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13a Early Quantum Theroy and the Atomic Model
Due: 11:00pm on Sunday, April 11, 2010
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The Bohr Atom
Description:
The problem applies quantization of angular momentum to the classical model of an electron orbiting
a nucleus.
Learning Goal:
To understand the Bohr model of the hydrogen atom.
In 1913 Niels Bohr formulated a method of calculating the different energy levels of the hydrogen atom. He did this
by combining both classical and quantum ideas. In this problem, we go through the steps needed to understand the
Bohr model of the atom.
Part A
Consider an electron with charge
and mass
orbiting in a circle around a hydrogen nucleus (a single proton)
with charge
. In the classical model, the electron orbits around the nucleus, being held in orbit by the
electromagnetic interaction between itself and the protons in the nucleus, much like planets orbit around the sun,
being held in orbit by their gravitational interaction. When the electron is in a circular orbit, it must meet the
condition for circular motion: The magnitude of the net force toward the center,
, is equal to
. Given these
two pieces of information, deduce the velocity
of the electron as it orbits around the nucleus.
Express your answer in terms of
,
,
, and
, the permittivity of free space.
Hint A.1
Electrostatic force
Recall that the force
between two charged particles is
,
where
and
are the charges on the particles,
is the separation of the particles, and
is the permittivity of
free space. This is the force that keeps the electron in a circular orbit, so
.
ANSWER:
=
Part B
The key insight that Bohr introduced to his model of the atom was that the angular momentum of the electron
orbiting the nucleus was quantized. He introduced the postulate that the angular momentum could only come in
quantities of
, where
is Planck's constant and
is a nonnegative integer (
). Given this
postulate, what are the allowable values for the velocity
of the electron in the Bohr atom? Recall that, in circular
motion, angular momentum is given by the formula
.
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