Document 6516263
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Document 6516263
Journal of Engineering Science and Technology Review 1 (2008) 100-150 Journal of Engineering Science and Technology Review 1 (2008) 100-150 Journal of Engineering Science and Technology Review 1 (2008) 100-150 JJ estr estr estr Journal of Engineering Science and Technology Review 1 (2008) 1-3 Journal of Engineering Science and Technology Review 1 (2008) 1-3 Journal of Engineering Science and Technology Review 1 (2008) 1-3 Lecture Note Lecture Note Lecture Note What is a surface excess? What is a surface excess? What is a surface excess? A. Ch. Mitropoulos* JOURNAL OF JOURNAL OF Science and Engineering Engineering Science JOURNAL OF and Technology Review Technology ReviewScience and Engineering Technology Review www.jestr.org www.jestr.org www.jestr.org A. Ch. Mitropoulos* A. Ch. Mitropoulos* Department of Petroleum Technology, Cavala Institute of Technology, St. Lucas 65 404 Cavala. Department of Petroleum Technology, Kavala Institute of Technology, St. Lucas 65 404 Kavala. 19Technology, October 2007; Accepted 14 January 2008 St. Lucas 65 404 Kavala. Department ofReceived Petroleum Kavala Institute of Technology, Received 19 October 2007; Accepted 14 January 2008 _______________________________________________________________________________________________ _______________________________________________________________________________________________ Received 19 October 2007; Accepted 14 January 2008 _______________________________________________________________________________________________ Abstract Abstract Abstract J. Willard Gibbs in his pioneering work on the influence of surfaces discontinuity upon the equilibrium of heterogeneJ.ous Willard Gibbs in his pioneering work on the of surfaces discontinuity upon the equilibrium heterogenemasses suggested for the measurement of influence the quantities of a system a geometrical surface dividingofthe interfacial ous masses suggested for the measurement of the quantities of a system a geometrical surface dividing the interfacial J. Willard Gibbs in his pioneering work on the influence of surfaces discontinuity upon the equilibrium of heterogenelayer. Surface excess is the difference between the amount of a component actually present in the system, and that layer. Surface excess is the difference between the amount of a component actually present in the system, and that ous masses suggested for the measurement of the quantities of a system a geometrical surface dividing the interfacial which would be present in a reference system if the bulk concentration in the adjoining phases were maintained up to which wouldSurface be present in a is reference system between if position the bulk concentration the adjoining phases were maintained up to layer. excess the determined difference thedividing amountsurface. of a incomponent actually present in the system, and that the arbitrary chosen but precisely in the arbitrary butpresent precisely position dividing surface. which chosen would be in determined a reference in system if the bulk concentration in the adjoining phases were maintained up to the arbitrary chosen but precisely determined in position dividing surface. Keywords: surface excess, Gibbs adsorption. Keywords: surface excess, Gibbs adsorption. _______________________________________________________________________________________________ _______________________________________________________________________________________________ Keywords: surface excess, Gibbs adsorption. _______________________________________________________________________________________________ volumes Vα α+Vβ β=V each one assumed to contain CαiαVα α=nαiα and Adsorption of a component at the phase boundary of a system β +V =V each αone assumed to contain Ci V =ni inand volumes Adsorption of a component at the phase boundary of a system moles αwithβ Cαi and Cβiβ the bulk concentrations the α CβiβVβ β=nβiV results to a different concentration in the interfacial layer than α α V =n moles with C and C the bulk concentrations C results to a different concentration in the interfacial layer than +V =V each one assumed to contain Cin =ni and volumes V Adsorption of a component at the phase boundary of a system i i i i i Vthe real system [5], respectively. More formally: that in the adjoining bulk phases. The adsorbate density and β β β α β real system [5], respectively. More formally: that in the adjoining bulk phases. The adsorbate density and V =n moles with C and C the bulk concentrations in the C results to a different concentration in the interfacial layer than i i i i composition profiles within that layer cannot be measured by composition within thatamounts layer cannot measured by and real system [5], respectively. More formally: that inprofiles the adjoining bulk phases. The be adsorbate density today’s technology; the actual adsorbed are not meantoday’s technology;profiles the actual amounts adsorbed are not meanwithin layerthis cannot be measured ingfulcomposition experimental variables. To that resolve problem Gibbs by ingful experimental variables. To resolve this problem Gibbs today’s technology; the actual amounts adsorbed are not mean[1] suggested that: “…it will be convenient to be able to refer [1] that: surface, “…it will be convenient to bethis ablecoincident to refer Gibbs ingful experimental variables. To be resolve problem to suggested a geometrical which shall sensibly towith a geometrical which shall be sensibly [1] that: “…it be convenient tocoincident be have able toa refer the suggested physicalsurface, surface of will discontinuity, but shall with the physical surface of discontinuity, but shall have a to a geometrical surface, which shall be sensibly coincident precisely determined position. For this end, let us take some precisely determined Forofthis end, letofus takeshall somehave a the discontinuity, but point with in or veryphysical nearposition. to surface the physical surface discontinuity, point in or very near to the position. physical surface ofend, discontinuity, precisely thisthrough letthis us point take some and imagine a determined geometrical surface toFor pass and imagine a geometrical surface to pass through this point point in or very near to the physical surface of discontinuity, and all other points which are similar situated with respect to and other points are matter. similar situated withthrough respectthis to point and imagine geometrical surface to pass the all condition of theawhich adjacent Let this geometrical surthe condition of the adjacent matter. Let this geometrical surand all other points which are similar situated with respect to face be called the dividing surface.” face be called the dividing surface.” the condition of the adjacent matter. Let this geometrical surSurface excess [2] is the difference between the amount of Surface [2] the difference betweenand the amount of face beexcess called theispresent dividing a component actually insurface.” the system, that which awould component actually present in the system, and that which Surface excess [2] is the difference between the amount be present in a reference system if the bulk concentra- of would be present in actually a reference system bulk concentraa component in ifthethesystem, and that which tion in the adjoining phases present were maintained up to a chosen tion inwould the adjoining phases were upthe to a interface chosen be present in a reference if concentrageometrical dividing surface [3]; maintained i.e. system as though thebulk geometrical surfacephases [3]; i.e. as maintained though the up interface individing the adjoining were to a chosen had notion effect. Schematically: had nogeometrical effect. Schematically: dividing surface [3]; i.e. as though the interface had no effect. ]surface==[n[ni]]real−Schematically: (1) − [n i ]reference . Figure 1. Concentration profiles of a binary system as a function of dis[n[ni i]surface [ni ]reference . (1) Figure 1. Concentration of aBold binary system as ainfunction of disi real tance normal to the phaseprofiles boundary. curved lines, both frames, are normal theConcentration phase boundary. Boldand curved lines, ininboth are of dis[ni ]surface =σ[ni ]real − [ni ]reference . (1)tance the concentration profiles of theprofiles solute solvent the real system, Figureto 1. of athe binary system asframes, a function where [ni]surface=nσi is the surface excess amount [4] of compo- the concentration profiles of the solute and the solvent in the real system, respectively, and again vertical lines are curved the concentrations the are tance normal to the phase broken boundary. Bold lines, in bothinframes, =ni is the surface excess amount [4] of compowhere [ni][n surface respectively, and again broken lines thethe concentrations the nent (i); i]real=ni is theσ total amount of that component in the reference (beingvertical actually thethe extend ofand the bulk concentrations upsystem, the system concentration profiles of soluteare solvent in theinreal a β =n amount that component thecomponent [ni]real[nand =ni total is =n the surfaceofexcess amount [4]inof where i is reference systemsurface). (being the extend of the i]surface to the dividing Chainvertical dotted lines indicate theconcentrations boundaries of up the in the real (i); system; [nthe respectively, and actually again broken linesbulk are the concentrations i]reference ai +nβi is the amount of the same tointerfacial the dividing surface). Chain indicate thethe boundaries the ]=n +n the amount theequal sametoin the real system; and ]arealireference theisystem total amount of thatofcomponent nent (i); [n reference i is layer. Bold horizontal linelines isthethe dividing surface and ofdotted i is =n reference system (beingdotted actually extend of bulk concentrations up component (i) ini[n having volume V a β interfacial layer. Bold horizontal line is the dividing surface and dotted of the component (i) in a reference system having volume V equal to horizontal line is another choice for the location of the dividing surface. to the dividing surface). Chain dotted lines indicate the boundaries ] =n +n is the amount of the same real system; and [n i reference byi a hypothetical i the real system which is divided surface into horizontal line is another choice for the location of the dividing surface. surface excess is the sumhorizontal of the shaded above and underand the dotted interfacial layer. Bold line isareas the dividing surface the realcomponent system which bysystem a hypothetical surface V into (i) inisadivided reference having volume equal The toThesurface excessline is the sum inofchoice the case shaded areas aboveof and under the dividing surface. Notice that the ofthe the solvent by choosing thesurface. horizontal is another for location the dividing the real system which is divided by a hypothetical surface into ______________ dividing surface. Notice that in the case of the solvent by choosing the upper dividing surface (a suitable one)of thethe resulted is zero, The surface excess is the sum shadedsurface areas excess above and under the upper dividing surface (a Notice suitable one) excessbyis choosing zero, whereas by choosing the lower one surface excess not zero. dividing surface. that inthe theresulted case ofissurface the solvent the whereasupper by choosing lower (a onesuitable surfaceone) excess not zero. dividingthesurface theisresulted surface excess is zero, ______________ * E-mail address: [email protected] * E-mail address: ISSN: 1791-2377 © [email protected] 2008 Kavala Institute of Technology. All rights reserved. ______________ ISSN: 1791-2377 © 2008 [email protected] Institute of Technology. All rights reserved. * E-mail address: ISSN: 1791-2377 © 2008 Kavala Institute of Technology. All rights reserved. whereas by choosing the lower one surface excess is not zero. 1 1 11 A. Ch. Mitropoulos/ Journal of Engineering Science and Technology Review 1 (2008) 1-3 A. Ch. Mitropoulos/ Journal of Engineering Science and Technology Review 1 (2008) 1-3 A. Ch. Mitropoulos/ Journal of Engineering Science and Technology Review 1 (2008) 1-3 niσσ = ni − C iaaV aa − C iββ V ββ . ni = ni − C i V − C i V . A fundamental equation for surfaces of discontinuity befundamental equation surfaces ofisdiscontinuity betweenA fluid masses at constantfortemperature given by Gibbs tween temperature is given by Gibbs having,fluid in themasses case ofataconstant binary system, the form: having, in the case of a binary system, the form: (2) (2) If As is the area of the interface [6], the areal surface excess of defined the interface If As is the area as: [6], the areal surface excess concentration Γiσ is concentration Γiσ is defined as: niσσ Γiσσ = ni . Γi = As . As dσ = −Γ1σσ dμι − Γ2σσ dμ 2 . dσ = −Γ1 dμι − Γ2 dμ 2 . (3) (3) (6) (6) where σ is the surface tension, and μ1 and μ2 are the chemical and μ2 are chemical where σ isofthethesurface tension, andBy μ1deciding potentials adjacent phases. to the choose a diσ phases. By deciding to choose a dipotentials of the adjacent viding surface such as Γ1σ=0, eq.(6) takes the form: viding surface such as Γ1 =0, eq.(6) takes the form: In a binary system, where (1) is e.g. the solvent and (2) is In a binary where (1) is e.g. the solvent and divid(2) is the solute and bysystem, (΄) is denoted a second choice for the the surface solute and (΄) is can denoted a second ing (seebyFig.1), be easily shownchoice that: for the dividing surface (see Fig.1), can be easily shown that: dσ = −Γ211 dμ 2 . dσ = −Γ2 dμ 2 . (7) (7) According to this equation [9] for a system involving a solvent According to there this equation [9] for a system involving aofsolvent and a solute, is an excess surface concentration solute (4) and a solute, there is anthe excess surface concentration of solute if the solute decreases surface tension and a deficient sur(4) if theconcentration solute decreases the surface and a deficient surface of solute if the tension solute increases the surface face concentration of solute if the solute increases the surface tension. tension. Since the location of the dividing surface was arbitrary chosen, Direct measurements of surface excess quantities were Since the location of can the dividing chosen, Direct measurements of surface excess[10] quantities werea the above equation only be surface true if was eacharbitrary side separately carried out by several investigators. McBain constructed the above equation berelative true if adsorption each side of separately carried out by several investigators. McBain [10] constructed equals a constant [7];can thatonly is, the compomicrotome device consisted of a sharp blade, Salley et al. [11]a equals a constant [7]; that the relative(1): adsorption of compo-a microtome device consisted of a sharp blade, is Salley et al. [11]a nent (2) with respect to is, component Γ211. Obviously, developed a tracer method where the solute labeled with Obviously, a nent (2) with respectoftoΓ12component (1): Γ2 . way developed a tracer method where the solute is labeled with exists too [8]. One to detersymmetric definition radioisotope, and Smith [12] studied the ellipticity of lighta toothe [8]. One way to detersymmetric definition Γof21 Γis12 toexists radioisotope, [12]ofstudied the ellipticity of light mine experimentally locate dividing surface at a reflection fromand theSmith thickness an adsorbed film on mercury. 1 mine experimentally to locate the dividing surface atthea reflection from the thickness of an adsorbed on mercury. reference system contains position where Γ1σσ=0;Γ2i.e.is the Application of Gibbs adsorption isotherm onfilm electrolyte solu=0; i.e. the(1)reference system the position whereofΓ1component Application of Gibbs adsorption isotherm on electrolyte solusame amount as the real one. contains In this case tion was introduced by Wagner [13]. same (1) as the one. where In thistocase tion was introduced Wagner [13]. is real not clear loeq.(4) amount reduces of to component Γ211=Γ2σσ. If however In this note an by elementary review on the concept of sur1however is not clear where to lo=Γ . If eq.(4) reduces to Γ 2 2 this was note given. an elementary review out, on the concept of even surcate the dividing surface, Γ21 may still be calculated by rearface In excess It must pointed however, that may still be calculated by rearcate the dividing surface, Γ 2 face excess was given. It must pointed out, however, that even ranging eq.(4) such as: in the frame of formal surface thermodynamics the characterisranging eq.(4) such as: in frame ofinvolved formal surface thermodynamics the characteristic the quantities need further specification in order to β a tic quantities involved need further specification in order for to a a became operational and again need different specification n 2 − C 2a V n1 − C1aV C 2a − C 2β 1 became operational and again need different specification for (5) Γ21 = n 2 − C 2 V − n1 − C1 V × C 2a − C 2β . each different type of interface (e.g. sold/gas [14], etc). (5) Γ2 = − × C1a − C1β . As As each different type of interface (e.g. sold/gas [14], etc). As As C1 − C1 Acknowledgments: The author would like to thank Archimedes Acknowledgments: The author would to thank Archimedes The quantities on the right hand side of eq.(5) are all directly research Project and INTERREG III like “Hybrid Technology for The quantities on the right hand side of eq.(5) are all directly research Project and INTERREG III “Hybrid Technology for measurable, provided that As is known. Separation” for funding this work measurable, provided that As is known. Separation” for funding this work ______________________________ ______________________________ References and Notes References and Notes ⎡ C 2a ⎡⎢ C a Γ2 − Γ1 ⎢⎣ C12aa ⎢⎣ C1 Γ2σσ − Γ1σσ − C 2ββ − C 2β − C1β − C1 ⎤ ⎡C a ⎤⎥ = Γ2′σσ − Γ1′σσ ⎡⎢ C 22a ⎥⎦ = Γ2′ − Γ1′ ⎢⎣ C1aa ⎥⎦ ⎢⎣ C1 − C 2ββ − C 2β − C1β − C1 ⎤ ⎤⎥ . ⎥⎦ . ⎥⎦ 5. 5. 6. 6. For liquid/vapour interfaces, at low vapor pressures, Ciβ, where β is For liquid/vapour at low vapor pressures, Ciβ, where β is the gaseous phase, interfaces, may be neglected. the gaseous may be neglected. Interface is phase, the plane ideally marking the boundary between two Interface is the plane ideally between phases. Interphase, however, is amarking differentthe termboundary which refers to thetwo inphases. however, is a different term refers to the interfacialInterphase, layer; being the inhomogeneous spacewhich region intermediate terfacial layer; being the ininhomogeneous spaceproperties region intermediate between two bulk phases contact, and where are signifibetween two bulk phases in contact, andproperties where properties are phases. significantly different from, but related to, the of the bulk cantly different from, but to, the properties When the interfacial layerrelated is regarded as a phaseofitthe is bulk calledphases. interWhen interfacial is regarded as a phase it is called interphase; the IUPAC: 58, 439layer (1986). 1 58, 439 (1986). 7. phase; Since ΓIUPAC: form the location of the dividing surface it is 2 is independent 7. possible Since Γ21tois dispense independent the location of the dividing it is the form geometric interpretation of excesssurface quantities possible to dispense the geometric interpretation of excess quantities and formulate a suitable algebraic method; i.e. without explicit referand a suitable algebraic i.e. and without explicit referenceformulate to a dividing surface. For bothmethod; geometric algebraic methods ence to a dividing surface. For both geometric algebraic methods see: Ref.2; R.S.Hansen, J.Phys.Chem. 66, 410 and (1962); F.C.Goodrich, see: Ref.2; R.S.Hansen, 410 (1962);inF.C.Goodrich, Trans. Faraday Soc. 64, J.Phys.Chem. 3403 (1968); 66, F.C.Goodrich, Surface and Trans. 64, 3403and (1968); F.C.Goodrich, in Surface and ColloidFaraday Science,Soc. E.Matijevic F.R.Eirich (eds), Wiley, New York Colloid Science, I.Prigogine, E.Matijevic A.Bellemans, and F.R.EirichD.H.Everett, (eds), Wiley, New York (1969); R.Delay, Surface Ten(1969); I.Prigogine, A.Bellemans, D.H.Everett, Surface Tension andR.Delay, Adsorption. Longmans, London (1966); R.J.Good, Thermosion and Adsorption. Longmans, London (1966); R.J.Good, dynamics of adsorption and Gibbsian distance parameters in Thermotwo and dynamics adsorptionPure and Appl. Gibbsian distance parameters two and three-phaseof systems, Chem. 48, 427 (1976);inR.J.Good, three-phase systems, Pure Appl.and Chem. 48, 427 (1976);Parameters: R.J.Good, Thermodynamics of Adsorption Gibbsian Distance Thermodynamics of Adsorption and Gibbsian The Pressure Coefficient of Interfacial Tension Distance in TernaryParameters: Two- and The Pressure Coefficient Interfacial TensionSci. in Ternary and Three-Phase Systems, J.ofColloid Interface 85, 128Two(1982); Three-Phase Systems, J. Colloid Interface 85, Distance 128 (1982); R.J.Good, Thermodynamics of Adsorption and Sci. Gibbsian PaR.J.Good, Thermodynamics of Adsorption and Gibbsian rameters: Interfacial Distances and the Surface ExcessDistance Volume,PaJ. rameters: Interfacial and theR.J.Good, Surface J. Excess Volume, J. Colloid Interface Sci.Distances 85, 141 (1982); Colloid Interface Colloid Interface Sci. 85, 141 (1982); R.J.Good, J. Colloid Interface 1. J.W.Gibbs, The Collected Works of J. W. Gibbs, Longmans, Green, 1. J.W.Gibbs, Collected J. W.also: Gibbs, Longmans, Green, New York, The 1931, Vol. I, Works p. 219.of See E.A.Guggenheim and New York, 1931, Vol. I, p. 219. See also:A139, E.A.Guggenheim and N.K.Adam, Proc.Roy.Soc. (London) 218 (1933); N.K.Adam, Proc.Roy.Soc. A139, (1933); E.A.Guggenheim, Trans.Faraday(London) Soc. 36, 397 (1940);218 R.C.Tolman, E.A.Guggenheim, Soc. 36, 397J.Chem.Phys. (1940); R.C.Tolman, J.Chem.Phys. 16, Trans.Faraday 758 (1948); R.C.Tolman, 17, 118 J.Chem.Phys. 16, 758 (1948); R.C.Tolman, J.Chem.Phys. 17, 253 118 (1949); E.H.Lucassen-Reynders, Prog.Surf.Membrane Sci. 10, (1949); E.H.Lucassen-Reynders, Sci. 10, 253 (1976); G.Schay, A comprehensiveProg.Surf.Membrane presentation of the thermodynam(1976); G.Schay, excess A comprehensive presentation of the48, thermodynamics of adsorption quantities, Pure Appl. Chem. 393 (1976); ics of adsorptionPhysical excess quantities, Chem. 48, 393 A.W.Adamson, ChemistryPure of Appl. Surfaces, Wiley, New(1976); York A.W.Adamson, Physical Chemistry of Surfaces, Wiley, New York (1982). 2. (1982). D.H.Everett, Definitions Terminology and Symbols in Colloid and 2. D.H.Everett, Definitions Symbols in D.H.Everett, Colloid and Surface Chemistry, Pure Terminology Appl. Chem. and 31, 577 (1972); Surface Chemistry, Pure Appl. Chem. 31, 577 (1972); D.H.Everett, Pure Appl. Chem. 53, 2181 (1981); D.H.Everett, in Colloidal DisperPure Chem. 53, 2181 (1981); Colloidal London Dispersion, Appl. J.W.Goodwin (ed), The RoyalD.H.Everett, Society of in Chemistry, sion, J.W.Goodwin The Royal Society of Chemistry, London (1982); D.H.Everett,(ed), Reporting Data on Adsorption from Solution at (1982); D.H.Everett, Reporting Data on Adsorption from Solution at the Solid/Liquid Interface, Document prepared for the IUPAC Comthe Solid/Liquid Interface, Document prepared for the IUPAC Commission meeting in Copenhagen (1985); D.H.Everett , Reporting data mission meeting in solution Copenhagen D.H.Everett , Reporting data on adsorption from at the(1985); solid/solution interface, Pure Appl. on adsorption solution at the solid/solution Pure Appl. Chem. 58, 967from (1986); D.H.Everett, Applicationinterface, of thermodynamics Chem. 58, 967 (1986); D.H.Everett, Application thermodynamics to interfacial phenomena, Pure Appl. Chem. 59, 45of (1987). interfacial Pure Appl. Chem.(GDS) 59, 45 (1987). 3. to Also referredphenomena, as Gibbs dividing surface or Gibbs surface; 3. Also referred Gibbs dividing surface (GDS) or Gibbs surface; IUPAC 31, 588as(1972). 588 is (1972). 4. IUPAC Surface 31, excess an algebraic quantity and may be positive (excess) 4. Surface excess is an algebraic quantity as andGibbs may adsorption: be positive (excess) or negative (deficiency). Also referred IUPAC: or (deficiency). as Gibbs adsorption: IUPAC: 31,negative 588 (1972). A formerAlso term referred for the surface excess was superficial 31, 588 (see (1972). A in former term for the surface excess was superficial density Gibbs Ref.1). density (see Gibbs in Ref.1). 2 2 2 A. Ch. Mitropoulos/ Journal of Engineering Science and Technology Review 1 (2008) 1-3 A. Ch. Mitropoulos/ Journal of Engineering Science and Technology Review 1 (2008) 1-3 8. 9. 10. 11. Monomolecular Layers, American Association for the Advancement of Science, Washington (1954). See also: K.Tajima, M.Muramatsu, and T.Sasaki, Bull.Chem.Soc. Japan 43, 1991 (1970); K.Tajima, Bull.Chem.Soc.Jap. 43, 3063 (1970); K.Sekine, T.Seimiya, and T.Sasaki, K.Tajima, M.Muramatsu, and T.Sasaki, Bull.Chem.Soc.Japan 43, 629 (1970); K.Tajima, M.Iwahashi, and T.Sasaki, Bull.Chem.Soc.Jap. 44, 3251 (1971); N.H.Steiger and G.Aniansson, J.Phys.Chem. 58, 228 (1954); K.Shinoda and K.Ito, J.Phys.Chem. 65, 1499 (1961); S.J.Rehfeld, J.Colloid Interface Sci. 31, 46 (1969). 12. T.Smith, J.Colloid Interface Sci. 28, 531 (1968). 13. C.Wagner, Phys. Z. 25, 474 (1924). See also: L.Onsager and N.N.T.Samaras, J. Chem. Phys. 2, 528 (1934); L.Yan, J.Stat.Phys. 110, 825 (2003). 14. See e.g. S.Sircar, Measurement of Gibbsian Surface Excess, AIChE J. 47, 1169 (2001); S.Sircar, Gibbsian Surface Excess for Gas Adsorptions, Ind.Eng.Chem.Res. 38, 3670 (1999); A.L.Myers, Thermodynamics of Adsorption in Porous Materials, AIChE J. 48, 145 (2002). Sci. 110, 298 (1986); K.J.Motomura, Comments on: Thermodynamics of Adsorption and Gibbsian Distance Parameters by R.J.Good, J. Colloid Interface Sci. 110, 294 (1986); K.J.Motomura, Thermodynamic Studies on Adsorption at Interfaces: General Formulation, J. Colloid Interface Sci. 64, 348 (1978); Z. Kiraly and I. Dekany, Algebraic and geometric interpretations of adsorption excess quantities, Colloid Polym. Sol. 266, 663 (1988); Z. Kiraly and I. Dekany, Interpretation of adsorption excess quantities: the absolute surface excess concentration, Colloid Polym. Sci. 268, 687 (1990). Other excess amount similar to the relative adsorption is the molar reduced adsorption Γ2n and the volumetric reduced adsorption Γ2V. For more details see: D.H.Everett, IUPAC manual of symbols and terminology for physicochemical quantities and units, Washington (1971). Referred also as Gibbs adsorption isotherm. J.W.McBain and C.W.Humphreys, J.Phys.Chem. 36, 300 (1932); J.W.McBain and R.C.Swain, Proc.Roy.Soc. (London) A154, 608 (1936). D.J.Salley, A.J.Weith Jr, A.A.Argyle, and J.K.Dixon, Proc.Roy.Soc. (London) A203, 42 (1950); J.K.Dixon, C.M.Judson, and D.J.Salley, 3 3