The Calculation of the Lattice Energy: The Born-Haber Cycle |

My blog in italian on this topics is very popular and for this reason I decided to add an English translation (when I have some free time, I will also translate the text in the Figure and Table). So be tune and more will come!

The stability of a crystal lattice at constant T and P conditions is linked to the Gibbs free energy of lattice formation by the relations

$M^+ (g) + X^- (g) \rightarrow MX (s) \hfill (1)$

$\displaystyle \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \hfill (2)$

If ${\Delta G^{\circ}}$ is more negative for the formation of the ${I}$ structure than for the ${II}$ structure, the ${II \rightarrow I}$ transition will be spontaneous and the solid will have that structure.

The lattice formation process starting from an ion in gas phase is so exothermic that at room temperature (298 K) the entropic contribution can be neglected (the entropic effect due to temperature becomes completely negligible only at temperatures close to absolute zero, $T = 0~K$ ). The reticular energetic can therefore be discussed solely on the basis of the lattice enthalpy.

The enthalpy of dissociation of the crystal lattice (or reticular enthalpy, ${\Delta H_c}$ , where $c$ stand for crystal) is the standard enthalpy variation that accompanies the formation of an ion gas starting from the corresponding ionic solid:

$\displaystyle MX (s) \rightarrow M^+ (g) + X^- (g) \qquad \Delta H_c \hfill (3)$

The dissociation of the crystal lattice is endothermic, therefore the values of $ latex {\Delta H_c} $ are always positive. The opposite reaction or the formation of the crystal

$\displaystyle M^+ (g) + X^- (g) \rightarrow MX (s) \qquad -\Delta H_c \hfill (4)$

it is highly exothermic and therefore the enthalpy value is negative.

The value of ${\Delta H_c}$ is derived from enthalpy data through the Born-Haber thermodynamic cycle. The Born-Haber cycle for the sodium chloride molecular consists of the thermodynamic cycle shown in Figure 1 and with the enthalpy values of the reactions listed in Table I.

Figure 1: Born-Haber cycle for caption sodium chloride

Table 1: The reactions of the Born-Haber for the NaCl cycle

Since enthalpy is a function of state, by the law of Hess the sum of the enthalpy variation along a complete cycle must be zero:

$\Delta H_6^{\circ} + \Delta H_2^{\circ} + \Delta H_3^{\circ} + \Delta H_4^{\circ} + \Delta H_5 ^ {\circ} - \Delta H_1 ^ {\circ} = 0 \hfill (5)$

Therefore, the sum of the enthalpies is given by $(\Delta H_6^{\circ} +791)~ \textrm{kJ mol}^{-1}$ from which it follows that $\Delta H_6^{\circ} = x = - 791~\textrm{kJ mol}^{-1}$ . The reticular enthalpy is the opposite of this enthalpy variation, $\Delta H_c = 791 \textrm{kJ mol}^{-1}$ .

Table 2 lists the reactions for the calculation of the energy of formation of the ${kCl}$ .

Table 2: The reactions of the Born-Haber for the KCl cycle.

In this case, the sum of the enthalpies is given by $(x +724) kJ mol^{- 1}$ from which $x = - 724 kJ mol^ {- 1}$ . The reticular enthalpy is the opposite of this enthalpy variation, $\Delta H_c = 724 kJ mol^{- 1}$ .

As a last example, in Figure 2, the cycle for the $MgCl_2$ is shown. In this case, the atomization enthalpy of the electron affinity of chlorine must be calculated twice.

Figure 2: Born-Haber cycle for $MgCl_2$ .

The reticular enthalpy allows to evaluate the nature of the bond present in the solid. If the value calculated with the ionic model agrees well with the experimental one, as in this example, it is a reasonable approximation to adopt the ionic model for the compound. Any discrepancy become significant when there is an amount of covalence in the bonds.

To calculate the reticular enthalpy of an ionic compound it is necessary to consider the different contributions to its energy, such as:

The electrostatic attractions and repulsions of the ions;
The dispersion interactions: for ions of limited polarizability this contribution is only about 1% of the electrostatic contribution.
Repulsive forces due to the overlap of electronic distributions. Repulsive interactions between ions are essential to the stability of the ionic solid; in the absence of repulsive interactions, the cations and anions would unite, ie they would collapse. A $T = 0 ~ K$ (when there is no thermal agitation and, therefore, there is no nuclear kinetic energy), the ions adopt distances such that the attractions compensate for the repulsive interactions.
The kinetic energy deriving from the vibration motions of the ions can be neglected (except for the contribution of the zero point energy) if we consider the solid at absolute zero.

In Table 3 some values of ${\Delta H_c}$ calculated by D. Cubicciotti (J. Chem. Phys., 31, 1646, 1959), using an accurate model model that takes into account the mentioned contributions are reported.

An acceptable agreement with the experimental data means that the compound is ionic; an unsatisfactory agreement means that we are facing a significant degree of covalence.

TABLE 3: Measured and calculated reticular enthalpies..

The percentage difference (last column) between the experimental and calculated value clearly shows that the alkali metal halides show a good agreement when the halogen is poorly polarizable (like the fluoride ion, ${F^-}$ ) and minor when it is very polarizable (like the iodide ione ${I^-}$ ).

The worst difference is found with combinations of a cation and anion both very polarizable, such as CuI, AgI and TlI. These compounds are substantially covalent, and therefore

Composto	${ \Delta H^c_c \ (kJ mol^{-1})}$	${\Delta H^s_c \ (kJ mol^{-1})}$	${100 \frac{(\Delta H^s_c-\Delta H^c_c)}{\Delta H^c_c}}$
	(calcolato)	(sperimentale)
LiF	1033	1037	0.4
LiCl	845	852	0.8
LiBr	798	815	2.1
LiI	740	761	2.8
NaF	915	926	1.2
NaCl	778	786	1.0
NaBr	739	752	1.7
NaI	692	705	1.8
KF	813	821	1.0
KCl	709	717	1.1
KBr	680	689	1.3
KI	640	649	1.4
RbF	778	789	1.4
RbCl	686	695	1.3
RbBr	659	668	1.3
RbI	622	632	1.6
CsF	748	750	0.3
CsCl	652	676	3.6
CsBr	632	654	3.4
CsI	601	620	3.1
CuCl	904	993	9.0
CuBr	870	976	10.9
CuI	833	963	13.5
AgF	920	969	5.1
AgCl	833	912	8.7
AgBr	816	900	9.3
AgI	778	886	12.2
TlCl	686	748	8.3
TlBr	665	732	9.2
TlI	636	707	10.0

M	T	W	T	F	S	S
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29	30	31

"… I seem […] only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me". – Isaac Newton.

The Calculation of the Lattice Energy: The Born-Haber Cycle

TABLE 3: Measured and calculated reticular enthalpies..

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TABLE 3: Measured and calculated reticular enthalpies..

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