Section 6. Thermodynamic Properties

All the four integral equations discussed in section THERMODYNAMIC PROPERTIES IN LIQUIDS become identical in this limit and can be solved... 

In this section thermodynamic properties of light diamondoids such as adamantane and diamantane are presented. 

For cubic crystals, which iaclude sUicon, properties described by other than a zero- or a second-rank tensor are anisotropic (17). Thus, ia principle, whether or not a particular property is anisotropic can be predicted. There are some properties, however, for which the tensor rank is not known. In addition, ia very thin crystal sections, the crystal may have two-dimensional characteristics and exhibit a different symmetry from the bulk, three-dimensional crystal (18). Table 4 is a listing of various isotropic and anisotropic sUicon properties. Table 5 gives values for the more common physical properties and for some of the thermodynamic properties. Figure 5 shows some thermal properties. 

Extensive tables of the viscosity and thermal conductivity of air and of water or steam for various pressures and temperatures are given with the thermodynamic-property tables. The thermal conductivity and the viscosity for the saturated-liquid state are also tabulated for many fluids along with the thermodynamic-property tables earlier in this section. 

Vapor pressure is the most important of the basic thermodynamic properties affec ting liquids and vapors. The vapor pressure is the pressure exerted by a pure component at equilibrium at any temperature when both liquid and vapor phases exist and thus extends from a minimum at the triple point temperature to a maximum at the critical temperature, the critical pressure. This section briefly reviews methods for both correlating vapor pressure data and for predicting vapor pressure of pure compounds. Except at very high total pressures (above about 10 MPa), there is no effect of total pressure on vapor pressure. If such an effect is present, a correction, the Poynting correction, can be applied. The pressure exerted above a solid-vapor mixture may also be called vapor pressure but is normallv only available as experimental data for common compounds that sublime. 

Data on the gas-liquid or vapor-liquid equilibrium for the system at hand. If absorption, stripping, and distillation operations are considered equilibrium-limited processes, which is the usual approach, these data are critical for determining the maximum possible separation. In some cases, the operations are are considerea rate-based (see Sec. 13) but require knowledge of eqmlibrium at the phase interface. Other data required include physical properties such as viscosity and density and thermodynamic properties such as enthalpy. Section 2 deals with sources of such data. 

Consider the pressure loss in a duct with straight, uniform cross-sectional area. The pressure loss is caused by friction. When different air sheets move against each other, friction is generated. The velocity and thermodynamic properties of air influence the friction. The duct wall has an overall roughness, which causes vortices to be formed with resulting friction in gas. The velocity has a pronounced effect in flow with low velocity, the vortices are small. Eor a straight duct the pressure loss Ap can be determined from... 

There are numerous possible applications for air curtains. For example, air curtains may be used to heat a body of linear dimensions (as used to move the fresh snow from the railway exchanges in Canada) to function as a partition between two parts of one volume to function as a partition between an internal room and an external environment, that have different thermodynamic properties and to shield an opening in a small working volume (see Section 10.4.6). 

In this section, we first discuss various experimental techniques that can be used to measure gas solubilities and related thermodynamic properties in ILs. We then describe the somewhat limited data currently available on gas solubilities in ILs. Finally, we discuss the impact that gas solubilities in ILs have on the applications described above (reactions, gas separations, separation of solutes from ILs) and draw some conclusions. 

In this section we describe some of the various experimental techniques that can be used to measure gas solubilities and related thermodynamic properties. 

In principle, the second law can be used to determine whether a reaction is spontaneous. To do that, however, requires calculating the entropy change for the surroundings, which is not easy. We follow a conceptually simpler approach (Section 17.3), which deals only with the thermodynamic properties of chemical systems. 

In addition to deciding on the method of normalization of activity coefficients, it is necessary to undertake two additional tasks first, a method is required for estimating partial molar volumes in the liquid phase, and second, a model must be chosen for the liquid mixture in order to relate y to x. Partial molar volumes were discussed in Section IV. This section gives brief attention to two models which give the effect of composition on liquid-phase thermodynamic properties. 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

Table A4.7 summarizes the thermodynamics properties of monatomic solids as calculated by the Debye model. The values are expressed in terms of d/T, where d is the Debye temperature. See Section 10.8 for details of the calculations. Tables A4.5 to A4.7 are adapted from K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 1995.

Estimation of parameters. Model parameters in the selected model are then estimated. If available, some model parameters (e.g. thermodynamic properties, heat- and mass-transfer coefficient, etc.) are taken from literature. This is usually not possible for kinetic parameters. These should be estimated based on data obtained from laboratory expieriments, if possible carried out isothermal ly and not falsified by heat- and mass-transport phenomena. The methods for parameter estimation, also the kinetic parameters in complex organic systems, and for discrimination between models are discussed in more detail in Section 5.4.4. More information on parameter estimation the reader will find in review papers by Kittrell (1970), or Froment and Hosten (1981) or in the book by Froment and Bischoff (1990). 

For a detailed description of spectral map analysis (SMA), the reader is referred to Section 31.3.5. The method has been designed specifically for the study of drug-receptor interactions [37,44]. The interpretation of the resulting spectral map is different from that of the usual principal components biplot. The former is symmetric with respect to rows and columns, while the latter is not. In particular, the spectral map displays interactions between compounds and receptors. It shows which compounds are most specific for which receptors (or tests) and vice versa. This property will be illustrated by means of an analysis of data reporting on the binding affinities of various opioid analgesics to various opioid receptors [45,46]. In contrast with the previous approach, this application is not based on extra-thermodynamic properties, but is derived entirely from biological activity spectra. 

It was shown in the previous section that different relatively stable conformations of a given molecule can result from internal rotation of a particular functional group. The possibility of the existence of various conformers is of extreme importance in many applications. It should be noted, for example, that the biological activity of an organic molecule often depends on its confonfia-tion - in particular the relative orientation of a specific functional grtmp. As another example, the thermodynamic properties of, say, an alkane are directly related to the conformation of its carbon skeleton. In this context the industrial importance of /sooctane is well-known. 

Since the interplay of theory and experiment is central to nearly all the material covered in this chapter, it is appropriate to start by defining the various concepts and laws needed for a quantitative theoretical description of the thermodynamic properties of a dilute solid solution and of the various rate processes that occur when such a solution departs from equilibrium. This is the subject matter of Section II to follow. There Section 1 deals with equilibrium thermodynamics and develops expressions for the equilibrium concentrations of various hydrogen species and hydrogen-containing complexes in terms of the chemical potential of hy-... 

A method for the estimation of thermodynamic properties of the transition state and other unstable species involves analyzing parts of the molecule and assigning separate properties to functional groups (Benson, 1976). Another approach stemming from statistical mechanics is outlined in the next section. 

The same type of polynomial formalism may also be applied to the partial molar enthalpy and entropy of the solute and converted into integral thermodynamic properties through use of the Gibbs-Duhem equation see Section 3.5. 

The formalism of the thermodynamics of solutions was described in Chapter 3. In this chapter we shall revisit the topic of solutions and apply statistical mechanics to relate the thermodynamic properties of solutions to atomistic models for their structure. Although we will not give a rigorous presentation of the methods of statistical mechanics, we need some elements of the theory as a background for the solution models to be treated. These elements of the theory are presented in Section 9.1. 

A large number of techniques have been used to investigate the thermodynamic properties of solids, and in this section an overview is given that covers all the major experimental methods. Most of these techniques have been treated in specialized reviews and references to these are given. This section will focus on the main principles of the different techniques, the main precautions to be taken and the main sources of possible systematic errors. The experimental methods are rather well developed and the main problem is to apply the different techniques to systems with various chemical and physical properties. For example, the thermal stability of the material to be studied may restrict the experimental approach to be used. 

The two parameters 8 and p for a given pair of substances can be obtained by application of the theorem of corresponding states to any suitable thermodynamic property. For example, in Section III we used the critical temperatures and pressures to determine the values of e and r for a series of substances, Kr being taken as reference. Of course all ratios of e and r for two substances, obtained from different thermodynamic properties, should agree closely the contrary would mean that the theorem of corresponding states is badly violated. 

As discussed in Section 5.1, the extension to non-isothermal conditions is straightforward under the assumption that the thermodynamic properties are constant. 

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