Question

Give a mathematical expression to find out the energy of different stationary states associated with Hydrogen like ions.

Answer 1

Bohr's radius for the nth Orbit for Hydrogen like Ions:
    n = principal quantum number , which gives the stationary energy state.

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

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

2) Angular momentum = m v R = n h / 2π          (integral multiple of  h/2π)
         =>  v = n h / (2 π m R)    --- (2)
   
3)  from (1) and (2):
        n² h² / (4π² m² R²) = K Z e² / (m R)
    => R = n² h² / (4π² m K e² Z)      --- (3)

4)  So speed of electron (linear along the circular orbit)  by substituting value of R,
    =>  v = (2 π  K e² Z) / (n h)

5)  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)

6) 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

7)  The total energy of the electron :  (a simple formula)
         KE + PE =  P.E / 2
         Total energy = - 13.6 Z² / n²  eV

For a Hydrogen like Ion:
       Total energy in nth stationary state =  - (13.6 Z² ) * 1/n²    electron Volts

The energy gaps between the stationary states  n and n+1 is:
         = - 13.6 Z² [ 1/(n-1)² - 1/ n² ]

The total energy can be expressed in terms of Rydberg constant also.
         = h c  * R_H * Z²/n²             where  R_H =  1.097 * 10⁷  m⁻¹