Factors Affecting Lattice Enthalpy

Spec Mapping — OCR H432 Module 5.2.1 — Factors affecting lattice enthalpy, covering the qualitative and semi-quantitative dependence of lattice enthalpy on ionic charge ($|q_+ q_-|$∣q+​q−​∣) and inter-ionic distance ($r_+ + r_-$r+​+r−​) via the simplified Coulomb expression $|\Delta_{LE} H^\ominus| \propto |q_+ q_-|/(r_+ + r_-)$∣ΔLE​H⊖∣∝∣q+​q−​∣/(r+​+r−​); trends across Group 1 halides (vary cation size at fixed charge), sodium halides (vary anion size), 2+/2- vs 1+/1- compounds (charge effect); comparison of theoretical (point-charge Coulomb) and experimental (Born–Haber) lattice enthalpies; introduction to Fajans's rules and covalent character; and application to melting points, hardness, thermal-decomposition trends of Group 2 carbonates, and solubility patterns (refer to the official OCR H432 specification document for exact wording).

Lesson 3 turned the Born–Haber cycle into a numerical machine for computing lattice enthalpies; this lesson explains why they vary so systematically across the periodic table. Two factors — ionic charge and inter-ionic distance — capture essentially all the variation, via a single simplified Coulomb relation that you should be able to write down on demand. The relation explains the four-fold jump from NaF to MgO (charge effect), the smooth fall from LiF to CsF down Group 1 (cation-size effect), the parallel fall from NaF to NaI across the halides (anion-size effect), and the more subtle deviations from pure ionicity in compounds like AgI, LiI, BeCl$_2$2​ (Fajans-rules covalent character). Beyond mechanism, the lattice-enthalpy magnitude controls macroscopic observables that the OCR examiner loves to test in synoptic mode: melting points of refractories (MgO at 2852 °C), thermal stability of Group 2 carbonates and nitrates (smaller cation → lower decomposition temperature), and the solubility trends developed in Lesson 5. Four worked comparison examples, the Fajans's-rules treatment with AgCl/AgI as the canonical covalent-character cases, and a tiered specimen question close out the lesson.

Key Equation: the simplified Coulomb relation governing the magnitude of lattice enthalpy is $|\Delta_{LE} H^\ominus| \propto \frac{|q_+ q_-|}{r_+ + r_-}$∣ΔLE​H⊖∣∝r+​+r−​∣q+​q−​∣​ where $q_+$q+​ and $q_-$q−​ are the formal cation and anion charges (in units of electron charge), and $r_+ + r_-$r+​+r−​ is the sum of ionic radii (in pm). A more negative (more exothermic) lattice enthalpy corresponds to (a) larger charge product and (b) smaller inter-ionic distance.

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Learning Objectives (OCR Spec 5.2.1)

By the end of this lesson you should be able to:

• Recall that the magnitude of lattice enthalpy depends on ionic charge ($|q_+ q_-|$∣q+​q−​∣) and inter-ionic distance ($r_+ + r_-$r+​+r−​)
• Predict the relative magnitude of lattice enthalpies for series of related compounds (Group 1 halides, sodium halides, Group 2 chlorides, etc.)
• Justify trends quantitatively using the simplified Coulomb expression $|\Delta_{LE} H^\ominus| \propto |q_+ q_-|/(r_+ + r_-)$∣ΔLE​H⊖∣∝∣q+​q−​∣/(r+​+r−​)
• Compare experimental (Born–Haber) and theoretical (point-charge Coulomb) lattice enthalpies, and attribute discrepancies to covalent character via Fajans's rules
• Relate lattice-enthalpy magnitudes to macroscopic physical properties: melting point, hardness, brittleness, thermal stability of Group 2 carbonates and nitrates
• Identify the dominant factor (charge or radius) in any qualitative comparison

The Electrostatic Foundation

The interaction energy between a single point cation (charge $+q_+$+q+​, in units of electronic charge $e$e) and a single point anion (charge $-q_-$−q−​) separated by a distance $r$r is given by Coulomb's law:

$E_{\text{pair}} = -\frac{e^2}{4\pi\epsilon_0} \cdot \frac{q_+ q_-}{r}$Epair​=−4πϵ0​e2​⋅rq+​q−​​

The negative sign reflects the attractive nature of the oppositely-charged interaction (the system descends in potential energy as $r$r falls). For an entire crystal lattice, summing the Coulombic interactions over all pairs of ions gives a more complex expression — Madelung sums for rock-salt geometry give $M = 1.7476$M=1.7476, for CsCl-type $M = 1.7627$M=1.7627, for ZnS-type $M = 1.6381$M=1.6381 — but the two scaling factors that emerge from any such treatment are the same as for the isolated pair:

1. The product $|q_+ q_-|$∣q+​q−​∣ — doubling either charge approximately doubles the magnitude of the lattice enthalpy. Going from 1+/1- to 2+/2- multiplies the charge product by 4 and, holding radius constant, multiplies the lattice enthalpy by approximately 4.
2. The sum $r_+ + r_-$r+​+r−​ — the inter-ionic distance, governing how closely the centres of charge can approach. Smaller ions sit closer, so the attraction is stronger and the lattice enthalpy more negative.

Hence the simplified Coulomb relation that captures essentially all the qualitative trends:

$|\Delta_{LE} H^\ominus| \propto \frac{|q_+ q_-|}{r_+ + r_-}$∣ΔLE​H⊖∣∝r+​+r−​∣q+​q−​∣​

Key Definitions:

• Charge product ($|q_+ q_-|$∣q+​q−​∣) — the product of the magnitudes of cation and anion charges. NaCl = 1, MgO = 4, Al$_2$2​O$_3$3​ has $|q_+ q_-| = 6$∣q+​q−​∣=6.
• Inter-ionic distance ($r_+ + r_-$r+​+r−​) — the sum of cation and anion ionic radii, in pm. Determines how closely the centres of charge can approach.
• Polarising power (cation) — proportional to charge/radius ratio. Small, highly charged cations are strongly polarising.
• Polarisability (anion) — increases with anion size. Large soft anions like I$^-$− have loosely-held outer electrons that are easily distorted.

Effect of Ionic Charge — the Dominant Factor

When charge product changes, the effect on lattice enthalpy is dramatic. Compare two near-isostructural rock-salt compounds with similar ionic radii:

| Compound | Cation | Anion | $r_+$r+​ / pm | $r_-$r−​ / pm | $r_+ + r_-$r+​+r−​ / pm | $|q_+ q_-|$∣q+​q−​∣ | $\Delta_{LE} H^\ominus$ΔLE​H⊖ / kJ mol$^{-1}$−1 | |----------|--------|-------|----------:|----------:|----------------:|:-----------:|------------------------------------:| | NaF | Na$^+$+ | F$^-$− | 102 | 133 | 235 | 1 | $-918$−918 | | MgO | Mg$^{2+}$2+ | O$^{2-}$2− | 72 | 140 | 212 | 4 | $-3791$−3791 |

Mg$^{2+}$2+ and O$^{2-}$2− have similar radii to Na$^+$+ and F$^-$− (the inter-ionic distances differ only by 10%), yet the lattice enthalpy of MgO is roughly four times that of NaF. The Coulomb relation predicts a factor of $\frac{|q_+ q_-|_{\text{MgO}}/(r_+ + r_-)_{\text{MgO}}}{|q_+ q_-|_{\text{NaF}}/(r_+ + r_-)_{\text{NaF}}} = \frac{4/212}{1/235} = 4.43$∣q+​q−​∣NaF​/(r+​+r−​)NaF​∣q+​q−​∣MgO​/(r+​+r−​)MgO​​=1/2354/212​=4.43, in excellent agreement with the observed factor of $3791/918 = 4.13$3791/918=4.13.

This is why doubly-charged ions form the most refractory ionic materials: MgO has melting point 2852 °C and is used to line high-temperature furnaces; CaO melts at 2613 °C and is the binder of cement; Al$_2$2​O$_3$3​ (charge product 6) melts at 2072 °C and forms corundum, ruby, and sapphire.

Effect of Ionic Radius — Smooth Trends Down a Group

When charge is fixed, lattice enthalpy varies smoothly with $1/(r_+ + r_-)$1/(r+​+r−​). Two clean cases:

Group 1 fluorides (vary cation radius):

Compound	$r_+$r+​ / pm	$r_+ + r_-$r+​+r−​ / pm	$\Delta_{LE} H^\ominus$ΔLE​H⊖ / kJ mol$^{-1}$−1	Ratio to LiF
LiF	76	209	$-1031$−1031	1.00
NaF	102	235	$-918$−918	0.89
KF	138	271	$-817$−817	0.79
RbF	152	285	$-783$−783	0.76
CsF	167	300	$-747$−747	0.72

The fractional decrease in lattice enthalpy down the group ($1.00 \to 0.72$1.00→0.72) tracks the inverse of the inter-ionic distance ($209/300 = 0.70$209/300=0.70) almost exactly — Coulomb's law is doing essentially all the work.

Sodium halides (vary anion radius):

Compound	$r_-$r−​ / pm	$r_+ + r_-$r+​+r−​ / pm	$\Delta_{LE} H^\ominus$ΔLE​H⊖ / kJ mol$^{-1}$−1	Ratio to NaF
NaF	133	235	$-918$−918	1.00
NaCl	181	283	$-787$−787	0.86
NaBr	196	298	$-751$−751	0.82
NaI	220	322	$-705$−705	0.77

Again the ratios track $1/(r_+ + r_-)$1/(r+​+r−​): $235/322 = 0.73$235/322=0.73, observed $0.77$0.77 — slight enhancement at the I$^-$− end from emerging covalent character (see Fajans below).

Combining Charge and Size — Which Wins?

For "compare-and-explain" questions where both charge and radius vary, charge product usually dominates: it appears as a multiplicative factor of 1, 2, 4, 6, ... while radius typically varies by 30% across a comparison.

Worked example A — CaO vs NaCl: Both are rock-salt structures.

• CaO: $|q_+ q_-| = 4$∣q+​q−​∣=4; $r_+ + r_- = 100 + 140 = 240$r+​+r−​=100+140=240 pm.
• NaCl: $|q_+ q_-| = 1$∣q+​q−​∣=1; $r_+ + r_- = 102 + 181 = 283$r+​+r−​=102+181=283 pm.

Coulomb ratio: $\frac{4/240}{1/283} = 4.72$1/2834/240​=4.72. Observed ratio: $\Delta_{LE} H^\ominus(\text{CaO}) / \Delta_{LE} H^\ominus(\text{NaCl}) = -3401/-787 = 4.32$ΔLE​H⊖(CaO)/ΔLE​H⊖(NaCl)=−3401/−787=4.32. The factor of 4 in charge product dominates over the slight (15%) difference in inter-ionic distance.

Worked example B — MgCl$_2$2​ vs CaCl$_2$2​ vs SrCl$_2$2​ vs BaCl$_2$2​: Charge product is fixed at 2; anion is fixed (Cl$^-$−, 181 pm). Only cation radius varies.

Compound	$r_+$r+​ / pm	$r_+ + r_-$r+​+r−​ / pm	$\Delta_{LE} H^\ominus$ΔLE​H⊖ / kJ mol$^{-1}$−1
MgCl$_2$2​	72	253	$-2526$−2526
CaCl$_2$2​	100	281	$-2258$−2258
SrCl$_2$2​	118	299	$-2156$−2156
BaCl$_2$2​	135	316	$-2056$−2056

Order: MgCl$_2$2​ (most exothermic) → CaCl$_2$2​ → SrCl$_2$2​ → BaCl$_2$2​ (least exothermic). The trend is smooth; the Coulomb relation predicts the inverse-radius ratio $253/316 = 0.80$253/316=0.80, observed $2056/2526 = 0.81$2056/2526=0.81. Beautiful agreement.

Worked example C — MgO vs SrO (an OCR favourite): Same charges (2+/2-), same anion. Mg$^{2+}$2+ (72 pm) is much smaller than Sr$^{2+}$2+ (132 pm).

$\Delta_{LE} H^\ominus(\text{MgO}) = -3791$ΔLE​H⊖(MgO)=−3791; $\Delta_{LE} H^\ominus(\text{SrO}) = -3223$ΔLE​H⊖(SrO)=−3223. Coulomb predicts the ratio $212/272 = 0.78$212/272=0.78; observed $3223/3791 = 0.85$3223/3791=0.85. Slight deviation from pure inverse-radius behaviour because the SrO Madelung sum and Born-repulsion exponent differ slightly from MgO — but the qualitative explanation (larger cation → longer inter-ionic distance → weaker Coulombic attraction → less exothermic lattice enthalpy) is exactly what the mark scheme rewards.

Worked example D — Where charge and radius fight: Compare LiF and CsI. LiF has small ions but charge product 1; CsI has large ions but the same charge product 1. Inter-ionic distances: LiF $\sim 209$∼209 pm, CsI $\sim 387$∼387 pm. Predicted ratio: $209/387 = 0.54$209/387=0.54. Observed: $\Delta_{LE} H^\ominus(\text{LiF}) / \Delta_{LE} H^\ominus(\text{CsI}) = -1031/-604 = 1.71$ΔLE​H⊖(LiF)/ΔLE​H⊖(CsI)=−1031/−604=1.71, i.e. LiF is more exothermic by a factor of 1.7 — matching the inverse-distance prediction.

graph TD
  A[Lattice enthalpy magnitude] --> B[Charge product q+ x q-]
  A --> C[Inter-ionic distance r+ + r-]
  B --> D[Larger charge product<br/>more exothermic]
  C --> E[Smaller ions<br/>more exothermic]
  D --> F[Coulomb pair: scales like q+q-/r]
  E --> F
  F --> G[Macroscopic properties:<br/>melting point, hardness,<br/>thermal stability]

Charge product dominates: 1 → 4 → 6 multiplies |ΔLE H| roughly 1 → 5 → 20× Inter-ionic distance plays a secondary, refining role — set the trend within each charge class

Theoretical vs Experimental — Fajans's Rules and Covalent Character

The simplified Coulomb relation works because for genuinely ionic compounds — most Group 1 halides, all the Group 2 oxides and halides except the smallest cations — the bonding really is dominated by point-charge electrostatics. But for compounds with small, highly polarising cations or large, polarisable anions, electron density is pulled into the inter-ionic region, creating partial orbital overlap. The bond acquires covalent character.

This is Fajans's rules (Casimir Fajans, 1923 — paraphrased):

• Cation polarising power ↑ with charge and ↓ with radius: small, highly charged cations are strongly polarising. Be$^{2+}$2+, Al$^{3+}$3+, Li$^+$+ are strongly polarising relative to their charge; d$^{10}$10 cations (Ag$^+$+, Cu$^+$+) are also strongly polarising.
• Anion polarisability ↑ with size: I$^-$− is more polarisable than Br$^-$− than Cl$^-$− than F$^-$−. Larger soft anions have loosely-held outer electrons.

When polarising cation and polarisable anion meet, the bond is strengthened beyond what point-charge Coulomb predicts. The experimental (Born–Haber) lattice enthalpy then exceeds the theoretical (point-charge) value: