Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup
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(3.22) |
More direct definitions of thermodynamic properties of the line can be obtained following the procedure for the surface in the following section.
3.3Defining thermodynamic variables for the surface
The relations for global equilibrium cannot be used to describe systems with gradients in μ, p, T or the electric field. We must therefore also formulate relations for local equilibrium. We need local forms of the Gibbs and Gibbs–Duhem equations. These will be given after we have first defined the surface variables.
An interface is the thin layer between two homogeneous phases. We restrict ourselves to flat surfaces that are perpendicular to the x-axis. The thermodynamic properties of the interface shall now be given by the values of the excess densities of the interface. The value of these densities and the location of the interface will be defined through the example of a gas–liquid interface. We shall correspondingly indicate these phases with the superscripts g and l
Figure 3.1 shows the variation in concentration of A, in a mixture of several components, as we go from the gas to the liquid phase. The surface thickness is usually a fraction of a nm. The x-axis of Fig. 3.1 has coordinates in nm, which has a molecular scale. A continuous variation in the concentration is seen. Gibbs [65] defined the surface of discontinuity as a transition region with a finite thickness bounded by planes of similarly chosen points. In the figure two such planes are indicated by vertical lines. The position a is the point in the gas left of the closed surface where cA(x) starts to differ from the concentration of the gas, cAg, and the position b is the point in the liquid right of the closed surface, where cA(x) starts to differ from the concentration of the liquid, clA. The surface thickness is then δ = b − a. It refers to component A. Other components may yield somewhat different planes.
Figure 3.1Variation in the molar density of a fluid if we go from the gas to the liquid state. The vertical lines indicate the extension of the surface. The scale of the x-axis is measured in nanometer.
Gibbs defined the dividing surface as “a geometrical plane, going through points in the interfacial region, similarly situated with respect to conditions of adjacent matter”. Many different planes of this type can be chosen. While the position of the dividing surface depends on this choice; it is normally somewhere between the vertical lines in Fig. 3.1. The planes that separate the closed surface from the homogeneous phases are parallel to the dividing surface. The continuous density, integrated over δ, gives the excess surface concentration of the component A as a function of the position, (y, z), along the surface:
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(3.23) |
where d is the position of the dividing surface. The surface concentration is often called the adsorption (in mol/m2). The Heaviside function θ is by definition unity when the argument is positive and zero when the argument is negative. It is common to choose d between a and b. Other excess variables than ΓA are defined similarly. All excess properties of a surface can be given by integrals like Eq. (3.23). The excess variables are the extensive variables of the surface. They describe how the surface differs from adjacent homogeneous phases.
Remark 3.3.It is clear from Fig. 3.1 that one may shift the position a to the left and b to the right without changing the adsorption. This shows why the precise location of a and b is not important for the value of the adsorption.
Gibbs defined the excess concentrations for global equilibrium. Equation (3.23) will be used in this book also for systems which are not in global equilibrium. In fact, ΓA, cA, cgA and clA as well as a, b and d may all depend on the time. For ease of notation this was not explicitly indicated.
Figure 3.2Determination of the position of the equimolar surface of component A. The vertical line is drawn so that the areas between the curve and the bulk densities are the same.
The equimolar surface of component A is a special dividing surface. The location is such that the surplus of moles of the component on one side of the surface is equal to the deficiency of moles of the component on the other side of the surface, see Fig. 3.2. The position d of the equimolar surface obeys
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(3.24) |
According to Eqs. (3.23) and (3.24), ΓA = 0, when the surface has this position, see Fig. 3.2. The shaded areas in the figure are equal. In a multicomponent systems, each component has its “equimolar” surface, but we have to make one choice for the position of the dividing surface. We usually choose the position of the surface as the equimolar surface of the reference component. This component has then no excess concentration, while another component B, like the one sketched in Fig. 3.3, has an excess concentration.
Remark 3.4.In homogeneous phases, it is common to use densities per mole or per unit of mass. For the surface this is possible, but not practical. The reason for this is that this would imply dividing the extensive properties of the surface by the excess molar density of the reference component. As we have just explained it is common to use the equimolar surface defined by this reference component. In that case, the excess molar density of the reference component is zero and we cannot divide by it. To avoid confusion, we will always use excess densities per unit of surface area.
Figure 3.3Variation in the density of component B across the surface. The excess surface concentration of component B is the integral under the curve in the figure.
Figure 3.4The equimolar surface plotted on a micrometer scale appears as a jump between bulk densities.
Thermodynamic properties of homogeneous systems are usually plotted on scales with greater dimension than nm. In Fig. 3.4, cA(x) is plotted on a μm scale. The fine details of Fig. 3.1 disappear, and the surface appears as a discontinuity. When plotted on a macroscopic scale, a non-zero excess surface density, like that of component B, appears as a singularity at the position of the dividing surface. On this scale, the possible choices of the dividing surface can no longer be distinguished, and the surface can be regarded as a two-dimensional thermodynamic system. The system has properties that are integrated out in the x-direction and are given per surface area. Dependence on the coordinates y and z remains. When excess surface