Other Surface Thermodynamic Relationships

The preceding material of this section has focused on the most important phenomenological equation that thermodynamics gives us for multicomponent systems—the Gibbs equation. Many other, formal thermodynamic relationships have been developed, of course. Many of these are summarized in Ref. 107. The topic is treated further in Section XVII-13, but is worthwhile to give here a few additional relationships especially applicable to solutions. 

Using the Gibbs convention for defining surface quantities, we define 

Alternatively, for the whole system (i.e., including the bulk phases), 

Integration of Eq. UI-87 holding constant the intensive quantities T, /i, and 7 gives 

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The choice of the size parameter d is somewhat ambiguous since even the relative values of d vary somewhat between solid, liquid, and gaseous salts because of the influence of interactions other than those represented by Eq. (7). For the case of a change of phase or for the description of phenomena where the environment of the ions changes drastically (as in the discussions of vapor pressure and surface tension), the influence of these other interactions is relatively large and other characteristic thermodynamic parameters (such as the melting temperature), which at least partly reflect these other interactions, should lead to more realistic relationships. Where there is no drastic change... 

We now want to use our thermodynamic relationships to describe this boundary or interface that we call the surface. As we do so, we should keep in mind that the change in properties in moving from one phase to another is not two-dimensional/ There is a small, but finite, thickness to the region in which the properties change from those on the interior of one phase to those on the interior of the other phase. There is every reason to believe that the thickness... 

The Gibbs equation contains three independent variables T, a, and p (defined either via concentration or pressure, c or p, respectively), and is a typical thermodynamic relationship. Therefore, it is not possible to retrieve any particular (quantitative) data without having additional information. In order to establish a direct relationship between any two of these three variables, it is necessary to have an independent expression relating them. The latter may be in a form of an empirical relationship, based on experimental studies of the interfacial phenomena (or the experimental data themselves). In such cases the Gibbs equation allows one to establish the dependencies that are difficult to obtain from experiments by using other experimentally determined relationships. For example, the surface tension is relatively easy to measure at mobile interfaces, such as liquid - gas and liquid - liquid ones (see Chapter I). For water soluble surfactants these measurements yield the surface tension as a function of concentration (i.e., the surface tension isotherm). The Gibbs equation allows one then to convert the surface tension isotherm to the adsorption isotherm, T (c), which is difficult to obtain experimentally. 

The equilibrium value of the polymer surface tension decreases with increase in temperature (see Fig. 2.12), which follows from basic thermodynamic relationships. Thus, the increase of the adhesion strength for cementing at higher temperatures is caused, apart from other factors, by the improvement of wetting of the substrate by adhesive due to the decrease of the adhesive surface tension. 

The same relationships also apply to the enthalpy and the entropy of electrosorption. The enthalpy of electro sorption turns out to be less (in absolute value) than the enthalpy of chemisorption of the same molecule on the same surface from the gas phase. On the other hand, the entropy of chemisorption from the gas phase is, as a rule, negative, since the molecule RH is transferred from the gas phase to the surface, losing in the process three degrees of freedom of translation. This is also true for electrosorption, but in this case n molecules of water are transferred from the surface to the solution, leading to a net increase of 3 (n— 1) degrees of freedom. As a result, the entropy of electrosorption is usually positive. Remembering the well known thermodynamic relationship... 

Similar types of relationships can be found between the other thermodynamic variables. In general, specifying two variables fixes the state of the third.y Thus specifying Vm and T fixes the value of Sm, specifying Hm and Gm fixes Um, and so on. As another example, Figure 1.4 shows the (Sm, p, T) surface for an ideal monatomic gas.z The entropy, Sm is restricted to values of p and T on the surface. 

As in so many other fundamental aspects of the thermodynamics of surfaces, we are indebted to Gibbs 6) for pointing out that for solids the surface tension (7) and surface free energy (F ) are not equivalent quantities. Nevertheless, as Shuttleworth (7) mentions in an excellent review, the two terms have been (8) and still are (9) confused. The relationship between the quantities is... 

It is now time to reconsider the simple case of a two-phase system that contains two different types of molecules. If molecules of phase a are polar and molecules of phase [3 are nonpolar, the introduction of amphiphilic molecules that are capable of associating with either one of the two bulk phase molecules will result in an accumulation at the interface. Hence, these molecules will have a true excess concentration at the interface. Figure D3.5.4 illustrates that once surfactants adsorb at interfaces, the concentration within the interface may be larger than in any of the other phases. In order to predict the influence that these adsorbed surfactant molecules can have on the properties of the bulk system, interfacial chemists must be able to quantify the number of molecules that are adsorbed at the interface, that is, they must be able to measure the interfacial coverage. Unfortunately, it is extremely difficult, if not impossible, to directly measure the concentration of surface-active molecules adsorbed in a two-dimensional plane. This is where the thermodynamic concepts discussed earlier prove to be very useful, because a relationship between the interfacial coverage (G) and the interfacial tension (y) can be derived. 

The remainder of this book applies thermodynamics to the description of a variety of systems that are of chemical interest. Chapter 12 uses thermodynamics to describe the effects of other variables such as gravitational field, centrifugal field, and surface area on the properties of the system. Most of the focus of the chapter is on surface effects. The surface properties of pure substances are described first, including the effect of curvature on the properties of the surface. For mixtures, the surface concentration is defined and its relationship to the surface properties is described. 

Of the three quantities (temperature, energy, and entropy) that appear in the laws of thermodynamics, it seems on the surface that only energy has a clear definition, which arises from mechanics. In our study of thermodynamics a number of additional quantities will be introduced. Some of these quantities (for example, pressure, volume, and mass) may be defined from anon-statistical (non-thermodynamic) perspective. Others (for example Gibbs free energy and chemical potential) will require invoking a statistical view of matter, in terms of atoms and molecules, to define them. Our goals here are to see clearly how all of these quantities are defined thermodynamically and to make use of relationships between these quantities in understanding how biochemical systems behave. 

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