gooEffective Nuclear Charge
Effective nuclear charge, the charge an electron experiences after accounting for the shielding due to other
electrons, increases from left to right across a given period, thus an electron in a 2p orbital of a nitrogen atom
experiences a greater Zeff (3.83) than an electron in a 2p orbital of a carbon atom (3.14). This trend makes sense
if we consider what happens to atomic structure as we move from left to right across a period. Every time we
move one group farther to the right, we add one more proton to the nucleus (i.e., the actual nuclear charge, Z,
increases by one) and we add one more electron to the valence shell. This additional e lectron will not be
perfectly shielded from the nucleus by the other electrons, nor will it do a perfect job of shielding the other
electrons from the nucleus. This means that S, the shielding constant, will increase by less than one and, as a
result, Zeff increases.
Effective nuclear charge is most useful when we are comparing electrons in the same shell and subshell for
two different atoms (for example we might compare Zeff for a 2p electron in nitrogen with Zeff for a 2p electron
in carbon). It is less useful to compare say, Zeff for a 2s electron of lithium with Zeff for a 3s electron of sodium,
and since we are rarely concerned with the core electrons of an atom, we will not generally compare Z eff for
atoms in different periods. We will use the concept of Zeff to rationalize the trends for some other physical
properties, so make sure you are comfortable with this explanation for the periodic trend for Zeff before
proceeding
Shielding and Effective Nuclear Charge
An atom of 1 H consists of a single proton surrounded by an electron that resides in a spherical 1 s orbital.
Recall that orbitals represent probability distributions meaning that there is a high probability of finding this
electron somewhere within this spherical region. This electron, being a negatively charged particle, is attracted
to the positively charged proton.
Now consider an atom of helium containing two protons (there are neutrons in the nucleus, too, but they are
not pertinent to this topic) surrounded by two electrons both occupying the 1s orbital. In this case, and for all
other many electron atoms, we need to consider not just the proton - electron attractions, but also the electron electron repulsions. Because of this repulsion, each electron experiences a nuclear c harge that is somewhat less
than the actual nuclear charge. Essentially, one electron shields, or screens the other electron from the nucleus.
The positive charge that an electron actually experiences is called the effective nuclear charge, Ze ff, and Zeff is
always somewhat less than the actual nuclear charge.
Zeff = Z - S, where S is the shielding constant
The two figures represent different instantaneous positions for the two electrons in an atom of helium. The
electrons both occupy a 1s orbital meaning that the time-averaged positions of the electrons can be represented
by a sphere. At any moment, however, we could envision the two elec trons as being on opposite sides of the
nucleus in which case they poorly shield each other from the positive charge. At a different moment, one
electron may be between the nucleus and the other electron, in which case the electron farther from the nucleus
is rather effectively shielded from the positive charge.
Considering helium, you might initially think that Zeff would be one for each electron (i.e., each electron is
attracted by two protons and shielded by one electron, 2 - 1 = 1), but Zeff is actually 1.69. To determine Zeff, we
do not subtract the number of shielding electrons from the actual nuclear charge (that would always result in a
value of Zeff of one for any electron in any atom!), but rather we subtract the average amount of electron density
that is between the electron we are concerned with and the nucleus. There are rules we can use to estimate the
value of S and thus determine Zeff, but we do not need to go into such details for this course.
Penetration
An electron in an s orbital has a finite, albeit very small, probability of being located quite close to the
nucleus. An electron in a p or d orbital on the other hand has a node (i.e., a region where there cannot be any
electron density) at the nucleus. Comparing orbitals within the same shell, we say that the s orbital is more
penetrating than the p or d orbitals, meaning that an electron in an s orbital has a greater chance of being
located close to the nucleus than an electron in a p or d orbital. For this reason, electrons in an s orbital have a
greater shielding power than electrons in a p or d orbital of that same shell. Also, because they are highly
penetrating, electrons in s orbitals are less effectively shielded by electrons in other orbitals. For example,
consider an atom of carbon whose electron configuration is 1s2 2s2 2p2 . The two electrons in the 1s orbital of C
will do a better job of shielding the two electrons in the 2p orbitals than they will of shielding the two electrons
in the 2s orbital. This means that for electrons in a particular shell, Zeff will be greater for s electrons than for p
electrons. Similarly, Zeff is greater for p electrons than for d electrons. As a result, within a given shell of an
atom, the s subshell is lower in energy than the p subshell which is in turn lower in energy than the d subshell.
A second trend that can be observed is a fall in ionisation energy on descending a given group. This is because
the valence electrons are in successively higher shells that are larger and less tightly held by the nucleus.
Another trend can be observed within single horizontal rows of the periodic table. The negatively charged
electron in the shells below the valence shell somewhat shield the valence electrons from the positive charge of
nucleus. This shielding isn't perfect, however, and the outer electrons feel what is called an effective nuclear
charge.
The idea of effective nuclear charge was put forward by JC Slater, and is the resultant charge felt by an electron
in a given orbital after shielding by inner electrons. The effective nuclear charge, Z* is can be calculated by the
following equation:
Z = Z* - σ
Z is the element's atomic number (the number of positively charged protons in its nucleus) and σ is a screening
factor dependent on the other electrons present in the atom. If the electron we are calculating Z * for is in an s or
p orbital:
Electrons in higher principle shells contribute 0 to the value of σ
Each electron in the same principle shell contribute 0.35 to σ
Each electron in the next principle shell down contribute 0.85 to σ
Each electron in shells deeper than this all contribute 1.0 to σ
If the electron we are calculating Z* is in a d or f orbital:
Electrons in higher principle shell contribute 0 to the value of σ
Each electron in the same principle shell contribute 0.35 to σ
All inner electrons contribute 1.0 to σ
As the atomic number increases from left to right across the row, or 'period', the effective nuclear charge felt by
the valence electrons also increases and so they are more tightly held, resulting in an increase in the first
ionisation energy (figure 1). This is not entirely linear, with a few peaks and troughs. As can also be seen in
figure 1, the ionisation energy increases from Li to Be but then falls as B, rises again until N, drops at O then
increases to the end of the row at Ne. This is because different types of orbital contain the highest energy
electrons for different elements. For Li and Be, this is the 2s orbital, while for the other elements it is the 2p.
s orbitals are said to be more penetrating than p orbitals, with most of the amplitude of its wave function
actually inside the nucleus. p, orbitals which are dumbbell shaped, have most of the amplitude, and therefore
electron density, to either side of the nucleus. Electrons in s orbitals, therefore, feel a greater effective nuclear
charge than p electrons. So, despite an increase in the overall nuclear charge, the p electron in a boron atom is
more easily removed than the s electron of a beryllium atom.
Nitrogen has three p electrons, one in each of the 2p orbitals. Oxygen has an extra electron in one of these
singly occupied orbitals. This results in an electrostatic electron-electron repulsion that makes ionisation much
more easy than might have been expected. The ionisation energy then increases to a maximum for the row when
we reach neon.
Atomic and Ionic Radii
There are three measurements that give an indication of the sizes of atoms:
1. Single-bond covalent radius (rcov )
2. Van der Waal's radius (rvdw)
3. Ionic radius (rion )
Values of the single-bond covalent radii are estimated from the known length of single bond in compounds
containing the element. Hence the covalent radius of carbon, for example, can be estimated as being half the
length of the C-C single bond in diamond.
The van der Waal's radius is the closest possible distance that the nucleus of a neutral atom can get to the edge
of another atom without the two bonding, and so will be the limit on the distance that two atoms in a solid can
come together.
The ionic radius is the radius of the charged atom in the lattice of an ionic solid. It is assumed that the distance
between the nuclei of a neighbouring anion and cation will be the sum of their ionic radii. This distance is found
by a technique called 'X-ray crystallography', but a starting assumption must be made as to where the
boundaries of the ions are - where one ion stops and the other begins. After one assumption is chosen and
applied, the ionic radii determined using this assumption will be self-consistent. For example, with an
estimation of the ionic radii in an ionic compound, such as sodium fluoride, the r adii of other ions can be found
by measuring the internuclear distances in the sodium and fluoride salts of other elements.
Table 2: Selected covalent, van der Waal's and ionic radii (numbers in brackets refer to the charge on the ion).
Ele ment
Single bond covalent Van de r Waal's Ionic radius,
radius, rcov (pm 1 ) radius, rvdw (pm) rion (pm)
Li
140
180
90 (1+)
Be
B
120
83
-
59 (2+)
-
C
77
170
-
N
73
155
-
O
70
140
126 (2-)
F
54
135
119 (1-)
Cl
97
180
167 (1-)
Br
114
190
187 (1-)
I
133
200
206 (1-)
Periodic trends in radii can be observed. On descending through a group of the periodic table, the radii increase
due to the use of successively higher principle shell to accommodate valence electrons (note the radii of the
halogens, F, Cl, Br and I, quoted in table 2). If we go across a row where the different elements house their
valence electrons in the same principle shell, a decrease in covalent and van der Waal's radii with increasing
atomic number can be seen. For example, the covalent radii of Li to F in table 2.
This is due to the increasing effective nuclear charge holding the valence electrons closer to the nucleus. The
ionic radii of cations are always observed to be smaller than the covalent radius for the parent atom. This is
because removal of an electron causes a reduction in the repulsions between the remaining electrons, and so
they are held closer by the positive charge of the nucleus.
In the cases where cation formation results in the loss of all valence electrons, as is the case for the alkali
metals, only the radius due to the inner closed shell is measured, which is smaller than that of the valence shell.
For anions, the ionic radius is larger than the covalent radius. This is because there is a greater repulsion when
extra electrons are added, giving a larger size.
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