CALCULATION OF EQUILIBRIA

The calculation is based on the rule of thermodynamics, which states that a system will be in equilibrium when the Gibbs free energy is at a minimum. Cl The objective then is the minimization of the total free energy of the system and the calculation of equilibria at constant temperature and volume or at constant pressure. It is a complicated and lengthy calculation but, fortunately, several computer programs are now available that considerably simplify the task. PI... 

In order to relate yx and xu the bubblepoint temperatures are found over a series of values of xv Since the activity coefficients depend on the composition of the liquid and both activity coefficients and vapor pressures depend on the temperature, the calculation requires a respectable effort. Moreover, some vapor-liquid measurements must have been made for evaluation of a correlation of activity coefficients. The method does permit calculation of equilibria at several pressures since activity coefficients are substantially independent of pressure. A useful application is to determine the effect of pressure on azeotropic composition (Walas, 1985, p. 227). 

AG is the thermodynamic quantity that is the most accurately measured from binding constants in solution, is subject to the least uncertainty in interpretation, and the most relevant in calculations of equilibria. 

For thermodynamic calculation of equilibria useful in petroleum science, combustion data of extreme accuracy are required because the heats of formation of water and carbon dioxide are large in comparison with those in the hydrocarbons. Great accuracy is also required of the specific heat data for the calculation of free energy or entropy. Much care must be exercised in selecting values from the literature for these purposes, since many of those available were determined before the development of modem calorimetric techniques. 

The same arguments apply with even greater force to rates of reactions. The calculation of rate constants is a much more difficult problem than the calculation of equilibria because we cannot determine the structures of transition states experimentally. We should in principle calculate the whole potential energy... 

There are no high-temperature thermodynamic data which would permit the calculation of equilibria for any of the above reactions. 

Increasingly higher pressures are being used in rockets and calculations of equilibria and flame temperatures must necessarily follow this trend. As mentioned previously, at these high pressures gases deviate from the ideal condition. Consequently the functions defining equilibrium that have been presented no longer hold, but must be modified. 

In the older organic chemistry it was assumed that a free rotation was possible about a single C—C bond because, for example, there exist no isolatable isomers of 1,2 dichloroethane. Accurate measurements of the specific heat of ethane at low temperatures and likewise the difference between determinations and calculations of equilibria of hydrocarbons have, however, shown that there is no question of a free rotation rotation is indeed possible but there is a potential barrier of about 3 kcal/mole which has to be surmounted. The state of lowest energy is that in which the two methyl groups, or the methyl and the CH2 group, are alternate. In 1, 2 dichloroethane etc. there appear to be two positions of (relatively) minimum energy, the trans position and an oblique position which lies 1.2 kcal higher than the former. 

In the few cases where the figures given refer to the formation of an aqueous solution the concentration units employed are molalities and not mole fractions the calculation of equilibria in solution will be taken up further in chap. XX. 

Other examples of the application of thermodynamics to the calculation of equilibria in technically important reactions are discussed by Rossini, f... 

Before we can know that precipitation has been subject to pollution, we must know what its natural composition is. This will depend on a variety of natural contributions and effects that are discussed in this chapter. Let us first examine the question What is the pH of natural rain To answer this question we will follow a method that applies to calculations of equilibria in natural waters, generally. The first question we ask is what information is given or assumed In this case we will assume 25°C, pure water, and an atmospheric CO2 pressure of co, jq-3.5 jjjg second question to ask is what chemical equilibria apply to the problem In this case the equilibria are... 

Predictions of phase diagrams, i.e., calculations of equilibria between solid-solid and solid-liquid phases as functions of temperature and pressure and the development of microscopic (atomic-level) mechanisms for the transitions between phases. 

The problem is to be attacked in quite another manner when our concern is simply to calculate chemical equilibria in this case it will of course be best to derive the chemical constants from actual chemical equilibria. I have naturally worked in both directions in my numerous calculations, of which I have published, of course, only a small fraction. In my publications I have laid less stress on the accurate calculation of equilibria than on the remarkable fact, that quantities like Trouton s coefficient, and in particular certain coefficients in my vapour-pressure formula, bore a dose relation to chemical equilibria. Those who are not very practised in thermodynamical calculations will hardly have recognized this distinction, and for this reason a repetition of the calculation of the ammonia equilibrium will be desirable. 

The data listed in the tables in Part VII also form the database for EQUITHERM, a software system authored by I. Barin, G. Eriksson, F. Sauert, M. Zeitler, B. Wittig, and W. Schmidt, which is available from the publishers of the book. EQUITHERM can be used to carry out thermodynamic calculations on multi-component, multi-phase systems made up of any of the substances listed in this work. Thus, the publishers are in the unique position of being able to offer both a printed compilation of thermodynamic data for pure substances and a software package for the calculation of equilibria in multi-species systems. 

The database and computing system equiTherm [5, 6] contains the computing program SOLGASMIX [4] coupled with the data set published in these Tables. equiTherm allows the calculation of equilibria using personal computers for systems of up to 15 elements and 20 phases (up to a total of 120 species). 

A. N. Krestovnikov, L. P. Vladimirov, B. S. Gulyanitskii, and A. Ya. Fisher, Handbook for Calculation of Equilibria in Metallurgical Reactions [in Russian], Metallurgizdat, Moscow (1963). 

Deiters, U.K. (1985) Calculation of equilibria between fluid and solid phases in binary mixtures at high pressures from equations of state. Fluid Phase Equilibria 20, 275-282. 

This convention applies to solutes, for example to salts dissolved in water. The standard state of species i is a solution of concentration Co,i = 1, which has the properties of an infinitely dilute solution. This does not refer to a real physical situation, but rather offers a formalism that is quite useful for the calculation of equilibria in solution. The concentration can be expressed in weight percent, molarity (mole per liter), or molality (mole per kg). 

Noddings and Mullet have provided a computational aid for the calculation of equilibria, which contains 587 tables of equilibrium constant against equilibrium compositions. It is claimed to cover about 90% of all chemical stoicheiometries. 

Rossini describes the calculation of equilibria, and the processes studied include isomerization, the production of toluene and of iso-octane, the syntheses of rubber and alcohols, and the transformation of graphite to diamond. 

The applicability of the resulting thermod)mamic functions is further exemplified by the calculations of equilibria in gas-phase reactions involving LaF (Hildenbrand and Lau, 1995) and LaCl (Chervonnyi and Chervonnaya, 2004b) molecules. Available experimental values of Kp for these reactions make it possible to trace changes in the enthalpies of reactions calculated by the third or second law (ArH°(0, III law) and ArH°(0, II law), respectively), as well as to compare the enthalpies of atomization derived from these values (AatH°(0, III law) and AatH° (0, II law), respectively) depending on the thermod)mamic functions used in the calculations. The results are summarized in Tables 65 and 66. 

To clearly show the differences in thermodynamic fimctions that arises due to the use of different methods of calculation of the electronic component, Tables 71 and 72 present the reduced Gibbs energies (third and fourth columns) and enthalpies (fifth and sixth columns) at three temperatures. As is known, these parameters are used in calculations of equilibria by the third and second laws of thermodynamics, respectively. The seventh column in these tables presents the differences ( i) between the values in the third and fourth columns, and the eighth column presents the differences ( 2) between the values of the fifth and sixth columns. 

Our findings show that the thermodynamic fimctions (Tables AlO and A12) of RCl and RF (R = Ce-Lu) in which the electronic contribution is determined from the excitation energies of R ions are more reliable for the description of high-temperature equilibria. The corresponding error, calculated from comparison of the AatH°(0, El) and AatH°(0, HI law) values (Table 75), is not larger than 3 J/(mobK). The thermod)mamic functions for LaCl and LaF (Table A9) are foimd by direct summation over the energy levels and recommended for use in thermod)mamic calculations of equilibria involving these molecules. 

In the step from the raw experimental data to data that can be treated by conventional computer programs, transformations are often necessary. For titrations, the raw data are EMF values as a function of titrant added (volume changes may be more or less important). These would typically be transformed to log[H ] (or pH), total acid concentration, and dilution factors (e.g., for FITEQL). With LAKE [32], raw data can be treated directly. An input file or the code itself must, in such a case, provide all the information necessary for the calculation of equilibria (e.g., LAKE must perform the transformations to obtain concentrations from the raw data). For the majority of eodes, however, pH or log[H ] as a function of titrant added is required. EMF values must be transformed to pH or log[H" ] using the apphed calibration procedures. Here, the calibration and, sometimes, assumptions in the data treatment may be important. Calibration can be performed on the eoneentration or on the activity scale. The concentration scale has the advantage that, a priori, no assumptions about activity coefficients are necessary the disadvantage is that such an approach is limited to one ionic medium (the constant ionic medium approach), although in work on suspensions, the variation of the ionic medium certainly is of importance. The activity scale requires assumptions concerning the treatment of activity coefficients it must be realized that sueh assumptions must lead to self-consistency between the finally presented experimental data and the model calculations. 

Occasionally, isotherms of binary and multicomponent exchanges are described using various empirical adsorption equations. These cannot be used for the prediction of multicomponent equilibria [88]. In fact, a closer inspection of these equations reveals that they have no in-built facility for true prediction (i.e. for the calculation of equilibria over ranges of different total solution concentrations for heterovalent exchanges). Thus these equations are useful in describing the observed isotherm in a mathematical form but the only pre-... 

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