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

by Samas 07.11.2015

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by Samas 07.11.2015

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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/(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

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) =

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:

The

=

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