Gibbs convention

Using the Gibbs convention for defining surface quantities, we define... 

Capillary Thermodynamics Without a Geometric Gibbs Convention... 

The introduction of the superscript (n, V) implies the adoption of a Gibbs convention algebraically expressed by Equations 5, which state that we are comparing the real system with a fictitious one consisting of two bulk phases in contact in the absence of an interfacial region, and with the added specification that the numbers of moles n and the total volume V of the fictitious system shall be the same as in the real one. The quantity S(r 7) (which is called SA° by Guggenheim (3) and sf by Defay (1)) is the entropy change per unit area of interface created when... 

We have conceptually arrived at the same point as we did in the argument leading to Equation 4 for Equation 11 is valid for arbitrary choice of multipliers x and y, and each such choice corresponds to a different Gibbs convention. The choice which leads to the Gibbs adsorption equation is that which makes the coefficients of dP and dm (conventionally defined to be the solvent) vanish ... 

Another excess quantity lV2) is defined by interchanging indices 1 and 2 in Equations 12, 13, and 14, and for this quantity the Gibbs convention is different, for we are now comparing the real system with a fictitious one defined to have the same total moles n2 of solute as has the real one. Despite the fact that lY2) and r2(1) are defined for different Gibbs conventions, they are algebraically related, as the reader may... 

As a final example, let us derive another set of excess quantities from Equation 11 by choosing the Gibbs convention... 

The Gibbs convention (Equation 17) states that we compare the real system with a fictitious one having the same total volume and total numbers of moles of all constituents as the real system. Under this convention the total moles of individual components 1 and 2 will differ between the real and the fictitious systems, but because the total of all moles of both components is the same, the surface excesses of each must sum to zero, and this is the meaning of Equation 20. 

Figure 3.1 Left In Gibbs convention the two phases a and (3 are separated by an ideal interface a which is infinitely thin. Right Guggenheim explicitly treated an extended interphase with a volume.

There are two common and widely used definitions of the interfacial excess enthalpy. We can argue that enthalpy is equal to the internal energy minus the total mechanical work 7A-PVa. Since in the Gibbs convention PVa = 0 we define... 

An adsorption isotherm is a graph of the amount adsorbed versus the pressure of the vapor phase (or concentration in the case of adsorption from solution). The amounts adsorbed can be described by different variables. The first one is the surface excess I in mol/m2. We use the Gibbs convention (interfacial excess volume Va = 0). For a solid surface the Gibbs dividing plane is localized directly at the solid surface. Then we can convert the number of moles adsorbed Na to the surface excess by... 

Fa, f7 Interfacial free (Helmholtz) energy in Gibbs convention (J) and interfacial free (Helmholtz) energy per unit area (J m-2), respectively / Force per unit area (N m-2)... 

S 7, sa Interfacial entropy in Gibbs convention(JK-1) and interfacial entropy per unit area (JK-1m-2), respectively T Temperature (K)... 

By assuming that the interface has no volume (Gibbs convention), the following equations can be written ... 

Table 1.2. Integral and differential characteristic functions for flat interfaces. The superscript o refers to surface excess (Gibbs convention, V s 0).

When surfactant molecules concentrate at the interface, some solvent molecules are displaced, so the surface solvent concentration is lower than the bulk solvent concentration. The Gibbs convention defines the dividing line between the two phases so that the (negative) surface excess of solvent equals zero. Then equation 4 gives the surface excess of (say) laurylsulfonic acid at the air-water interface. When the actual interfacial concentration of surfactant is needed, the situation is more complicated. Methods for handling these complications have been discussed (1,7). 

This equation is also based on the Gibbs convention (r = 0 for the surface tension of liquid a). For many-component systems U depends on the location of this plane, unlike the surface tension. This difference corresponds with our findings in sec. 2.2. 

All functions have been derived on the basis of the Gibbs convention (V s 0), except for G which is invariant with respect to the position of the dividing plane. 

To calculate surface pressure-area isotherms, following the Gibbs convention, we used the adsorption equation relating surface tension to surface excess, T, and chemical potential, /a, in the form... 

Since Vs = 0 in the Gibbs convention, we will have, Fs = yAs + ifnf so that, unless... 

From the Gibbs convention where Vs = 0 is assumed, we know that V=Va+ V, where V is the total volume of the system (or solution). If we substitute [V = V - V ] into Equation (413), we have... 

If the Gibbs convention is also adopted so that the Gibbs dividing plane is located such that the surface excess of the water sub-phase is zero ( w = 0), we may then write... 

For a saturated monolayer (T = l/co j), the dividing surface defined by Eq. (16) coincides with the dividing surface of the Gibbs convention, for which rQ=0. For Fj = 0, however, the convention of Eq. (16) shifts the dividing surface towards the bulk solution by the distance A = (co j cq ) as compared to the Gibbs convention (40). For large molecules, such as proteins (co coq), the value of A becomes negligibly small, and therefore for any adsorption the Lu-... 

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