Physical Chemistry Third Edition

19.6 Atoms with More Than Two Electrons 817

row of the periodic table. All of these elements have ground-level configurations with

only filled subshells (beryllium, magnesium, zinc, and mercury) or with a half-filled

valence subshell (nitrogen and phosphorus). As with the ground configurations, this

corresponds to the idea that a filled subshell or a half-filled subshell is relatively stable.

The electron affinity is defined as the energy given off in forming a negative ion from

a gaseous atom and a free electron. It is also equal to the energy required to remove an

electron from a gaseous, singly charged negative ion. It is positive if a gaseous atom

spontaneously attracts an electron, and most elements have positive values. A graph of

the electron affinity has the same general trend as the graph of the ionization potential,

except that the graph is shifted one column to the left, corresponding to the fact that a

singly charged negative ion has one more electron than the neutral atom. The elements

of column VIIA (the halogens) have the greatest electron affinity, whereas the inert

gases have the largest ionization potential. The inert gases have the lowest electron

affinity, whereas the metals in column IA (the alkali metals) have the lowest ionization

potentials. Accurate values of the electron affinities of beryllium, magnesium, and the

inert gases are apparently not available, but these are probably fairly small and negative.

Just as the elements in column VA (nitrogen, phosphorus, etc.) have larger ionization

potentials than their immediate neighbors, the elements in column IVA (carbon, silicon,

etc.) have larger electron affinities than their neighbors. This corresponds to half-filled

subshells in the negative ions, just as the ionization potentials correspond to half-filled

subshells in the neutral atoms.

Atomic size is also correlated with position in the periodic table, with values

generally decreasing from left to right and increasing from top to bottom. Other chem-

ical and physical properties, including melting temperature and electronegativity, can

be correlated with electron configuration and thus with position in the periodic table.

These correlations are discussed in general chemistry textbooks.

Exercise 19.6

Give explanations for the trends in atomic size.

Unsöld’s theoremasserts that the charge distribution in a filled hydrogen-like sub-

shell is spherically symmetric (independent ofθandφ).

EXAMPLE19.7

For hydrogen-like orbitals, show that

|ψ 2 p 1 |^2 +|ψ 2 p 0 |^2 +|ψ 2 p,− 1 |^2

is independent ofθandφ, as asserted by Unsöld’s theorem.

Solution

|ψ 2 p 1 |^2 +|ψ 2 p 0 |^2 +|ψ 2 p,− 1 |^2 R^221

[

Θ^211 |Φ 1 |^2 +Θ^210 |Φ 0 |^2 +Θ^2 1,− 1 |Φ− 1 |^2

]



R^221

2 π

[

3

4

sin^2 (θ)+

6

4

cos^2 (θ)+

3

4

sin^2 (θ)

]



(

R^221

2 π

)(

6

4

)

[sin^2 (θ)+cos^2 (θ)]

(

R^221

2 π

)(

6

4

)