Surfaces, thermodynamics curvature, effect

A very important thermodynamic relationship is that giving the effect of surface curvature on the molar free energy of a substance. This is perhaps best understood in terms of the pressure drop AP across an interface, as given by Young and Laplace in Eq. II-7. From thermodynamics, the effect of a change in mechanical pressure at constant temperature on the molar h ee energy of a substance is... 

In the last two sections the formal theory of surface thermodynamics is used to describe material characteristics. The effect of interfaces on some important heterogeneous phase equilibria is summarized in Section 6.2. Here the focus is on the effect of the curvature of the interface. In Section 6.3 adsorption is covered. Physical and chemical adsorption and the effect of interface or surface energies on the segregation of chemical species in the interfacial region are covered. Of special importance again are solid-gas or liquid-gas interfaces and adsorption isotherms, and the thermodynamics of physically adsorbed species is here the main focus. 

For small particles with large curvature the surface has, as previously stated, a significant effect on the thermodynamics, and the concepts developed apply to all types of interfaces between solid, liquid and gas. 

In addition to capillarity, another important consequence of the pressure associated with surface curvature is the effect it has on the thermodynamic activity of substances. As a consequence, phase equilibria (including dissolution of chemical species in the different phases) are affected by the presence of interfaces. In this section we consider a few such cases. 

Equation (42) provides a thermodynamically valid way to determine y for an interface involving a solid. The thermodynamic approach makes it clear that curvature has an effect on activity for any curved surface. The surface free energy interpretation of y is more plausible for solids than the surface tension interpretation, which is so useful for liquid surfaces. Either interpretation is valid in both cases, and there are situations in which both are useful. From solubility studies on a particle of known size, y5 can be determined by the method of Example 6.2. 

To date we have limited our discussion to plane surfaces of pure substances. We now look at the effect of curvature on the properties of these surfaces. Specifically, we will derive relationships to obtain the difference in the thermodynamic properties of a flat surface or the bulk phase, and of small droplets. In doing so, we will assume 7 is not affected by surface curvature. This assumption is justified as long as the radius of curvature is large compared with the thickness of the surface layer. We mentioned earlier that usually the surface has a thickness of only a few molecules that is, it is of the order of a micrometer. Thus, curved surfaces with a radius greater than a few micrometers should follow the equations we derive. 

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. 

Let us now examine the effect of curvature of the surface on the molar Gibbs energy of a liquid substance. From the thermodynamics it follows that the change of the molar Gibbs energy by a change of pressure at constant temperature is... 

Another pore filling model based upon capillary equilibrium in cylindrical pores has recently been proposed in which the condition of thermodynamic equilibrium is modified to include the effects of surface layering and adsorbate-adsorbent interactions [135-137]. Assuming that the vapor-liquid interface is represented by a cylindrical meniscus during adsorption and by a hemispherical meniscus during desorption, and invoking the Defay-Prigogene expression for a curvature-dependent surface tension [21], the equilibrium condition for capillary coexistence in a cylindrical pore is obtained as... 

Assume that Biot numbers for the inner and outer surfaces satisfy the conditions Ajii/ki < 0.1 and hz jjkt < 0.1, and lumped conditions prevail. Also, in view of the fact that Si and So are small compared with the mean radius R of the brake system, neglect the effect of curvature. For the lumped system shown in Fig. 3.6, the first law of thermodynamics combined with Newton s law for the outer and inner surfaces results in the governing equation for the brake system,... 

Surface curvature also leads to important thermodynamic effects such as solubility enhancement and melting point depression in very small (typically nanometer-scale) particles. 

In this chapter the effects of surface curvature on the thermodynamics of various types of systems are described. Some examples are as follows. 

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