# 2. The Bohr Model

advertisement ```The Bohr Model
(of the Hydrogen atom)
Planetary Model
• Force Applied to the electron:
• One deduces: v = e / (√4πε0mr)
• Total energy with K = (1/2)mv2:
Failure of the classical
(planetary) atomic model
• Atom (neutral) = nucleus (+q) + q electrons
• Assuming the Hydrogen atom:
– The electron is attracted by the
nucleus
– Even in circular motion around the
nucleus, the electron loses energy:
• Radial acceleration: ar = v2/R
• Classical e.m. theory: an accelerating
charge continuously radiates energy, r
decreases…
The electron would eventually crash into the nucleus
Failure of the planetary model
• Doomed because the electron radiates energy,
while orbiting around the nucleus
• But:
– There is some “truth” to it, since Rutherford was
successful in describing the scattering experiment
• In 1913, Niels Bohr 1885-1962 postulates that
the electrons may be in stable (non-radiating)
circular orbits, called stationary orbits
Niels Bohr’s postulates
• Existence of “stationary” states for the orbiting
electrons
• Transitions between stationary states:
E = E1 – E2 = hν
• Classical laws of physics do not apply to transitions
between stationary states
• The electron can only exist in orbits for which its
angular momentum is given by:
L= | r x p | = mvr = nh/2π with n=1,2,3…
Bohr Radius
• From Bohr’s postulate: v = nħ/mr
• Since: v = e / (√4πε0mr) [Planetary Model]
• One can deduce the diameter of the hydrogen atom for
stationary states :
with a0, the Bohr radius
Calculation
With: h = 6.6 x 10-34 J.s
ε0 = 8.85 x 10-12 C2/N.m2
m = 9.1 x 10-31 kg
e = 1.6 x 10-19 C
• Diameter of the Hydrogen Atom:
n=1: ground state (lowest energy) orbit of the hydrogen atom
Energy Levels of the Hydrogen atom
• Energy of the stationary states:
with E0 = -13.6 eV
Emission of light: hν = Enu – Enl
Using: 1/λ = ν/c,
1
λ
R∞, the Rydberg constant for Hydrogen
Spectral Lines of Hydrogen
n&gt;l,
l=1 &amp; n=2,3,4,… Lyman Series
l=2 &amp; n=3,4,5,… Balmer Series
l=3 &amp; n=4,5,6,… Pashen Series
By 1913, some of these lines were not observed.
Still they were predicted by Bohr’s model
Emission &amp; Absorption
• When shining a white light (all visible frequencies),
one observes a spectrum containing dark lines 
Absorption
• These lines are the same than those observed in
the Emission spectrum.
• Hydrogen in its ground state:
– Transition induced by incoming photon hν
 Absorption
– Then, the atom goes back to its ground state by emitting
photon(s)
 Emission
Hydrogen Absorption/Emission
The Sun
Quantum to Classical Mechanics ?
• The Correspondance Principle (Bohr):
In the limits where classical and quantum
theories should agree, the quantum theory
must reduce to the classical result.
Limitations
•
The Bohr model was a great step of the new
quantum theory, but it had its limitations:
Works only to single-electron atoms.
Could not account for the intensities or the fine structure of
the spectral lines.
Could not explain the binding of atoms into molecules.
Exercise
The electron in a hydrogen atom in the n=2 state absorbs a photon of
wavelength λ=327 nm. Does the hydrogen atom become ionized ?
Energy of the n=2 state: En = -13.6/n2 eV  E2 = -13.6/(2)2 = -3.4 eV
The H atom will ionize, if the photon brings an energy higher than 3.4 eV.
Ephoton
= hν = hc/λ = (6.6 x 10-34) x (3.0 x 108) / (327 x 10-9) = 6.1 x 10-19 J
with 1 eV = 1.6 x 10-19 J,
Ephoton = 6.1 x 10-19 / 1.6 x 10-19 = 3.8 eV
The Hydrogen atom will become ionized.
Exercise
Calculate the 3 first transitions of:
- The Lyman series (l=1):
n=2
n=3
n=4
λ=
λ=
λ=
n=3
n=4
n=5
λ=
λ=
λ=
- The Pashen series (l=3):
n=4
n=5
n=6
λ=
λ=
λ=
- The Balmer series (l=2):
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