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In Class Exercise #5
Shell Structure of Atoms
Charged particles particles are either attracted to, or repelled from, one
another. The potential energy of the attraction (repulsion) is given by
Coulomb’s law:
k × q1 × q 2
d
d = distance between particles
k = proportionality constant
q1 = charge on particle one
Potential Energy = V =
where
q 2 = charge on particle two
An electron has a charge of -1 and a proton has charge +1. The sign of
the proportionality constant, k, is positive.
1. Two interacting particles with the same charge sign (e. g., proton-proton,
eletron-electron) repel one another. Is the potential energy between repelling
particles positive or negative? Explain.
What is the sign of the potential energy between two particles that are attracted to
one another?
What is the sign and magnitude for the potential energy of a neutron interacting
with an electron?
As the distance between two particles increases, what happens to the absolute
value of the potential energy?
What is q for the nucleus of the B atom?
The Ionization Energy (IE) of an atom is the energy needed to remove
an electron from the atom to a position very far away.
2. Consider the H atom. List all of the attractive and repulsive interactions. (Hint
there is only one pair of charged particles).
Now consider the He+ ion. List all the attractive and repulsive interactions (Hint
there are three).
The nuclear model of the atom, describes a very small nucleus in which
the neutrons and protons occupy a very small volume and are bound
together by very strong and short range attractive nuclear forces.
Although protons experience Coulomb repulsion, the attractive nuclear
forces are so strong that the nucleus is tightly bound together. The
electrons occupy most of the atom’s volume. Electrons are confined to
the atom by their Coulombic attraction to the positive charge of the
nucleus.
Which do you think has the strongest (most negative) potential energy of
attraction, the single electron in H or the single electron in He+. Explain.
For which is the IE energy the largest, H or He+? Explain.
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There are a variety of ways of measuring the ionization energy (IE) of a
atom. Commonly energy is added either as light or as a high speed
electron and the energy required to remove an electron is measured. If
an atom has many electrons, one can measure the energy needed to
remove the first electron (to give an ion with +1 charge), the second
electron (to give an ion with +2 charge), and so on. These energies are
called the first IE, second IE, etc. The IE for the H atom has been
determined to be 2.178 x 10-18 J/atom. We could write this as
H(g) + 2.178 x 10-18 J/atom  H+ (g) + e3. What is the energy required to remove the electron from one mole of H atoms?
Is this number more or less than the energy required to break a H-H bond?
As the nuclear charge (Z) of an atom increases do you think that the IE will
increase or decrease? Why?
Some first ionization energies for gas phase elements are provided in the
table below. Note that all of the energies are per mole of atom and use
the units of MJ/mole.
Element
H
He
Z
1
2
IE(MJ/mole)
1.31
2.37
Element
Na
Mg
Z
11
12
IE(MJ/mole)
0.50
0.74
3
Li
Be
B
C
N
O
F
Ne
3
4
5
6
7
8
9
10
0.52
0.90
0.80
1.09
1.40
1.31
1.68
2.08
Al
Si
P
S
Cl
Ar
K
Ca
13
14
15
16
17
18
19
20
0.58
0.79
1.01
1.00
1.25
1.52
0.42
0.59
4. Consider the elements H, Li, Na, and K which are all members of the first group
or column of the periodic table. Do the first ionization energies change with Z in
the way that you predicted in part 3? If not, how is it different?
Now consider the general trend for elements across the second period, or row, of
the periodic table: Li, Be, B, C, N, O, F, Ne. Do the first ionization energies
change with Z in the way that you predicted in part 3? If not, how is it different?
The Shell Model of the Atom
The atoms H and He follow the expected trend: as Z increases the first
IE increases. Overall this reflects the greater attraction of the electrons to
the nuclei with higher charge. This is mitigated somewhat by e-e
repulsion; thus the first ionization energy of He=2.37 is less than twice
the value for H, 2 x 1.31=2.62. These data are consistent with the
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electrons occupying a spherical shell around the nucleus for both H and
He, with similar average electron-nuclear distances.
5. How do the IE data cause us to propose that H and He have similar average
electron-nuclear distances? (Hint: what is the ratio of Z for H and He? What is the
ration of the IE’s?).
However, a problem with this model occurs when we reach Li (Z=3).
Not only is the IE less than that for He, it is also less than that for H.
Let’s consider two hypothetical models for Li as shown below.
Model 1
Model 2
Li
Z=3
Li
Z=3
However, the principles of quantum mechanics and other experimental
data demonstrate that only Model 1 can be correct! Some of the types
of data (such as emission lines from excited atoms) are discussed in the
textbook. One key finding of the quantum mechanical shell model is
that the first, innermost shell can hold at most two electrons.
The outermost electron for the correct Li shell model (Model 1) is
attracted to the nucleus but repelled by the electrons in the inner shell.
To a crude approximation the two electrons of the inner shell lie very
close to the nucleus. Thus the average distance between the outermost
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electron and the electrons of the inner shell are nearly the same as the
average distance between the outermost electron and the nucleus.
6. If the two electrons of the innermost shell of Li were located in, or very close,
to the nucleus, what would be the net charge of the nucleus + 2 inner shell
electrons? What charge would the electron in the outermost shell feel?
One can describe the inner shell electrons as shielding or screening the
outermost electron from the full +3 of the Li nucleus. We could say that
the effective nuclear charge felt by the outer electron of Li is closer to
ca. +1 than +3. In general the effective nuclear charge is the charge felt
by the electrons of the outermost shell. Numerically, a convenient
quantity is the core charge, qcore= # of protons - # of inner shell
electrons. The core charge is generally greater than the true effective
nuclear charge but is easily calculated and lends insight to important
atomic properties.
The electron shells commonly are represented by the principal quantum
number (n), with the shell closest to the nucleus having n=1. The shell
Be
Z=4
core
q ≈2
C
Z=6
core
q ≈4
Ne
Z=10
qcore≈8
structures for Be, C, and Ne are shown below.
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7. Based on the changes in qcore as one proceeds from left to right across the
second period (row), should the IE increase or decrease? What does the data table
show?
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