = kZe²Cot(θ/2) / ½ mv²

    Here k = 1/4πε₀ = 9x10⁹

         Z = atomic No.

 Sol.             Ex  E₃ = -13.6 = -1.51 eV

     E₄ = -0.85 eV

   i) K.E of electron in 2nd excited state = -E₃ = 1.51 eV.

   ii) P.E of electron in 2nd = 2E₃ = -3.02 eV

   iii) E₄-E₁ = -0.85 - (-13.6) = 12.25 eV

       = hc/λ = 12.95 ⇒ λ = 970 Å

SOL.              E = 12400/ λ = 12.7 ev

                 Energy of electron in excited state

                     En = E1+127

                      =-13-6 +12.7 = -0.9 ev

     and E₁=-13.6

      :. n² = -13.6 / En

        = -13.6 / -0.9 = 15.11

       n=4

     No. of spectral lines emitted = n(n-1)/2

          = 4x3 / 2

          = 6

     Longest wavelength will correspond to the transition

     n= 4 to n= 3 level.

Sol.      For shortest wave length of Balmer series,

                             n₁=2, n₂ = ∞

                         1/ λ₁= R(1/22 – 1/infinity)

      = R/4

     1/R =912 λ1=4/R = 4/1.1 X 107

       λ₁ = 3.637 X 10-7 m

Sol.      The line of Balmer series of H-atom is obtained when electron transition occurs from n₂ = 3 to n₁ = 2

                                  :. λ= R(1/22-1/23) = 5R/3R λ= 36/5R

Sol.          (i) we know freq. α 1/n₁² - 1/n₂²

                         :. ν₁ α 1/2² - 1/2²

                          ν₁ α 3/4 -①

  (ii) ν₂ α 1/2² - 1/∞² => ν₂ α 1/4

     :. ν₁/ν₂ = 3/4 x 4/1 = 3/1

Sol.      For shortest wavelength of Paschen series n₁ =3, n₂=∞

                        1/λ = R (1/3² - 1/∞)

     λ = R/9

     λ = 9/R = 1.097x10⁷ = 8204.1 A°

Sol.              Total energy= -13.6 ev

                         ∵K.E. = -E

                                   = 13.6 ev

                         ∵P.E = -2K.E

                                   = -27.2 ev.

Sol.                         Here r₁ = 5.3 x 10^-11 m.

  we know rₙ = n²r₁

      :. r₂ = (2)²r₁

       = 4 x 5.3 x 10-11 m

           = 2.12 x 10-10 m

      r₃ = (3)² x 5.3 x 10-11

      r₃ = 4.77 x 10^-10 m

Note : for any spectral series of H-atom:

    1) n₂ = n₁+1 is called Hα line (minimum freq)

    2) n₂ = n₁+2 is called Hβ line

    3) n₂ = n₁+3 is called Hr line

    4) n₂ = ∞ is called series limit (maximum freq.)

✤ REVISION OF FORMULAS