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 (383) than an electron in a 2p orbital of a carbon atom (314) 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 (ie 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 Zeff 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 1H consists of a single proton surrounded by an electron that resides in a spherical 1s 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 charge 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 Zeff 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 (ie each electron is attracted by two protons and shielded by one electron 2 - 1 = 1) but Zeff is actually 169 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 (ie 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 1s22s22p2 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 isnt 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 elements 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 035 to σ

Each electron in the next principle shell down contribute 085 to σ Each electron in shells deeper than this all contribute 10 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 035 to σ All inner electrons contribute 10 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 Waals 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 Waals 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 radii 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 Waals and ionic radii (numbers in brackets refer to the charge on the ion)

Element Single bond covalent

radius rcov (pm 1) Van der Waals

radius rvdw (pm) Ionic radius

rion (pm)

Li 140 180 90 (1+)

Be 120 - 59 (2+)

B 83 - -

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 Waals 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

httpwwwchmdavidsoneduronuttche115ZeffZeffhtm

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 elements 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 035 to σ

Each electron in the next principle shell down contribute 085 to σ Each electron in shells deeper than this all contribute 10 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 035 to σ All inner electrons contribute 10 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 Waals 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 Waals 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 radii 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 Waals and ionic radii (numbers in brackets refer to the charge on the ion)

Element Single bond covalent

radius rcov (pm 1) Van der Waals

radius rvdw (pm) Ionic radius

rion (pm)

Li 140 180 90 (1+)

Be 120 - 59 (2+)

B 83 - -

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 Waals 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

httpwwwchmdavidsoneduronuttche115ZeffZeffhtm

The van der Waals 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 radii 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 Waals and ionic radii (numbers in brackets refer to the charge on the ion)

Element Single bond covalent

radius rcov (pm 1) Van der Waals

radius rvdw (pm) Ionic radius

rion (pm)

Li 140 180 90 (1+)

Be 120 - 59 (2+)

B 83 - -

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 Waals 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

httpwwwchmdavidsoneduronuttche115ZeffZeffhtm