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

1
by Samas

Log in to add a comment

by Samas

Log in to add a comment

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

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

=

The

=