Electrons in Molecules

An electron can be transferred from one neutral (uncharged) atom to another. When this happens, the atom from which the electron is transferred becomes a positive ion or cation and the atom to which the electron is transferred becomes a negative ion or anion.

The transfer of an electron to or from an atom involves energy. The energy required to remove one electron from an atom is characteristic of the atom and is its ionization energy. The energy released when an electron is acquired by an atom is its electron affinity. The electron transfer can take place with a release of energy only when the electron affinity of the receiving species is greater than the ionization energy of the donating species, which is rarely the case, or, most often, when the resulting ions associate themselves into a new configuration of lower energy. Most electron transfer reactions occur when the ions once formed associate themselves into structured lattices called ionic crystals.

Energy Considerations in Ionic Structures

The energy required to produce any cation from an atom, the ionization energy of the atom, is always larger, and usually much larger, than the energy which is released when an electron is added to any atom, the electron affinity. This is true even if an electron is being removed from an atom which loses it relatively easily, like sodium, and is being added to an atom which readily acquires it, such as chlorine. The ionization energy of sodium is +496 kJ/mole while the electron affinity of chlorine is only -349 kJ/mole. The reaction Na(g) + Cl(g)  Na+(g) + Cl-(g) would require +147 kJ/mole to proceed while the reaction Na(s) + 1/2 Cl2(g) Na+(g) + Cl-(g)  requires +377 kJ/mole. Electron affinity alone cannot provide sufficient energy to form ions or ionic structures; the energy must come from the assembly of isolated ions into stable multi-ion structures.

Ions with the same type of charge repel each other, but ions of opposite charge attract each other. The simplest possible ionic structure which might be stable is the gas-phase ion pair, which consists of one cation and one anion held together by electrostatic attraction. It is relatively simple to calculate how much energy would be gained by this association using the Coulomb law of electrostatic attraction. The energy of the attraction is given by

E = (2.31 x 10-16 J-pm) Z+Z-/d

where Z is the charge on the cation and anion and d is the distance between the ions, in pm.  The energy of the two associated ions will be less than the energy of the two isolated ions by this amount if the ions are of opposite charge. For sodium ion the ionic radius is 97 pm and for chloride ion it is 181 pm so the distance of separation of the centers of the two ions is 278 pm. The energy for one ion pair, multiplied by the Avogadro number NA, gives the molar energy of [Na+Cl-](g) relative to the molar energy of the isolated ions as:

E = -8.31 x 10-19 J/molecule x 6.022...x 1023 molecules/mole

This is -500 kJ/mole, so the standard molar enthalpy of formation of the ion pair estimated using the Coulomb law is -123 kJ/mole (-500 kJ/mole + 377 kJ/mole). Even for a single sodium ion and chloride ion in the gas phase, it is the lower energy available through association of ions of opposite charge that drives the formation of ionic compounds.

The ionic radii used in the calculation above were the radii of sodium and chloride ions found in ionic crystals. They are, however, very similar to the radii of these ions under other conditions. The actual distance between the ions in Na+Cl-(g) has been measured and found to be 236.1 pm.

Association of ions of opposite charge is not normally into ion pairs. It is far more common to find ions in the form of the solid ionic crystals, which are large ordered three-dimensional arrays of ions.

The diagram below is the Born-Haber cycle for the formation of an ionic compound from the reaction of an alkali metal (Li, Na, K, Rb, Cs) with a gaseous halogen (F2, Cl2). The Born-Haber thermo chemical cycle is named after the two German physical chemists, Max Born and Fritz Haber, who first used it in 1919.

The energies of the cycle above to get energy tables needed for all the alkali metal halides.

The enthalpy change in the formation of an ionic lattice from the gaseous isolated sodium and chloride ions is -788 kJ/mole. That enthalpy change, which corresponds to the reaction Na+(g) + Cl-(g)  NaCl(s), is called the lattice energy of the ionic crystal. Although the lattice energy is not directly measurable, there are various ways to estimate it from theoretical considerations and some experimental values. For all known ionic crystals, the lattice energy has a large negative value. It is ultimately the lattice energy of an ionic crystal which is responsible for the formation and stability of ionic crystal structures.

For sodium chloride, the Born - Haber cycle is:

A cycle of this type is an example of Hess's Law.  It can be used to calculate any of the six enthalpies, given the other five.

Lattice Energy: The Born-Haber cycle

Table of contents

Ionic solids tend to be very stable compounds. The enthalpies of formation of the ionic molecules cannot alone account for this stability. These compounds have an additional stability due to the lattice energy of the solid structure. However, lattice energy cannot be directly measured. The Born-Haber cycle allows us to understand and determine the lattice energies of ionic solids.

Introduction

This module will introduce the idea of lattice energy, as well as one process that allows us to calculate it: the Born-Haber Cycle. In order to use the Born-Haber Cycle, there are several concepts that we must understand first.

Lattice Energy

Lattice Energy is a type of potential energy that may be defined in two ways. In one definition, the lattice energy is the energy required to break apart an ionic solid and convert its component atoms into gaseous ions. This definition causes the value for the lattice energy to always be positive, since this will always be an endothermic reaction. The other definition says that lattice energy is the reverse process, meaning it is the energy released when gaseous ions bind to form an ionic solid. As implied in the definition, this process will always be exothermic, and thus the value for lattice energy will be negative. Its values are usually expressed with the units kJ/mol.

Lattice Energy is used to explain the stability of ionic solids. Some might expect such an ordered structure to be less stable because the entropy of the system would be low. However, the crystalline structure allows each ion to interact with multiple oppositely charge ions, which causes a highly favorable change in the enthalpy of the system. A lot of energy is released as the oppositely charged ions interact. It is this that causes ionic solids to have such high melting and boiling points. Some require such high temperatures that they decompose before they can reach a melting and/or boiling point.

Born-Haber Cycle

There are several important concept to understand before the Born-Haber Cycle can be applied to determine the lattice energy of an ionic solid; ionization energy, electron affinity, dissociation energy, sublimation energy, heat of formation, and Hess's Law.

• Ionization Energy is the energy required to remove an electron from a neutral atom or an ion. This process always requires an input of energy, and thus will always have a positive value. In general, ionization energy increases across the periodic table from left to right, and decreases from top to bottom. There are some excepts, usually due to the stability of half-filled and completely filled orbitals.
• Electron Affinity is the energy released when an electron is added to a neutral atom or an ion. Usually, energy released would have a negative value, but due to the definition of electron affinity, it is written as a positive value in most tables. Therefore, when used in calculating the lattice energy, we must remember to subtract the electron affinity, not add it. In general, electron affinity increases from left to right across the periodic table and decreases from top to bottom.
• Dissociation energy is the energy required to break apart a compound. The dissociation of a compound is always an endothermic process, meaning it will always require an input of energy. Therefore, the change in energy is always positive. The magnitude of the dissociation energy depends on the electronegativity of the atoms involved.
• Sublimation energy is the energy required to cause a change of phase from solid to gas, bypassing the liquid phase. This is an input of energy, and thus has a positive value. It may also be referred to as the energy of atomization. 
• The heat of formation is the change in energy when forming a compound from its elements. This may be positive or negative, depending on the atoms involved and how they interact.
• Hess's Law states that the overall change in energy of a process can be determined by breaking the process down into steps, then adding the changes in energy of each step. The Born-Haber Cycle is essentially Hess's Law applied to an ionic solid.

Using the Born-Haber Cycle

The values used in the Born-Haber Cycle are all predetermined changes in enthalpy for the processes described in the section above. Hess' Law allows us to add or subtract these values, which allows us to determine the lattice energy.

Step 1

Determine the energy of the metal and nonmetal in their elemental forms. (Elements in their natural state have an energy level of zero.) Subtract from this the heat of formation of the ionic solid that would be formed from combining these elements in the appropriate ration. This is the energy of the ionic solid, and will be used at the end of the process to determine the lattice energy.

Step 2

The Born-Haber Cycle requires that the elements involved in the reaction are in their gaseous forms. Add the changes in enthalpy to turn one of the elements into its gaseous state, and then do the same for the other element.

Step 3

Metals exist in nature as single atoms and thus no dissociation energy needs to be added for this element. However, many nonmetals will exist as poly atomic species. For example, Cl exists as Cl2 in its elemental state. The energy required to change Cl2 into 2Cl atoms must be added to the value obtained in Step 2.

Step 4

Both the metal and nonmetal now need to be changed into their ionic forms, as they would exist in the ionic solid. To do this, the ionization energy of the metal will be added to the value from Step 3. Next, the electron affinity of the nonmetal will be subtracted from the previous value. It is subtracted because it is a release of energy associated with the addition of an electron. 

*This is a common error due to confusion caused by the definition of electron affinity, so be careful when doing this calculation.

Step 5

Now the metal and nonmetal will be combined to form the ionic solid. This will cause a release of energy, which is called the lattice energy. The value for the lattice energy is the difference between the value from Step 1 and the value from Step 4. Below is another representation of the Born-Haber Cycle.