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Thermodynamics of Ionic Crystal Formation: Born-Haber Cycle and Lattice Energy, Lecture notes of Thermodynamics

Information on the thermodynamics of ionic crystal formation, focusing on the concepts of standard enthalpies and free energies of formation, the Born-Haber cycle, and lattice energy. It includes examples of various ionic compounds and their corresponding values, as well as an explanation of the factors that influence a more stable crystal lattice.

What you will learn

  • What is the significance of negative standard enthalpies and free energies of formation for ionic crystals?
  • What factors contribute to a more stable crystal lattice?
  • How does the Born-Haber cycle explain the exothermic and spontaneous formation of ionic solids?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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Thermodynamics of Crystal Formation
!All stable ionic crystals have negative standard enthalpies of formation,
ΔHo
f, and negative standard free energies of formation, ΔGo
f.
Na(s) + ½ Cl2(g) ÷ NaCl(s)ΔHo
f = –410.9 kJ ΔGo
f = –384.0 kJ
Cs(s) + ½ Cl2(g) ÷ CsCl(s)ΔHo
f = –442.8 kJ ΔGo
f = –414.4 kJ
Mg(s) + ½O2(g) ÷ MgO(s)ΔHo
f = –385.2 kJ ΔGo
f = –362.9 kJ
Ca(s) + C(s) + 3/2 O2(g) ÷ CaCO3(s)
ΔHo
f = –1216.3 kJ ΔGo
f = –1137.6 kJ
!The exothermic and spontaneous formation of ionic solids can be
understood in terms of a Hess's Law cycle, called the Born-Haber cycle.
!The lattice energy is the most important factor in making the formation
of ionic crystals exothermic and spontaneous.
!Lattice energy, U, is defined as the enthalpy required to dissociate one
mole of crystalline solid in its standard state into the gaseous ions of
which it is composed; e.g.,
NaCl(s) ÷ Na+(g) + Cl(g)U = +786.8 kJ
UDefined in this way, lattice energy is a positive (endothermic)
quantity.
USometimes lattice energy is defined by the reverse reaction, in which
case the values are negative (exothermic).
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Thermodynamics of Crystal Formation

! All stable ionic crystals have negative standard enthalpies of formation, Δ H o f , and negative standard free energies of formation, Δ G o f.

Na( s ) + ½ Cl 2 ( g ) ÷ NaCl( s ) Δ H o f = –410.9 kJ Δ G o f = –384.0 kJ

Cs( s ) + ½ Cl 2 (g) ÷ CsCl( s ) Δ H o f = –442.8 kJ Δ G o f = –414.4 kJ

Mg( s ) + ½O 2 ( g ) ÷ MgO( s ) Δ H o f = –385.2 kJ Δ G o f = –362.9 kJ

Ca( s ) + C( s ) + 3 / 2 O 2 ( g ) ÷ CaCO 3 ( s ) Δ H o f = –1216.3 kJ Δ G o f = –1137.6 kJ

! The exothermic and spontaneous formation of ionic solids can be understood in terms of a Hess's Law cycle, called the Born-Haber cycle.

! The lattice energy is the most important factor in making the formation of ionic crystals exothermic and spontaneous.

! Lattice energy, U , is defined as the enthalpy required to dissociate one mole of crystalline solid in its standard state into the gaseous ions of which it is composed; e.g.,

NaCl( s ) ÷ Na+^ ( g ) + Cl–( g ) U = +786.8 kJ

U Defined in this way, lattice energy is a positive (endothermic) quantity.

U Sometimes lattice energy is defined by the reverse reaction, in which case the values are negative (exothermic).

Na+( g ) + Cl-( g )

NaCl( s )

 H

o

f = -410.9 kJ

 H

o

sub = +107.7 kJ^2 D^ = +121.7 kJ

Na( s ) + 2 Cl 2 ( g )

Na( g ) + Cl( g )

I = +496 kJ A = -349 kJ

- U =?

Born-Haber Cycle for NaCl( s )

Na( s ) 6 Na( g ) Δ H osub = 107.7 kJ Na( g ) 6 Na+^ ( g ) + e –^ I = 496 kJ ½Cl 2 ( g ) 6 Cl( g ) ½ D = 121.7 kJ Cl( g ) + e –^6 Cl–( g ) A = –349 kJ Na+^ ( g ) + Cl–( g ) 6 NaCl( s ) – U =? Na( s ) + ½Cl 2 ( g ) 6 NaCl( s ) Δ H o f = –410.9 kJ

Y Δ H o f = Δ H osub + I + ½ D + AU

ˆ U = Δ H osub + I + ½ D + A – Δ H o f = 107.7 kJ + 496 kJ + 121.7 kJ + (–349 kJ) – (–410.9 kJ) = 787 kJ

Calculating Lattice Energy

U In principle, the lattice energy for a crystal of known structure can be calculated by summing all the attractive and repulsive contributions to the potential energy.

! For a pair of gaseous ions

where Z +^ , Z –^ = ionic charges r o = distance between ions e = electronic charge = 1.602 × 10–19^ C 4 πεo = vacuum permittivity = 1.11 × 10–10^ C^2 @J –1@m–

! Potential energy is negative for the attraction of oppositely charged ions and positive for repulsion of like-charged ions.

! The potential energy arising from repulsions and attractions acting on one reference ion can be calculated.

! Scaled up to a mole of ion pairs (and with a change of sign) this should equal the lattice energy of the crystal.

Calculating U for NaCl

Consider the potential energy arising from attractions and repulsions acting on a central Na+^ ion of NaCl.

Neighbors Distanc e 6 Cl–^ r o 12 Na+

8 Cl–

6 Na+

24 Cl–

@@@ @@@

L The series in parentheses converges at a value that defines the Madelung constant, M.

Born-Landé Equation

U At r = r o the potential energy must be a minimum, so

U Solving for B gives

U Substituting for B in the equation for the Coulombic and Born contributions to potential energy gives the Born-Landé equation ,

! The value of n can be calculated from measurements of compressibility or estimated from theory.

L For NaCl, n = 9.1 from experiment, and the Born-Landé equation gives U o = -771 kJ/mol.

! In the absence of experimental data, Pauling's approximate values of n can be used.

Ion configuration He Ne Ar, Cu+^ Kr, Ag +^ Xe, Au+ n 5 7 9 10 12

Born-Mayer Equation

! Born-Landé values are approximate.

! Mayer showed that e– r / ρ , where ρ is a constant dependant on the compressibility of the crystal, gives a better repulsion term than 1/ r n.

! Using this improved repulsion term leads to the Born-Mayer equation :

  • ρ = 30 pm works well for all alkali metal halides and other simple cases when r o values are in pm.

! Further refinements involve terms for van der Waals (dispersion) energy and evaluation of the zero point energy.