Introduction to Inorganic Chemistry/Ionic and Covalent Solids - Energetics

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  Chapter 9: Ionic and Covalent Solids - Energetics[edit]

In Chapter 8, we learned all about crystal structures of ionic compounds. A good question to ask is, what makes a compound choose a particular structure? In addressing this question, we will learn about the forces that hold crystals together and the relative energies of different structures. This will in turn help us understand in a more quantitative way some of the heuristic concepts we have learned about in earlier chapters, such as hard-soft acid-base theory.

   9.1 Ionic radii and radius ratios[edit]

Ionic Radii

Atoms in crystals are held together by electrostatic forces, van der Waals interactions, and covalent bonding. It follows that arrangements of atoms that can maximize the strength of these attractive interactions should be most favorable and lead to the most commonly observed crystal structures.

Radius ratio rules. Early crystallographers had trouble solving the structures of inorganic solids using X-ray diffraction because some of the mathematical tools for analyzing the data had not yet been developed. Once a trial structure was proposed, it was relatively easy to calculate the diffraction pattern, but it was difficult to go the other way (from the diffraction pattern to the structure) if nothing was known a priori about the arrangement of atoms in the unit cell. It was (and still is!) important to develop some guidelines for guessing the coordination numbers of bonding geometries of atoms in crystals. The first such rules were proposed by Linus Pauling, who considered how one might pack together oppositely charged spheres of different radii. Pauling proposed from geometric considerations that the quality of the "fit" depended on the radius ratio of the anion and the cation.


The basic idea of radius ratio rules is shown at the left. We consider that the anion is the packing atom in the crystal and the smaller cation fills interstitial sites ("holes"). Cations will find arrangements in which they can contact the largest number of anions. If the cation can touch all of its nearest neighbor anions, as shown for the small cation in a triangular hole, then the fit is good. If the cation is too small for a given site, that coordination number will be unstable and it will prefer a lower coordination structure. The table below shows the ranges of cation/anion radius ratios that give the best fit for a given coordination geometry.

Coordination number Geometry ρ = rcation/ranion
0 - 0.155
0.155 - 0.225
0.225 - 0.414
square planar
0.414 - 0.732
0.414 - 0.732
0.732 - 1.0

There are unfortunately several challenges with using this idea to predict crystal structures:

  • We don't know the radii of individual ions
  • Atoms in crystals are not really ions - there is a varying degree of covalency depending electronegativity differences
  • Bond distances (and therefore ionic radii) depend on bond strength and coordination number (remember Pauling's rule D(n) = D(1) - 0.6 log n)
  • Ionic radii depend on oxidation state (higher charge => smaller cation size, larger anion size)

We can build up a table of ionic radii by assuming that the bond length is the sum of the radii (r+ + r-) if the ions are in contact in the crystal. Consider for example the compounds MgX and MnX, where X = O, S, Se. All of the crystallize in the NaCl structure:

                      bond distance (rMX)

MgO  2.10      MgS  2.60      MgSe  2.73 Å
MnO  2.24      MnS  2.59      MnSe  2.73 Å

For the two larger anions (S2- and Se2-), the unit cell dimensions are the same for both cations. This suggests that the anions are in contact in these structures. From geometric considerations, the anion radius in this case is given by:


and thus the radii of the S2- and Se2- ions are 1.84 and 1.93 Å, respectively. Once the sizes of these anions are fixed, we can obtain a self-consistent set of cation and anion radii from the lattice constants of many MX compounds.

How well does this model work? Let's consider the structures of tetravalent metal oxides (MO2), using Pauling radii and the predictions of the radius ratio model:

Oxide MO2 Radius ratio Predicted coord. no. Observed coord no. (structure)
2 (linear molecule)
4 (various tetrahedral structures)
4 (silica-like structures)
6 (rutile)
6 (rutile)
7 (baddleyite)
8 (fluorite)
8 (fluorite)

Note that cations have different radii depending on their coordination numbers, and thus different radius ratios are calculated for Ge4+ with coordination numbers 4 and 6, and for Zr4+ with coordination numbers 6 and 8.

For this series of oxides, the model appears to work quite well. The correct coordination number is predicted in all cases, and borderline cases such as GeO2 and ZrO2 are found in structures with different coordination numbers. The model also correctly predicts the structures of BeF2 (SiO2 type), MgF2 (rutile), and CaF2 (fluorite).

What about the alkali halides NaCl, KBr, LiI, CsF, etc.? All of them have the NaCl structure except for CsCl, CsBr, and CsI, which have the CsCl (8-8) structure. In this case the radius ratio model fails rather badly. The Li+ salts LiBr and LiI are predicted to have tetrahedral structures, and KF is predicted to have an 8-8 structure like CsCl. We can try adjusting the radii (e.g., making the cations larger and anions smaller), but the best we can do with the alkali halides is predict about half of their structures correctly. Since the alkali halides are clearly ionic compounds, this failure suggests that there is something very wrong with the radius ratio model, and its success with MO2 compounds was coincidental.

  9.2 Structure maps[edit]


Structure maps, which plot structures against properties such electronegativity, are more consistent than radius ratio rules in correctly predicting coordination numbers and crystal structures. One of the early examples of this approach was published by Mooser and Pearson in 1959.[1]

A Mooser-Pearson diagram maps crystal structures according to the average principal quantum numbers of the atoms and their electronegativity difference. The basic ideas behind such a plot are:

  • The greater the electronegativity difference, the more ionic is the compound. Higher ionicity results in higher coordination numbers because anions like to surround cations (and vice versa).
  • Higher principal quantum numbers result in less s-p hybridization, less directional bonding, and therefore higher coordination number. We saw this trend before with the structures of elements in group IV: descending the group the coordination number increases progressively from 3-4 (carbon) to 12 (Pb).

The lines in the Mooser-Pearson diagram separate MX compounds with CsCl, NaCl, and tetrahedral (wurtzite and zincblende) structures. Note that wurtzite has higher ionicity than zincblende in the plot, consistent with our discussion of the "boat" and "chair" ring structures in Chapter 8. Diamorphic compounds tend to fall on the boundaries. On the whole, the Mooser-Pearson diagram makes far fewer errors in predicting structures than the radius ratio rule. There are similar diagrams for MX2 structures, in which the order of ionicity is CaF2 (8:4 coordination) > rutile (6:3) > silica structures (4:2).

  9.3 Energetics of crystalline solids: the ionic model[edit]

The NaCl crystal structure is the archetype for calculating lattice energies and computing enthalpies of formation from Born-Haber cycles.

Let's start looking at the lattice energies from the sense of chemical bonding. There are mainly two kinds of force that determine the energy of an ionic bond.

A graphical representation of Coulomb's law

1) Electrostatic Force of attraction and repulsion (Coulomb's Law): Two ions with charges q1 and q2, separated by a distance r, experience and attractive force -F:

|\mathbf F|=-e^2{|q_1q_2|\over r^2}\qquad

The Coulombic potential energy, V, is then given by


|\mathbf V|=\int_\infty^r F(r)\,\mathrm{d}r=-e^2{|q_1q_2|\over r}\qquad

2) Closed-shell repulsion. When electrons in the closed shells of one atom overlap with those of another atom, there is a repulsive force comes from the Pauli exclusion principle. A third electron cannot enter an orbital that already contains two electrons. This force is short range, and is typically modeled as falling off exponentially or with a high power of the distance r between atoms. For example, in the Born approximation, B is a constant and n is a number ranging from 5~12

|\mathbf V|={B\over r^n}\qquad

The energy of the ionic bond between two atoms is then calculated by the combination of net electrostatic attraction and the closed-shell repulsion energies, as shown in the figure at the right. In the crystal lattice, the equilibrium distance between ions is determined by the minimum in the total energy curve. At this distance, the net force on each ion is zero.

We can calculate the electrostatic energy of a crystal lattice using Coulomb's law as follows:

Consider a row of anions and cations in the NaCl structure.

A chain of anion and cation

We can see that the nearest neighbor interaction (+ -) is attractive, the next nearest neighbor interactions (- - and + +) are repulsive, and so on. In the NaCl structure, counting from the ion in the center of the unit cell, there are 6 nearest neighbors (on the faces of the cube), 12 next nearest neighbors (on the edges of the cube), 8 in the next shell (at the vertices of the cube), and so on:

|\mathbf V|=-6{e^2|q_1q_2|\over r}\qquad

|\mathbf V|={12\over \sqrt{2}}{{e^2|q_1q_2|\over r}\qquad}

|\mathbf V|={-8\over \sqrt{3}}{{e^2|q_1q_2|\over r}\qquad}

|\mathbf V|={6\over \sqrt{4}}{{e^2|q_1q_2|\over r}\qquad}

The net attractive energy between the cation and anion in this infinite series will then result in a function:

|\mathbf V|={-{e^2|q_1q_2|\over r}}(6- \frac{12}{\sqrt{2}}+ \frac{8}{\sqrt{3}}- \frac{6}{\sqrt{4}}+\cdots)

Generalizing this formula for any three-dimensional ionic crystal we get a function:

|\mathbf V|=-e^2{|q_1q_2|\over r}NA\qquad

where A is called the Madelung constant. The Madelung constant depends only on the geometrical arrangement of the point charges so it varies between different types of crystal structures, but within a structure type it does not change. Thus MgO and NaCl have the same Madelung constant because they both have the NaCl structure.

Having that formula in hand we can then combine the two forces into an equation which gives us the lattice energy.

\mathbf E_L=-e^2{|q_1q_2|\over r}NA+ \frac{BN}{r^n}

Lattice Energy is equivalent to the equilibrium bond length so....


with \mathbf B=\frac{e^{2}q_1q_2Ar^{n-1}}{n}.

\mathbf E_L={-{e^2|q_1q_2|NA\over r_e}}(1-\frac{1}{n})

and by expressing the repulsion term in exponential form we result in the Born-Mayer equation:

where p is the fudge factor~0.35Å

\mathbf E_L={{e^2|q_1q_2|NA\over r_e}}(1-\frac{p}{r_e})

  9.4 Born-Haber cycles for NaCl and silver halides[edit]

Which the equation and the derivation mentioned above we know that lattice energy is equivalent to its heat of formation. Lets consider the formation of table salt (NaCl)

Na+(g)+Cl-(g) -------> NaCl(s)

Born-Harber cycle:

Born-Haber Cycle of NaCl

S= Sublimation

IP= ionization of Na(g)

D= Dissocation energy

EA= Electron affinity

EL=Lattice energy

R= Gas constant

t= Temperature

From Hess' Law:

dHf = S + 1/2D + IP + EA + EL + 2RT = -92.2

Acutal dHf= -98.3

The error is only about 3% off. The result is promising because we neglected the van der Waals term.

But....Why did we get away when Van der Waals term is neglected?

This is because we used energy minimization during the calculation. This results in an overestimation of the repulsion force and underestimation of the attraction force. The two errors compensate each other out.

Silver Halides

Silver Halide Calculated Cycle Difference
AgF 220 228 8
AgCl 199 217 18
AgBr 105 215 20
AgI 186 211 25

Looking at the table, we know that the ionic model didn't work for AgI from section 9.1 and 9.2 due to the electronegativity different dx=0.6. However we are still acquiring answers within ~10% error. How do we account for the partial covalent part of the EL? The answers follows the same rule explained above. The ionic model over estimates the Madelung energy and this compensate for neglect of covalency.

  9.5 Kapustinskii equation[edit]

The Lattice energy, EL, for an ionic crystals are difficult to determine experimentally . In 1956 a Russian chemist, Anatoli Fedorovich Kapustinskii, published a formula that allow us to calculate EL for any compound if we know the univalent radii. The Formula was later named The Kapustinskii Formula.

A. F. Kapustinskii noticed that the Madelung constant, A, is proportional to the number of ions in formula unit, n. showing:

\frac{A}{n}= invariant

Structure A/n
NaCl 0.874
CsCl 0.882
Rutile 0.803
Fluorite 0.800

The difference in ionic radii between M+ and M2+ compensates for the difference in A/n for monovalent (NaCl, CsCl) and divalent (Rutile, CaF2) structure.

This gives the Kapustinskii formula in (\frac{kcal}{mol}) as:

\mathbf E_L= {290.1Z_+Z_-n \over r_++r_-}(1-\frac{0.345}{r_++r_-})

The formula allows Neil Bartlett to fill out the fissing thems in the Born-Harder cycle, in this case EA, which is hard to measure and discovered the first Noble Gas.

  9.6 Discovery of noble gas compounds[edit]

In 1962 at the University of British Columbia, Neil Bartlett synthesized PtF6 and accidentally noticed the compound’s reaction with O2 to generate O2+PtF6-.
EA can be calculated by following step. EA = -38 - 278 + 134 = -182 kcal/mol
Bartlett noticed Xe has ionization energy of +280 kcal, which is identical to the ionization energy of O2, so he concluded that Xe+ should be about the same size as O2+ and that Xe+PtF6- should be a stable compound. The successful synthesis of Xe+PtF6- demonstrated that similarities in lattice energies can predict the stability of unknown compounds. In the formula shown below detailing the reaction between Xe and PtF6, Xenon prefers the +2 oxidation state. PtF6Formula.png
For example, CuF and AuF are unknown compounds, but AgF is a known, stable compounds, so from the Born Haber cycle for CuF, the compound should be marginally stable with respect to the elements.

  9.7 Stabilization of high and low oxidation states[edit]

To predict the stability of a certain oxidation state of a given ion, we can apply the lattice energy theories. Let's see two cycles showing the formation of CuF2 which is known to exist, and CuF which is unknown for its existence.
From the first cycle, CuF should be marginally stable with respect to the elements. The second cycle is for CuF2 which is known to exist.

Combining the two cycles shows that disproportionation is spontaneous (Same for 3AuF → AuF3 + 2Au). AgF is stable because the second IP is very high (495 kcal vs. 468 kcal for Cu, 473 for Au). Note that the difference in EL (690-230=460 kcal) drives the disproportionation. This difference is bigger for smaller ions. For example, in fluorides, CuF is unstable but CuF2 is stable. However, in iodides, CuI is stable while CuI2 is unstable. As a conclusion: small anions (O,F) stabilize high oxidation states, large anions (S, Br, I...) stabilize low oxidation states.

  9.8 Alkalides and electrides[edit]

Another interesting consequence of lattice energies involves the formation of certain salts containing Na- + e- as anions.

18-Crown-6 Ether

Complexing Na+ (K+, Rb+, Cs+) with crown ethers stabilizes the M+ form of the metal ("salt" form) without generating a large EL cost due to large atomic size of the anion and "particle in a box" effect.


  9.9 Resonance energy of metals[edit]

Consider the stability of Na (metal) relative to Na+ e-. The Na (metal) is more stable relative to the Na+ e- "salt" due to the fact that valence electrons are shared by all atoms in the crystal. In the "salt" form, electrons are localized and the additional kinetic energy (ie. "particle in a box") adds to the total energy.

KE = \frac{h^2 n^2}{8mL^2}

h = Plank's constant;
n = energy level, assume to be the lowest, n = 1
m = electron mass
L = size of box

For a 3Å box:
KE = \frac{(6.626\cdot10^{-34})^{2}(6.022\cdot10^{23})}{8\cdot (9.1\cdot10^{-31})(3\cdot10^{-10})^{2}} = +96 \frac{kcal}{mol}
So the Na+e- "salt" is unstable. The calculation is not very accurate because e- potential is not zero, and the "box" size is not so well defined.

  9.10 Lattice energies and solubility[edit]

Lattice energies can also help predict compound solubilities.


E_{L} \propto \frac{1}{r_{+}+r_{-}}

E_{H} \propto \frac{1}{r_{+}^{2}}+\frac{1}{r_{-}^{2}}

For large anions, EL doesn't change much with respect to r+; EH changes since EH is proportional to \frac{1}{r_{+}^{2}}. For example, with sulfate salts, MgSO4 (epsom salts) are soluble, while the larger BaSO4 (Ksp = 10-10)is insoluble.

Figure A: sulfate salts EL diagram. This example shows how large anion, SO42- affects solubility of dications with different sizes.Figure B: fluoride ion EL diagram. This example shows how small anion, F- affects solubility of cation with different sizes

For small anions, EL is proportional to r+, while EH does not depend on r+ as strongly. For fluorides and hydroxides, LiF is slightly soluble while CsF is vastly soluble, and Mg(OH)2 is insoluble where Ba(OH)2 is very soluble. In both cases, the solubility depends on the size.

Combining the trends listed above:
1) Increasing size mismatch between the anion and cation leads to greater solubility, so CsF and LiI are the most soluble alkali halides.

2) Increasing covalency leads to lower solubility in the salts (due to larger EL. For example, AgCl, AgBr, and AgI exhibit lower solubility as the atoms move down the row.

  • AgCl(Ksp = 10-10) > AgBr> AgI(Ksp = 10-17)

3) Increasing charge on the anion generates lower solubility due to the increased effect on the EL compared to the EH.

4) Small, polyvalent cations (having EH) generate soluble salts with large, univalent anions such as I-, NO3-, ClO4-, PF6-, and acetate.

  • Lanthanides
  • Ln3+: Nitrate salts are soluble, but oxides and hydroxides are insoluble.
  • Fe3+: Perchlorate is soluble, but sulfate is insoluble.

5) Multiple charged anions such as O2-, S2-, PO43-, and SO42- generate insoluble salts with most M2+, M3+, and M4+ metals.

  9.11 Discussion questions[edit]

  • Explain why lattice energy calculations are very accurate for NaCl and CaCl2, but less accurate (by about 10%) for AgCl and PbCl2. Does the Born-Mayer equation under- or overestimate the latter values?
  • Fluorine is more electronegative than oxygen. However, for many transition metals, we can make higher oxidation states in oxides than we can in fluorides. For example, Mn(IV) is stable in an oxide (MnO2), but MnF4 is unstable relative to MnF3 and fluorine.[2] Can you explain this in terms of lattice energies?

  9.12 Problems[edit]

1. Use lattice energies to explain why MgSO4 decomposes to magnesium oxide and SO3 at a much lower temperature than does BaSO4.

2. Solid MgO might be formulated as Mg+O- or Mg2+O2-. Use the thermochemical data below (some of which are irrelevant) and Kapustinskii's formula to determine which is more stable. The lattice constant for MgO (NaCl structure) is 4.213 Å. While the idea of an O- ion might seem strange, note that the second electron affinity of O and the second ionization potential of Mg (in the table below) are both quite endothermic.

Reaction ∆Ho, kcal/mol
Mg(s) = Mg(g) 35.3
Mg(g) = Mg+(g) + e- 176.6
Mg+(g) = Mg2+(g) + e- 347.0
O2(g) = 2 O(g) 119.0
O(g) + e- = O-(g) -33.7
O-(g) + e- = O2-(g) 188.9

3. From the heat of formation of solid NH4Cl (-75.2 kcal/mol) and gaseous NH3 (-11.0), the bond dissociation energies of H2 (104.2) and Cl2 (58.2), the ionization potential of atomic hydrogen (313.4), and the electron affinity of atomic chlorine (-83.4), calculate the gas-phase proton affinity of NH3. The lattice energy of NH4Cl may be estimated from Kapustinskii's formula using rN-Cl = 3.50 Å.

4. Bottles of aqueous ammonia are often labeled “ammonium hydroxide.” We will test this idea by using a lattice energy calculation to determine whether the salt NH4+OH- can exist.

The heats of formation of gaseous OH- and H2O are respectively -33.7 and -57.8 kcal/mol. Assuming that NH4+ is about the same size as Rb+, and OH- about the same size as F-, using Kapustinskii's formula, ionic radii, and the NH3 proton affinity calculated in problem 3, determine whether NH4+OH- should be a stable salt relative to NH3 and H2O. At what temperature should NH4+Cl- be unstable relative to NH3 and HCl, if ΔHfo for HCl is -22.0 kcal/mol and ΔSo (NH4Cl --> NH3 + HCl) = 67 cal/mol K?

5. (a) Do you expect BaSO4 or MgSO4 to be more soluble in water? (b) Is LiF more soluble than LiClO4? Explain.

6. Which polymorph of ZnS (zincblende or wurzite) would you expect to be more stable on the basis of electrostatic energy?

Diagrama de Pourbaix del Arsènic.jpg

7. Arsenic contamination of ground water is a serious problem in Bangladesh, Chile, Argentina, and other parts of the world including the western United States. Arsenic poisoning been widespread in the Ganges river delta, where tube wells bring contaminated water up from 20-100 meters below the surface. One simple treatment that has been proposed is to precipitate the arsenic by aeration of the well water, which also contains high concentrations of Fe2+. Referring to the Pourbaix diagram of arsenic at the right and the Pourbaix diagram of iron in Chapter 4, identify the iron and arsenic species that are present in aerated water at neutral pH. What insoluble compound precipitates to lower the concentration of arsenic? (Hint: which compound would have the largest lattice energy?)

  9.13 References[edit]

  1. E. Mooser and W. B. Pearson, On the Crystal Chemistry of Normal Valence Compounds, Acta. Cryst. 12, 1015 (1959).
  2. K. O. Christe, "Chemical synthesis of elemental fluorine," Inorg. Chem. 1986, 25, 3721–3722. DOI: 10.1021/ic00241a001