Atomic Radii and Periodic Properties:

Problem 5.82:  Why do atomic radii increase going down a group of the periodic table?

Down a group, atomic radii increase because the electron shells are further away from the nucleus.  Also, there are many more electrons from one period to the next and these are separated from the nucleus by completed inert gas shells.

Across a period, even though the number of electrons is increasing, the nucleus is also gaining charge.  Since the net shielding is about the same across a period, the electrons are more strongly held and, as a result, the atomic radii decrease.  The decrease is not completely smooth, however.  As an example of this, look at the following graph.

As we go from left to right in the periodic table, we see a large increase in atomic radius as we reach each new 1A element (Li, Na, K, Rb, Cs, Fr).  The progression from 1A to 8A elements is mostly smooth and decreasing except for some "blips."  For instance, Zn is larger than its predecessors; the reason for this is that it has a fully filled d subshell.  Similar irregularities can be seen in other parts of the progression.  For instance, element #64, gadolinium deviates because of a half-filled f subshell and element 70, ytterbium does a similar thing because of its filled f subshell.  But, for the most part, the atoms decrease in size as we go through a period.  Note that the noble gases do not appear in the graph.  That is because the radii of these atoms obey quite different rules and they do not fit into the progression in any orderly or discernible fashion.

The next figure we show is perhaps even more useful in comparing atomic radii.  Here we see the actual sizes depicted graphically.

Here we have the advantage of seeing these put into the periodic table.  We can use this picture to help in the next question (although periodic trends should be clear even without its help).

Here we get the answers using periodic trends.  Our conclusions can then be checked using the figures just presented above.

Problem 5.86:  The amount of energy that must be added to remove an electron from a neutral atom to give a cation is called the atom's ionization energy.  Which would you expect to have the larger ionization energy, Na or Mg?  Explain.

Mg has a higher ionization energy than Na since it has a higher Z_eff coupled with a smaller size.  The electrons are held more tightly in Mg.

This is an excellent thought problem and we shall answer each part in turn.

(a)  To analyze this, we simply apply our new rule to l while remembering that the rules for n and m stay the same.  Also each orbital would have two electrons because the rules for s remain the same (s = + 1/2).  We summarize the results in the following table:
 

Row 1:	n = 1	l = 0	1s	2 elements
		l = 1	1p	6 elements
		l = 2	1d	10 elements
Row 2:	n = 2	l = 0	2s	2 elements
		l = 1	2p	6 elements
		l = 2	2d	10 elements
		l = 3	2f	14 elements

Summing the numbers in the right column, we see that there would be 50 elements in the first two rows (18 in the first and 32 in the second).

(b)  With 18 elements in the first row, the fifth element in the second would have the atomic number Z = 23.

(c)  For Z = 12 (in the first row), the orbital filling diagram would look like this:

As an aside we note that in some theoretical computations, artificial orbitals such as 1p are generated to add to what is called a basis set.  These extra orbitals refine the precision of some atomic and molecular orbital calculations.  How this is done is too technical to discuss here but such things are possible when refining calculations.