H.2 Euler-Maclaur in Sum Formula
13.1.1 Gibbs Dividing Surface Model
Following Gibbs, we replace the actual system by a model system consisting of two strictly homogeneous phases separated by a single mathematical plane, known as the Gibbs dividing surface. In this model system, the homogeneous phases extend uniformlyuntil they meet at the dividing surface. This plane is similarly situated with respect to the transition region. For the moment, we assume that it is locatedanywherein the system, not necessarily in the transition layer, and discuss later the implications of its actual location.
One then definessurface excess quantitiesby subtracting the extensive properties of the homogeneous parts of the model system from the corresponding actual parts:
Uxs:=U−Uα−Uβ; (13.2)
Sxs:=S−Sα−Sβ; (13.3)
Nixs:=Ni−Niα−Niβ; (13.4)
0 :=V−Vα−Vβ. (13.5)
3Note that this treatment avoids discussion of the pressure of the subsystemL. In fact, that subsystem is inhomogeneous, so on a microscopic scale it could be characterized by a pressure tensorpij. If thezdirection is perpendicular to the walls of the layer, thenpzzmust be uniform and equal to the common pressurepof the homogeneous phases. The componentspxx=pyywill vary frompnear the homogenous phases to negative values within the discontinuity itself, giving rise to a surface tensionσ=
(p−pxx)dz>0, where the integration includes the region of discontinuity. See [27, p. 44] for a derivation.
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Equation (13.5) is different from the previous three equations because there is no excess volume, due to the fact that the homogeneous phases of the model system meet at the dividing surface, which has no thickness. Since the temperature is uniform, one can also define excesses of the thermodynamic potentials, such as the Helmholtz free energy F=U−TS, for which
Fxs:=F−Fα−Fβ. (13.6)
It follows that all excess quantities follow the same algebra as their bulk counterparts.
Although these excess quantities can be defined, they usually donot have physical significance because they depend on the location of the dividing surface. This is easily illustrated for the case of a single component material in which one bulk phase is a liquid having molar densitynand the other is a gas having molar densityng. Then if the dividing surface is located such that the gas has volumeVgand the liquid has volumeV−Vg, where V is the total volume, it follows that
Nxs=N−nV+(n−ng)Vg. (13.7) Forn−ng > 0, the sum of the first two terms on the right is negative and independent of the location of the dividing surface whereas the last term on the right is positive and depends linearly onVg and hence linearly on the position of the dividing surface.
Thus,Nxsvaries with the position of the dividing surface and can be positive, negative, or zero. Therefore, Nxs has no physical significance. One could fix the position of the dividing surface by convention by choosing its location so thatNxs=0; this is known as the equimolar surface. Nevertheless, this choice is still artificial. Moreover, in a multi- component system one could only choose the dividing surface to be equimolar relative to one of the components. Similarly, it follows thatUxs,Sxs, andFxsdepend on the location of the dividing surface.
On the other hand, the excess of the Kramers potential4K=F−
iμiNi, namely Kxs:=K−Kα−Kβ=Fxs−
i
μiNixs (13.8)
turns out to be independent of the location of the dividing surface. This can be seen by noting for a bulk phase thatF−
iμiNi=F−G= −pV, soKα= −pVαandKβ= −pVβ. Therefore
Kxs=K+p(Vα+Vβ)=K+pV, (13.9) where Eq. (13.5) has been used. The right-hand side of Eq. (13.9) isindependent of the location of the dividing surface, so Kxs is also independent of that location and has
4This is also called the grand potential and is often denoted by.
physical meaning. We can therefore divide by the areaAof the dividing surface to define the surface free energy5(per unit area of interface)
γ :=Kxs
A =Fxs−
iμiNixs
A =K+pV
A , (13.10)
which will be independent of the choice of the location of the dividing surface for a planar region of discontinuity.
We now approach the same problem from a different vantage point by considering small reversible changes of the same planar system in contact with a thermal reservoir at temperatureT, a pressure reservoir at pressurep, and chemical reservoirs at potentialsμi. In particular, we allow the system to undergo an infinitesimal change in which its length is unchanged but its cross-sectional area changes by an amount dA. In order to account for work done “by the surface” we write the reversible work done by the system in the form δW=pdV−σdA, wherepdVis the usual quasistatic work done by the pressure andσdA is the extra work doneonthe system because of the surface of discontinuity. The quantity σis the surface (interfacial) tension, which is a force per unit length that must be applied by an external agent to extend the surface. Thus
dU=TdS−pdV+σdA+
i
μidNi. (13.11)
We shall proceed to show thatσ =γ. Indeed, for the bulk systems we have dUα=TdSα−pdVα+
i
μidNiα; (13.12)
dUβ=TdSβ−pdVβ+
i
μidNiβ. (13.13)
We subtract both of these equations from Eq. (13.11) to obtain dUxs=TdSxs+
i
μidNixs+σdA. (13.14)
Equation (13.14) illustrates thatUxs can be regarded as a function of Sxs, Nixs, and A.
Moreover, by considering systems that have the same values ofT,μi, andσ but simply different cross-sectional areas, we deduce that
Uxs(λSxs,λNixs,λA)=λUxs(Sxs,Nixs,A) (13.15)
5The name surface free energy is commonly used, but it is important to remember that the relevant free energy is the Kramers potential. For the case of planar interfaces that is treated here, the pressure is the same in both bulk phases so one can define a Gibbs free energyG=U−TS+pVand note thatK+pV=G−
iμiNi. Then γAcan be thought of as an excess Gibbs free energy relative to a homogeneous system. For curved surfaces, the pressures in the bulk phases are not equal, so one must resort to the Kramers potential. For the very special case of a single component material with the dividing surface chosen to be the equimolar surface, one hasγ=Fxs/A which is the surface excess of the Helmholtz free energy. The name surface tension is perfectly applicable for γfor a surface of discontinuity between fluids because it can be shown to be the force per unit length needed to extend the surface. For solids, this would be a misnomer because surface can be created but also stretched elastically.
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for any positiveλ. Thus by the Euler theorem of homogeneous functions of degree one (see Eq. (5.39)), we deduce that
Uxs=TSxs+
i
μiNixs+σA. (13.16)
Equation (13.15) can be solved forσ to deduce σ =Uxs−TSxs−
iμiNixs
A =γ. (13.17)
Equation (13.17) together with Eq. (13.11) show that the reversible work associated with an increase in surface area is justγdA, whereγis the surface excess of the Kramers potential.