Using Bohr atomic model, drive expression for calculating the radius of orbits (5)in He+. Using this expression, calculate the radius of fourth orbit of He+ion.
Answers
Bohr's radius for the nth Orbit: (n =
principal quantum number)
Let
R = Bohr' radius for an atom of atomic number Z,
n = orbit number = principal quantum number = 4
h = Planck's constant = 6.626 * 10⁻³⁴ units
K = 1/(4πε₀) = 9 * 10⁹ N-m²/C² =
Coulomb's constant
Z = 2 for Helium
m = mass of an electron = 9.1 * 10⁻³¹ kg
e = charge on the electron = 1.602 * 10⁻¹⁹ C
a) centripetal force = electrostatic attraction between an electron and
protons.
m v² / R = K (Z*e) * e / R²
=> v² = K Z e² / (m R) --- (1)
b) Angular momentum = m v R = n h / 2π
(integral multiple of h/2π)
=> v = n h / (2 π m R)
--- (2)
c) from (1) and (2):
n² h² / (4π² m² R²) = K Z e² / (m R)
=> R =
n² h² / (4π² m K e² Z) --- (3)
= n² * 6.626² *
10⁻⁶⁸ / [ 4π² * 9.1 * 10⁻³¹ * 9* 10⁹ * 1.6² * 10⁻³⁸ * Z ]
= 0.529 * n² / Z °A
= 0.529 * 4² / 2 °A
= 4.232 °A
==========
We can also calculate
other physical quantities as follows:
d) Linear Speed of electron (in the circular orbit) by substituting
for R
=> v = (2 π K e² Z) / (n h)
e) 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)
f) 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