Equation for computing thermodynamic

Working equations for computing thermodynamic properties of interest are derived from (4-102) and (4-103) and the equations of Table 4.3, in the same manner as for the original R-K equation. The resulting expressions are applicable to either the liquid or vapor phases provided that the appropriate phase composition and compressibility factor are used. 

Equations for Computing Thermodynamic Functions of Gases and Solids... 

We must state, however, that the thermodynamic dissociation constants have a relatively small practical importance. It is true that they remain constant with changing ionic strength but to use them it is necessary to know the activity coefficients of the several components at different electrolyte concentrations. The simple Debye-Huckel equation for computing activity coefficients is valid only at very small ionic strengths. At larger ionic strengths it is preferable to determine empirically the stoichiometric dissociation constants for various types of electrolytes. 

In the previous section, we have derived the equations for the thermodynamic quantities of ideal-gas mixtures. We could also compute all of these quantities from the knowledge of the molecular properties of the single molecules. Once we get into the realm of liquid densities, we cannot expect to obtain that amount of detailed information on the thermodynamics of the system. 

An isochoric equation has been developed for computing thermodynamic functions of pure fluids. It has its origin on a given liquid-vapor coexistence boundary, and it is structured to be consistent with the known behavior of specific heats, especially about the critical point. The number of adjustable, least-squares coefficients has been minimized to avoid irregularities in the calculated P(p,T) surface by using selected, temperature-dependent functions which are qualitatively consistent with isochores and specific heats over the entire surface. Several nonlinear parameters appear in these functions. Approximately fourteen additional constants appear in auxiliary equations, namely the vapor-pressure and orthobaric-densities equations, which provide the boundary for the P(p,T) equation-of-state surface. 

In 4.1 we introduce ideal gases and their mixtures, and we derive equations for computing their thermodynamic properties. Then, we use the rest of the chapter to develop expressions for computing deviations from ideal-gas values the difference measures in 4.2, the ratio measures in 4.3. 

In this section, the first and second laws of thermodynamics are used to derive useful equations for computing the lost work of any process. A general energy balance (first law of thermodynamics) can be written for a system bounded by the control volume shown in Fisure 9.18. Streams at certain fixed states flow at fixed rates into or out of the control volume, heat and work are transferred at fixed rates across the boundaries of the control volume, matter within the control volume undergoes changes in amount and state, and the boundaries of the control volume expand or contract. The energy balance for such a control volume over a period of time. A/, is A(/wC7)... 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

Such information has been stored within a database in the ECES system. Then, using user input in the form of equation (1), ECES writes the expression for computing the thermodynamic equilibrium constant as a function of temperature to a file where it will eventually become part of a program to solve the many equilibria that might describe a complex system. 

Careri (Ref 7b) suggested a procedure for computing exactly the thermodynamic functions of a nonequilibrium system. The state of the system was then varied, at fixed volume and temperature, so as to give a minimum Helmholz free energy, consistent with such conditions as are imposed to permit the exact computation. The condition under which this method leads to self-consistent equations is discussed in detail. The method is then applied in a way that is very close to the Lennard-Jones and Devonshire cell method, bur with cells of variable size. The distribution within a cell is assumed to be Gaussian. Mayer Careri claimed that the method is easier to apply than the cell method, but it seems to be rather complicated... 

Values of n are available for numerous gases and gas mixtures. Burning velocities have been measured on burner flames, and flame temperatures, Tb, can be computed thermodynamically. It is thus possible to put Equation 5 to a test by comparing experimental values of minimum spark-ignition energies with values calculated from data on quenching distances, burning velocities, heat conductivities, and flame temperatures. 

The input of the problem requires total analytically measured concentrations of the selected components. Total concentrations of elements (components) from chemical analysis such as ICP and atomic absorption are preferable to methods that only measure some fraction of the total such as selective colorimetric or electrochemical methods. The user defines how the activity coefficients are to be computed (Davis equation or the extended Debye-Huckel), the temperature of the system and whether pH, Eh and ionic strength are to be imposed or calculated. Once the total concentrations of the selected components are defined, all possible soluble complexes are automatically selected from the database. At this stage the thermodynamic equilibrium constants supplied with the model may be edited or certain species excluded from the calculation (e.g. species that have slow reaction kinetics). In addition, it is possible for the user to supply constants for specific reactions not included in the database, but care must be taken to make sure the formation equation for the newly defined species is written in such a way as to be compatible with the chemical components used by the rest of the program, e.g. if the species A1H2PC>4+ were to be added using the following reaction ... 

The formal similarity between adsorption and complexation reactions can be exploited to incorporate adsorbed species into the equilibrium speciation calculations described in Sections 2.4 and 3.1. To do this, a choice of adsorbent species components (SR r in Eq. 4.3) must be made and equilibrium constants for reactions with aqueous ions must be available. A model for computing adsorbed species activity coefficients must also be selected.8 Once these choices are made and the thermodynamic data are compiled, a speciation calculation proceeds by adding adsorbent species and adsorbed species (SR Mp(OH)yHxLq in Eq. 4.3) to the mole-balance equations for metals and ligands, and then following the steps described in Section 2.4 for aqueous species. For compatibility of the units of concentration, njw) in Eq. 4.2 is converted to an aqueous-phase concentration through division by the volume of aqueous solution. 

Using these methods to describe an aqueous electrolyte system with its associated chemical equilibria involves a unique set of highly nonlinear algebraic equations for each set of interest, even if not incorporated within the framework of a complex fractionation program. To overcome this difficulty, Zemaitis and Rafal (8) developed an automatic system, ECES, for finding accurate solutions to the equilibria of electrolyte systems which combines a unified and thermodynamically consistent treatment of electrolyte solution data and theory with computer software capable of automatic program generation from simple user input. 

For both hypothetical and pseudo components, physical properties are computed by the same equations, which are based upon the correlations given in the Technical Data Book of the American Petroleum Institute (1). Equivalent molar quantities of these petroleum components are added to the amounts of the discrete (methane, etc.) components, to obtain the complete mixture for the thermodynamic calculations that follow. 

Each of the concentration-dependent G terms in the equations presented earlier can have a temperature dependency given by Eq. 11.38. Similar equations can be written for other thermodynamic properties, from which the Gibbs energy can be computed, such as the enthalpy of formation, entropy, and heat capacity. Equation 11.38 is a much more efficient way of incorporating information into a software database than tables contaming discrete values, which is important for minimizing computer resource requirements. 

Finally, we must select appropriate methods of estimating thermodynamic properties. lime (op. cit.) used the SRK equation of state to model this column, whereas Klemola and lime (op. cit.) had earlier used the UNIFAC model for liquid-phase activity coefficients, the Antoine equation for vapor pressures, and the SRK equation for vapor-phase fugacities only. For this exercise we used the Peng-Robinson equation of state. Computed product compositions and flow rates are shown in the table below. 

For very hot flames (that is, Tb > 1500 K) such computations, which can be made from tables of thermodynamic data, can become extremely laborious, since no explicit equation for Tb can be written. Instead, Tb itself depends on the various equilibria among the product species at the temperatures concerned, the product species may include appreciable concentrations of free radicals. Such calculations are described by S. R. Brinkley, Jr., in Combustion Processes, vol. II, chap. C, Princeton University Press, Princeton, N.J., 1956. 

Thermodynamic data, whether determined through calorimetry or solubility studies, are subject to refinement as more exact values for the components in the reaction scheme, or more complete description of the solution phases, become available. Many of the solubility studies on clays were done before digital-computer chemical equilibrium programs were available. One such program, SOLMNEQ, written by one of the authors ( ) solves the mass-action and mass-balance equations for over 200 species simultaneously. SOLMNEQ was employed in this investigation to convert the chemical analytical data into the activities of appropriate ions, ion pairs, and complexes. 

---