Langmuir 1995,11, 3598-3600
3598
Robert A. Alberty
The following derivation differs from the Bett-Rowlinson-Saville derivation by starting with the fundamental equation for the Gibbs energy of the whole threephase system.
Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
dG = -S dT V dP pla dnla p Z adnza + pf d n f p l d n t p(1(1dn," p,O dn,O y dA, (1)
On the Derivation of the Gibbs Adsorption Equation
Received December 13, 1994. In Final Form: March 13, 1995
The adsorption at an interface between phases can be treated like a system at chemical equilibrium; that is, chemical work terms are introduced in the fundamental equation of thermodynamics for each species in each phaseel Equilibrium conditions of the form pia = pip for independent equilibria between phases can be used in the same way as the independent conditions for chemical equilibria. The introduction of these conditions leads to the definition of amounts of components, which are independentvariables and can be used as natural variables for the Gibbs en erg^.^-^ The importance of a natural variable is that if a thermodynamic potential can be determined as a function of its natural variables, all of the thermodynamic properties of the system can be determined. When phases are in equilibrium, the fundamental equation for the whole system can be written in terms of amounts of components, rather than species, and this is important because the amounts of components are natural variables. The fundamental thermodynamic equations for systems involving surfaces have been discussed by Rowlinson and W i d ~ m .Bett, ~ Rowlinson, and Savilles have discussed the thermodynamics of surfaces of binary solutions in greater detail. The treatment here differs from that of Bett, Rowlinson, and Saville in that it deals with components, rather than species, and properties of the whole system, rather than properties of the interface. Nevertheless,the same result is obtained for liquid-vapor equilibrium. The system considered consists of a binary liquid solution in contact with its vapor or an immiscible binary solution in a container with variable surface area between the phases. The effects due to the surface ofthe container are ignored. The system is analyzed by use of the Gibbs convention that a plane is drawn through the surface and the compositions of the bulk phases are assumed to be constant up to this surface. The differences between the amounts in the system calculated in this way and the actual amounts are referred to as surface excess amounts and are designated by nP. The surface can be located so that the surface excess amount of one of the species (the solvent) can be set equal to 0, but that is not done in making the derivation. In the Gibbs convention the surface does not have a volume. Nevertheless, the interface is referred to here as a phase because it has amounts of species that have the same chemical potentials as in the bulk phase. The form of the phase rule when the interface is counted as a phase is considered in the Discussion. (1)Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory anddlgorithms;Wiley-Interscience: New York,1982. (2) Beattie, J. A.; Oppenheim, I. Principles of Thermodynamics; Elsevier Publ. Co.: Amsterdam, 1979. 13)Alberty, R. A.; Oppenheim, I. J. Chem. Phys. 1988,89, 3689. (4)Alberty, R. A. Chem. Rev. 1994,94, 1457-1482. ( 5 )Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982. (6)Bett, K. E.; Rowlinson, J. S.; Saville, G. Thermodynamics for Chemical Engineers; MIT Press: Cambridge, MA, 1975.
+
+
+ +
+ +
+
Here ni and pi are the amounts and chemical potentials of species i, y is the interfacial tension, and A, is the interfacial area. The superscripts a and B indicate the two bulk liquid phases, and the CJ superscripts indicate properties of the surface phase. This equation is general and applies even when the surface and bulk phases are not at equilibrium, provided that they are homogeneous, isothermal, and isobaric. If the three phases are at equilibrium, we can write the independent equilibrium relations in the same way as for chemical reactions: pla = plU,pla = pf, pza = pzU,and pza = p8. Since p l a = p1@ = plu = p l and p Z a= pl@= pZu= p 2 , the superscripts on the chemical potentials can be dropped when the system is at equilibrium. Thus, eq 1can be written as
+
+
+
+
+
dG = -S dT V dP pl(dn; d n f dn,") p2(dnZa d n f dn,") y dA, = -S dT V dP pldnlf + p 2 dn,' + y dA, (2)
+
+
+
+
+
where the amounts of the two componentsare represented by nl' and n2':
+ n f + nlu n2/ = nZa+ n,B + n2' n,' = nla
(3)
(4)
At equilibrium, the natural variables for G are T,P , n i , n2', andA.. This is represented by G(T,P , n i , n2',As).The components are independent variables because n i and n i are the amounts added to the system. The amounts of speciesin a phase are not independentvariables because they are determinedby the equilibriumthat is established. In eq 2, the chemical work terms are written in the same way as they would be for a chemical reaction ~ystem.l-~ Integration ofeq 2 at constant T ,P, and compositionyields
G = plnl'
+ pzn2' + yA,
(5)
One way to obtain the Gibbs-Duhem equation for the system is to make the following Legendre transform4 to define a transformed Gibbs energy G that has T, P, pl, p2, and y as its natural variables.
G = G - plnl' - ,uzn2' - yA,
(6)
It is important that this equation containn{ and n2', rather than nla, ma,nip, n28, nl', and nZu,because nl' and n2' are independent variables (natural variables) for the equilibrium system. Comparison of eq 5 with eq 6 shows that G = 0. Taking the differential of G and substituting eq 2 yields 0 = -S dT
+ VdP - n,' dpl- nidp2 - A ,
dy ( 7 )
Since eq 6 is a complete Legendre transform, eq 7 is the Gibbs-Duhem equation for the system,but the advantage of deriving it this way is that eq 6 reminds us to use components in the Legendre transform, rather than species. Thusthere is a relation between the five intensive variables for the system. The corresponding equations
0743-7463/95/24l1-3598$09.00/00 1995 American Chemical Society
Notes
Langmuir, Vol. 11,No. 9, 1995 3599
for a two-phase binary system without a significant contribution for the interface are given by Kirkwood and Oppenheim7in their equation 9.28. Equation 7 is like the Gibbs-Duhem equation for a chemical reaction system at chemical equilibrium,and it provides a relation between the intensive variables of the system, rather than the interface. It is like equation 6.154 of Bett, Rowlinson, and Saville except that it applies to the whole system, and it is written in terms of components. As described by Bett, Rowlinson, and Saville, the chemical potentials of the two species are functions of the temperature and the composition of the a phase, so the differentials of the chemical potential of species 1in the liquid can be written
-SF dT +
(%) &2
+
E) 2
s = nlaSla+ n2'SZa+ S"
(16)
n 2 a ~ l=a n 1'x 2a
(17)
and eq 13 becomes
Since
eq 12 reduces to
Dividing by A, and holding the pressure constant yields Bett-Rowlinson-Saville equation 6.162, d r = -[(S"/A,) - SFrl - S2*r2] dT
+
(9)
T
These equations can be used to eliminate from eq 7 and obtain
A, dy = -(S - nlfSla- n2'S2') dT
(15)
(14)
(8)
T
The correspondingequation for the second speciesin phase a is
dp, = -S2a dT
+ n2' = n2a+ n2" n,' = nla nl"
Glaand dpza
where the adsorptions of the components are given by rl = nla/As and rz = nf/A,. This indicates that two derivatives can be determined experimentally.
+ VdP and
As shown by Bett, Rowlinson, and Saville, the GibbsDuhem equation for the a phase at constant T and P can be used to eliminate (apl/&za)~from this equation. The Gibbs-Duhem equation for the liquid phase can be written
(11)
+ V dP +
where
s = sa+ SP
r = ~~~r~ - XZarl
(22)
can be determined. These approximations do not apply when there is an interface between two immiscible liquids.
Using this equation in eq 10 yields
A, dy = -(S - nl'SF - niS2a)dT
Thus
+ S" = nFSF + n2aS2a+ n,BSt + n j S j + S " (13)
Equation 12is general and applies to a system containing two liquid phases as well as to liquid-vapor equilibrium. However, eq 12 is complicated, so we apply it to liquidvapor equilibrium. In liquid-vapor equilibrium,the vapor phase has a very low density; therefore, the system can be set up so that the amounts of 1 and 2 in the L,? phase can be neglected, leading to (7) Kirkwood, J. G.; Oppenheim, I. Chemical Thermodynamics; McGraw-Hill Book Co.: New York, 1961.
Discussion The Gibbs adsorption equation has been derived by the use of the fundamental equation for the Gibbs energy for the whole three-phase system. The introduction of the equilibrium conditions makes it possible to write the fundamental equation in terms of amounts ofcomponents, which are independent variables for the system, whereas amounts of species are not. Making a complete Legendre transform yields the Gibbs-Duhem equation for the whole system. This equation can be written in terms of the differential amount of a species in one of the bulk phases, and if the density of one of the bulk phases is very low, the Gibbs adsorption equation is obtained in the form derived by Bett, Rowlinson, and Saville. Since the fundamental equation for a system with an interface has a surface work term, the number of degrees of freedom F is given by F = C - p 3, where C is the number of components and p is the number of phases, including the interface. The number of components C is equal to N - R, where N is the number of species (a species in a different phase counts as a separate species) and R
+
Notes
3600 Langmuir, Vol. 11, No. 9, 1995 is the number of independent reactions or independent equilibrium conditions for species between phases. The 3 is used in the phase rule because there are three terms in addition to the chemical terms in eq 7. In the system discussed here, N = 6 and R = 4, so that C = 2 and F = 2 - 3 3 = 2, which can be taken at T and P. Thus the number of degrees of freedom is the same as for the twocomponent liquid system, ignoring the interface. Fixing T and P fixes the compositions of the three phases and the surface tension. There is another way to look at this, and that is to use the phase rule in the form F = C - p 2 and not count the interface as a phase. The Gibbs adsorption isotherm is of interest in connection with mixtures that separate into two liquid phases. The Flory-Huggins thee$ was developed for nonaqueous systems containing polymers. A number of theoretical and experimental advances have subsequentlybeen made in the field, as indicated by the recent studiesby Xi, Franck, and W i d ~ m Aqueous .~ systems containing high polymers such as polyethylene glycols and dextrans may separate
+
+
(8)Flory, P. J. Principles ofPolymer Chemistry; Come11 University Press: Ithaca, NY, 1952;Chapter 12.
into two phases that are useful for separating biologically active macromolecules, as shown by Guan et al.1° The fact that these systems may separate into two phases at as low as 5% by weight ofthe water-solublepolymers raises the question as to whether phase separation may occur spontaneously in a living cell where water-soluble polymers are produced. Such a phase separation would permit the orientation of adsorbed molecules in the interface. Liu et al." have demonstrated phase separation in multicomponent aqueous protein solutions and have shown how to predict the coexisting points along the binodal curve. Acknowledgment. This research was supported by NIH- 1-RO1-GM48358-01Al. LA941002Y (9)Xia, K. Q.;Franck, C.;Widom, B.J.Chem. Phys. 1992,97,14461454. (10)Guan, Y.;Lilly, T. H.; n e w , T. E. J. Chem. SOC.,Faraday Trans. lB93,89 (24),4283-4298. (11)Liu, C.; Lomakin, A.; Thurston, G. M.; Hayden, D.; Pande, A.; Ogun, 0.;Asherie, N.; Benedek, G. B. J.Phys. Chem. 1996,99,454461.
