I. PROKS. Institute of Inorganic Chemistry, Slovak Academy Bratislava. Received 2 January 1977

Size: px

Start display at page:

Download "I. PROKS. Institute of Inorganic Chemistry, Slovak Academy Bratislava. Received 2 January 1977"

Garry Payne
5 years ago
Views:

1 Calculation of thermodynamic excess functions in binary systems on the basis of isobaric phase equilibria and calorimetric measurements of changes in enthalpy I. PROKS Institute of Inorganic Chemistry, Slovak Academy Bratislava of Sciences, Received 2 January 1977 Dedicated to Professor M. Gregor, DrSc, Corresponding Member of the Slovak Academy of Sciences, on his 75th birthday he general procedure of calculation of mixing parameters AS* ix and AG^U in binary systems the components of which do not form compounds is presented. he excess functions can be calculated in this way assuming that the phase diagram of the system in question is known and that the corresponding enthalpies of mixing and of phase transitions are known in the chosen temperature range as well. В работе дан общий ход вычисления функций смешивания ^S, ix и AG^ix в бинарных системах компоненты которых не образуют соединения. Избыточные функции можно рассчитать этим способом только в случае известной фазовой диаграммы данной системы и известных соответствующих теплот смешения и теплот фазовых превращений в избранном температурном интервале. On the basis of thermodynamic excess functions AH^u, AS^ix (and therefore also AG^ix) of a binary system and from their isobaric temperature functions the values of some energetic and structural properties of the system can be calculated [1, 2] and compared with those of model solutions [3]. Knowing the isobaric temperature functions of heats of mixing and isobaric phase diagram of the studied system the thermodynamic molar excess functions can be calculated as follows. 1. Calculation of excess entropy of a binary solution which coexists with crystals of component A or B. It is assumed that the concentration of the second component in the crystals is very low and that the components do not form compounds. Chem. zvesti32 (4) (1978) 433

2 I. PROKS Let us have two phases a and ß which coexist in equilibrium. he components A and В are present in both phases. hen it holds and (л А (х, Т ец (х, y)) = [i A,ß(y, Т сц (х, у)) (1) ß B. a (x, Т^(х, у)) = (л В ф(у, cq (x, у)) (2) x and у are equilibrium mole fractions of component В in the phase a and ß, respectivelynd Т сц (х, у) is the temperature at which both phases are in equilibrium (Fig. 1). he phase a is a solution for which in the concentration range 0 x cut and at coexistence of both phases хшу and in the concentration range *cui 1 У =x (*eut being the mole fraction of component В in the eutectic solution). If the concentration у of component В in the phase ß is very low, the equilibrium between the almost pure crystal of the component A and the solution may be approximately described by the relation (for x^x eut ) (Лл (х, cq = (г Аф (0, Т сц (х, 0)) (3) p '. (0.0) /í Г. ч (1.1) Тц(*.у) т t \ ß i i i Л ~х(у) Fig. 1. Phase diagram of a binary system A B. his relation can be used for calculation of excess entropy of a solution at the temperature exp (the temperature for which the experimental data are known). From eqn (3) it follows 434 Chem. zves i 32 (4) (1978)

3 HERMODYNAMIC EXCESS FUNCIONS H A (x, eq - eq (x, 0) Š A (x, eq = or = Н Аф (0,Т ец (х, 0))- eq (x, 0) S A,,(0, C4 (A:, 0)) (4) Q ( Y (, rm Ш^1 ľz *2 Q))-^A.,(0, cc, ЬлАх, le q {x,ö))- cq (x,0) + S Aß (0, e4 (x,0)) (5) he value of the partial molar entropy of component A in the solution at the temperature exp equals č / r ч_рл (х, cq -Я А,Д0, eq C (x,) + ~ A -" K f ''ď (6) l cq (x,0) and thus for the partial molar entropy of mixing it follows ^^ггпх, А,<Д-» * expj ^A.aV,-^? -* expj ^A.o^i * expj _Я А (*, eq -Я А,р(0, r eq + ^ -d- S A.ß(0, eq + 4p ^d + Ařf tr, A ( eq (0,0)) Г«р guo) 1 + a i eq (0,0) г J J f4(ao) U J he term in brackets equals S A (0, exp ). ЛЯ, г. А (Т сч (0, 0)) is the change in enthalpy at the phase transition of ß in a at equilibrium temperature. After rearranging eqn (7) we obtain (for x?áx eut ) ДО.HAJX, -Я,Д0, (Y eq А cq ^*kj mix,a,a\-*'? * exp/ rj-t /-ч\ ' «p č A, (x,) H г т -ч<»-»> c A, ß (o, т) ДЯ,, (Т н т г А сс,(0, 0)) / eq (jt,0) * JC4(X.Ü) /Т ем (х.(» i Xeq^U, VJ) т»- Сд. а (0, Г) d (8) eg(0, 0) he right side of eqn (8) can be expanded by adding zero differences of the quantities; then Chem. zvesti 32 (4) (1978) 435

4 I. PROKS ^^mix.a.«^? -»exp,) HAJX, Т СЦ (Х, 0))-Н А,Д(), Т СЦ (Х, 0)) + Я А (0, eq (jc, 0))-Н А, н (0, Ľq Т сч (х, 0) + - <,,0) C A,,(0,) d + Г-^ > C A, t,(q,) d r_ fv' 0 ) C A (Q,) d _ ^H, r, A ( Ľq (0,0)) f -p С А, ц (0,Г) 7^(0,0) J ei((, 0) Q i * ^ After arithmetic operations we obtain the final relationship ~, ЛН,. (х, + Aff, ( л ( = ты А а eq tr A cq ХАДХ expj ' Г еч (х, 0) _ - - Ač mlx.ux, ) d+ f > > лс, г, А (о,т) d_ Т сч (дс.о) * J C4 (X.<» 4H, r, A ( eq (0,0)) Гея(0, 0) (10) In the same way we can calculate the partial molar entropy of mixing of component В in the range хшх си nd at the temperature exp. (he same assumption about coexistence of almost pure substance В with melt as in the former case has to be fulfilled.) *. н Лх,тихЛ))-НвАит (ха)). (Y V в сц ъ^ымлх, icxpj- Ľq (x,l),' - č B Ax,) d r^(1,) c B,,(i,) d M lr, fl (r eq (i,i)) *.l) J C4 (JC.1) cq (l,l) or c B (i,) d ( n ) 1) * ~ л f rj, v _^H,, (x, (x, 1)) + ЛЯ,,в(Т mix B a eq г СС) (дс, 1)) ^^mix.b.ul^' * exp) rp / л \ I r - 4č mix, B, (*,r) d r^1-, Mc, r, B (i,) d ля (г, в (т еч (1Д)) (12) J C4 (X.I) * J C4 U.I) eq (l,l) 436 Chem. nesti 32 (4) (1978)

5 HERMODYNAMIC EXCESS FUNCIONS At the eutectic temperature both almost pure crystal components A and В coexist in equilibrium with a eutectic solution and therefore it is possible to calculate partial molar entropies of mixing. for both components (AŠ mix. Afl (x eut, cxp ) and /15 m i X.B. a (x eu,, ex p). he values AŠ mixm. tt (x, Г схр ) for x ^jc cut and AŠ mix. A. (/ (x,up) for хшх см are calculated using relevant integrated Gibbs Duhem equations. For jc =jc eut Aömix_B.uyX, i exp) A j mix.b.«(.^eut - ' exp) - ( ^«M-^W 1-х das mix. A (x, cxp ) J^S mix. A, (x ĽU,. Ľxp ) X (/3) and for x^x ev Aj m i x, Ajt \.X, 1 expj ^-i'jmix.a.tí^cut i -»exp/ r "^Š AS,,x.B<*-. (JC. cxp> 1 Г - r 3-dzlS m, x.b. < Xx,r i;xp ) (/4) he excess entropy of a solution of components A and В at the temperature cxp is given by the relation AS* ix. a (x, exp ) = AS mix. a (x, cxp ) - AS mix(id), (*, ĽXP ) = = (1 -x) AŠ mix. A. (í (x, Ľxp ) +xaš mix,b, u (x, Гехр) + + R[(l-jc)ln(l-jc) + jclnjc] (Í5) 2. Approximate calculation of excess entropy at the formation of liquid phase a in a binary system A В the components of which in the phase ß form solid solutions with limited solubility but do not form any compounds. he equilibrium between phases a and ß is described precisely by eqns (/) and (2) from which the following relationships can be derived and H A Ax,^(x, y))-^(x,y) S A Ax, eq (x,y)) = H A4} (0, eq (x,y))- - Ľ4 (;t, у) S A.p-(0, cq (x,y)) + ЯТ сч (х,у) In а Аф (у, cq (x,y)) (16) Я в (х,т сч (х,у))-т сч (д:,у)5 в. и (л:,т сч (х,у)) = Я в,д1,г еч (х,>;))- - L4 (x,y)s ß.,(l,r c4 (x,y)) + Ä 04 (x,y)lna ß.,(y,r eq (x,y)) (/7) Chem. zvesti32 (4) (1978) 437

6 I. PROKS If the value of у in the phase ß with prevailing component A (resp. В) is near to zero (resp. to one) the activity coefficient may be considered in both cases as unit and the activity of component A (resp. В) in the phase ß with prevailing component A (resp. В) may be replaced by mole fraction of the component. hen treating eqns (16) and (17) in the same way as described in the previous paragraph we obtain for л:= х еи1 л <г / v x.^h A (x, cli (x,y))-h Aß (0, C4 (x,y)) ^^mix(fts).a.a \ X > l exp) ~ (Y v} ( - Č AM (x, ) d f -.<»-"> C A,,(0,) d AH^jJOM Т еч (д:.у) "» J^X.y) * Т сц (0,0) -Г С±фТ±аТ-R In z( eq (x,y)) (18) J,. 4 (O.<>) -Í where z = l y. For JC =JC eu t л^ / v w HB. (*,ľ cq (*,y))-h B, ß (1,Т ец (х,у)) t-*ijmix(lis).b.«yx, -» exp) y^ ч 1" + r- č B(< (x,) d f - (U,) CsAU) d ЛН 1ГМ (ТМЛ)) )т сц (х.у) ]т с (.х,у) eq (l,l) Г- c Ba (i,r) d _ i?ln^(r^>;)) (i9) j e4 (l,l) i he quantities having the index (ßs) denote the system in which a solution with low solubility of components is formed. Further treatment is similar as in the paragraph 1. Eqns (18) and (19) can be written in the simplified form; e.g. for ^ ^ mix(lis).b.<t( X > I exp) AŠ mixm), BM (x, Г ехр ) = ÁŠ mix.b.a(x, cxp ) - R In y( eq (x,y)) (20) 3. Calculation of AG ты- he values of ágl are calculated using the definition equation ( exp = const) AGl ix = 4HL - cxp z\s* ix = AH mix - exp 4S^ix (21) Eqns (8, io 12, 15, 18 20) can be derived also from the temperature dependence of the changes in Gibbs energy, or more efficiently, from the Planck function Ф 438 Chem. zvesti 32 (4) (1978)

7 HERMODYNAMIC EXCESS FUNCIONS and AG mx, (t (x,) дт элф тш1 (х,ту дт _AH m U*,) 2 (22) длф 1г (ту дт cq Э Г AGŮV]' = R дт - eq dln a ß ЭТ _ eq AH ir () 2 (23) he second relationship is a special case of the van't Hoff reaction isobar (the LeChatelier Shreder equation). 4. Measurements of AH mix. he excess quantities AS^, ix and AG ix (AH^, X = AH mix ) can be calculated using eqns (15 21) assuming that the phase diagram of the system, heats of mixing and heats of phase transitions are known in the studied temperature range For the systems in which mixing of pure components is sufficiently fast AH mix can be measured directly at chosen temperature using calorimetry for mixing Dealing with the systems which do not obey the condition of sufficiently fast mixing AH mix must be determined using the following relation AH mix (x, exp ) = H a (x, cxp )-(í -x) Я А (0,Г ехр )-хяв (1,Т схр ) = = H rc, (x, ĽXp )-(l -x) Я гс1. А (0,Т схр )-^Я ге1, в.а(1,г С]! р ) (24) where all quantities H Cl are increments in enthalpy measured with respect to the convenient reference state (which is the same for all phases) Dealing with the systems in which equilibrium is quickly achieved after cooling the convenient reference state can be the mechanical mixture of crystals of pure components at laboratory temperature. his condition is fulfilled e.g. in the case of some systems consisting of inorganic salts. he values Я гс, can be measured in this case using drop calorimetry For the systems in which equilibrium after cooling is not achieved in reasonable time (e.g. metal alloys, systems known from silicate technology) as Chcm. zvt-sti32 (4) (1У7К) 439

8 I. PROKS standard state can be recommended the state of the system in an appropriate solution or in a mixture of its combustion products at chosen temperature, etc. In this case the values H rc, in eqn (24) must be replaced by the quantities *~»rel = (^-"ccx)! I /j/i so (comb)) \2j ) 4H ax,i is the change in enthalpy measured at cooling the sample in drop calorimeter and 4Я 5(), спть) is heat of dissolution (or combustion) of the sample which was cooled up to laboratory temperature in the drop calorimeter. In this experimentally most complicated case heat of mixing can be obtained by the method of "double calorimetry of the same sample", i.e. using drop calorimetry and subsequently determining heat of solution or combustion. References 1. Prigogine, I. and Defay, R., Chemical hermodynamics, p Longmans, Green, London, Malinovský, M., Chem. Zvesti 25, 92 (1971). 3. Holm, J. L., Chem. Zvesti 30, 759 (1976). ranslated by P. Fcllner 440 Chem. zvesti 32 (4) (1978)

Determination of Molar Gibbs Energies and Entropies of Mixing of Melts in Binary Subsystems of the System CaO Si0 2. Si0 2

Determination of Molar Gibbs Energies and Entropies of Mixing of Melts in Binary Subsystems of the CaO Si CaO Al Si CaO Al Si I. Theoretical Part. CaO Al Si CaO Al Si I. PROKS, J. STREČKO, L KOSA, I. NERÁD,