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PHYS 221 Recitation
Kevin Ralphs
Week 13
Overview
• Atomic Theory
– Pre-Quantized Atomic Models
– Atomic Spectra
– The Bohr Model
– Wave Mechanics and the Hydrogen Atom
– Multielectron Atoms
– Periodic Table
– Applications
Atomic Theory
• Pre-Quantized Atomic Models
– Plum-Pudding Model
• Experiments first showed that atoms contained removable
negative electric charges, electrons
• However, non-ionized atoms are electrically neutral so there must
be some amount of positive charge that balances the electron’s
charge
• It was originally thought that the positive
charge was evenly distributed throughout
the volume of the atom
• Confirmation of this model was sought
after by Rutherford using alpha particle
scattering which implied that the positive
charge was centrally located
Atomic Theory
• Pre-Quantized Atomic Models (cont.)
– The Planetary Model
• The existence of the atomic nucleus and the discovery of the
proton are actually two separate discoveries, although both by
Rutherford’s scattering experiments
• He showed that he could “knock” hydrogen nuclei out of other
atoms with Z, the atomic number, describing how many hydrogen
nuclei an atom had
• We now know that the nucleus is made up of both protons and
neutrons with the neutron stabilizing the nucleus
• The planetary model assumes that the
electrons in an atom move around it like
planets held in orbit by the Coulomb force
• Although it was wrong, it was an important
stepping stone to us understanding the
atom
Atomic Theory
• Atomic Spectra
– A normal blackbody spectrum has no
sudden dips in it so we would expect
the sun, an approximate blackbody,
to have such a spectrum, but this is
not the case. The gas in the sun and
our atmosphere is absorbing that
light
– Additionally, we noticed that if we electrically excited gases that
they only emitted light at specific frequencies
– The interesting thing we found was that the absorption and
emission lines for a gas line up
– This was a problem for the planetary model because an electron
orbit can have a continuous range of energies so why did
absorption and emission have discrete spectra?
Atomic Theory
• The Bohr Model
– Bohr assumed that the angular momentum of
electron orbits was evenly spaced giving the following
quantization:
𝐿=
ℎ
𝑛
2𝜋
– Using this with the planetary model and assuming
circular orbits, he was able to derive an expression for
the different electron radii using Newtonian
mechanics
2
ℎ
2 (5.29 × 10−11 𝑚)
𝑟𝑛 = 𝑛2
=
𝑛
4𝜋 2 𝑚𝑘𝑒 2
Atomic Theory
• The Bohr Model (cont.)
– From here, the energy of the electron can be calculated
2𝜋 2 𝑘 2 𝑒 4 𝑚 1
13.6 𝑒𝑉
𝐸𝑛 = −
=−
2
2
ℎ
𝑛
𝑛2
– This correctly predicts the ionization energy and the
energy spectrum of hydrogen
– De Broglie reasoned that this was due to electrons needing
to form standing waves in their orbits
– It should be noted that not everything is quantized in
quantum theory. If a photon has more than 13.6 eV in
energy, the electron will be ejected so any photon above
this can be absorbed giving a continuous energy spectrum
Atomic Theory
• Wave Mechanics and the Hydrogen Atom
– Eventually in the 1920s, the nascent quantum
mechanics was applied to the hydrogen atom
– Two versions, wave mechanics proposed by Erwin
Schrodinger and matrix mechanics put forth by
Heisenberg were used to explain things
– The mathematics, of course, are beyond the scope
of this course, but it led to certain numbers that
could be used to classify electrons as they were
bound to atomic nuclei
Atomic Theory
• Wave Mechanics and the Hydrogen Atom (cont.)
– There are four quantum numbers that we can use to completely
classify the state of a bound electron in an atomic nuclei (ie the
electron orbital)
• n, the principle quantum number: n begins at 1, and gives the energy
of an electron when there are no external electric or magnetic fields
present; it also can be viewed as describing the average distance of
the electron from the nucleus and is referred to as an “n shell”
• l is the orbital quantum number: This describes the amount of orbital
angular momentum the electron has and can go from 0 to n-1. Letters
are often used to describe this: l = 0 is an s state, while l = 1 is a p
state.
• m aka 𝑚𝑙 is the orbital magnetic quantum number which goes from m
= -l to m = l can be thought of as describing the quantization of the
direction of the angular momentum
• s aka 𝑚𝑠 is the spin quantum number and describes the direction of
the angular momentum of the spin
Atomic Theory
• The hydrogen wave function can be broken
into two parts: one that gives the angular
distribution and one that gives the radial
distribution. When combined they are the
following:
Atomic Theory
• Multielectron Atoms
– When there are multiple electrons bound to an atom, the
same quantum numbers apply, but only one electron can
have a specific set of quantum numbers
– This is known as the Pauli Exclusion Principle and has to do
with the fact that the electron has an odd half multiple for
its spin
– Electron configurations are denoted by the number of
electrons in each combination of n, l numbers
Electron configuration for C: 1𝑠 2 2𝑠 2 2𝑝2
Atomic Theory
• Periodic Table
– The periodic table was developed before quantum mechanics and
organizes elements by the kinds of chemical reactions they take part in
– The elements in each group (column) behave similarly and are then
sorted in the group by mass
– The gaps in the table had no real explanation until the full quantum
treatment of the atom
– We now know that it is the valence
electrons in an open shell that participate
in chemical reactions
– It is extremely difficult to remove
electrons from a closed shell so they
are normally inert
Quiz Question
How much energy is needed to ionize a
hydrogen atom that is initially in the n = 4 state?
a)
b)
c)
d)
3.4 eV
0.85 eV
6.8 eV
13.6 eV
Quiz Question
What is the total number of electron states in
the n = 3 shell?
a)
b)
c)
d)
e)
4
8
9
16
18
Quiz Question
What are the allowed values of 𝑙 for the n = 5
shell?
a)
b)
c)
d)
0, 1, 2, 3, 4
0, 1, 2, 3, 4, 5
-4, -3, -2, -1, 0, 1, 2, 3, 4
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
Quiz Question
Give the electron configuration for Silicon
(Z = 14) in the ground state
a)
b)
c)
d)
e)
1𝑠 2 2𝑠 2 2𝑝4 3𝑠 2 3𝑝4
1𝑠1 2𝑠1 2𝑝3 3𝑠1 3𝑝3 4𝑠1 3𝑑 4
1𝑠 2 2𝑠 2 2𝑝8 3𝑠 2
1𝑠 2 2𝑠 2 2𝑝6 3𝑠 2 3𝑝1
1𝑠 2 2𝑠 2 2𝑝6 3𝑠 2 3𝑝2