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 Enthalpy calculations using the Born–Haber Cycle

Part 2 Enthalpy changes – lattice enthalpy: Born Haber Cycle – how to set up for problem solving calculations of an unknown enthalpy

2.2a–m INDEX includes Born Haber Cycle calculations – trends in lattice enthalpies and comparing experimental and theoretical data and discussions on trends in melting points linked to ionic radii

[Author ©  Dr Phil Brown PhD: Doc Brown's exam revision notes suitable for students of advanced pre–university level theoretical–physical chemistry courses: Energetics–Thermodynamics: Born–Haber cycles to calculate lattice enthalpies [updated RE-EDIT]

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Energetics–Thermochemistry–Thermodynamics Notes INDEX

Sub–index for this section on Lattice Enthalpy and the Born–Haber cycle

2.2a An introduction to the Born–Haber cycle (this page)

2.2b The Born–Haber cycle for sodium chloride and all ∆H definitions explained (this page)

All the rest of this section on the Born-Haber Cycle and lattice enthalpies are on separate pages

2.2c The Born–Haber Cycle calculating the lattice enthalpy of sodium chloride NaCl

2.2d The Born–Haber Cycle calculating the lattice enthalpy of potassium bromide KBr

2.2e The Born–Haber Cycle calculating the lattice enthalpy of potassium iodide KI

2.2f. The Born-Haber Cycle calculating the lattice enthalpy of a group 2 metal halide MX2

2.2g Born–Haber cycle enthalpy level diagrams for magnesium chloride MgCl2, MgCl and MgCl3

2.2h. Born-Haber Cycle  for calculating the lattice enthalpy of a group 2 metal oxide MO or sulfide MS

2.2i Born–Haber Cycle energy level diagram for magnesium oxide MgO and magnesium sulfide MgS

2.2j Born–Haber cycle energy level diagram for Group 1 metal oxides and sulfides e.g. sodium oxide Na2O and sodium sulfide Na2S

2.1k Comparison of experimental and theoretical lattice enthalpies (more on this in 2.2l)

2.2l Data table of lattice enthalpies, ionic radii (nm) for group 1/2 metal halides, oxides and sulfides and more discussion of trends and comparing theoretical/experiment lattice enthalpies

2.2m Comparing melting points (K) and ionic radii (nm) for group 1 and group 2 halides, oxides and sulfides and discussion of melting point trends

INDEX of ALL advanced level pages on thermochemistry and thermodynamics

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ALL my advanced A level theoretical chemistry revision study notes

2.2a Page introduction to the 'more complex' Born–Haber Cycle

Another application of Hess's Law Cycle

This page shows you how to construct and do calculations with Born–Haber cycles from enthalpy data AND comparing experimental and theoretical lattice enthalpies for evidence of covalent character

PLEASE note that delta H/S/G values vary slightly from source to source, so I apologise in advance for any inconsistencies that may arise as I've researched and developed each section.

However, any discrepancy should not detract from learning how to construct and calculate using Born–Haber Cycles.

I hope this page will help you to be able to construct Born–Haber cycles by the various methods described and do calculations to calculate an unknown enthalpy, usually the lattice enthalpy, since this is the only enthalpy value not obtainable by direct experiment.

It is useful to have studied section

2.1a–c Introducing enthalpies of ion hydration, solution, atomisation, lattice energy, electron affinity, solvation before tackling this section, in which there is much overlap.

You need to know all the definitions with clinical precision and apply them correctly in using the Hess's Law principle applied to the Born–Haber cycle to calculate a lattice enthalpy.

You need to know all about the following five enthalpies

enthalpy of formation of an ionic compound (from elements in standard states)

enthalpy of atomisation of an element (half of bond enthalpy for diatomic gas molecules)

enthalpy of ionisation (to form positive ions from gaseous atoms/ions by electron loss)

enthalpy of electron affinity (to form negative ions from gaseous atoms/ions by electron gain)

lattice enthalpy (to form an ionic lattice from all the constituent gaseous ions of the formula)

and they are all fully explained and defined in section 2.2b for the first example – sodium chloride

and some involve 1st, 2nd etc. enthalpy change for a more specific enthalpy definition.

Principles of a Born-Haber Cycle

Above is the sort of diagram you have to come to terms with!, i.e. Fig 3. is a Hess's Law way of thinking to solve the cycle for one unknown enthalpy component in a Born–Haber cycle presented in the style of a multiple energy change diagram.

The Born-Haber Cycle is a complex way of breaking down the formation of a compound from its elements into various stages involving the enthalpy changes listed above.

2.2b The Born Haber Cycle – introduction to all the definitions

Starting with sodium chloride and all the ∆H definitions needed for other cycles, so I will not keep on repeating the definitions.

This section looks at the application of Hess's Law to the theoretical formation of an ionic lattice from its constituent elements in their standard states i.e. most stable state at 298K (25oC), 101kPa (1 atm).

Diagrams of the Born–Haber Cycle for sodium chloride

Born-Haber cycle for sodium chloride NaCl using Hess's Law

Brown arrows = exothermic  and  Blue arrows = endothermic

Electrons not shown in either diagram of the Born-Haber Cycle, they are not needed to solve a problem, but they should be included if you have to write half-equations for ionisation or electron affinity (examples are given below) and I have included them in sections 2.2c to 2.2m (see sub-index).

Fig 1. Represents the simplest and easiest way to learn to construct and present a Born–Haber cycle, in this case for sodium chloride NaCl.

I always find the simplest approach is to start with ΔH1 = ΔHformation for NaCl as the top line of the cycle.

Fig 2. Is the same Born–Haber cycle but presented as enthalpy level diagram.

I find that Fig. 1 is by far the best way of solving a Born-Haber Cycle to determine lattice enthalpy.

All the delta H values are defined and their use explained below.

You need to be able to recognise any of ΔH 1–6 and be familiar with the 'styles' of figs 1–2 and be able to complete/construct a cycle and solve a problem to obtain an unknown value.
  • In any enthalpy value with a superscript theta, (θ), i.e. ΔHθx = ??, the θ denotes a standard enthalpy value, which usually means 1 atm. pressure (101 kPa) and a 298 K temperature (25oC).
  • x is usually shorthand for the full name of the specific enthalpy use in the Born-Haber Cycle.
  • Where relevant, any such criteria need to be added to the definitions below if quoted as standard values.
  • ΔH1 = ΔHθf(NaCl) the enthalpy of formation of sodium chloride:
    • Na(s) + 1/2Cl2(g) ==> NaCl(s)  ΔHf(NaCl) = –411 kJ mol–1
    • The standard enthalpy of formation is defined as the energy released or absorbed when 1 mole of a compound is formed from its constituent elements in their normal stable states at 298K and 1atm.
  • ΔH2 = ΔHθatom(Na)  the enthalpy of atomisation of sodium:
    • Na(s) ==> Na(g)  ΔHatom(Na) = +107 kJ mol–1
    • The standard enthalpy of atomisation is defined as the energy absorbed when 1 mole of gaseous atoms is formed from the element in its normal stable state at 298K and 101kPa.
      • other examples, but not needed here are ...
        • 1/2O2(g) ==> O(g)    ΔHatom(O) = +249 kJ mol–1
        • 1/2Br2(l) ==> Br(g)    ΔHatom(Br) = +112 kJ mol–1
        • 1/2I2(l) ==> I(g)    ΔHatom(I) = +107 kJ mol–1
        • S(s) ==> S(g)  ΔHatom(S) = +279 kJ mol–1
        • Note that the enthalpy of atomisation for gaseous species is related to the 'gaseous' bond enthalpy value
        • e.g. the enthalpy of atomisation of oxygen is half the bond enthalpy of the O=O double bond and similarly for fluorine or chlorine.
        • Also note the physical state of the halogens, do NOT assume they are all gases!
  • ΔH3 = ΔHθatom(Cl2) the enthalpy of atomisation of chlorine:
    • 1/2Cl2(g) ==> Cl(g)  ΔHatom(Cl) = +121 kJ mol–1
    • Already defined above and is also half the bond enthalpy for a chlorine molecule.
      • Cl2(g) ==> 2Cl(g)  ΔHBE(Cl2) = +242 kJ mol–1
      • so take care with how the data is presented.
  • ΔH4 = ΔHθel.affin.(Cl)  the 1st electron affinity of chlorine:
    • Cl(g) + e– ==> Cl–(g)  ΔHelec. affin.(Cl) = –355 kJ mol–1
    • The standard enthalpy change for the first electron affinity is defined as the energy released or absorbed when one mole of gaseous neutral atoms gain one electron each to form one mole of singly charged negative gaseous ion at 298K and 1atm/101kPa.
      • I wouldn't worry about the 2nd of chlorine, the Cl2– ion will be too endothermic to be form in a chemical change and is electronically very unstable.
        • The 2nd electron affinity would be defined as the energy absorbed or released when 1 mole of singly charged negative ions gain one electron each to form 1 mole of doubly charged negative ions.
      • However in a Born–Haber Cycle for the formation of an ionic metal oxide e.g. for magnesium oxide MgO you need two electron affinities
        • 1st electron affinity of oxygen is exothermic
        • O(g) ==> O–(g)  ΔH1st elec. affin.(O) = –142 kJ mol–1
        • but the 2nd is very endothermic
        • O–(g) + e– ==> O2–(g)  ΔH2nd elec. affin.(O) = +844 kJ mol–1
        • because of the O–... e– repulsion, but must be considered in the cycle for MgO etc.
    • The 2nd electron affinity (not needed here for NaCl) would be defined as:
      • The energy released or absorbed when one mole of gaseous singly charged negative ion gain one electron each to form one mole of doubly charged negative gaseous ions at 298K and 1atm.
      • See equations above for oxygen which is a good comparison of the two electron affinities.
      • These are needed for the Born–Haber cycle for magnesium oxide (Mg2+O2–), in fact any Group 1 or Group 2 oxide )ignoring other peroxides or super-oxides).
      • Similarly, you would need two electron affinities for gaseous sulfur atoms to solve the Born-Haber Cycle for a Group 1 Group 2 sulfide.
  • ΔH5 = ΔHθ1st IE(Na)  the 1st ionization enthalpy of sodium:
    • Na(g) ==> Na+(g) + e–  ΔH1st IE(Na) = +502 kJ mol–1
    • The standard first ionisation energy is defined as the energy absorbed when the most loosely bond electron is removed from one mole of neutral gaseous atoms to form one mole of singly positively charged gaseous ions at 298K and 1atm/101kPa.
    • Again, in a B–H cycle for MgO etc. the 2nd ionisation energy must be taken into consideration and would be defined as:
      • The energy absorbed when the most loosely bond electron is removed from one mole of singly positively charged gaseous ions to form one mole of doubly positively charged gaseous ions at 298K and 1atm.
      • For group 2 metals you also have to consider the 2nd ionisation energy
      • e.g. Mg+(g) ==> Mg2+(g) + e–  ΔH2ndIE(Mg) = +1471 kJ mol–1
      • There is no need to consider the much to high, for stable chemical combination, 2nd ionisation energy of the Group 1 alkali metals.
  • ΔH6 = ΔHθLE(NaCl)  the lattice enthalpy of sodium chloride:
    • Na+(g) + Cl–(g) ==> NaCl(s)  ΔHLE(NaCl) = –786 kJ mol–1
    • The standard enthalpy change, known as the lattice enthalpy (lattice energy) is defined as the energy released when 1 mole of an ionic compound is formed from its constituent gaseous positive and negative ions at 298K and 1atm/101kPa.
    • Factors affecting the value of lattice enthalpy – lattice enthalpy
      • The greater the force of attraction between the ions, the greater the energy release, in coming together to the point of minimum potential energy in forming the most stable ionic crystal lattice.
      • From the laws of electrostatics, the force of attraction is proportional to ...
        • +ve charge x –ve charge/(distance between the centres of the charges)2,
        • translating this into the ionic lattice situation, the force of attraction is proportional to the charge on cation x charge on anion/(radius of cation + radius of anion)2,

        • F c+ x c– / d2, (Note that d = r+ + r–

        • where d = total of the two ionic radii, assuming a simple lattice structure, which means the sum of the cation and anion radius = the distance between the positive and negative centres of attraction.

          •
        • So. quite simply, the smaller the radius of an ion or the greater the charge on an ion, the greater will be the lattice enthalpy because the electric field intensity is increases, hence the force of attraction is stronger as the ions are closer together, hence the greater amount of energy released when the ions come together.
        • From this simple trends can be noted e.g. in terms of +∆LE (in kJmol–1) ...
        • LiF (1022) > LiCl (846) > LiBr (800) > LiI (744), from L to R, the anion radius becomes larger,
        • Al2O3 (15916) > MgO (3889) > Na2O (2478), from L to R, the cation charge decreases (and the cation radius increases),
        • MgO (3889) is over 4 x NaCl (780), charges 2+ x 2– versus 1+ x 1–, ignoring ionic radii differences.
    • Now everything is defined and explained, so,
      • next in 2.2c we put it all together to illustrate how a lattice enthalpy can be calculated from a Born–Haber Cycle,
      • again using sodium chloride as an example illustrating a direct application of Hess's Law,
      • but no repetitions of the enthalpy definitions from above,
      • I have assumed you will know them, since you need to know them for the exam!

SPARE DATA TABLE matrix

LATTICE ? ? ? ? ? ? ? ? ? ? ? ?
IONS↓→ F– F– Cl– Cl– Br– Br– I– I– O2– O2– S2– S2–
Li+                        
Na+                        
K+                        
Rb+                        
Cs+                        
Be2+                        
Mg2+                        
Ca2+                        
Sr2+                        
Ba2+                        
                         
                         

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QUICK INDEX for Energetics-Thermochemistry-Thermodynamics:

INDEX of ALL advanced level pages on thermochemistry and thermodynamics

Part 1a–b ΔH Enthalpy Changes 1.1 Advanced Introduction to enthalpy changes of reaction, formation, combustion etc. : 1.2a & 1.2b(i)–(iii) Thermochemistry – Hess's Law and Enthalpy Calculations – reaction, combustion, formation etc. : 1.2b(iv) Enthalpy of reaction from bond enthalpy calculations  : 1.3a–b Experimental methods for determining enthalpy changes and treatment of results and calculations : 1.4 Some enthalpy data patterns : 1.4a The combustion of linear alkanes and linear aliphatic alcohols : 1.4b Some patterns in Bond Enthalpies and Bond Length : 1.4c Enthalpies of Neutralisation : 1.4d Enthalpies of Hydrogenation of unsaturated hydrocarbons and evidence of aromatic ring structure in benzene : Extra Q page A set of practice enthalpy calculations with worked out answers ** Part 2 ΔH Enthalpies of ion hydration, solution, atomisation, lattice energy, electron affinity and the Born–Haber cycle : 2.1a–c What happens when a salt dissolves in water and why? : 2.1d–e Enthalpy cycles involving a salt dissolving : 2.2a–c The Born–Haber Cycle *** Part 3 ΔS Entropy and ΔG Free Energy Changes : 3.1a–g Introduction to Entropy : 3.2 Examples of entropy values and comments * 3.3a ΔS, Entropy and change of state : 3.3b ΔS, Entropy changes and the feasibility of a chemical change : 3.4a–d More on ΔG, free energy changes, feasibility and applications : 3.5 Calculating Equilibrium Constants from ΔG the free energy change : 3.6 Kinetic stability versus thermodynamic feasibility – can a chemical reaction happen? and will it happen?

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