# 7.1 Atomic Radius ## Atomic Radius Definition

It is difficult to measure the radius of a single atom as the boundary of an atom is 'soft' and technically orbitals don't actually 'stop' at a certain distance from the nucleus but just have a probability that asymptotically approaches zero as the distance from the nucleus increases.  The typical illustration of an atomic orbital is therefore usually cut off to the 90% or 95% probability interval (a 100% probability interval would have to include the entire universe!!!).

Therefore, the atomic radius of an atom is typically measured as half the bond length between two identical atoms.  This is not the only convention used for atomic radius, but the most common one used for describing the periodic trend and is often referred to as the 'bonding atomic radius.' ## Atomic Radius Trend

Atomic radius increases down a group and right to left across a period. ### Atomic Radius Trend: Atomic Radius Increases Down a Group

Atomic radius increases down a group on the periodic table.  This is fairly intuitive as this increase in radius is explained by an increase in the number of shells of electrons as one moves down a group on the periodic table.

### Atomic Radius Trend: Atomic Radius Increases Right to Left Across a Period

Atomic radius increases right to left across a period on the periodic table.  Initially this seems counter-intuitive as this increase in radius is accompanied by a general decrease in atomic mass.  But if we keep in mind that the atomic mass is concentrated in the nucleus whereas the atomic radius is related more to the size of the 'electron cloud' it can be understood that these trends do not need to align.

This trend in atomic radius is best understood in terms of the effective nuclear charge experienced by the valence electrons.  The effective nuclear charge (abbreviated Zeff) is measure of the strength of the attraction of a valence electron to the nucleus.  It accounts for both the attraction to the protons in the nucleus and the repulsion from the other electrons in the atom.

The effective nuclear charge can be approximated by the following simplified formula:

Zeff = Z - S

where Z is the number of protons and gives the charge of the nucleus and S represents the number of 'shielding' or 'screening' electrons.  For a valence electron the number of shielding electrons is simply equal to the number of core electrons.  This methodology says that valence electrons don't experience repulsion from each other.  This is not technically true but a decent approximation for our purposes as the repulsion from another valence electron should be considerable less than from a core electron.  It turns out that not all core electrons will contribute the same repulsion either as described in Slater's Rules, but the rough approximation being used here will be sufficient to explain the trend in atomic radius.

Sodium and magnesium are a good comparison to demonstrate the trend.  Sodium has 11 protons (Z = 11) and 10 core electrons (S = 10) and its effective nuclear charge is

Zeff = Z - S = 11 - 10 = +1

Magnesium has 12 protons (Z = 12) and 10 core electrons (S = 10) and its effective nuclear charge is

Zeff = Z - S = 12 - 10 = +2

Magnesium's valence electrons experience a higher effective nuclear charge which indicates a greater attraction to the nucleus explaining why it would have a smaller atomic radius (130pm for magnesium vs 154pm from sodium).

Overall, as one proceeds from left to right across a period there is an increase in the number of protons in the nucleus while the number of shielding electrons remains constant leading to an increasing effective nuclear charge.  We've now established that effective nuclear charge increases left to right across a period and see how it explains the corresponding decrease in atomic radius.  The atomic radii and effectively nuclear charges for the 2nd period elements is show below. The ionic radius differs from the atomic radius in that the ionic radius is the radius of a charged species, and the atomic radius is the radius of a neutral atom.  Ionic radii are typically determined using measurements from the crystal lattice structures of ionic compounds.

We should start of by stating that metals typically form cations and nonmetals typically form anions.  To form a cation a metal will lose electrons, and for the main group elements they typically lose all the electrons in their valence shell.  Losing electrons will decrease the size of the electron cloud and result in a decrease in the radius.  As often the entire valence shell is lost this is not a small reduction but can be pretty significant.  Take for example potassium which has an atomic radius of 196pm.  Its corresponding cation, K+, has an ionic radius of 133pm, a reduction of over 30%.  A similar decrease is present for calcium.  Calcium has an atomic radius of 174pm.  Its corresponding cation, Ca2+, has an ionic radius of 99pm, a reduction of over 40%. In contrast to cations, anions are formed by the addition of electrons to a neutral atom.  These additional electrons increase the electronic repulsions between all the electrons in an atom which further spread out to reduce those repulsions.  This increase in size can, once again, be pretty significant.  Take for example sulfur with an atomic radius of 102pm.  Its corresponding anion, S2-, has an ionic radius of 184pm, an increase of over 80%.  A similar increase is present for chlorine.  Chlorine has an atomic radius of 99pm.  Its corresponding anion, Cl-, also has an ionic radius that is also more than 80% larger at 181pm. ## Bond Length

The length of a covalent bond can be approximated by the sum of the two atoms' atomic radii, and the length of an ionic bond as the sum of the two ions' ionic radii.  Ultimately, larger atoms/ions form longer bonds.  As a result the trend for atomic radius can be used to compare bond lengths.  For example students could be asked to rank C-F, C-Cl, C-Br, and C-I bond lengths in increasing or decreasing order.  They all have carbon in common.  As fluorine is the smallest of the other atoms the C-F bond would be the shortest, and as the iodine is the largest of the other atoms the C-I bond would be the longest. The actual bond lengths match fairly well with those predicted using the atomic radii and match up with what would be expected regarding the relative bond lengths:  C-F < C-Cl < C-Br < C-I.

## Isoelectronic Series

'Iso' means 'the same' and an iselectronic series is a series of atoms/ions with the same number of electrons and the same electronic configuration.  S2-, Cl-, Ar, K+, and Ca2+ comprised just such an isoelectronic series. They all have 18 electrons and are isoelectronic with argon. The difference in radius here isn't due to any difference in the electrons, but due to the number of protons in the nucleus.  In this series Ca2+ has the greatest number of protons (20) in its nucleus and the greatest nuclear charge and should therefore be the smallest.  S2- has teh fewest number of protons (16) in its nucleus and the lowest nuclear charge and should therefore be the largest. And if asked to rank this isoelectronic series in order of decreasing size you should answer as follows:

S2-  >  Cl-  >  Ar  >  K+  >  Ca2+

But if you look at the illustration below it actually appears that argon is smaller than Ca2+.  But you should also note that argon is the only neutral atom listed and its atomic radius is being compared to the other ions' ionic radii.  There is a difference in how they're determined that is difficult to properly account for with the noble gases.  And regardless of what the data appear to say, you should still rank them as listed above.   ## Slater's Rules

Slater's rules allow for a better approximation of the screening constant, S, and of the resulting effective nuclear charge.  Ultimately, Slater's rules assign a screening value for as experienced by an electron specific to the electrons that are screening.  The sum of all these screening values yields the screening constant, S, used to calculate the effective nuclear charge:

Zeff = Z - S

It should be first understood that electrons with a higher principal quantum number do not contribute to screening according to Slater's Rules.  After that Slater's rules can be summarized into 3 groups of electrons experiencing the screening.

1. 1s Electrons

The screening constant experienced by a 1s electron from the other 1s electron is 0.30.

2. [ns, np] electrons

The screening value of an electron in this group from any other electrons in this group is 0.35.

The screening value of an electron with principal quantum number (n-1) is 0.85.

The screening value of an electron with principal quantum number ≤(n-2) is 1.

3. [nd] or [nf] electrons

The screening value of an electron in this group from any other electrons in this group is 0.35.

The screening value for all other electrons with principal quantum number ≤n is 1.

SLATER'S RULES
Same
Group
Same n
(lower l)
(n - 1) ≤ (n - 2)
[1s] 0.30 - - -
[ns,np] 0.35 - 0.85 1
[nd] or [nf] 0.35 1 1 1