(i) The energy associated with the first orbit in the hydrogen atom is -2.18 * 10^{-18}J atom^{-1} . What is the energy associated with the fifth orbit?
(ii) Calculate the radius of Bohr’s fifth orbit for hydrogen atom.

NCERT Class XII
Chemistry - Main Course Book I

Chapter 2. Structure of Atom

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2015-05-30T18:27:14+05:30

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N = 1        Z = 1  for Hydrogen atom...

energy = - Z² 13.6 / n²  eV 
         = - 13.6 * 1² * 1.602 * 10⁻¹⁹ / n²    Joules
         =  - 2.178 * 10⁻¹⁸ / n²    Joules         

the energy is inversely proportional to the square of n or orbit or shell number..

for the 5 th orbit:
energy =   - 2.178 * 10⁻¹⁸ / 25  Joules   =  - 8.712 * 10⁻²⁰  Joules
           =  0.544  eV
===============================

Bohr's radius for the 5th Orbit:

   m v² / R  =  K Ze * e / R²        =>  v² = K Z e² / (m R)      --- (1)

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

              => R = n² h² / (4π² m K e² Z)      --- (3)

  R = Bohr' radius for an atom of atomic number Z,
   n = orbit number = principal quantum number  =   5
   h = Planck's constant = 6.626 * 10⁻³⁴ units
   K = 1/(4πε₀) = 9 * 10⁹ units  = Coulomb's constant
   Z = 1  for Hydrogen
   m = mass of an electron = 9.1 * 10⁻³¹ kg
   e = charge on the electron = 1.602 * 10⁻¹⁹ C

Radius for the first orbit n = 1 is:

 R = 0.00529 * 10⁻⁸  meters  = 5.29 * 10⁻¹¹ meters
         = 0.529 Angstroms:  °A

Radius for the n = 5  orbit is:

   R = 5² * 5.29 * 10⁻¹¹ m =  13.227 °A

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