Oxford, 1972. 4.2 Electrocapillarity in .NET

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Oxford, 1972. 4.2 Electrocapillarity
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The interfacial tension at an interface is a force acting on a unit length of the interface against an increase in the interface area. The region around the interface in which interfacial tension is produced is very narrow. Tolman, Kirkwood and Buff state that this distance is only about 0.10.3 nm for the liquid-vapour interface, corresponding to a monolayer at the surface of the liquid, still affecting the next nearest layer. In order to derive a relationship that would qualitatively characterize the formation of interfacial tension at the interface between two homogeneous singlecomponent phases, the following cycle will be considered. First a free surface area of A is formed in each of phases a and / . These surfaces are then brought into contact, forming the interphase s. The work required to separate these two phases and to return the system to the original state is yA, where y is the interfacial tension. Thus, yA = AG(or) + AG(j8) - AG(s) (4.2.1) where AG(ar) and AG(/J) are the Gibbs energies required for reversible separation (i.e. the formation of two free surfaces A) of the phases a and jS both divided by two. AG(s) is the Gibbs energy required for interrupting the contact between phases a and jS. The AGs are given by the equations
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AG(j8) = (-K03) - n.08)] ^
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AG(s) = [ec(s) + ev(s)]N(sM
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where no(a) and no((i) are the numbers of nearest neighbours that the particle has in the bulk of phases a and jS, respectively, ns(a) and ns(fi) are the numbers of nearest neighbours from phase a or / that the particle has at the interface, ec(a) and ec(/J) are the bond energies between this particle and its nearest neighbour, r's are the energies required for rearrangement of the particle in the interphase in order to be able to form a bond to a
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particle in the neighbouring phase, ew(a), ev(j8), and v(s) are the changes in the thermal vibrational energies of the particle at the interface as a result of formation of the free surface and contact between the phases, N(a) and N(p) are the numbers of particles per unit surface area for the free surface of phases a and /3, c(s) is the bond energy between the particles of phases a and jS in the interphase and N(s) is the number of such bonds per unit surface area at the interphase. If we assume that N(a) = N((3) = N(s)9 that species at the surface have only one free bond (n0-ns=l) and that ec er, v, then (s) (4.2.3)
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This expression can be roughly interpreted as the difference between the Gibbs energy of adhesion of the two phases and the sum of the Gibbs energies of cohesion for the two phases. These considerations can also be used to derive the Dupre equation, where AG(s) is the Gibbs energy of adsorption of the solvent per unit area of the metal surface: AG(s) = y m / a + y 1 / a - y (4.2.4)
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where ym/a is the interfacial tension of the metal-air interface, y1/a is the interfacial tension of the solution-air interface and y is the interfacial tension of the metal-solution interface. Now the relationship between the interfacial tension and the composition of the two phases in contact will be analysed thermodynamically by using the approach of J. W. Gibbs. The interphase can be considered as a particular phase s of thickness h. This phase differs from the homogeneous phases only in that the effect of pressure is accompanied by the effect of the interfacial tension y. Consider a rectangle with sides h (perpendicular to the interphase) and / (parallel with the interphase) located perpendicular to the interphase. The force acting on the rectangle is not equal to the product phi (as for an area in the bulk of the solution) but phi yl. If the volume of the interphase V(s) is increased by dV(s) by increasing the thickness of the interphase by dh, then area A = V(s)/h increases by cL4. The overall work, Wy connected with this process consists of volume work accompanying the increase in the thickness of the interphase and volume and surface work connected with an increase of the surface area: W(s) = -pA dh + {-ph + y) cL4 = -p dV(s) + y dA (4.2.5)
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Because of this formation, a different definition of the enthalpy must be introduced for the interphase, differing from the usual expression for a homogeneous phase: H(s) = U(s) + pV(s) - yA (4.2.6)
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where U(s) is the internal energy of the interphase. For the differential Gibbs energy of the interphase we have dG(s) = d[//(s) - TS(s)] = dU(s) + p dV(s) - V(s) dp - y dA - A dy - T dS(s) - 5(s) dT + 2 ft d^i(s) (4-2.7) /=o where nf-(s) is the amount of the ith component of the system in phase s and n is the total number of components. As dU(s) + p dV(s) -ydA-T it follows that dG(s) = ~S(s) dT + V(s) dp - A dy + J) pg dnt(s)
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dS(s) = 0
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