Application of thermodynamics to interlacial phenomena

Transcription

1 Pure & App!. Chem., Vol. 59, No., pp , 987. Printed in Great Britain. 987 IUPAC Appliation of thermodynamis to interlaial phenomena Douglas H. Everett Department of Physial Chemistry, Shool of Chemistry, University of Bristol, BS8 TS, U.K. Abstrat This paper sets out to identify some of the soures of onfusion whih have arisen in the past onerning the appliation of thermodynamis to interfaial phenomena and to indiate their origin and how they an be resolved. Various approahes to the onept of 'surfae exess properties' will be outlined and their role in relating observed phenomena to theoretial models will be disussed. Attention is direted to some of the outstanding problems in the field. Although there is an extensive literature on the thermodynamis of interfaes, there still seems to be a need to larify ertain aspets. To establish a perspetive and to explain the relationship between the approahes used by different authors, it is neessary to restate some well-known results and to indiate how they may be arrived at by alternative routes. The first objetive, as in all lassial thermodynami treatments, is to desribe systems involving interfaes in terms of experimentally observable quantities, and then to derive equations whih enable one to relate the properties of a system under one set of onditions to those in different irumstanes. The equations obtained may be used in several ways They may enable the thermodynami onsisteny of experimental tehniques and methods of analysis of data to be heked, or make it possible to derive from one set of experimental data, information on a system whih, while in priniple observable, may be diffiult or inonvenient to obtain by diret experiment. For example, for a liquid/vapour interfae it is exeedingly diffiult to measure adsorption aurately, but this information an be obtained from surfae tension measurements as a funtion of solution omposition. Conversely, for liquid/solid or vapour/solid interfaes, diret measurement of surfae tension is generally impossible, but values relative to a standard state an be alulated from readily performed adsorption measurements. Furthermore, the analysis of experimental data in thermodynami terms very often presents a piture whih is strongly suggestive of a partiular moleular dynami interpretation, and hene leads to the development of statistial mehanial theories whih, to a lesser or greater extent, provide an interpretation of the thermodynami parameters. Thermodynami relationships again have two roles. For example it is often simpler to derive theoretial equations for adsorption effets from whih surfae tensions an be derived for omparison with experiment. Seondly they provide a means of heking theories for thermodynami onsisteny. We reall first that in the bulk thermodynami desription of multiphase systems, the system is represented by a subdivision of the spae it oupies into volumes, separated by geometrial boundaries, the omposition and other intensive variables being uniform within eah volume, or phase. This desription is entirely adequate provided that the areas of the interphase boundaries do not hange, and/or that the fration of the system within a few moleular diameters of a boundary is negligibly small. When the interfaial area beomes very large (e.g. when one phase is subdivided into regions having linear dimensions less than about pm) interfaial effets play a dominant role as they do for example in the domain of olloid siene. In these irumstanes bulk thermodynamis beomes invalid and the observed properties are found to depend on the interfaial area. In moleular terms this is beause moleules near a boundary are subjeted to fores different from those in the interior of a bulk phase, and make different ontributions to the thermodynami properties of the system. The differene between the behaviour of a system as predited by bulk thermodynami arguments, in whih the intensive properties of eah phase are supposed to remain onstant up to the phase boundaries, and the observed behaviour is a measure of the influene of the presene of interfaes: it is thus possible to define 'exess quantities' (whih may be positive or negative) whih quantify the interfaial ontributions to the properties of the system. 45

2 46 D. H. EVERETT The first of these is the surfae or interfaial tension whih in the framework of ontinuum thermodynamis is a onsequene of the fat that the isotropi hydrostati pressure in the bulk fluid is perturbed lose to an interfae and has to be replaed by a stress tensor (Ref.l). The differene between the mehanial properties of an interfae alulated on the assumption that the bulk hydrostati pressures in the bulk phases adjoining the interfae remain onstant to that interfae, and the observed mehanial properties is measured by the surfae tension (a). The position of the interfae whih satisfies these mehanial onditions is the 'surfae of tension'. It follows that the interfae perturbs the energy of the system by an amount GA5 where A5 is the area of the interfae onerned. Suh an interpretation is assoiated with the so-alled 'quasi-thermodynami' desription of a system and in essene an be traed bak to Thomas Young (Ref.2). An alternative phenomenologial justifiation for the inlusion of the GA term in energy equations omes from the Laplae equation relating the pressure differene aross a urved surfae to the surfae tension and the urvature of the surfae. This may be illustrated by a simple example, presented by Defay and Prigogine (Ref.3). We onsider a spherial drop of liquid of radius r, and volume V, suspended and in equilibrium with vapour, of volume v, (gravity is negleted) ontained in a ylinder of total volume V at a pressure g The pressure within the drop, p, is given by the Laplae equation. The work done in an infinitesinl ompression is then dw = = = dv + (p = - p dv + (2'/r)dV (from Laplae's equation) = - pdv + () The work done on the system an then be split into three terms, two arising from volume hanges of the bulk phases, and the third from the hange in interfaial area. A more omplex ase is that of a liquid onfined by rigid solid walls (Ref.4). Making use of Gauss' equation for the effet of a shift of the 2/v interfae on the 5/v, v/s and s/ interfaial areas the ontributions of the surfae area hanges to the energy of the systej are shown to b9 of the form Y x (area). These arguments are to be preferred over the more onventional justifiation for the inlusion of a surfae term whih requires one to arry out 'thought-experiments' with surfae pistons; only in the speial ase of the Langmuir trough do these have a real meaning. In even more general terms one may start by asserting that it is self-evident that the independent variables desribing a system must inlude the interfaial area, and define Y as the intensive fator onjugate with the area. A thermodynami approah requires us also to be able to desribe the material state of the system ontaining interfaes. Again we onsider the effet of the interfaes on the material omposition by omparing the ontent of different substanes alulatedqas though the interfaes had no effet, i.e. assuming onstant bulk ompositions () up to the boundary between phases and, with the amounts of these substanes atually present in the system (n). These differenes are alled 'surfae exess amounts' (n) n. n -n. -n. n. -.V -.V 2 where and V are the volumes of the two phases. A diffiulty arises, however, beause while the phrase 'up to the phase boundary' sounds innouous, it turns out that in real physial systems the values to be asribed to the surfae exess amounts are extremely sensitive to the loation hosen for the surfae defining the interfae and hene to the values of Va and For the above definition to have any operational meaning it would be neessary to loate this dividing surfae with an experimentally unattainable preision. As is well known this problem was first addressed by Gibbs (Ref.5), although over a entury later his method of solution, employing a so-alled Gibbs dividing surfae (G.D.S.) is still widely misunderstood, and in the opinion of some, the ause of major diffiulties. It is said to have 'bedevilled the student' and been a 'soure of endless onfusion'(ref.6); and by others to imply an impossible physial situation. We shall first onsider briefly the Gibbs method, and then two alternative approahes whih do not appeal diretly to the onept of a dividing surfae. and onsider F. = n/ae, the For simpliity we onsider a plane interfae of area A, surfae exess onentration or areal surfae exess (Lf.7). The diviáing surfae is plaed a distane z from an arbitrary plane whih is parallel to the physial surfae. It is readily seen that the hange in the value of F if the dividing surfae is moved by a

3 Appliation of thermodynamis to interfaial phenomena 47 distane z in the diretion of the a-phase is given by SF. = ( )Sz. = Ad5z. (3) Graphs of F and F. as a funtion of the hosen position z of the G.D.S. therefore have the form shown in Figure. ( ) 2 F (n) 2 z (n) z Figure The problem is to define a proedure for loating the G.D.S. whih an be diretly related to experimental quantities and provide an unambiguous desription of the material state of the system. One way of doing this is to loate the G.D.S. at zat whih F = 0. The value of F at is then alled the relative adsorption of i with respet to omponent F(). Simple geometry then shows that if we do not know where to loate the G.D.S., but make an arbitrary hoie then we an still alulate F2 through the equation: F - F i A (4) where F. and F are defined with respet to the same but arbitrarily hosen G.D.S. It is perhaps'this dual interpretation of FS' whih sometimes auses onfusion. The experimental determination of F' follows immediately by writing eqn. () for omponent i and, and taking these with a third equation V = Va+V. By elimination of Va and V from these three equations, and rearranging we obtain 0() = AF' = A[F.F. -] = ni] [ - i]v] = [(n. - V) - (nl - V) (5) and F are All quantities on the right hand side are diretly measurable so that experimental quantities,provided A is known,independent of the hoie of G.D.S. Figure suggests an alternative definition onvenient when a binary system is onsidered. We may then hoose the G.D.S. at zsuh that F2(n) = - F(n) Alternatively, if F and F2 are defined with respet to an arbitrary G.D.S. = A2 2 '2A (6) * More generally we hoose lf = 0

4 48 D. H. EVERETT where F = F + F2 and A ( ) - (+). omponent 2. is alled the redued adsorption of The equation for n' in terms of experimental quantities is then A n' = AF = (n. V) - (n V). (7) Sine the experimental measurement of and ny(n)involves no mention of a dividing surfae, it is relevant to ask whether he formal'definitions of these quantities need to involve suh surfaes. Before disussing this point we go a little further in developing the thermodynamis. Other surfae exess quantities are defined in an analogous way e.!;. o U =U-uV oaa -uv, (8) where and are the energy densities in the bulk phases. Thus starting from the basi equation for the whole system. du = TdS - pdv + OdA + J.dn., (9) i=l and subtrating the orresponding equations for the bulk phases we have (remembering that V v + V)* du = dh0 = Td50 + OdA +. (0) a. = U0 of ourse depends on the hoie of the position of the G.D.S. Following the onventional proedure of defining a Gibbs energy of the surfae = H0 - TS0 () and integrating the resulting differential, do0, at onstant intensive quantities leads to G = GA + J.n. (2) il whih on division by A gives = a +if. (3) Sine 0 is a physial quantity, while F. depends on the hoie of the G.D.S., depend on this hoie. must also It is important to stress that 0 is not equal to the surfae Gibbs energy per unit area (areal Gibbs energy), exept in the speial ase of a one omponent system when, hoosing the G.D.S. suh that F 0, o = go(l), the relative areal Gibbs energy. Following the same mathematial proedure as that used to derive the Gibbs-. Dubem equation for bulk systems, we obtain its surfae analogue, the Gibbs adsorption equation: do = - -, (4) i=l \0 0 where s = S /A, or at onstant temperature, do Z i= F.Ii., (5) * Sine no volume term appears in eqn. (0) there is no distintion between surfae energy and surfae enthalpy, nor between the Helmholtz and Gibbs surfae free energies.

5 Appliation of thermodynamis to interfaial phenomena 49 the Gibbs adsorption isotherm. If the surfae is in equilibrium with both bulk phases, the dli. '5 are not all independent but must satisfy the bulk Gibbs - Duhem equations for the adjaent phases. When this ondition is imposed it follows that do = F (6) i=2 where is now the relative adsorption of i with respet to. Alternatively we ould have obtained (6) more simply from (5) by deiding to hoose the G.D.S. suh that = 0 and so reduing the number of independent variables by one. The Gibbs adsorption isotherm in the form of eqn. (6) is the fundamental equation from whih all surfae thermodynami properties an be derived. Let us now examine alternative approahes whih do not involve diret mention of a dividing surfae (Ref. 8), limiting the disussion, for simpliity, to a binary system. The Gibbs-Duhem equation for the whole system is SdT - Vdp + Ado + ndp + n2dl2 = 0, (7) where n and n2 are the total amounts of omponents and 2, while, for the individual phases the Gbs-Duhem equations an be written in the intensive forms, by dividing through by V0 and V respetively, ox x a s dt - dp + dl + 2dlJ2 = 0, (8) os dt-dp+dp+2dj2o, (9) where,a, onentrations. are the entropy densities in the bulk phases and et. are the bulk We onsider isothermal onditions, multiply eqn. (8) by x, eqn. (9) by y and subtrat them from eqn. (7) giving - (V-x-y)dp + AdO + (n-x-y)di+ (n2-x-y)dl2 = 0 (20) Here x and y are introdued as arbitrary multipliers. However, if we hoose x and y to satisfy the onditions x+y=v (2) and (n-x-y) = 0 (22) then AdO = - (n2-x-y)dp. (23) If eqn. (2) and (22) are solved for x and y and the results inserted in equation (23) we obtain Ado = - (n2-v)- (n-v) J dp2. (24) The term in square brakets is seen immediately to be n as defined in eqn. (5). We thus rederive eqn. (6). Although x and y are initially arbitrary, eqn. (2) requires them to subdivide the total volume into two regions in just the same way as implied by a dividing surfae, while the ondition (22) further restrits the volume subdivision to be that whih (f. eqn. ()) makes the adsorption of omponent zero. While some authors (Ref. 9) have preferred to use this method, it would appear that the onept of a dividing surfae is introdued impliitly rather than expliitly. The redued adsorption is obtained by the same proedure exept that ondition (22) is replaed by x(+) + y(+) = n + n2, (25) whih is just the ondition that x and y be hosen so that the total adsorption is zero.

6 50 D. H. EVERETT The most general phenomenologial approah is that presented by Wagner (Ref. 9 ). We start from the fundamental eqn. (9) and set up a harateristi funtion GW = H - TS - p.dn. (26) so that dg = - SdT + Vdp + ada + pdn - ndi.. C i=2' (27) Cross-differentiation with respet to A and p. then gives r r r n. = - L. (28) LASP i = L ] The last term is therefore the amount of omponent i whih has to be added to the system to maintain the intensive state of the system onstant when, keeping n and onstant, the area is inreased by da5. The Gibbs adsorption isotherm then appears immediately in the form: n. ia = dp., (29) i=2 s. int, where the subsript mt means that the intensive variables are onstant. An alternative definition of the relative adsorption is therefore: n ' = [;--].. (30) int, n,p.. Here the onept of a dividing surfae appears to be ompletely absent. The appliation of the above definition may be illustrated in the ase of adsorption at the solid/liquid interfae. If we hoose the surfae of the solid as G.D.S., then provided the solid is impermeable to all omponents of the liquid, ti. = and eqn. (7) redues to T(n) o. 2 o n =n n =n nx. (3) 2 Z We note that the total volume V disappears from this expression. If the experiment is onduted by taking an amount n of solution of initial mole fration x2, then n2 fl X2 and nh)= n0ax, (32) where Ax x - x, the hange in mole fration when the solution is ontated with the solid and adsorption equilibrium set up. Alternatively, applying the same proedure to eqn. (5) one obtains T() 0 0(n) n = n L\x2/x = n2 /x. (33) The operational appliation of equation (30) involves an experiment in whih, after ontating the solution with the solid, x is returned to its initial value x by the addition of an amount Ln2 of omponent 2: i.e. the initial intensive state is re-established after the surfae area has inreased by - A 5 Now n2 = n x + An; n = no + An2 (34) and n2 0(n) = n x2 + - An2 (n + L\n2)x2 = - An2( x) = An2x (35)

7 Appliation of thermodynamis to interfaia! phenomena 5 Thus G(l) = G(n) 0 n2 n2 /x = Ln2 (36). = 2 (37) and F is seen to equal the amount of omponent 2 whih has to be added for unit inrease in surfae area to re-establish the original mole fration. We note that the total volume of the system is not involved, nor is it neessary to know the initial amount of solution, n' A further interesting point arises in the ase of liquid/solid systems. It has been argued that for a -omponent liquid mixture in ontat with a solid, the system should be regarded as a (+l)-omponent system. Then, in general, if an arbitrary hoie of G.D.S. is made, there will, formally, be a surfae exess, or defiit, (Fe) of solid. In partiular, for the hoie of a surfae whih makes F, zero, F5 will not be zero. The diffiulty is readily overome by hoosing by G.D.S. whih makes F5 = 0 i.e. the surfae of the solid. (n) Then da = - E F dp.. (38) i i=l However, from the Gibbs-Duhem equation so that d. = - E xdp x i=2 i = i=2' - F)]diil= Fdii.. (39) Whether we regard the systeri as a -omponent system under the influene of an external field, or a (+l)-omponent system inluding the solid, the same result is obtained. In the following attention will be limited to the liquid/solid interfae. The objetive of experimental studies should be to establish via measurements of F!m) or r) and integration of equation (6), the funtion G(x,T), G - x F (n) x2i2 d(x2y2) (40) x2l where is the value of a2 for solid in ontat with pure omponent 2 and is the ativity oeffiient of 2 in the solution. When this has been done it is then possible to obtain the orresponding enthalpy and entropy funtions. Defining surfae enthalpies and entropies through the equation one obtains = jy(n) - = a + (4) 3(a/T) ja(n) (h - h') (42) L (/T)j 2 2 x2 where h and h are the partial molar enthalpies in the bulk liquid. It may then be shown (Ref. ) that the right hand side is the enthalpy of immersion of unit area of solid in a volume of mixture large enough for the resulting hange in onentration to be negligible: [J = = a(n) - F(h - h) (43) Here again ii(m) is seen to depend on the standard states hosen for the enthalpies. Similarly, the entropy of immersion is given by: x2 = =,G(n) - Fh)(s - s) = a + TAh (44)

8 52 D. H. EVERETT Although Ah is a diretly measurable experimental quantity, only differenes in J are aessible via equation (40). Comparison of enthalpies of immersion derived from alorimetri measurements and adsorption measurements must therefore be made through the equat ions or [(a_)/t,\* Ah2 - Ah (45) L 2(l/T) * * 2, [ (G2 )/T,* = Ah2 - Ah (46) [ 3(l/T) The latter equation may be used if enthalpies of immersion are available only for the pure liquids. Finally having established the relationship between experimental quantities, the question remains of the theoretial interpretation of the quantities so derived. At present only relatively simple theoretial models are available. The ultimate objetive will be to be able to alulate the profile of loal omposition of the liquid phase as a funtion distane from the solid surfae. If this an be established as a funtion of temperature then the above thermodynami equations will allow theoretial estimates to be made of the surfae exess quantities (-G) Ah, for omparison with experiment. However, it is more usual, and often more reliable, to ompare the measured surfae exess isotherms with those derived from a theoretial model. However, it is less easy to derive, in a general ase, theoretial values for the enthalpy of immersion, sine are has to be taken to establish the standard states for the enthalpies of the two omponents. It is important to stress that monolayer models annot be regarded as anything more than very rude approximations appliable only to near-ideal systems. If the influene of intermoleular potentials or moleular size differenes are to be taken into aount, then some form of multilayer theory must be developed sine monolayer models are then thermodynamially inonsistent. So far those available are based on or equivalent to lattie models and learly will need refinement before they an be applied to real systems. Attention must also be drawn to the fat that most existing models take no expliit aount of entropy effets arising from hanges in the moleular partition funtion aused by modifiation of the rotational degrees of freedom of adsorbed moleules. That these are important is beoming inreasingly lear from experimental studies (Ref. 2). The heterogeneity of the solid surfae also plays a major role in determining the adsorption behaviour of real systems, and muh work still has to be done to provide an adequate theoretial basis for the analysis of suh systems. REFERENCES. e.g. S. Ono and S. Kondo, in Enylopaedia of Physis (S. Fliigge, Ed) 960, 0, Thomas Young, Phil.Trans.Roy.So., London, 805, 95, R. Defay, I. Prigogime, A. Bellemans and D.H. Everett, Adsorption and Surfae Tension, Longmans, London, D.H. Everett, Pure Appl.Chem., 980, 52, J.W. Gibbs, Colleted Works., Vol., Longmans Green, New York 928, p F.C. Goodrih, in Surfae and Colloid Siene (E. Matijevi and F. Eirih, Eds) Wiley- Intersiene, New York Vol., 969, p.. 7. IUPAC Manual: Reporting Data on Adsorption from Solution at the Solid/Solution Interfae, Pure Appl.Chem. 986, 58, R.S. Hansen, J.Phys.Chem., 962, 66, 40; P.C. Goodrih, Trans. Faraday So., 968, 64, 3403; G. Shay, in Surfae and Colloid Siene (E. Matijevi and F. Eirih, Eds) Wiley-Intersiene, New York, Vol.2, 969, p F. Seelih, Monats.Chem. 948, 79, 338; C. Wagner, ivah.akad.wiss.g3ttingen II, Math.Phys. Xl. 973, G. Shay, J.Coll.Interfae Si., 973, 42, 478. S. Sirar, J. Novosad amd A.L. Myers, I and EC Fundamentals, 972,, 249; C.E. Brown and D.H. Everett in Colloid Siene, Speialist Periodial Reports (D.H.Everett, Ed) The Chemial Soiety, London, 975, Vol.2, hap.2; D.H. Everett and R.T. Podoll, in Colloid Siene, Speialist Periodial Reports (D.H. Everett, Ed) The Chemial Soiety, London, 979, Vol.3, hap.2.. D.H. Everett in Adsorption from Solution (R.H. Ottewill, C.H. Rohester and A.L. Smith, Eds) Aademi Press, London, 983, p..