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

1
by Samas

2015-11-13T00:59:19+05:30

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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/(4πε
₀) = 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 e
⁴ ] / (n² h²)

6) Kinetic energy of electron:
=> 1/2 * m * v² = (π m * R e² Z ) / (n h)
= [ 2 π² K² Z² e
⁴ 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

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