Name: predicting ionic stability with lattice energy
Uploaded: 2020-09-03T13:38:28.000Z
Duration: 2 min 44 s
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The Born-Haber Cycle

Formation of ionic bonds requires an electron transfer from a metal to a nonmetal atom - a process that is often endothermic. However, when elemental sodium and chlorine react to form a solid sodium chloride crystal, it&#39;s a highly exothermic process. But where is the energy coming from?

According to Coulomb&rsquo;s law, cations and anions are attracted to each other by strong electrostatic forces into a solid array&nbsp;or lattice. The resulting lattice structure is stabilized by decreasing the potential energy, which is released as heat, an exothermic reaction.&nbsp;

The total energy associated with the formation or breakdown of a crystalline lattice into its gaseous constituents is called lattice energy.&nbsp;

In a solid ionic compound, a large number of charged particles interact with each other making it difficult to determine the exact value of the lattice energy experimentally. However, it can be calculated using Hess&rsquo;s law in a hypothetical series of steps called the Born&ndash;Haber cycle, which represents the formation of an ionic compound from its constituent elements.

For example, the Born-Haber cycle for sodium chloride formation considers two alternate routes, one direct and another indirect. The direct route represents the standard enthalpy of formation of NaCl from elemental sodium and chlorine.&nbsp;

The indirect route has five steps. In the first step, solid-state sodium is converted into its gaseous form. Next, diatomic chlorine molecules dissociate into gaseous chlorine atoms.&nbsp;

The third and fourth steps account for the electron transfer to form ions. An electron is removed from the gaseous sodium to form a sodium cation. The electron is then taken up by the gaseous chlorine to form a chloride anion.&nbsp;

In the final step, the electrostatic attraction between the gaseous ions leads to the formation of the lattice structure.

Hess&rsquo;s law states that the change in the overall enthalpy of a stepwise process is the sum of the enthalpy changes of each step. Meaning the enthalpy value of the direct route equals the sum of enthalpies of the five steps.

By solving the equation for lattice energy, a large&nbsp;negative value is determined, which signifies an exothermic reaction.

the born-haber cycle

<h4>Lattice Energy&nbsp;</h4>

An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a compound is a measure of the strength of this attraction. The lattice energy (&Delta;Hlattice) of an ionic compound is defined as the energy required to separate one mole of the solid into its component gaseous ions.&nbsp;

Here, the convention is used where the ionic solid is separated into ions, meaning the lattice energies will be endothermic (positive values). Another way is to use an equivalent, but opposite convention, wherein the lattice energy is exothermic (negative values) and described as the energy released when ions combine to form a lattice. Thus, make sure to confirm which definition is used when looking up lattice energies in another reference.&nbsp;

In both cases, a larger magnitude for lattice energy indicates a more stable ionic compound. For sodium chloride, &Delta;Hlattice = 769 kJ. Thus, it requires 769 kJ to separate one mole of solid NaCl into gaseous Na+ and Cl&ndash; ions. When one mole each of gaseous Na+ and Cl&ndash; ions form solid NaCl, 769 kJ of heat is released.

<h4>Determination of Lattice Energy of an Ionic Compound</h4>

It is not possible to measure lattice energies directly. However, the lattice energy can be calculated using a thermochemical cycle. The Born-Haber cycle is an application of Hess&rsquo;s law that breaks down the formation of an ionic solid into a series of individual steps:

<table style="margin-left: auto; margin-right: auto;">
	<tbody style="border: 1px solid black;">
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Enthalpy of sublimation of Cs (s)</td>
			<td colspan="2" style="color: black; border: 1px solid black; text-align: center;">Cs (s) &rarr; Cs (g)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&Delta;H = &Delta;Hs&deg; = 76.5 kJ/mol</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">One-half of the bond energy of F2</td>
			<td colspan="2" style="color: black; border: 1px solid black; text-align: center;">&frac12; F2 (g) &rarr; F (g)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&Delta;H = &frac12; D = 79.4 kJ/mol</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Ionization energy of Cs (g)</td>
			<td colspan="2" style="color: black; border: 1px solid black; text-align: center;">Cs (g)&nbsp; &rarr; Cs+ (g) + e&minus;</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&Delta;H = IE = 375.7 kJ/mol</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Electron affinity of F</td>
			<td colspan="2" style="color: black; border: 1px solid black; text-align: center;">F (g) + e&minus; &rarr; F&minus; (g)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&Delta;H = EA = &minus;328.2 kJ/mol</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Negative of the lattice energy of CsF (s)</td>
			<td colspan="2" style="color: black; border: 1px solid black; text-align: center;">Cs+ (g) + F&minus; (g) &rarr; CsF (s)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&Delta;H = &minus;&Delta;Hlattice = ?</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Enthalpy of formation of CsF (s), add steps 1&ndash;5</td>
			<td colspan="2" style="color: black; border: 1px solid black; text-align: center;">&Delta;H = &Delta;Hf&deg; = &Delta;Hs&deg;+ &frac12; D + IE + (EA) + (&minus;&Delta;Hlattice) 
			Cs (s) + &frac12; F2 (g) &rarr; CsF (s)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&Delta;H = &minus;553.5 kJ/mol</td>
		</tr>
	</tbody>
</table>

<ol>
	<li>Consider the elements in their most common states, Cs (s) and F2 (g).&nbsp;</li>
	<li>The &Delta;Hs&deg; represents the conversion of solid cesium into a gas (sublimation), and then the ionization energy converts the gaseous cesium atoms into cations.&nbsp;</li>
	<li>In the next step, the energy required to break the F&ndash;F bond to produce fluorine atoms needs to be accounted for.&nbsp;</li>
	<li>Converting one mole of fluorine atoms into fluoride ions is an exothermic process, so this step gives off energy (the electron affinity)&nbsp;</li>
	<li>Now, one mole of Cs cations and one mole of F anions is present. These ions combine to produce solid cesium fluoride. The enthalpy change in this step is the negative of the lattice energy, so it is also an exothermic quantity.&nbsp;</li>
	<li>The total energy involved in this conversion is equal to the experimentally determined enthalpy of formation, &Delta;Hf&deg;, of the compound from its elements. In this case, the overall change is exothermic.</li>
</ol>

Lattice energies calculated for ionic compounds are typically much higher than bond dissociation energies measured for covalent bonds. Whereas lattice energies typically fall in the range of 600 &ndash; 4000 kJ/mol (some even higher), covalent bond dissociation energies are typically between 150 &ndash; 400 kJ/mol for single bonds. Keep in mind, however, that these are not directly comparable values. For ionic compounds, lattice energies are associated with many interactions, as cations and anions pack together in an extended lattice. For covalent bonds, the bond dissociation energy is related to the interaction of just two atoms.

<h4>Lattice Energy as a Function of Ion Radius and Charge&nbsp;</h4>

The lattice energy of an ionic crystal increases rapidly as the charges of the ions increase, and the sizes of the ions decrease. When all other parameters are kept constant, doubling the charge of both the cation and anion quadruples the lattice energy. For example, the lattice energy of LiF (Z+ and Z&ndash; = 1) is 1023 kJ/mol, whereas that of MgO (Z+ and Z&ndash; = 2) is 3900 kJ/mol (Ro = the interionic distance defined as the sum of the radii of the positive and negative ions, is nearly the same &mdash; about 200 pm for both compounds).

Different interatomic distances produce different lattice energies. For example, compare the lattice energy of MgF2 (2957 kJ/mol) to that of MgI2 (2327 kJ/mol), which demonstrates the effect on lattice energy of the smaller ionic size of F&ndash; as compared to I&ndash;.

<h4>Other Applications of the Born-Haber Cycle</h4>

The Born-Haber cycle may also be used to calculate any one of the other quantities in the equation for lattice energy, provided that the remainder is known. For example, if the relevant enthalpy of sublimation &nbsp;&Delta;Hs&deg;, &nbsp;ionization energy (IE), bond dissociation enthalpy (D), lattice energy &Delta;Hlattice, and standard enthalpy of formation &nbsp;&Delta;Hf&deg; &nbsp;are known, the Born-Haber cycle can be used to determine the electron affinity of an atom.

This text is adapted from <a href="https://openstax.org/books/chemistry-2e/pages/7-5-strengths-of-ionic-and-covalent-bonds">Openstax, Chemistry 2e, Section&nbsp;7.5: Strengths of Ionic and Covalent Bonds.</a>

Lattice Energy&nbsp;

An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a ...

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predicting ionic stability with lattice energy

Lattice energy helps explain why ionic compounds are stable. It measures the strength of attraction between positive and negative ions in a crystal. In one common definition, lattice energy is the energy required to separate one mole of an ionic solid into gaseous ions, so it has a positive value. Another definition uses the opposite sign and describes the energy released when gaseous ions form a lattice. When checking data, it is important to confirm which convention is being used. In either case, a larger lattice energy magnitude means a more stable ionic compound.

Sodium chloride gives a simple example. Its lattice energy is 769 kJ/mol, which means 769 kJ is needed to pull one mole of solid NaCl apart into Na+ and Cl- ions in the gas phase. The same amount of heat is released when those ions come together to form solid NaCl. This value shows how strongly the ions attract each other in the crystal.

Lattice energy cannot be measured directly. Instead, it is found with a thermochemical cycle. The Born-Haber cycle uses Hess’s law to break the formation of an ionic solid into separate steps. For cesium fluoride, these steps include sublimation of Cs(s) to Cs(g), half of the F2 bond energy to make F(g), ionization of Cs(g) to Cs+(g) plus an electron, electron affinity of F(g) to make F-(g), and formation of solid CsF from the gaseous ions. The enthalpy change for forming the lattice is the negative of the lattice energy.

For CsF, the overall reaction Cs(s) + 1/2 F2(g) -&gt; CsF(s) has a standard enthalpy of formation of -553.5 kJ/mol. The cycle combines the sublimation enthalpy of cesium, 76.5 kJ/mol, one-half of the F2 bond energy, 79.4 kJ/mol, the ionization energy of cesium, 375.7 kJ/mol, and the electron affinity of fluorine, -328.2 kJ/mol, with the lattice term. Because the total is exothermic, energy is released overall when the solid forms from the elements in their most common states, Cs(s) and F2(g).

Lattice energies for ionic compounds are usually much larger than bond dissociation energies for covalent bonds. Ionic lattice energies often fall between 600 and 4000 kJ/mol, and some are even higher. Single covalent bond dissociation energies are usually about 150 to 400 kJ/mol. These values are not directly comparable because lattice energy reflects many attractions in an extended crystal, while a covalent bond energy describes the interaction of just two atoms.

Ion charge and ion size strongly affect lattice energy. When all other factors stay the same, larger charges give much larger lattice energies, and smaller ions also increase lattice energy. Doubling the charge on both ions can quadruple the lattice energy. For example, LiF has a lattice energy of 1023 kJ/mol, while MgO is 3900 kJ/mol, even though their interionic distances are nearly the same at about 200 pm. Size also matters in compounds such as MgF2 and MgI2, where the smaller F- ion leads to a higher lattice energy than the larger I- ion.

The Born-Haber cycle can also solve for other unknowns in the lattice energy equation. If the other terms are known, the cycle can be used to find a missing quantity such as electron affinity. This makes the method useful for linking measured enthalpy values to the energy changes inside ionic compounds.

Predicting Ionic Stability with Lattice Energy

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Lattice energy shows how strongly ions hold an ionic solid together. It is the energy required to separate one mole of a solid into its gaseous ions. For sodium chloride, this process requires 769 kJ per mole. The opposite convention may also be used, where lattice energy is the energy released when gaseous ions form the solid. In either case, a larger magnitude means a more stable ionic compound. For ionic crystals, lattice energy follows a pattern based on charge and ion size. The...

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Learn how lattice energy, ion charge, ion size, and the Born-Haber cycle help explain ionic stability and calculate values for compounds like CsF.

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Predicting Ionic Stability with Lattice Energy

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