Question

Using Bohr atomic model, drive expression for calculating the radius of orbits (5)
in He+
. Using this expression, calculate the radius of fourth orbit of He+
ion.

Answer 1

Bohr's radius for the nth Orbit:  (n = principal quantum number)

Let
   R = Bohr' radius for an atom of atomic number Z, 
   n = orbit number = principal quantum number  =   4 
   h = Planck's constant = 6.626 * 10⁻³⁴ units
   K = 1/(4πε₀) = 9 * 10⁹ N-m²/C²  = Coulomb's constant
   Z = 2  for Helium
   m = mass of an electron = 9.1 * 10⁻³¹ kg
   e = charge on the electron = 1.602 * 10⁻¹⁹ C

a) centripetal force = electrostatic attraction between an electron and protons.
         m v² / R  =  K (Z*e) * e / R²
      => v² = K Z e² / (m R)      --- (1)

b) Angular momentum = m v R = n h / 2π          (integral multiple of  h/2π)
         =>  v = n h / (2 π m R)    --- (2)
    
c)  from (1) and (2):
        n² h² / (4π² m² R²) = K Z e² / (m R)

 => R = n² h² / (4π² m K e² Z)      --- (3)
         = n² * 6.626² * 10⁻⁶⁸ / [ 4π² * 9.1 * 10⁻³¹ * 9* 10⁹ * 1.6² * 10⁻³⁸ * Z ]
         = 0.529 * n² / Z    °A
         = 0.529 * 4² / 2     °A
         = 4.232 °A

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We can also calculate other physical quantities as follows:

d)  Linear Speed of electron (in the circular orbit)  by substituting for R
    =>  v = (2 π  K e² Z) / (n h)

e)  Potential energy of the electron:
         We ignore gravitational potential energy here.
       PE = - K * Z * e * e / R  = - K Z e² / R  --- (4)
               = - [4 π² m K² Z² e⁴ ] / (n² h²)

f) Kinetic energy of electron:
         => 1/2 * m * v² = (π m * R e² Z ) / (n h)
               = [ 2 π² K² Z² e⁴ m ] / (n² h²)  =  - P.E / 2