A Lattice Energy Calculation for LiH

Transcription

1 A Lattice Enegy Calculation fo LiH Fank Riou Lithium hyie is a white cystalline soli with the face-centee cubic cystal stuctue (see lattice shown below). The moel fo LiH(s) popose in this stuy constists of the following elements:. The boning in LiH(s) is completely ionic. The lattice sites ae occupie by the spheical, two-electon ions, Li + an H -.. The electons of Li + an H - occupy hyogenic s atomic obitals with ajustable scale factos an, espectively. Epesse in atomic units the wavefunctions have the fom, (,) = s()s() = ( /)ep[-( + )] The scale facto etemines how apily the wavefunction (an, theefoe, the electon ensity) iminishes as the istance fom the nucleus inceases. an ae, theefoe, invesely elate to the atomic aius. The lage an, the smalle the ionic aii ae.. The aveage istance of an electon fom the nucleus, <>, in a scale s obital is./. Theefoe, it seems easonable to take <>, o / as the effective ionic aius in the soli. It is easy to show that 94% of the chage is containe within this aius. (See Appeni) 4. Van e Waals inteations between the electon clous of the ions an the quantum mechanical zeo-point enegy of the lattice ae neglecte. To check the valiity of this moel the lattice enegy of LiH(s) will be calculate an compae to the value obtaine by a Bon-Habe analysis. The lattice enegy is efine as the enegy equie to bing about the following pocess, LiH(s) ----> Li + (g) + H - (g) The etemination of the lattice enegy on the basis of the popose moel, theefoe, pocees by calculating the goun state enegies of Li + (g) an H - (g). an subtacting fom them the goun state enegy of LiH(s). Since tems fo the kinetic enegy of the ions ae not inclue, the calculations efe to absolute zeo. Li + (g) an H - (g) The calculations fo the goun-state enegies of Li + (g) an H - (g) ae simila to that of He. The enegy opeatos consist of five tems: kinetic enegy opeatos fo each of the electons, electon-nuclea potential enegy opeatos fo each of the electons, an an electon-electon potential enegy opeato. H Li = H H = When the tial wavefunction an the appopiate enegy opeato is use in the vaiational integal, E = Ψ( ) HΨ( ) τ τ

2 the following epessions esult (see Appeni fo etails): E Li = 6 8 E H = Minimization of the enegy with espect to the scale factos to obtain the goun state enegies of the gas-phase ions is the net step. Calculation of the enegies of the gas phase ions: 8 See value fo the cation scale facto: Calculate the enegy an aius of the gas phase cation: E Li ( ).7 Minimize E Li.687 E Li ( ) 7.7 E Li E Li ( ) R Li R Li.6 See value fo the anion scale facto: Calculate the enegy an aius of the gas phase anion: E H ( ).7 Minimize E H.687 E H ( ).477 E H E H ( ) R H R H 4.66 Lithium hyie soli - LiH(s) As note above, LiH has the face-centee cubic stuctue shown below. The goun state enegy of LiH(s) consists of thee tems: the intenal enegy of Li +, the intenal enegy of H -, an the coulombic inteaction enegy of the ions occupying the lattice sites. E LiH = E Li E H E coul Fom the esults of the pevious section an the knowlege that lithium hyie has the face-centee cystal stuctue, this equation can be witten whee E LiH =.7.7 E coul E coul = fo R c /R a >=.44 E coul = fo R c /R a <.44 R c R a R a

3 Hee.748 is the Maelung constant fo the face-centee cubic stuctue fo singly chage ions. R c an R a ae the aii of the cation an anion. (R c + R a ) is the inte-ionic sepaation fo situations (R c /R a >=.44) in which thee is cation-anion contact, while.44r a is the inte-ionic sepaation fo those cicumstances (R c /R a <.44) in which thee is only anion-anion contact. On the basis of assumption of the moel, R c an R a ae eplace by / an /, the effective ionic aii of the cation an the anion. The coulombic contibution now has the fom E coul = fo / >=.44 E coul = fo / <.44 Minimization of the enegy of the soli simultaneously with espect to an is outline below. Enegy of the soli assuming anioncation contact. Enegy of the soli assuming anionanion contact an that the cation attles in the octaheal hole. f ( ).7 g( ) Composite epession fo the enegy of the soli using a conitional statement E LiH ( ) if.44 f ( ) g ( ) Minimization of the enegy of LiH with espect to the paametes an Minimize E LiH.687 R c R c.6 R c (epeimental) =.4.89 R a R a.76 R a (epeimental) =.9 E LiH ( ) 8. E LiH E LiH ( ) Compaison of gas-phase an soli-state ion enegies (see Appeni fo intepetation): Cation: E Lis.7 E Lis 7.7 E Li 7.7 Cation enegy oesn't change. Anion: E Hs.7 E Hs.4 E H.477 Anion enegy inceases. Coulomb enegy in soli state: E LiH E Lis E Hs.68 The calculate lattice enegy fo LiH(s): U Lattice E Li E H E LiH U Lattice.7

4 This esult in atomic units is equivalent to a lattice enegy epesse in SI units of 86 kj/mol. A Bon-Habe analysis (see below) yiels a lattice enegy of 9 kj/mol. Thus, the calculate esult of the popose moel is in eo by only 6%. The eos fo the soli-state ionic aii ae.6% (cation) an 4.6% (anion). Given the simplicity of the moel these compaisons with epeimental ata ae encouaging. Fo futhe etails on this moel see the efeence cite below. Hfom 9.4kJ Hsub kj IE = kj + - LiH(s) Li(s) + H ( g) Li(g) + H(g) Li (g) H (g) BDE=8kJ EA= 7kJ F. Riou, "Simple Calculation of the Lattice Enegy of Lithium Hyie," Jounal of Chemical Eucation 4, (977). Appeni: π ep( ) 4π 9.8 % The table below povies a summay of the lattice enegy calculation caie out in this tutoial. Popety Cation CationRaius CationEnegy Anion AnionRaius AnionEnegy InteIon CoulombEnegy TotalEnegy LatticeEnegy GasPhase SoliState Fom the table it is clea that in the fomation of LiH soli, the hyie anion contacts significantly fom its gas-phase size. This inceases its enegy ( ). The incease in anion enegy is moe than offset by the attactive inte-ion coulombic enegy (-.68). In othe wos, the anion suffes a moest incease in enegy by shinking in size so that it can be on-aveage close to the cation, theeby inceasing the coulombic attaction between the ions an leaing to a stable ionic soli. Most of the integals equie in the analysis above ae now evaluate. Pevious memoy of an values is cleae: Tial one-electon wavefunction: Ψ( ) π ep( )

5 Demonstate that it is nomalize: Ψ ( ) 4π assume Calculate the aveage value of the electon's istance fom the nucleus: R( ) Ψ ( ) Ψ( ) 4π assume Calculate the aveage value of the kinetic enegy of the electon: T( ) Ψ( ) ( Ψ( ) ) 4π assume Calculate the aveage value of the electon-nucleus potential enegy: V( Z) Z ( ) Ψ( ) 4π assume Ψ Z Calculate the aveage value of the electon-electon potential enegy in two steps:. The electostatic potential at ue to electon is: Φ( ) Ψ( ) 4π Ψ( ) 4π assume e e simplify. The electostatic inteaction between the two electons is: V EE ( ) Φ( ) Ψ( ) 4π assume simplify 8 To summaize, the tial wavefunction chosen fo two electon systems lea to the following epession fo the enegy. E( Z) = Z 8 = Minimization of the enegy with espect to the vaiational paamete yiels: = Z Z 6 6 Goun state enegy: E( Z) = Z Ionic aius: R Z = 6 Z 6