Chapter 7 Vacancies 魏茂國 物理冶金

Transcription

1 Chapter 7 acacies Thermal behaior of metals Iteral eergy Etropy Spotaeous reactios Gibbs free eergy Statistical mechaical defiitio of etropy acacies acacy motio Iterstitial atoms ad diacacies

2 Thermal ehaior of Metals Thermal behaior of metals - Thermodyamics is ot directly cocered with what happes o a atomic scale; it is more cocered with the aerage properties of large umbers of atoms, ad mathematical relatioships are deeloped betwee such thermodyamical fuctios as temperature, pressure, olume, etropy, iteral eergy, ad ethalpy, without cosideratio of atomic mechaisms. It makes computatios easier ad more accurate, but, ufortuately, it tells us othig about what makes thigs happe as they do. - Equatio of state for a ideal gas P RT (7.) where P is the pressure, is the olume, is the umber of moles of gas, R the uiersal (gas) costat, ad T the absolute temperature. o explaatio as to the reasos for the existece of this relatioship is gie. 2

3 Thermal ehaior of Metals - Kietic theory attempts to derie relatioships such as the equatio of state, startig with atomic ad molecular processes. The aerage kietic eergy of the atoms is directly proportioal to the absolute temperature. - Statistical mechaisms applies the realm of statics to heat problems. This approach is feasible because most practical problems iole quatities of matter cotaiig large umbers of atoms or molecules. It is thus possible to thik i terms of the behaior of the group as a whole, as does a life isurace actuary who predicts the ital statistics of a large populatio. - It should be metioed that thermodyamics ad statistical mechaisms are oly applicable to problems iolig equilibrium, ad caot predict the speed of a chemical or metallurgical reactio. This latter is the special proice of the kietic theory. 3

4 Iteral Eergy Iteral eergy ( 內能 ) - Iteral eergy, U, represets the total kietic ad potetial eergy of all the atoms i a material body, or system. I the case of crystal, a large part of this eergy is associated with the ibratio of the atoms i the lattice. - Whe the temperature of the crystal is raised, the amplitudes of the elastic waes icrease, with a correspodig icrease i iteral eergy. The itesity of the lattice ibratios is therefore a fuctio of temperature. 4

5 Etropy Etropy ( 火商, 亂度 ) - I thermodyamics, the etropy chage, S, may be defied by the followig equatio: dq S S S (7.2) T where S is the etropy i state, S is the etropy i state, T is the absolute temperature, ad dq is the heat added to system. The itegratio is assumed to be take oer a reersible path betwee the 2 equilibrium states ad. - The itegral of this quatity (dq/t) oer a irreersible path does ot equal the etropy chage. dq for a irreersible path betwee ad. re S S S T irre ds ds dq T dq T (reersible chage) (irreersible chage) (7.3) (7.4) (7.5) 5

6 Spotaeous Reactios Spotaeous reactios ( 自發反應 ) - Reactios that occur spotaeously are always irreersible. - Liquid water will trasform to ice at -C ( 263 K), but the reerse reactio is, of course, impossible. - It is importat to kow the coditios that brig about spotaeous reactios, ad to hae a yardstick for measurig the driig force of this type of reactio. The yardstick that is most aluable for this purpose is called Gibbs free eergy. 6

7 Gibbs Free Eergy Gibbs free eergy - Gibbs free eergy is defied by the followig equatio: G U P TS (7.6) where G is Gibbs free eergy, U is iteral eergy, P is pressure, is olume, T is absolute temperature, ad S is etropy. - The special symbol H is called the ethalpy. H U P (7.7) G H TS (7.8) -Let G 2 be the free eergy of a mole of solid water (ice), ad G that of a mole of liquid water. Whe a mole of water chages ito ice, the free eergy chage is G G2 G ( H 2 TS2) ( H TS) (7.9) where H 2 ad H are the ethalpies of the solid ad liquid, respectiely, S 2 ad S their respectie etropies, ad T the temperature. G H TS (7.) 7

8 Gibbs Free Eergy - Water freezig to ice at the equilibrium freezig poit, T e, is a reersible reactio, ad uder these coditios we hae see that the etropy chage is gie by S S T e dq T e Q (7.) (7.2) where Q is the latet heat of freezig for water. y the first law of thermodyamics, du dw dq (7.3) where du is the chage i iteral eergy, dw is the work doe o system, ad dq is the heat added to system. - The first law may be writte i terms of the ethalpy istead of the iteral eergy. dh du Pd dp (7.4) where at costat pressure dp is by defiitio zero ad Pd is equialet to dw i Eq

9 Gibbs Free Eergy - I the freezig of water, the oly exteral work that is doe is agaist the pressure of the atmosphere due to the expasio whe water chages from liquid to solid. This ca be eglected because of its small size, so settig dw equal to zero, we obtai dh du Q (7.5) Q dg Q Te Q Q (7.6) Te The free-eergy chage i this reersible reactio (the freezig of water at C) is zero. - If liquid water is ow cooled to a temperature well below 273 K ad allowed to freeze isothermally, the trasformatio will be made uder irreersible coditios. Q S (7.7) T TS Q (7.8) 9

10 Gibbs Free Eergy - The free-eergy equatio tells us that for this reactio G H TS (7.9) where H agai equals Q as i the preious example. If T S is greater tha Q, howeer, G must be egatie. - The free-eergy chage for this spotaeous reactio is egatie, which meas that the system reacts so as to lower its free eergy. This result is true ot oly i the aboe simple system, but also for all spotaeous reactios. spotaeous reactio occurs whe a system ca lower its free eergy. - While the free eergy tells us whether or ot a spotaeous reactio is possible, it caot predict the speed of the reactio. Ex: graphite is the phase with the lower free eergy, ad diamod should trasform spotaeously ito graphite. The rate is so slow that there is o eed to cosider it.

11 Statistical Mechaical Defiitio of Etropy Statistical mechaical defiitio of etropy - Let us take a 2-chamber box, each chamber filled with a differet mooatomic ideal gas. Gas is i chamber I, ad gas i chamber II. Let the partitio betwee the chambers be remoed ad the gases will mix by diffusio. Such mixig occurs at costat temperature ad costat pressure if the gases hae the same origial temperature ad pressure. o work is doe ad o heat is trasferred to or from the gases; therefore, the iteral eergy of the gaseous system does ot chage. This fact is i agreemet with the law of coseratio of eergy (first law of thermodyamics), which is expressed i the followig equatio: dh dq dw (7.2) dh (7.2) where dq = = heat absorbed by the system (gases), dh = chage of the ethalpy of the system (gases), dw = = work doe o gases by the surroudigs. fudametal chage i the system occurs as a result of the diffusio.

12 Statistical Mechaical Defiitio of Etropy - Like the freezig of water at temperatures below 273 K, the mixig of gases is a spotaeous, or irreersible reactio, The free eergy must decrease. dg dh Tds (7.22) dh dg TdS (7.23) decrease i free eergy ca oly mea that ds must be positie. I other words, the etropy of the system has icreased by the mixig of the gases. The etropy icrease ioled i this reactio is kow as a etropy of mixig. Whe the etropy of a system icreases, the system becomes more disordered. - oltzma itroduced the followig equatio: S k l P (7.24) where S is the etropy of a system i a gie state, P is the probability of the state, ad k is oltzma s costat ( J/K). 2

13 Statistical Mechaical Defiitio of Etropy - The chage i etropy (mixig etropy) resultig from the mixig of gas ad may be expressed i terms of the oltzma equatio: S S 2 S k l P2 k l (7.25) P S k l 2 (7.26) P where S is the etropy of umixed gases, S 2 is the etropy of mixed gases, P is the probability of umixed state, ad P 2 is the probability of mixed state. P 3

14 Statistical Mechaical Defiitio of Etropy - The probability of fidig the atoms i the umixed, or segregated, state is computed as follows: Let = the olume origially occupied by the atoms of gas = the olume origially occupied by the atoms of gas = the total olume of the box If oe atom is itroduced ito the udiided box, the probability of fidig it i is /. If a secod atom is ow added to the box, the chace of fidig both at the same time i is ( /) 2. third atom reduces the probability of fidig all atoms i to ( /) 3, ad if is the total umber of atoms of gas, the probability of fidig all i is ( /). Fially, the probability of fidig all atoms of gas i, while all atoms of gas i is P (7.27) where is the umber of atoms, ad is the umber of atoms. 4

15 5 物理冶金 - experimetal homogeeous mixture is ot oe with a perfect costat ratio of to atoms, but oe with a ratio that does ot ary sufficietly from the mea alue to be detectable. oth atom forms, whe preset i ery large umbers, will seek a mea distributio i which the atoms are uiformly distributed i the box. Deiatios from this mea alue will be ery small ad the probability of a experimetally homogeeous mixture extremely high. It ca be assumed, for the purposes of calculatio, that this probability is equal to oe. - Returig to the mixig-etropy equatio gie i Eq. 7.26, P 2 ca be take as uity. Statistical Mechaical Defiitio of Etropy 2 2 l l l P k P k P P k S P k k k k S k S P l l l l l

16 Statistical Mechaical Defiitio of Etropy - Sice we hae assumed ideal gases at the same temperature ad pressure, the olumes occupied by the gases must be proportioal to the umbers of atoms i the gases. ad where is the total umber of atoms of both kids. - The ratios / ad / are the mea chemical fractios of atoms ad i the box, (7.28) (7.29) where is the mole fractio of ad = ( ) is the mole fractio of. S k l S k l k S k l k k l l l (7.3) 6

17 Statistical Mechaical Defiitio of Etropy - If it is ow assumed that we hae oe mole of gas, the the umber of atom becomes equal to where is ogadro s umber. R k (7.3) where R is the uiersal gas costat (8.3 J/mol) ad is ogadro s umber. k k R - We therefore write the mixig-etropy equatio i its fial form S k S R S R l l l k l l l (7.32) 7

18 相變態 Gibbs Free Eergy of iary Solutios - The free eergy of the system will ot remai costat durig the mixig of the ad atoms ad after step (2) the free eergy of the solid solutio G 2 ca be expressed as G 2 G G mix (.2) G G 2 G H H mix G 2 G mix TS TS 2 G H H H T S S H mix = H 2 H : heat absorbed or eoled durig step 2, S mix = S 2 S : differece i etropy betwee the mixed ad umixed states. 2 mix 2 TS mix Gmix Hmix TSmix (.2) 2 8

19 相變態 Ideal Solutios Ideal solutios -Whe H mix =, the resultat solutio is said to be ideal. Gmix H mix TSmix Gmix TSmix (.22) - I statistical thermodyamics, etropy is quatitatiely related to radomess by the oltzma equatio, i.e. S k l (.23) k: oltzma s costat, : a measure of radomess. - There are two cotributios to the etropy of a solid solutio- a thermal cotributio S th ad a cofiguratioal cotributio S cofig. If there is o olume chage or heat chage durig mixig the the oly cotributio to S mix is the chage i cofiguratioal etropy. efore mixig, the ad atoms are held separately i the system ad there is oly oe distiguishable way i which the atoms ca be arraged. k l - Stirlig s approximatio If l! l S S mix S 2 9

20 相變態 Gibbs Free Eergy of iary Solutios - ssumig that ad mix to form a substitutioal solid solutio ad that all cofiguratios of ad atoms are equally probable, the umber of distiguishable ways of arragig the atoms o the atom sites is cofig : umber of atoms, : umber of atoms. - Sice we are dealig with mol of solutio,!!! (.24) a : ogadro s costat. a a a 2

21 相變態 l l l l l l l S cofig cofig cofig cofig cofig cofig cofig ( l Gibbs Free Eergy of iary Solutios l l l a a a a a l l l l )! l(!! a a a a l a a l l a a l l a a l l a l a a l l a l a l l l R l l 2 k lcofig ak S l l mix l a )! l Smix R( l l ) (.25) R = a k: uiersal gas costat. a l! l a! G T mix S mix G RT l l ) < (.26) mix ( Fig..9 Free eergy of mixig for a ideal solutio. 2

22 acacies acacies - Metal crystals may cotai may defects. Oe of the most importat is called a acacy. - Diffusio i crystals is explaied, i terms of acacies, by assumig that the acacies moe through the lattice, thereby producig radom shifts of the atoms from oe lattice positio to aother. The basic priciple of acacy diffusio is illustrated i Fig I order to make the jump, the atom must oercome the et attractie force of its eighbors o the side opposite the hole. Work is required to make the jump ito the hole, or, as it may also be stated, a eergy barrier must be oercome. () () (C) Fig. 7. Three steps i the motio of a acacy through a crystal. 22

23 acacies - The higher the temperature, the more itese the thermal ibratios, ad the more frequetly are the eergy barriers oercome. acacy motio at high temperatures is ery rapid ad the rate of diffusio icreases rapidly with icreasig temperature. - Let us assume that i a crystal cotaiig atoms there are acat lattice sites. The total umber of lattice sites is, accordigly, +, or the sum of the occupied ad uoccupied positios. - Suppose that acacies are created by moemets of atoms from positios iside the crystal to positios o the surface of the crystal (Fig. 7.2). Whe a acacy has bee formed i this maer, a Schottky defect is said to hae bee formed. () () Fig. 7.2 The creatio of a acacy. (C) 23

24 acacies - Let the work required to form a Schottky defect be represeted by the symbol w. crystal cotaiig acacies will hae a iteral eergy greater tha that of a crystal without acacies by a amout w. The free-eergy icremet may be writte as follows: G H TS (7.33) where G is the free eergy due to acacies, H is the ethalpy icrease due to the acacies, ad S is the etropy due to the acacies. H w G w TS (7.34) - ow the etropy of the crystal is icreased i the presece of acacies for 2 reasos. First, the atoms adjacet to each hole are less restraied tha those completely surrouded by other atoms ad ca ibrate i a more irregular or radom fashio tha the atoms remoed from the hole. Each acacy cotributes a small amout to the total etropy of the crystal. The other etropy form arisig i the presece of acacies is a etropy of mixig. 24

25 acacies - Let us desigate the ibratioal etropy associated with oe acacy by the symbol s. The total icrease i etropy arisig from this source is s, where is the total umber of acacies. While a cosideratio of this ibratioal etropy is importat i a thorough theoretical treatmet of acacies, it will be omitted i our preset calculatios. - The etropy of mixig for the mixig of 2 ideal gases is expressed by the equatio Sm S k l l (7.35) where S m is the mixig etropy, is the total umber of atoms ( + ), k is the oltzma s costat, is the cocetratio of atom = /, ad ( - ) is the cocetratio of atom = /. 25

26 acacies - If there are occupied sites ad uoccupied, the umixed state will correspod to oe i which a lattice of + sites has all positios o oe side filled ad all o the other empty (Fig. 7.3). The correspodig mixed state i the box is show i Fig ( ) where is the cocetratio of acacies, ad is the cocetratio of occupied lattice positios. () () Fig. 7.3 The box aalogy of a crystal. () acacies ad atoms i the segregated state. toms to the left, acacies to the right. () The mixed state. 26

27 27 acacies 物理冶金 l l l l l l l l l l l l l l l k S k S k S k S k S m m m m m (7.36) l l l kt w G TS w G m (7.37) - This free eergy must be a miimum if the crystal is i equilibrium; that is, the umber of acacies ( ) i the crystal will seek the alue that makes G a miimum at ay gie temperature. s a result, the deriatie of G with respect to must equal zero with the temperature beig held costat.

28 28 acacies 物理冶金 kt w e kt w kt w kt w d dg kt w G / l l l l l l l (7.38) kt w e / (7.39) where w is the work to form oe acacy i J, k is the oltzma s costat i J/K, T is the absolute temperature i K, is the umber of acacies, ad is the umber of atoms.

29 acacies - If both the umerator ad the deomiator of the expoet of Eq are multiplied by, ogadro s umber ( ), the equatio will be ualtered with respect to the fuctioal relatioship betwee the cocetratio of acacies ad the temperature. H f w R k e w/ kt e w/ kt where H f is the heat of actiatio; that is, the work required to form oe mole of acacies, i joules per mole, is the ogadro s umber, k is the oltzma s costat, ad R is the gas costat = 8.3 J/molK. e H f / RT (7.4) 29

30 acacies - Ex: the experimetal alue for the actiatio ethalpy for the formatio of acacies i copper is ~83,7 joules per mole of acacies. H f / RT / 837/8.3T 72 T e e e t absolute zero the equilibrium umber of acacies should be zero. 72/ t 3 K, approximately room temperature e e 72/3 e Howeer, at 35 K, six degrees below meltig poit, e 72/35 e e Therefore, just below the meltig poit there is approximately oe acacy for eery atoms

31 acacies - Fig. 7.4 shows the cures of the fuctios G, w, ad -TS m as fuctios of the umber of acacies. I this figure, the temperature is assumed close to the meltig poit. t low cocetratios the etropy compoet (-TS) icreases ery rapidly with, but less ad less rapidly as grows larger. t the alue marked i the figure, the 2 quatities w ad -TS become equal, ad, at this poit, the free eergy G, the sum of ( w-ts), equals zero. I the regio of egatie free eergy, a miimum occurs correspodig to the cocetratio marked e i the figure. This is the equilibrium cocetratio of acacies. Fig. 7.4 Free eergy as a fuctio of the umber of acacies,, i a crystal at a high temperature. 3

32 acacies - Fig. 7.4 shows a secod set of dashed lie cures for a lower-temperature cures (T < T ). The lowerig of the temperature does ot chage the w cure, but makes all ordiates of the -TS cure smaller i the ratio T /T. s a result, both ad e of the free-eergy cure (G ) are shifted to the left. Thus, with decreasig temperature, the equilibrium umber of acacies becomes smaller because the etropy compoet (-TS) decreases. Fig