The classic ionic bonding picture is that cations donate electrons to anions in order that each species can obey the octet rule.

i.e.    Li    +   F   ----------> Li⁺ + F^-

_{According to the Coulomb Energy expression the attractive energy between two ions is maximized if the charge difference is maximized and the distance, r, is minimized. What limits these quantities ?}

_{The oxidation state is determined by the group in the periodic table}

_{Alkalis ; +1
Alkaline Earths ; +2
Rare Earths ; +3 (with some exceptions)
Halogens ; -1
Chalcogens ; -2 (unless bonded to itself, always assume O^2-)
etc.}

_{The oxidation states of the transition metals are typically variable, especially toward the middle of the transition metal series.}

_{Examples of Oxidation States}

BaO          Ba²⁺ O^2-
SiO₂          Si⁴⁺ O^2-
LiNbO₃     Li⁺ Nb⁵⁺ O^2-
CuO          Cu²⁺ O^2-
FeTiO₃      Fe²⁺ Ti⁴⁺ O^2-

_{One way to approach this problem is to examine the bond distances, which relate back to the bond order (or bond valence). This is similar to distinguishing a carbon-carbon single bond from a double bond. We will return to this point in greater detail later when we discuss the bond valence concept.}

_{Another possible way to answer the question is to consider the reduction potentials of the ions. This can be useful, but with a note of caution.  Reduction potentials are measured for coordinated ions in solution, these values do not necessarily translate to the environments found in crystals.  Nonetheless, reduction potentials can be used in a qualitative sense.}

_{It is important to understand that the oxidation states tell us something about the ratio in which elements combine, and the expected coordination environment but they do not necessarily represent actual charges on each element. For example,}

_{Compound    Oxidation States     Estimated Real Charges}
   SiO₂              Si⁴⁺ / O^2-                  Si⁺² O^-
   CuO              Cu²⁺ / O^2-                Cu^+1.6 O^-1.6

_{The distance between ions is limited by electron-electron repulsions, because ions are not true point charges.}

_{The repulsion term is modelled by the Born approximation}

_{where B is the Born constant and n is the Born exponent.  The value of the Born exponent is dependent upon the electronic configuration of the colsed shell ion:}

_{Ion Configuration    n
         He                 5
         Ne                 7
      Ar, Cu⁺             9
      Kr, Ag⁺            10
      Xe, Au⁺           12}
 

_{The repulsive energy can be modeled more accurately using the relationship}

_{Calculating the Coulombic interactions for a compound with the rock salt structure}

_{The first term in the parentheses originates from the 6 nearest neighbors located about the central ion at a distance r, the next term is repulsive (hence the opposite sign) and originates from 12 next nearest neighbors (of same sign as the central ion) located at a distance of r sqrt(2), and so on....}

_{The series shown in parentheses converges to a value of 1.748. This is known as the Madelung constant, A. The value of A is different for each structure.}
 

_{Structure Type   Madelung constant, A      CN
 Rock Salt                     1.748                     6
 CsCl                            1.763                     8
 Wurtzite                       1.641                     4
 Sphalerite                     1.638                     4
 Fluorite                        5.039                     8
 Rutile                           4.816                     6}

_{Note the Madelung constant increases as we increase CN.}

_{To get the largest possible A we want high symmetry, high CN, minimize A-X distances and maximize A-A and X-X distances.}

_{To get the total lattice energy we combine the Coulomb and ion-ion repulsive terms}

_{To find the minimum energy (most stable configuration)}

Rearranging we can solve for the Born constant in terms of the other variables:

Substituting this value back into the Madelung energy expression gives:

in kJ/mol, where n is the number of ions per formula unit (2 for NaCl, 3 for TiO₂) and r_o = r_c + r_a  is the anion-cation distance in picometers.
 
 

Born-Haber cycle
 

_{The concept behind the Born-Haber cycle is based on Hess' Law, which follows from the first law of thermodynamics.}

_{Hess's Law  :  If a reaction is carried out in a series of steps, DH for the reaction will be equal to the sum of enthalpy changes for the individual steps.}
 

Ti(s) -----> Ti(g)                       DH_A
Ti(g) -----> Ti⁴⁺(g)                   DH_IE
O₂(g) -----> 2O(g)                    DH_A
2O(g) -----> 2O^2-(g)                 DH_EA
Ti⁴⁺(g) + 2O^2-(g) -----> TiO₂     U

Ti(s) + O₂(g) -----> TiO₂          DH_f

Values for these quantities are as follows:

DH_A ~ DH_S (Ti) = 471 kJ/mol
DH_IE (Ti -> Ti⁴⁺) = 4,174.6 + 2,652 + 1,310 + 659 = 8,796 kJ/mol
DH_A (O₂) = 249 kJ/mol
DH_EA (O^-> O^2-) = 2 x (-141 + 780) = 1,278 kJ/mol

DH_f = DH_A (Ti) + DH_IE (Ti -> Ti⁴⁺) + DH_A (O₂) + DH_EA (O^-> O^2-) + U
       = 471 + 8,796 + 2 (249 + 639) – 11,915
       = -872 kJ/mol

Experimental DH_f (TiO₂) = -944.7 kJ/mol

The same calculation for TiO (rock salt structure) gives (all values in kJ/mol)
 

                                  TiO₂            TiO
DH_A (Ti)                       471              471
DH_IE (Ti)                    8,796           1,969
DH_A (O₂)                      498              249
DH_EA (O^- --> O^2-)     1,278              639
U                             -11,915          -4,296
DH_f                              -872             -968

Accoding to this prediction both TiO₂ and TiO are stable with respect to formation from the elements.
However, TiO is predicted to be more stable than TiO₂.

If we use more sophisticated calculations of the lattice enthalpy, we obtain :

        U                         -12,150          -3,832
        DH_f                        -1,107            -504

Here we obtain the opposite (and correct) prediction, TiO₂ is more stable than TiO.

Note that the value of U obtained by the simple Madulung energy estimates is off by 1.9% in the case of TiO₂, and 12.1% in the case of TiO.  This is partially a reflection of the fact that TiO is not really an ionic compound, but in fact has metallic conductivity.

The take home messages of this example are as follows:

• The DH_f values of competing phases are often very similar (in some cases they may only differ by 0.1%), so the calculations must be very good if they are to be used predictively.

• The fact that a compound is stable with respect to the elements does not insure that it will form, it may be less stable than an alternative structure that you have not yet considered.

• Keeping these limitations in mind one can use the Born-Haber cycle to estimate the stability of hypothetical compounds.  In general the difficult step is to accurately estimate the lattice enthalpy.

• If the enthalpy of formation of a compound is measured experimentally, then the Born-Haber cycle can be used to determine the experimental value of the lattice enthalpy.