Ions, Ionization Energy, and Electron Affinity:

Problem 6.36:  What are the likely ground-state electron configurations of the following cations?
 
  (a)  La3+
  (b)  Ag+
  (c)  Sn2+

Problem 6.37:  What are the likely ground-state electron configurations of the following anions?
 
  (a)  Se2-
  (b)  N3-

Problem 6.39:  Identify the element whose 2+ ion has the ground-state electron configuration [Ar]3d10.

This is obviously Zn, element #30.  (28 electrons in ion + 2 lost = 30 at start.)
Problem 6.40:  Do ionization energies have a positive sign or a negative sign?  Explain.
Positive sign.  It takes energy to remove an electron and this energy must be added to the atom.  Hence--and obviously--positive!
Problem 6.41:  Do electron affinities have a positive sign or a negative sign?  Explain.
When an atom absorbs an electron, energy is released or, to put it another way, subtracted from the atom's total energy.  Thus, EA's must be negative.
Problem 6.44:
 
  (a)  Which of the following has the smaller second ionization energy, K or Ca?
Look at the ground state configurations of these atoms:

Calcium has the lower second ionization energy.  Removing a second electron from K would involve removing an electron from the inert gas core, [Ar].  This is very difficult!  Also, think about the experimental evidence:  Ca2+ ions most definintely exist.  But, ever heard of a compound such as KF2?  I think not!
  (b)  Which has the larger third ionization energy, Ga or Ca?
As we did earlier, let us first look at the electronic configurations (to make the answer easier to see):

Obviously, it is easier to take 3 electrons from gallium than from calcium.  In the former case, we are left with an argon core and a fully-filled d-subshell.  This configuration is quite stable and "happy."  However, to get Ca3+, we would have to strip an electron from an argon core.  This cannot be done easily, to say the least!

Problem 6.45:
 
  (a)  Which has the smaller fourth ionization energy, Sn or Sb?
Sn would have the smaller fourth ionization energy; antimony is smaller and, thus, has a larger effective charge, Zeff.
  (b)  Which has the larger sixth ionization energy, Se or Br?
This would be bromine.  It is smaller than selenium and, thus, would have a smaller Zeff.

Problem 6.50:  What is the relationship between the electron affinity of a monocation such as Na+ and the ionization energy of a neutral atom?

They would be the same.  Adding an electron to the ion would have the same energy (but opposite in sign) from removing the electron from the neutral atom.  In other words, one process is the exact opposite of the other.
Problem 6.51:  What is the relationship between the ionization energy of a monoanion such as Cl- and the electron affinity of the neutral atom?
The relationship between the ionization energy of a univalent anion and the electron affinity of the neutral atom is that they have the same magnitude but are opposite in sign.
Problem 6.54:  Why is energy usually released when an electron is added to a neutral atom but absorbed when an electron is removed from a neutral atom?
Energy is usually released when an eletron is added to a neutral atom but absorbed when an electron is removed from a neutral atom because Zeff is positive.
Problem 6.56:  Which element in each of the following pairs has the larger (more negative) electron affinity?
 
  (a)  F or Fe F; nonmetals have more negative electron affinities than metals.
  (b)  Ne or Na Na; Ne, being a noble gas, has a positive electron affinity.
  (c)  Ba or Br Br; bromine is a nonmetal and barium is a metal (cf. (a) above).

Problem 6.57:  According to the data in Figure 6.6, zinc, cadmium, and mercury all have near-zero electron affinities.  Explain.

We first look at the figure.

All these elements have filled s and d-subshells.  An additional electron would have to go into the higher energy p-subshell.  This is unfavorable and, thus, we have near-zero electron affinities.