Uses of the thermodynamic functions

We consider a change of state involving a chemical reaction, and wish to calculate the equilibrium conditions. We assume that the change of state is isothermal and that the standard states of all of the reacting substances are the pure substances at the temperature and 1 bar pressure. When the pressure is not 1 bar, the appropriate corrections are easily made. When the change of state is written as v,, - = 0, the condition of equilibrium becomes 

The calculation then reverts to the determination of iv,/ e (()), or Xt v He(298)f. These can be obtained from the knowledge of the change of enthalpy for the same change of state because 

The concepts of the standard changes of enthalpy of formation and standard changes of the Gibbs energy of formation are developed in Chapters 9 and 11. These functions can be combined with the set of functions based on the third law. From the definitions of the standard changes on formation, 

The quantity Af (0)j required in Equation (15.30) is thus obtained from Equation (15.31). If the basis of the enthalpy functions is at 298 K, the equivalent of Equation (15.30) is 

Equations (15.25) and (15.26) afford a means of determining the validity of experimental data when values of [Ge(T) — H°(0)] or [Ge(7 ) — He(298)] are known for each of the reacting substances. Assume that values of AGe (T) have been determined at various temperatures for some standard change of state from the experimental measurement of the equilibrium conditions of the system. Then the value of viHe(0)i or viHe(298)i is calculated from the value of AG°(T) for each experimental point. All of the values of 

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Further, and more concrete, examples of the use of the thermodynamic functions are contained in the discussion of galvanic cells immediately following. 

Data were obtained using the thermodynamic functions calculated by the method of Mayer and Goeppert-Mayer with inclusion of the electronic excitation energies for LaQ. The use of the thermodynamic functions calculated by the direct summation method changes the enthalpy value by 0.5-0.8 kJ/moL Same calculation method but with inclusion of the electronic excitation energies of the free La+ ion. 

The determination of the heat capacity of a substance as a function of the temperature is by itself a very important application of DSC, because it may lead to values of the thermodynamic functions S%, //-" — //q, and Gy, mentioned in chapter 2. An example is the study of C6o carried out by Wunderlich and co-workers [271], The application of DSC in the area of molecular thermochemistry has been particularly important to investigate trends in transition metal-ligand bond dissociation enthalpies. The typical approach used in these studies, and its limitations, can be illustrated through the analysis of the reaction 12.27, carried out by Mortimer and co-workers [272] ... 

The surface tension of the aqueous solution of dode-cylaitunonium chloride (DAC) — decylairanonium chloride (DeAC) mixture was measured as a function of the total molality m of surfactants and the mole fraction X of DeAC in the total surfactant in the neighborhood of the critical micelle concentration (CMC). By use of the thermodynamic equations derived previously, the mole fraction in the mixed adsorbed film was evaluated from the y vs. X and m vs. X curves. Further, the mole fraction in the mixed micelle was evaluated from the CMC vs. X curve. By comparing these values at the CMC, it was concluded that the behavior of DAC and DeAC molecules in the mixed micelle is fairly similar to that in the mixed adsorbed film. 

With the discussion of the free-energy function G in this chapter, all of the thermodynamic functions needed for chemical equilibrium and kinetic calculations have been introduced. Chapter 8 discussed methods for estimating the internal energy E, entropy S, heat capacity Cv, and enthalpy H. These techniques are very useful when the needed information is not available from experiment. 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

Although the choice of standard states is arbitrary, two choices have been established by convention and international agreement. For some systems, when convenient, the pure component is chosen as the substance in the standard state. For other systems, particularly dilute solutions of one or more solutes in a solvent, another state that is not a standard state is chosen as a reference state [19]. This choice determines the standard state, which may or may not be a physically realizable state. The reference state of a component or species is that state to which all measurements are referred. The standard state is that state used to determine and report the differences in the values of the thermodynamic functions for the components or species between some state and the chosen standard state. When pure substances are used in the definition of a standard state, the standard state and the reference state are identical. 

The state of a single-phase, one-component system may be defined in terms of the temperature, pressure, and the number of moles of the component as independent variables. The problem is to determine the difference between the values of the thermodynamic functions for any state of the system and those for the chosen standard state. Because the variables are not separable in the differential expressions for these functions, the integrations cannot be carried out directly to obtain general expressions for the thermodynamic functions without an adequate equation of state. However, each of the thermodynamic functions is a function of the state of the system, and the changes of these functions are independent of the path. The problem can be solved for specific cases by using the method outlined in Section 4.9 and illustrated in Figure 4.1. 

Occasionally the problem arises of converting values of various thermodynamic functions of the components of a solution that have been determined on the basis of one reference state to values based on another reference state. To do so we equate the two relations of the thermodynamic function of interest obtained for the two reference states, because the value of the function at a given temperature, pressure, and composition must be the same irrespective of the reference state. We also equate the relation for the thermodynamic function for the component in the new reference state expressed in terms of the new reference state to that for the same state expressed in terms of the old reference state. The desired relation is obtained when the chemical potentials of the component in the different standard states are eliminated from the two equations. For examples, we use only the chemical potentials and discuss three cases. 

The energy and entropy functions have been defined in terms of differential quantities, with the result that the absolute values could not be known. We have used the difference in the values of the thermodynamic functions between two states and, in determining these differences, the process of integration between limits has been used. In so doing we have avoided the use or requirement of integration constants. The many studies concerning the possible determination of these constants have culminated in the third law of thermodynamics. 

The Soave modification of the Redlich-Kwong equation is the basis for the fourth thermodynamic properties method. This equation of state is applied to both liquid and vapor phases. Binary interaction coefficients for these applications are from Reid-Prausnitz-Sherwood (13) and the mathematical derivations used here are from Christiansen-Michelson-Fredenslund (14). Temperature and composition derivatives of the thermodynamic functions are included in the later work. These have applications in multistage calculations. 

Here, AH(A-B) is the partial molar net adsorption enthalpy associated with the transformation of 1 mol of the pure metal A in its standard state into the state of zero coverage on the surface of the electrode material B, ASVjbr is the difference in the vibrational entropies in the above states, n is the number of electrons involved in the electrode process, F the Faraday constant, and Am the surface of 1 mol of A as a mono layer on the electrode metal B [70]. For the calculation of the thermodynamic functions in (12), a number of models were used in [70] and calculations were performed for Ni-, Cu-, Pd-, Ag-, Pt-, and Au-electrodes and the micro components Hg, Tl, Pb, Bi, and Po, confirming the decisive influence of the choice of the electrode material on the deposition potential. For Pd and Pt, particularly large, positive values of E5o% were calculated, larger than the standard electrode potentials tabulated for these elements. This makes these electrode materials the prime choice for practical applications. An application of the same model to the superheavy elements still needs to be done, but one can anticipate that the preference for Pd and Pt will persist. The latter are metals in which, due to the formation of the metallic bond, almost or completely filled d orbitals are broken up, such that these metals tend in an extreme way towards the formation of intermetallic compounds with sp-metals. The perspective is to make use of the Pd or Pt in form of a tape on which the tracer activities are electrodeposited and the deposition zone is subsequently stepped between pairs of Si detectors for a-spectroscopy and SF measurements. 

Numerous organic species are known to lead to the crystallization of the MFI-type structure (7). but the tetrapropylammonium cations can be considered to be the most specific. To our knowledge no thermodynamic data such as standard formation enthalpies (AfH°) and stabilization energies attributed to the organic species have been published to corroborate this experimental observation. The published thermodynamic data (AfG°, AfH°, AfS°. Cp) are for natural zeolites (8-11) or for organic-free synthetic zeolites. However. some data have been obtained by calculations using lattice energy minimization and extended Hiickel theory (1 2) or by semi-empirical methods based on addition of the thermodynamic functions of the oxide compo-... 

However, there is no harm in carrying along sets of terms that ultimately vanish, and to continue using (1.24.1) with dn — dNi + dN. Similar commentary applies to expressions for the remaining differentials of the thermodynamic functions H, F, and G. 

There are various approaches, in addition to Boltzmann s equation, that can be taken to calculate thermodynamic parameters (e.g., through the use of the paitition function). 

The solubilities of gases in binary, ternary or more complex multicomponent solvents are good examples in which the Kirkwood-Buff theory of solutions provides exceUent results that cannot be obtained using the methods of traditional thermodynamics. Thermodynamics cannot provide explicit pressure, temperature, and composition dependence of the thermodynamic functions, such as the activity coefficients of the components. Therefore, various assumptions regarding the activity coefficients must be made. In contrast, the Kirkwood-Buff theory of solution allows one to establish, in some cases, relations between multicomponent... 

Because the entropy, like the energy, is a single-valued function of the state of the system, dS, like dE, is a complete differential. This fact adds considerably to the thermodynamic usefulness of the entropy function. The entropy of a system, like the energy, is an extensive property, dependent upon the amount of matter in the system. For example, if the amount of matter is doubled, the heat quantities required for the same change of state will also be doubled, and the entropy will clearly increase in the same proportion. Another consequence of the entropy being an extensive property is that when a system consists of several parts, the total entropy change is the sum of the entropy changes of the individual portions. 

The path used for evaluation of the thermodynamic functions of mixtures in terms of temperature, pressure, and composition is shown in Fig. 14-3. The standard state of each component is taken to be the pure component in the perfect gas state T0 and P0. Again, in the limit as P is allowed to approach zero, the container at P and T (see Fig. 14-3) becomes a convertor in the sense that a perfect gas mixture enters and an actual gas mixture leaves. The resulting expressions for the thermodynamic functions are presented in Table 14-1. 

The restatement of the equations given in Table 14-2 in terms of the residual work function is presented in Table 14-3. The use of the reformulated form of the equations for the evaluation of the thermodynamic functions is a useful technique which appears to have been suggested first by Benedict et al.6... 

The language of Statistical Mechanics evolved over a considerable period of time. For example, the term "ensemble" is used to denote a statistical population of molecules "partition function" is the integral, over phase-space of a system, of the exponential of -E/kT [where E is the energy of the system, k is Boltzmann s eonstant, and T is the temperature in °K]. From this "function", all of the thermodynamic functions can be derived. The definitions that we shall need are given as follows in 2.5.1. on the next page. 

This is the value of the rotational partition function for unsymmetrical linear molecules (for example, heteronuclear diatomic molecules). Using this value of we can calculate the values of the thermodynamic functions attributable to rotation. 

A pump designed for hydrogen at its liquid density is not properly designed to make maximum use of the thermodynamic expansion process. In order to use the thermodynamic energy, the significant reduction in inlet fluid density occurring upstream of the impeller must be considered. The magnitude of the density is a function of the initial saturation tank pressure. 

Using our and published values of the entropies of the components and the heat of fusion of gallium [12], we calculated the standard values of the thermodynamic functions associated with the formation of gallium phosphide. 

An emf study was made of the thermodynamic functions relating to the formation (free energy, entropy, and enthalpy) of gallium selenide having the composition GajScs. The experimental data were used to calculate the standard values of the free energy, entropy, and enthalpy of formation at 298 K. 

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