Chapter 3 Crystal Binding

Transcription

1 Chapte 3 Cystal Binding The intenal enegy of a cystal Ionic cystals The Bon theoy of ionic cystals Van De Waals cystals Dipoles Inet cases Induced dipoles The lattice enegy of an inet-gas solid The Debye fequency The zeo-point enegy Dipole-quadupole and quadupole-quadupole tems Molecula cystals Refinements to the Bon theoy of ionic cystals Covalent and metallic bonding 1

2 Cystal Binding Intoduction - Cystalline solids ae empiically gouped into 4 classifications: (1) ionic, () van de Waals, (c) covalent, and (d) metallic.

3 Intenal Enegy of a Cystal Intenal enegy ( 內能 ) - The intenal enegy of a cystal is consideed to be composed of pats. 1. Thee is the lattice enegy U ( 晶格能 ) that is defined as the potential enegy ( 位能 ) due to the electostatic attactions and epulsions that atoms exet on one anothe.. Thee is the themal enegy ( 熱能 ) of the cystal, associated with the vibations of the atoms about thei equilibium lattice positions. 3

4 Ionic Cystals Ionic cystal ( 離子晶體 ) - Fig. 3.1 shows the lattice stuctue of sodium chloide cystal, which is simple cubic with altenating lattice positions occupied by positive and negative ions. This lattice can be pictued as two intepenetating face-centeed cubic stuctues made up of positive and negative ions espectively. In the sodium chloide lattice, the coesponding coodinate numbe is 6. Fig. 3.1 The sodium chloide lattice 4

5 Ionic Cystals - Fig. 3. shows the lattice of cesium chloide. Hee each ion of a given sign is suounded by 8 neighbos of the opposite sign. - Fig. 3.3 shows the lattice of zincblende ( 閃鋅礦 ). The coodination numbe is 4. - Ideally, ionic cystals ae fomed by combining a highly electopositive metallic element with a highly electonegative element. Fig. 3. The cesium chloide lattice. Fig. 3.3 The zincblende lattice, ZnS. 5

6 Bon Theoy of Ionic Cystals Bon theoy of ionic cystals - Bon and Medelung assumed that the ions ae electical chages with spheical symmety and that they inteact with each othe accoding to simple cental-foce laws. - In ionic cystals, these inteactions take basic foms, one long ange and the othe shot ange. - The fist is the electostatic, o coulomb, foce that vaies invesely with the squae of the distance between a pai ions, ke1e f (3.1) 1 whee e 1 and e ae the chages on the ions, 1 is the cente-to-cente distance between the ions, and k is a constant. If cgs units ae used, k = 1 dyne (cm )/(statcoulombs), and if mks units ae employed, k = Nm /C. e 1 e 1 6

7 Bon Theoy of Ionic Cystals The coesponding coulomb potential enegy fo a pai of ions is ke1e (3.) 1 - The othe type of inteaction is a shot-ange epulsion that occus when ions ae bought so close togethe that thei oute electon shells begin to ovelap. When this happens, vey lage foces ae bought into play that foce the ions away fom each othe. In a typical ionic cystal, such as NaCl, both the positive and negative ions have filled electon shells chaacteistic of inet gases. - Accoding to the Bon theoy, the total potential enegy of a single ion in an ionic cystal of the NaCl type, due to the pesence of all the othe ions, may be expessed in the fom (3.3) M R whee is the total potential enegy of the ion, M is its enegy due to coulomb inteactions with all the othe ions, and R is the epulsive enegy. 7

8 Bon Theoy of Ionic Cystals If one uses the cgs system of units whee, in Eqs. 3.1 and 3., k = 1 this expession may be witten Az e Be (3.4) n whee e is the electon chage, z is the numbe of electonic chages on the ions, is the distance between centes of an adjacent pai of negative and positive ions (Fig. 3.4), n is a lage exponent, usually of the ode of 9, and A and B ae constants. Fig. 3.4 Inteionic distances in the sodium chloide lattice. 8

9 Bon Theoy of Ionic Cystals If we now think in tems of the potential enegy of a cystal containing one mole of NaCl, the peceding equation becomes, ANz e NBe U (3.5) n whee N is Avogado s numbe ( ) and U is the total lattice potential enegy. The fist tem on the ight-hand side of this equation epesents the electostatic enegy due to simple coulomb foces between ions, wheeas the second tem is that due to the epulsive inteactions that aise when ions closely appoach each othe. It is a basic assumption of the Bon theoy that the epulsive enegy can be expessed as a simple invese powe of the inteionic distance. 9

10 Bon Theoy of Ionic Cystals - The cohesive enegy U ( 內聚能 ) is obtained as a function of by summing the cuves of the individual tems. It is done in Fig. 3.5 fo an assumed value of 9 fo the exponent n. Note that the epulsive tem, because of the lage value of the exponent, detemines the shape of the total enegy cuve at small distances, wheeas the coulomb enegy, with its smalle dependence on, is the contolling facto at lage value of. The cohesive enegy shows a minimum, U 0, at the inteionic distance 0, whee 0 is the equilibium sepaation between ions at 0 K. Fig. 3.5 Vaiation of the lattice enegy of an ionic cystal with the spacing between ions. 10

11 Bon Theoy of Ionic Cystals - Let us conside the coulomb enegy tem of the Bon equation, which fo a single ion is z e A M (3.6) Assumed that we ae specially inteested in a sodium chloide cystal whee thee is a unit chage on each ion and z = 1. e A M (3.7) Because the coulomb enegy vaies invesely as the fist powe of the distance between chaged ions, coulomb inteactions act ove lage distances, and it is not sufficient to conside only the coulomb enegy between a given ion and its immediate neighbos. 11

12 Bon Theoy of Ionic Cystals In Fig. 3.4, the coulomb enegy of a single ion equals a seies of tems of the fom M 6e 1 1e 8e 3 6e 4 4e 5... (3.8) M Ae e Fo sodium chloide, the Madelung numbe (A) is and the coulomb, o Madelung enegy, fo one ion in the cystal is, accodingly, e M (3.10) (3.9) Fig

13 Bon Theoy of Ionic Cystals - Fo a single ion, the epulsive enegy tem is Be R (3.11) n In this tem the quantities, B and n, must be evaluated. This can be accomplished with the aid of expeimentally detemined quantities: 0, the equilibium inteionic sepaation at 0 K; and K 0, the compessibility of the solid at 0 K. At 0 the net foce on an ion due to the othe ions is zeo, so that the fist deivative of the total potential enegy with espect to the distance, which equals the foce on the ion, is also zeo o d d Ae Be 0 n (3.1) d d 0 Since the quantity A is aleady known, the peceding expession poduces an equation elating n and B. The equilibium sepaation of ions 0 can be expeimentally obtained fom X-ay diffaction measuements of the lattice constant extapolated to 0 K, In the NaCl 13 cystal, this quantity equals 0.8 nm.

14 Bon Theoy of Ionic Cystals - The compessibility is a function of the second deivative of the cohesive enegy (d /d ) =0 at = 0. The compessibility is defined by the expession V K 1 0 (3.13) V p T whee K 0 is the compessibility, V is the volume of the cystal, and (V/P) T is the ate of change of the volume of the cystal with espect to pessue at constant tempeatue. The compessibility is a quantity capable of expeimental evaluation and extapolation to 0 K. When the calculations outlined above ae made, it is found that the Bon exponent fo the sodium chloide lattice is 8.0. The computed cohesive enegy is Kcal ( J) pe mole. The latte is actually the enegy of fomation of a mole of solid NaCl fom a mole of Na + ions, in the vapo fom, and a mole of gaseous Cl - ions. The expeimental value of U fo the NaCl lattice tuns out to be 188 Kcal ( J) pe mole. 14

15 Bon Theoy of Ionic Cystals - The geneal equation fo the foce between electic chages is f ke1 e / 1, whee k is a constant with units foce distance chage. In the electostatic o cgs system of units, k = 1 dynecm /(statcoulombs). Pove that k = Nm /C in the intenational (mks) system of units. (1 dyne = 10-5 N, 1 statcoulomb = 10-9 C/3) k 1dyne cm statcoulombs N 10 m 9 N C / 3 C m 15

16 Van de Waals Cystals Van de Waals ( 凡得瓦 ) cystals - We shall conside anothe type of bonding that makes possible the fomation of cystals fom atoms o even molecules that ae electically neutal and possess electon configuations chaacteistic of inet gases. - The foces that hold this type of solid togethe ae usually quite small and of shot ange. They ae called van de Waals foces and aise fom nonsymmetical chage distibutions. The most impotant component of these foces can be ascibed to the inteactions of electical dipoles. 16

17 17 Dipoles 物理冶金 Dipole ( 偶極 ) - An electical dipole is fomed by a pai of oppositely chaged paticles (+e 1 and e 1 ) sepaated by a small distance (a). Because the chages ae not concentic, they poduce an electostatic field that is capable of exeting a foce on othe electical chages. - In Fig. 3.6, let l 1 and l be the espective distances fom the chages of the dipole to a point in space at a distance fom the midpoint of the dipole. If is lage compaed to a, then the potential at the point p, using cgs units, is given by Fig. 3.6 An electical dipole. cos cos a e a e l e l e V (3.14) 1 cos 1 cos 1 cos 1 a a a e V (3.15)

18 Dipoles Since (a/) <1, Eq simplifies to a cos e1 V (3.16) a 1 cos V e1 a cos e1a cos (3.17) The components of the electic field intensity in the adial and tansvese diections ae then given by V e1a cos E 3 (3.18) 1 V e1a sin E 3 whee is the distance fom the dipole to the point p, and is the angle between the axis of the dipole and the diection. It is customay to call the quantity e 1 a, the poduct of one of the dipole chages and the distance between the dipole chages, the 18 dipole moment () ( 偶極矩 ).

19 Dipoles e1a cos cos E 3 3 e1a sin sin E The component of the foce, which would act on a chage of magnitude e if placed at point p, ae accodingly o using mks units with k = Nm /C Note that the electic field intensity due to a dipole vaies as the invese cube of the distance fom the dipole, wheeas the field of a single chage vaies as the invese squae of the distance. F F F F ee ee e cos 3 esin 3 kecos 3 kesin 3 (3.19) (3.0) 19

20 Inet Cases Inet cases - It is inteesting that the inet-gas atoms cystallize (at low tempeatue) in the facecenteed cubic system. - Because of thei closed-shell stuctues, we can conside that ove a peiod of time the negative chages of the electons ae distibuted about the nucleus with complete spheical symmety. The cente of the negative chage on a time-aveage basis theefoe coincides with the cente of the positive chage on the nucleus, which means that the inet gas atoms have no aveage dipole moment. - They do, howeve, have an instantaneous dipole moment because thei electons, in moving aound the nuclei, do not have centes of gavity that instantaneous coincide with the nuclei. 0

21 Induced Dipoles Induced dipoles ( 感應偶極, 誘導偶極 ) - When an atom is placed in an extenal electical field, its electons ae displaced fom thei nomal positions elative to the nucleus. This chage edistibution may be consideed equivalent to the fomation of a dipole inside the atom. Within limits, the size of the induced dipole is popotional to the applied field, I E (3.1) whee I is the induced dipole moment, E is the electic field intensity, and is a constant known as the polaizability ( 極化率 ). - Fig. 3.7 epesents inne-gas atoms of the same kind sepaated by the distance. Let it be assumed that the atom on the left possesses an instantaneous dipole moment due to the movement of the electons aound the nucleus. This moment will poduce a field E at the position of the second atom, which, in tun, induces a dipole moment in the latte as given by Eq

22 Induced Dipoles - In Fig. 3.7, conside the foce exeted by the left dipole on the ight dipole. The induced dipole in the ight-hand atom is equivalent to the pai of chages e and +e sepaated by the distance a. Accoding to this, the induced dipole moment is e a. Let E be the field intensity due to the instantaneous dipole on the left atom at the negative chage (-e ) of the induced dipole. The coesponding field at the position of the positive chage (+e ) of the induced dipole is E+a de/d. The total foce on the induced dipole to the field of the othe dipole is f e' E e' E a' E I f I de d de d e' a' E de d de d I de d (3.) (3.3) Fig. 3.7 Dipole-dipole inteaction in a pai of inet-gas atoms.

23 Induced Dipoles In geneal the field of a dipole is popotional to the invese cube of the distance E 3 (3.4) E e1a cos cos 3 3 f E de d d 3 d (3.5) The enegy of a pai of inet-gas atoms due to the dipole inteactions can be evaluate as follows 1 6 d (3.6) 6 6 The van de Waals enegy between a pai of inne-gas atoms due to dipole inteactions vaies as the squae of the dipole moment and the invese sixth powe of thei distance of sepaation. Ove a peiod of time the aveage dipole moment fo an inet-gas atom must be zeo. The squae of this quantity does not equal zeo, and it is on this basis that inet-gas 3 atoms can inteact.

24 Lattice Enegy of an Inet-Gas Solid Lattice enegy of an inet-gas solid - When the atoms of a ae-gas solid have thei equilibium sepaation, the van de Waals attaction is counteed by a epulsive foce. The latte is due to the inteaction that occus when closed shells of electons stat to ovelap. - The cohesive enegy of an inet-gas solid may be expessed in the fom A B U 6 n (3.7) whee A, B, and n ae constants. 4

25 Debye Fequency Debye fequency - In a cystalline solid, an atom is fee to vibate independently in 3 othogonal diections. A cystal of N atoms is consideed as equivalent to 3N oscillatos of vaious fequencies v. - Debye assumed that the foces of inteaction between a neighboing pai of atoms wee oughly equivalent to a linea sping. Pushing the atoms togethe would have the effect of compessing the sping, and in so doing, a estoing foce would be developed that would act to etun the atoms to thei est positions. Pulling the atoms apat would poduce an equivalent opposite esult. On this basis, Debye concluded that the entie lattice might be consideed to be a 3-dimensional aay of masses inteconnected by sping. In fact, assuming a simple cubic cystal, each atom would be held in space by a set of 3 pais of spings, as indicated in Fig. 3.8B. A one-dimensional cystal will be consideed, as indicated in Fig. 3.8B. 5

26 (A) Debye Fequency (B) Fig. 3.8 (A) Debye model of a simple cubic cystal pictues an atom as a mass joined to its neighbos by spings. (B) A one-dimensional cystal model. - The vibation modes of such an aay ae analogous to the standing waves that can be set up in a sting. Accoding to Debye, as shown in Fig. 3.9, the minimum wavelength o the mode of maximum fequency is obtained when neighboing atoms vibate against each othe. Fig. 3.9 The highest fequency vibation mode fo an aay of 4 masses. 6

27 Debye Fequency - The minimum wavelength coesponds to twice the spacing between the atoms, o min = a, whee a is the inteatomic spacing. The (maximum) vibational fequency associated with this wavelength is v m v (3.8) whee v is the velocity of the shotest sound waves. This latte is nomally of the ode of m/s. At the same time, the inteatomic spacing in metals is ~0.5 nm, so that v min m a v 0.5nm m / s m 0.50nm s Hz (3.9) The value, v m = Hz, is often used in simple calculations to epesent the vibation fequency of an atom in a cystal. 7

28 Zeo-Point Enegy Zeo-point enegy - The zeo-point enegy of a cystal is its themal enegy when the atoms ae vibating in thei lowest enegy states. - In a 3-dimensional cystal of N atoms, each atom inside the cystal can undego tansvese vibations in 3 independent diections (Fig. 3.8A), it is possible to show that thee ae 3N independent tansvese modes of vibations. - In a linea cystal (Fig. 3.8B), the density of vibational modes is the same in any fequency inteval dv. Howeve, in a 3-dimensional aay o cystal the Fig. 3.8(A) Fig. 3.8(B) vibational modes ae 3-dimensional, and the multiplicity of the standing wave pattens inceases with inceasing fequency. As a esult, in the 3-dimensional case the numbe of modes possessing fequencies in the ange v to v+dv is given by 9N f ( v) dv v dv 3 (3.30) vm whee f(v) is a density function, N is the numbe of atoms in the cystal, v is the vibational fequency of an oscilllato, and v m is maximum vibational fequency. 8

29 Zeo-Point Enegy - Fig is a schematic plot of the Debye density function f(v) as a function of v. The aea unde this cuve fom v = 0 to v = v m equals 3N, the total numbe of oscillatos. Accoding to the quantum theoy, the zeo-point enegy of a simple oscillato is hv/. The total vibational enegy of the cystal at absolute zeo is, accodingly, 3 4 m vm hv vm 9N hv vm 9Nhv 9Nhv 9 EZ f ( v) dv v dv dv Nhv (3.31) 0 m 0 3 vm 0 3 v 3 m 8v m The coection due to the zeo-point enegy is ~31% o ~0.5 kj/mol in the case of neon, so that the lattice enegy U 0 should be ~.47 kj/mol athe than 1.88 kj/mol, as shown in Table 3.1. v Fig Fequency spectum of a cystal accoding to Debye. The maximum lattice fequency is v m. 9

30 Dipole-Quadupole and Quadupole-Quadupole Tems Dipole-quadupole and quadupole-quadupole tems - Moden quantum mechanical teatments geneally use an expession fo the van de Waals attactive enegy (expessed in tems of a single ion) of the type ( ) c c c (3.3) whee c 1, c, and c 3 ae constants. - The fist of this expession is the dipole-dipole inteaction aleady consideed. - The second tem is the invese eighth powe of the distance is called the dipolequadupole tem because the inteaction between a dipole on one atom and a quadupole on anothe will lead to an enegy that vaies as the invese eighth powe of the distance. The dipole-quadupole tem equals ~16% of the total attactive enegy. - The last tem, vaying as the invese tenth powe of the distance, is called the quadupole-quadupole tem. It is, in geneal, small and amounts to less than 1.3% of the total van de Waals attactive enegy fo all of the inet-gas solids. 30

31 Dipole-Quadupole and Quadupole-Quadupole Tems Electic quadupole ( 電四極 ) - A geneal distibution of electic chage may be chaacteized by its net chage, by its dipole moment, its quadupole moment and highe ode moments. - An elementay quadupole can be epesented as two dipoles oiented antipaallel. Souce: 31

32 Molecula Cystals Molecula cystals - Many molecules fom cystals which ae held togethe by van de Waals foces. Among these ae N, H, and CH 4 ; these ae typical covalent molecules in which the atoms shae valence electons to effectively obtain closed shells fo each atom in the molecule. These molecules ae nonpola molecules; they do not have pemanent dipole moments. - Thee ae also pola molecules, such as wate (H O), which possess pemanent dipoles. The inteaction between a pai of pemanent dipoles is, in geneal, much stonge than that between induced dipoles. 3

33 Refinements to the Bon Theoy of Ionic Cystals Refinements to the Bon theoy of ionic cystals - The foces of van de Waals exist in othe solids, but when the binding due to othe causes is stong they may contibute only a small faction of the total binding enegy. This is geneally tue in ionic cystals, although some types, like the silve halides, may have van de Waals contibution of moe than 10%. 33

34 Covalent and Metallic Bonding Intoduction - In Fig. 3.11, the coodination numbe of the diamond stuctue is 4. - Pauli exclusion pinciple: two electons can occupy the same quantum state only if the spins ae oppositely diected. - In geneal, covalent cystals follow what is known as the (8-N) ule, whee N is the numbe of valence electons, and the facto (8-N) gives the numbe of neaest neighbos in the stuctue. Fig The diamond stuctue. Each cabon atom is suounded by 4 neaest neighbos. Note: this stuctue is the same as that of zincblende (ZnS). Fig. 3.3, except that this lattice contains one kind of atom, instead of two. 34

35 Covalent and Metallic Bonding Covalent bonding ( 共架鍵結 ) - Suppose that hydogen atoms ae made to appoach othe. Then thee ae cases to be consideed: when the spins of the electons on the atoms ae paallel, and when the spins ae opposed. - Spins ae opposed As the atoms come close and close togethe, the electon on eithe atom begins to find itself in the field of the chage on the nucleus of the othe atom. Since the spins of the electons ae opposed, each nucleus is capable of containing both electons in the 1s gound state. Unde these conditions, thee is a stong pobability that the electons will spend moe time in the neighbohood of one nucleus than the othe, and the hydogen molecule becomes a pai of chaged ions-one positive and the othe negative. This stuctue is unstable, as may be estimated fom the enegy equied to fom a positive and a negative pai of hydogen ions (-137 kj/mol). At the nomal distance of sepaation of the atoms in a hydogen molecule, the ionic stuctue exists fo limited peiods of time and contibutes ~5% of the total binding enegy. 35

36 Covalent and Metallic Bonding A much moe impotant type of electon intechange occus when both electons simultaneously exchange nuclei. The esulting shifting of the electons back and foth between the nuclei, which occus at a vey apid ate, is commonly known as a esonance effect, and ~80% of the binding enegy of the hydogen molecule is attibuted to it. In addition to the ionic and the exchange, o esonance enegy, thee ae othe moe complicated electostatic inteactions between the two electons and the two potons. These contibute the emaining 15% of the binding enegy of the hydogen molecule. Quantum mechanics shows that, on the aveage, the electons spend moe of thei time in the egion between the two potons than they do on the fa sides of the potons. Fom a vey elementay point of view, we may conside that the binding of the hydogen molecule esults fom the attaction of the positively chaged hydogen nuclei to the negative chage which exists between them. 36

37 Covalent and Metallic Bonding Attention is called to the inteelation between space and time implied in this discussion concening the time-aveage position of the electons. This is a coodination system that includes both the positions of the paticles and thei momenta (that is, velocities). Thus, fo a single paticle fee to move along a single diection (the x-axis), thee will be dimensions in phase space: its position along the axis, and its momentum. Fo n paticles capable of moving in a single diection, thee will be n linea dimensions in the phase space associated with these paticles. These ae x 1, x, x n, the positions of the paticles; and p 1, p, p n, the momenta of the paticles. Fo paticles capable of moving in 3 dimensions, thee will be 6n degees of feedom and theefoe a coesponding numbe of dimensions in phase space. An impotant theoem in statistical mechanics that beas on the subject of phase space states that the position and momentum aveage in phase space coincide with the same aveage ove an infinite time. 37

38 Covalent and Metallic Bonding - Spins ae paallel Now conside the case whee hydogen atoms with paallel spins ae made to appoach each othe. Hee, as the electon on eithe of the atoms comes within the ange of effectiveness of the field on the nucleus of the othe atom, it is found that the enegy level it would nomally occupy is aleady filled. The nomal electonic obits become badly distoted, o else the second electon moves into a highe enegy state, such as s 1. In eithe case, binging togethe hydogen atoms with electons that have paallel spins inceases the enegy of the system. A stable molecule cannot be fomed in this fashion. This is shown in Fig. 3.1, whee the uppemost cuve epesents the hydogen molecule with paallel spins and both electons in 1s obits. Fig. 3.1 Inteaction enegy of two 38 hydogen atoms.

39 Covalent and Metallic Bonding The lowe cuve is fo opposed electon spins, and it can be obseved that this cuve has a ponounced minimum, indicating that in this case a stable molecule can be fomed. Fig

40 Covalent and Metallic Bonding Metallic bonding ( 金屬鍵結 ) - In a metal the valence electons ae able to move at will though the lattice, while in a covalent cystal the electons fom diected bonds between neighboing atoms. - Metals tend to cystallize in close-packed lattices (face-centeed cubic and closepacked hexagonal stuctues) in which the diectionality of the bonds between atoms is of seconday impotance, while covalent cystals fom complicated stuctues so that the bonds between neighboing atoms will give each atom the effect of having a closed-shell configuation of electons. - Inside a metal the valence electons ae fee. - A metal consists of an odeed aay of positively chaged ions between which the valence electons move in all diections with high velocities. Ove a peiod of time, this movement of electons is equivalent to a moe o less unifom distibution of negative electicity which might be thought of as an electon gas that holds the assembly togethe. 40

41 Covalent and Metallic Bonding The coopeative inteaction between the electon gas and the positively chaged nuclei foms a stuctue that is stable. The binding foces that hold a metallic cystal togethe can be assumed to come fom the attaction of the positively chaged ions fo the cloud of negative chage that lies between them As the distance between nuclei is made smalle (as the volume in metal is deceased), the velocities of the fee electons incease with a coesponding ise in thei kinetic enegy. This leads to a epulsive tem which becomes lage when a metal is compessed. 41

42 Covalent and Metallic Bonding Bon-Habe cycle - Thee ae seveal impotant concept to undestand befoe the Bon-Habe Cycle can be applied to detemine the lattice enegy of an ionic solid; ionization enegy, electon affinity, dissociation enegy, sublimation enegy, heat of fomation, and Hess's Law. - Ionization enegy is the enegy equied to emove an electon fom a neutal atom o an ion. This pocess always equies an input of enegy, and thus will always have a positive value. In geneal, ionization enegy inceases acoss the peiodic table fom left to ight, and deceases fom top to bottom. Thee ae some excepts, usually due to the stability of half-filled and completely filled obitals. - Electon affinity is the enegy eleased when an electon is added to a neutal atom o an ion. Usually, enegy eleased would have a negative value, but due to the definition of electon affinity, it is witten as a positive value in most tables. Theefoe, when used in calculating the lattice enegy, we must emembe to subtact the electon affinity, not add it. In geneal, electon affinity inceases fom left to ight acoss the peiodic table and deceases fom top to bottom. 4

43 Covalent and Metallic Bonding - Dissociation enegy is the enegy equied to beak apat a compound. The dissociation of a compound is always an endothemic pocess, meaning it will always equie an input of enegy. Theefoe, the change in enegy is always positive. The magnitude of the dissociation enegy depends on the electonegativity of the atoms involved. - Sublimation enegy is the enegy equied to cause a change of phase fom solid to gas, bypassing the liquid phase. This is an input of enegy, and thus has a positive value. It may also be efeed to as the enegy of atomization. -The heat of fomation is the change in enegy when foming a compound fom its elements. This may be positive o negative, depending on the atoms involved and how they inteact. - Hess s law states that the oveall change in enegy of a pocess can be detemined by beaking the pocess down into steps, then adding the changes in enegy of each step. The Bon-Habe Cycle is essentially Hess's Law applied to an ionic solid. 43 Souce: _The_Bon-Habe_cycle

44 Covalent and Metallic Bonding Using the Bon-Habe cycle - The values used in the Bon-Habe Cycle ae all pedetemined changes in enthalpy fo the pocesses descibed in the section above. Hess' Law allows us to add o subtact these values, which allows us to detemine the lattice enegy. Na + (g) + Cl(g) Enegy Na(g) + Cl(g) 1 IE 0 H d Na(g) +1/ Cl (g) 0 H a step 4 step 3 step Na(s) +1/ Cl (g) EA step 4 Na + (g) + Cl - (g) 0 H lattice step 5 0 H f step 1 NaCl(s) 44 Souce: _The_Bon-Habe_cycle

45 Covalent and Metallic Bonding - Step 1: Detemine the enegy of the metal and nonmetal in thei elemental foms. (Elements in thei natual state have an enegy level of zeo.) Subtact fom this the heat of fomation of the ionic solid that would be fomed fom combining these elements in the appopiate ation. This is the enegy of the ionic solid, and will be used at the end of the pocess to detemine the lattice enegy. - Step : The Bon-Habe Cycle equies that the elements involved in the eaction ae in thei gaseous foms. Add the changes in enthalpy to tun one of the elements into its gaseous state, and then do the same fo the othe element. -Step 3: Metals exist in natue as single atoms and thus no dissociation enegy needs to be added fo this element. Howeve, many nonmetals will exist as polyatomic species. Fo example, Cl exists as Cl in its elemental state. The enegy equied to change Cl into Cl atoms must be added to the value obtained in Step. 45 Souce: _The_Bon-Habe_cycle

46 Covalent and Metallic Bonding - Step 4: Both the metal and nonmetal now need to be changed into thei ionic foms, as they would exist in the ionic solid. To do this, the ionization enegy of the metal will be added to the value fom Step 3. Next, the electon affinity of the nonmetal will be subtacted fom the pevious value. It is subtacted because it is a elease of enegy associated with the addition of an electon. This is a common eo due to confusion caused by the definition of electon affinity, so be caeful when doing this calculation. - Step 5: Now the metal and nonmetal will be combined to fom the ionic solid. This will cause a elease of enegy, which is called the lattice enegy. The value fo the lattice enegy is the diffeence between the value fom Step 1 and the value fom Step Souce: _The_Bon-Habe_cycle

47 Covalent and Metallic Bonding - The diagam below is anothe epesentation of the Bon-Habe Cycle. Na(s) +1/ Cl (g) 0 H a 1 0 H d Na(g) + Cl(g) 0 H f IE EA NaCl(s) 0 H lattice Na + (g) + Cl - (g) 47 Souce: _The_Bon-Habe_cycle