dimrcamx
April, 1960
OF THE
GIBHSASD GIBBS-DGHEM EQUATIONS
40 1
APPLICATION O F GIBBS AKD GIBBS-DUHEM EQUATIONS TO TERNARY ASD MULTICOMPONENT SYSTEMS BY SEV A. GOKCEX Contribution from the University of Pennsylvania, Philadelphia, Penna. Received June 29, 1969
A simple method of application of the Gibbs equation for the calculation of partial molar properties of components in ternary and multicomponent systems, from the known partial molar property of one component, is presented. It is also shown that Wagner’s, SlcKay’s and Schuhmann’s equations for the calculation of thermodynamic properties of ternary systems may be rederived in a systematic and concise manner either from the Gibbs free energy function or the Gibbs-Duhcm equation, and thnt Wagner‘s and Schuhmann’s methods consist of the integration of the same cross differential equation along the same path. I n addition, these methods are extended to multicomponent systems.
The Gibbs-Duhem equation has been used for deriving the partial molar properties of ternary and multicomponent systenis from the experzmental data The preceding relationship is also applicable to o n the partial molar propertg of one ~ o m p o n e n t . ‘ - ~ the implicit function (Sn, J , y) = 0. Substitution For this purpose. Darken1 obtained an equation of eq. 3 in eq. 2 gives by integration to express the molar property of = a multicompoiieiit system from which the unknown am ay 3~ partial molar properties can be obtained by difThe usefulness of eq. 2 and 4 may be illustrated ferentiation. Wagner,* hlcKay3 and S ~ h u h m a n n , ~ as follows. If 3-i. is a property that cannot be as however, first differentiated the Gibbs-Duhem equation for a ternarg system and then integrated conveniently measurable as m, 5 and g J it can be it to obtain the appropriate equations for the obtained by integrating either eq. 2 or 4, whichpartial molar properties. The purpose of this paper ever is best suited to the available data. Equations 2 and 4,usually called cross differenis (a, to present a simple method of application of the Gibbs :Lnd Gibbs-Duhem equations for the cal- tials, are very useful in deriving numerous thermoculation of partial molar properties of components dynamic relations. As a classical exaniple the from the known partial molar property of one com- derivation of the well-known hlaxwell re1a t’ions ponent, (b) to rederive TT7agner’s, RIcKay’s, and may be cited. Application to Ternary Systems.-Let nl, n2, n 3 Schuhmann’s equations in a simple and concise manner either hy using the Gihbs free energy represent the numbers of moles of components 1, function or the Gihbs-Duhem relation, and to show 2 and 3 in a ternary system; F the corresponding that their equations are based on the same relation- Gibbs free eneigy; p1, ~2 and 1 3 the chemical poship obtained hy cross differentiation, and (c) tentials of components. The complete differential to extend their equations, as such applicable to of the Gibbs free energy a t constant piessure and the ternary systems, to the multicomponent temperature is d F = yi dni pr dnr p3 dna systems. (5) Equations-The equations used in deriving the where p, = d F / b n , at8 constant pressure, tempcranecessary relationships are based on the following ture and ni, n2, . . . . . . ni-l. If the e.rperirncnta1 properties of exact differentials. Let u,m and 3-i. i d u e s of p l at various compositions are known, be some functions of z and g. An expression such and the evaluation of p2 and p3 from pl is desired, as m d X 4-3-i. dy is an exact differential if it is then a suitable cross differential can be used. equal to tlie cunplete differential of a function u; Thus imposing the restriction that n 3 be constant i.e. in eq. 5 and then applying eq. 2 yields
(”>,
(”)
+
du
=
mdx
+ xdy
Since the sequence of differentia tion of 21 i q immaterial
+
(1) IL
with z and
I t is important to note the symmrtrjj of sirbscripts in eg. 5 and 6 , and to renzembcr thnt in obtaining eg. 6 f o r the other chemical potrntials, e . g . , dpa/bnlr all the subscripts 2 and 3 must be interchanged so that eq. 5 remains unchanged. Thi. generalization is it c a n be showii readily that also applicable to all the succeeding relatioiiships and, further, it simplifies the derivation of any equation for multicomponent systems. For any explicit function m of 5 and ?/, i . e . , Since it is more convenient to deal with mole = )“(xtioii,respectively, it. i q pwferatJk t o mbstitute t,he excess partial l i d a r free energy 1;;1 (excess) = R1‘ In y1 or merely In 71 for 1-11 in eq. 11. I n the resulting equation the and 713
+
-
J -
+
403 ni
’ I = n?
+ + n4 + . . . ni n3
(15)
Part’ialdifferentials of nl and n2from these equations can now k’e expressed in terms of x and y and then substituted iiit,o eq. 13. The result is identical with eq. 0. The inclependent variables expressed by eq. 14 and 15 are not unique; any other set would yield an equation eit,tier identical with or similar t,o eq. 9. For cs:an1plc n4
.c = TI]
+ ns + n4 + . .
I
(16)
ni
and n4 nr+n3+n4+
?= , --
(17) . . . ni n-ould yield th9 ssnie equat’ioii as 9. Xevertheless
the choice of eq. 14 and 15 has the merit that these variables make the foregoing treatment general, hence also applicable to a biliary system. Thus for such a system i = 2 aiid x = nz/nl; = 1, or y is no longer a variable, and eq. G assumes the familiar form cl@? = -
d p i / ~=
E’ig. 2.-Ilepresentation of a yuaterriary syhtem by means of a tetrahedron. On the intersecting plane l-%-TV, n d / n lor LY3/.Y4is a constant.
- (ni/n?) dpi
Representation of Multicomponent Systems.Integrat,ion of eq. 9 or 11 for the multicomponent systems requires t’herepresentation of such systems on appropriate coordinates. In this respect, consideration of quaternary systems is helpful in devising a general method. ’The tetrahedron in E’ig. 2 represents a quaternary sy,$tenifor which ey. 14 and lt5may be written as
and
(N,+ N), Fig. 3.-Itcpreseritvtiuri of the scmtion 1-2-11’ in Fig. 2 o n triangular coordinatw. -4t 1V, sum of t h r mule fractiuris arid ‘Y4is unity with a constant value of .\‘.9/AV4 throiighout. Straight, line d-H-(’ is tiirigent to :L cunstmt h,-wrve a t ‘VI.
The chemical potentials are int~ensiveproperties: hence they may be mrittmenas some functioiis of Hence, n3 7z1 may be held constant without making two independent composition variables, or any .u and conitants. Xiiy con\tant ratio n3/n4 two independent mole ratios such as nl;inq and is the same as the ratio of mole fractions N i N , , n*/n3. It is therefore evident that the constancy ;~iicl in the plane 1-2-TI’ this ratio i5 conitant. of n2 aiid 123 for the left-hand side of eq. 20 is equivalent t,o that of )22/n3. Further, t’he eliminaAlong an) liiie 1-T7, (-Y, S,) tion of nl and n9from the right-hand side of ecr. 20 i. also constant: therefore, 9 cordingly, the line 1-T’ repreqento the variation of oiily x along the path of integration. The plane 1-2-17’ may be repreiented on ordinary triangular coordinates as shown in Fig. 3 . The corner ITr of unity for N? S,in the = &TI7 /&IV in Fig. 2 iierit sy-tem, the common con-taiit ratios for r and y arc A T ~ / L ~ IAYd’AYl, , . . . .; where the constancy of 123 on the right hide is iiow therefore 13n the corner TY, all the mole fractioiis redundant since the mole fractions are independent, and 2 must be grouped together ill of 123. Equations 20 and 21 may also he obtained atisfyiiig the constancy of such molar immediately liy applying eq. 4 t o the Gibhsratiob. For each chemical potential p3, p4. . p,, nuhem relatioil arranged in the following exact the appropriate planes of representation may he differential form c~lio~cii i n :L biniilar manner. Other Methods for Ternary Systems.-Tho (‘io++d1ffeielltl;d of tllP typP reprc.cntcd I,v cq 4 5 to oht:rii~ n i ~ yiion lie :tppliccl t o n4
71.4
+
+
(20)
NEV A. GOKCEN
404
and as a result N1/N2 = nl/n2 becomes constant; consequently (bnl/bnz),, or N~ reduces to N , / N 2 . It. will be seen later that eq. 20 is thus completely general, and applicable to any system of any number of components. In Wagner’s treatment the independent variables expressing the chemical potentials are chosen5 as N 1 and y = LjT3/(N2 N 3 ) . Elimination of N z and 1\r3 from eq. 21 by using y and N1 N 2 N 3 = 1, then noting that the constancy of n2/n3is the same as that of y, and multiplying both sides with ( 3 ~ gives
+
+ +
(23)
Dividing both sides with (dNl), and comparing the first term on the right with eq. 3 shows that
Substitution of this relation in eq. 23 yields Wagner’s equation; i.e.
Yol. 64
substituting (br13)., n3 and (br23)?z,,for the corresponding partial differentials of nl and n2 in Eq. 6 and integrating from r13 = 0 to a finite value a t constant T23. However, it was pointed out in connec%ionwith eq. 7 and 8 that r13 = Nl/-v3 and r23 = N z / N ~become infinite v-hen the mole fraction of component 3 becomes zero. Kevertheless McKay’s equation is simple and very useful for the particular solutions wherein the molar ratios are An excellent summary of his ~ ) ~ and the correlation of various thermodymethod namic properties of strong electrolytes are presented by Harned and Owen.’ I n their succeeding treatments, Wagner used an analytical integration of eq. 24, and Schuhmann, a direct graphical integration of eq. 20; in both methods the integration is along the path of coiistant y (or hr2/N3),and from ;\rl in the vicinity of unity to any value of N1. Multicomponent Systems.-The foregoing relationships may now be extended to the multicomponent systems in the follon-ing manner. Imposing the restrictions that 713, n4. . . n, be constant, and then applying eq. 4 to ey. 1%yields
(24)
The corresponding relation for p3 may be obtained either by starting with eq. 20 for p3, or eliminating bpz irom eq. 24 by means of eq. 22; however a much shorter procedure consists of interchanging the subscripts 2 and 3 i n ey. 24 and in y, and then observing that y and d y become 1 - y and - d y , and the last term remains unchanged. The foregoing method makes quite evident the necessity of choosing y as one of the variables because y must be any function of the mole ratio n2/n3which appears on the left-hand side of eq. 21 so that the constancy of y is the same as that of n2/n3. Therefore, any simple variable satisfying this condition may be selected; e.g., N3/N2, or N2/(Nz N3), the latter simply being equal to 1 - y used in deriving eq. 24. However it is convenient to use an expression for y which, unlike lv3/NZ, should remain finite for all values of N 2 and N3. This provides the justification for Wagner’s choice of lV3/(lv2 N a ) . Further when y = 0, eq. 24 becomes applicable to the binary system 1-2, and when y = 1 the relationship for (bP3/bl\;l)y becomes the expression for the binary system 1-3. The foregoing treatment shows that eq. 24 may be obtained in a simple and concise maimer by applying the cross differential 4 either to eq. 5 or to eq. 22 and then substituting the variables and y, and utilizing eq. 3. Further it is seen that eq. 20 is also that obtained by Schuhmann4; hence both methods are based on the same cross-diff erential. It is interesting to note that eq. 20 has been presented earlier by lIcI be ex d u I t t d t i t l i L r q r a i i l i i d l > 01 ai1 I!\ Pcliuhrninn’s intrgration b\ tiif iiietliod of trtngcnt i i i t ( I C L ts ~ 15 p u i i l y (0) €3
(7) H
graphical.
April, 1SGO
, ~ P P L I C A T I O SOF THE
GIBBSA S D GIBBS-DCHEN EQUATIOUS
reduce the system into a binary system so that the resulting equation would assume the simple form. The relationship satisfying these conditions is
It is therefore obvious that for a multicomponent system the numerator in this equation contains i - 2 terms, and the denominator i - 1 terms. I n the tetrahedron of Fig. 2 , representing a quaternary system, the plane 1-2-W is that on which n4/n3= N4/iiV3 = a is a constant. Along any line from 1 to any point T‘, the ratio n2/n3is also a constant. Integration of eq. 26 will therefore be carried out along the line 1-1- on which there is one independent compoqition variable, i.e., N1. From eq. 27, and from LV4/LV3= a,and 2iYi = 1, it followsthat Elimination of ATzand A V 3 from eq. 26 by means of the preceding relations gkes
This relationship is the same as eq. 23 except for the additional restriction n3/n4on both sides; hence it may be transformed into the form represented by cq.24, i.e.
405
surfaces are represented by the constant p1 curves. The intercept of the tangent - A with the edge 1-2 yields, on the basis of the expansion of the right hand side of eq. 26 aiid the values of dNl/d2\lT3 and b.+r2/b~Tr3 from the triangle, the relations
The right-hand side of eq. 31 may thus be evaluated graphically from a plot of the values of eq. 32, us. p l a t constant n2/n3and n3/n4. For the chemical potential p3, eq. 26 assumes the form
Hence the evaluation of bnl/bn3must be made on a plane containing the edge 1-3 and intersecting 2-4, so that the restriction of constant n2/na and n8/n4 on the right side is satisfied. For a ternary system, i.e., N 4 = 0 in Fig. 3, the intercepts A , B and C are such that from the implicit function f(n1,n2, n3)pI = 0 and from eq. 3 they satisfy the relationship
where all segments, except TI’ - Coutside the triangle, are positive. (See also Schuhmann.) Equation 34 constitutesa statementof Jlenelaus aiid Ceva theorems in g e ~ m e t r y . ~ I n a quaternary system the intercepts B and C are of no useful significance since they do not correspond to any relevant partial derivative.’O The relationships parallel to eq. 34 may be obtained by means of the intercepts of a (constant y and nJ/n4) (30) plane tangent to the constant MI-surface in the where the lower integration limit p2 (y, n3/n4, quaternary tetrahedron. In view of the fact that S1-.l) is, in view of vanishing concentrations of all p1 is a homogeneous function of zeroth degree in the solutes, the same as p 2 (Sl+l)and may be ob- number of moles of all components, there are four tained from the binary system 1-2 as pointed out possible implicit functions, and their derivatives in by Wagner. the form represented by eq. 3, i e . For a multicomponent system, obviously there are additional restrictions n4/n5, etc. on both sides .fi(ni,n2,723)pl,na = 0, ( 3 5 ) of eq. 29. The relationships for other chemical potentials can be obtained by interchanging the subscripts in eq. 29 and in the parameter y. Graphical Method.-Equation 25 may be inte= - 1 (36%) grated for a quaternary system by extending the graphical method for the ternary systems described by Schuhmnnn. The integration path is again represented by the constant ratios n2/n3 and n3/n4, i.e., along 1-T‘ in Figs. 2 and 3. Thus, the integration from Arl = h to S I , for a quaternary system is given t q Integration of this equation is then carried out a t constant y and n3/n4from Nl+l to N1 as outlined by Wagner. It is interesting to note that the integration of the left-hand side of eq. 29, after multiplication by bN1, is
(2)(2)(2)
I n a quaternary system, constant u1 is represented by an appropriate surface in the tetrahedron; the intersections of I-2-W with the constant p1-
(9) 0. Veblen and J. W.Young. “Projective Geometry.” Ginn and Co., New York, N. Y., 1918, p. 89. (10) A somewhat parallel situation occurs when a partial molar property GI is obtained from a plot of t h e molar property G v s . .VI. I n a binary system t h e intercepts are GI s n d 02, in a ternary system when G a t a chosen ratio N?/.\‘I is clotted us. N I , the intercept at .\TI = 1 is 81b u t t h e intercept a t .VI = 0 is not a useful thermodynamic property (see eq. 1 in Darken’).
I n eq. 35a-:Ba, p 1 is held constant throughout, arid the n’q other than those in the parentheses are also held constant. A tetrahedron has six edges, each of which, or its extension in space, is generally intersected by a plane. The intercepts of the tangent plane with six sides must obey these equations. Multiplying cq. 8.ia-38a and taking the square root gives
where the derivatives in parentheses represent the molar ratioq st the intercepts on the four edges 1-2, 2-3, 3-4 and 4-1, respertively. Equation 39 reprebents the extension of llenelnus and Ceva theorem-$ t o an equal tetrahedron. Since there are four chemical potentials, there are three more qets of equations similar to eq. 35-38 and three more relations represented by eq. 39. A h o t h e r interesting relationship may he derived by the application of eq. 3 to fs(fLl>n2,n3)nl.n4 /R(lll,fLz,nna)n*
R4
fi(n1,n?,~)n3.n4
= 0 = 0 =
0
other at the point where the roiiitaiit p l , p2 aiid p3-surfaces n l , interPect ~ oiie another at one point. For a ternary system nl = 0 and ,211 tangents are on the same plane. Eqzratzons 3&& arc not essential hit thc!j are ziscfiil in checking the results of the graphical cnlcidations within the accuracij with which fhr tnngcnts ran be drawn. It should br pointed out that the mcthods of f l i e author, Wagner, V c K a y and Schuhmann, are also applicable to any e.ctensiiie property G = f (nl, n 2 . . . . .\, because the complete di$crcntial of such a function has the f o i m represented by q. 12. Consistency of Calculations.-Internal consistcncy of ca1cul:btioiis cnii be tested a i follom. For n ternary system the complete diffcrentinl of the molar free energy. I;, = plYl pL?\-? p&3. i-
+
dF,
(pi
+
- f ~ ? d-Vi ) t (pz - p ) < I S ?
( G)
Cross differentiation of this equation. as shown by cq. 2 . gives
(40) (41) (42)
and multiplying the results side by side and then cancelling out the six derivatives b y means of the relationships similar to eq. 13
For the activity coefficients, this ecpition heconies
In all the foregoing methods, the integration was carried out along the line. from thc coriierq interqecting the opposiiig edge of the trimgle. Equation 47, however, represents the parti‘il differentials along the lines parallel to the edge.; hence it is a useful relationship in testing the internal coilsistency of the calculations by plotting In(nir3) us. .Vz along a chosen line of constant S1,parallel t o 2-3, aiid liltewi-e plotting In(y2 y3) SIat eonqtaiit .V2; at the intersection of coiiqtant XI and S2lilies the slope!: must be equal. .In alternative method consists of the graphical intcgratioii of eq. 47 along the entire leiigth of either :L constant SI or a constant LV2line. A relationship similar to eq. 4 i for the section of the quateriittry system qhowri in Pig. 3 may be obtained hy uiiiig the constant ratio S d / S a = a, and zN, = 1 to elimiiiate N 3 and N a , and then by expresing the coefficients of d-171 and dS2 in eq. 45. Acknowledgment.-This research has been sponsored by the U. S. ,Itomic Energy Commission. Colltlact ,IT (30-1) 1076. 11s.
Sumerous similar relations may be obtained by following a consistent pattern of sequence in the subscripts of variahles as shown in eq. 40-42. Division of eq. 43 by eq. %a yields
(44)
,Igain iiutiiwous such relations readily may be obtained. The quantities in the first set of brackets in the preceding equation refer to the tangents on the constant ns/n4plane, and the others to those on the ronstant iz2;n4plane. I11 eq 4:; and 44 all tangents intersect one an-
