Thermodynamic integration

For any solution the three thermodynamic properties of major importance are the enthalpy, the entropy, and the free energy of the solution. Usually these properties are given for the formation of one mole of the solution at constant temperature and pressure, and they are referred to as integral thermodynamic properties. The free energy of mixing (AGM) is related to the corresponding enthalpies (AHM) and entropies (ASM) by the equation... 

Simple polynomial expressions constitute the most common analytical model for partial or integral thermodynamic properties of solutions ... 

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 electromotive force (EMF) generated by electrochemical cells can be used to measure partial Gibbs energies which, like vapour pressure measurements, distinguishes these methods from other techniques that measure integral thermodynamic quantities. Following Moser (1979), a typical cell used to obtain results on Zn-ln-Pb is represented in the following way ... 

Similar methods have been used to integrate thermodynamic properties of harmonic lattice vibrations over the spectral density of lattice vibration frequencies.21,34 Very accurate error bounds are obtained for properties like the heat capacity,34 using just the moments of the lattice vibrational frequency spectrum.35 These moments are known35 in terms of the force constants and masses and lattice type, so that one need not actually solve the lattice equations of motion to obtain thermodynamic properties of the lattice. In this way, one can avoid the usual stochastic method36 in lattice dynamics, which solves a random sample of the (factored) secular determinants for the lattice vibration frequencies. Figure 3 gives a typical set of error bounds to the heat capacity of a lattice, derived from moments of the spectrum of lattice vibrations.34 Useful error bounds are obtained... 

More interesting is, however, the question concerning the effect of the total surface inhomogeneity on the integral thermodynamic characteristics values. Here it should be noted that the calculations made in [11] for a homogeneous graphite surface were performed with the atom- atom potential well depth reduced by 13% with respect to the values estimated from the quantum mechanical expressions (compare last two entries in both Tables 1 and 2) this enabled a correspondence to be achieved between the calculated and experimental thermodynamic quantities. We believe however (see [14]) that this... 

Another way to obtain integral thermodynamic functions is provided by the Duhem-Margules equation. The partial molar function AZ is plotted versus y (the composition of the alloy is represented by the formula A B) and then is integrated from... 

Thus, if the compensative effect for integral thermodynamic functions is interpreted, one should conclude that characteristic temperature (this is tana = AH/AS) is a negative value, but that is, of course, devoid of physical sense. 

The application of the ATHAS has produced a large volume of critically reviewed and interpreted heat capacity data on solid and liquid homopolymers. This knowledge is helpful in the determination of the integral thermodynamic functions which are also part of the data bank. Even of greater importance is the help these basic data give in the separation of nonequilibrium enthalpy and heat capacity effects as will be illustrated in a number of examples. 

Figure 4.47 shows that fullerene has an additional small transition at 256 K. Analysis of the entropy of transition of 27 J K" mol" and mobility by solid-state NMR proved a crystal to plastic-crystal transition as is described in Sect. 2.5.3. Fullerene is the plastic crystal material with the widest known temperature range. Its practical applications have not been fully explored. Figure 4.48, shows the integral thermodynamic functions that can be derived from the measured heat capacities. 

Besides interpretation of the heat capacity by itself, the heat capacity, Cp can also be used to derive the integral thermodynamic functions, enthalpy, H, entropy, S, and free enthalpy, G, also called Gibbs function ... 

Let us illustrate the principle of integral thermodynamic characteristics of chemical equilibrium in mixed solvent based on the example of ionic association process of methylsulfuric acid HSO3CH3 and its tetraalkylammonium salt CH3(C8H 7)jN + OSO3CH3 in mixed solvent - methanol - n-butanol. The dependencies of ionic association constants for these electrolytes on temperature and permittivity are described by equations based [9.56] the relationship ... 

A graphic user interface (GUI) was built to make the C-V device software user friendly. The GUI shows the measured temperature, electrical conductivity, acoustic velocity, and the determined salt and inhibitor concentrations. By clicking on HSZ on the menu bar, a HSZ window will appear and the determined salt and inhibitor concentrations will be automatically fed to the integrated thermodynamic model. The hydrate phase boundary is determined by the model after the hydrocarbon composition is loaded. The determined hydrate phase boundary can be shown in a plot or saved in a Microsoft Excel data file. The hydrate safety margin can be calculated if the operating temperature and pressure are known. 

Basic to the thermodynamic description is the heat capacity which is defined as the partial differential Cp = (dH/dT)n,p, where H is the enthalpy and T the temperature. The partial differential is taken at constant pressure and composition, as indicated by the subscripts p and n, respectively A close link between microscopic and macroscopic description is possible for this fundamental property. The integral thermodynamic functions include enthalpy H entropy S, and free enthalpy G (Gibbs function). In addition, information on pressure p, volume V, and temperature T is of importance (PVT properties). The transition parameters of pure, one-component systems are seen as first-order and glass transitions. Mesophase transitions, in general, were reviewed (12) and the effect of specific interest to polymers, the conformational disorder, was described in more detail (13). The broad field of multicomponent systems is particularly troubled by nonequilibrium behavior. Polymerization thermodynamics relies on the properties of the monomers and does not have as many problems with nonequilibrium. 

Integral thermodynamic values are derived for the a-phase by Allen (1991). The pressure-composition data are unanticipated and raise questions about the nature of the solution. Enthalpies of formation for the solid and liquid solutions are comparable with changes of about — 84 J for each 0.01 mol of H dissolved in the metal. As evidenced by the absence of a resolvable temperature dependence for the InP versus H/Pu isotherms, the derived entropies of formatiom for the solutions are essentially zero. This result is inconsistent with a ASf value near — 130 + 20 J/mol K Hj observed for condensed hydride phases (see tables 4a, 7 and 8) and implies that the disorder of H atoms in solution with plutonium is comparable to that for gaseous H2. 

Winters, BA., 1996. Analysis of the solar thermal upper stage technology demonstrator liquid acquisition device with integrated thermodynamic vent system. In AIAA-96-2745 32nd Joint Propulsion Conference, Lake Buena Vista, FL, July 13. 

Beginning with Fig. 2.14 it is first shown how the integral thermodynamic functions can be derived from the experimental heat capacity. Enthalpy,... 

The integral thermodynamic functions can be calculated with help of Gibbs-Duhem or Gibbs-Duhem- Margules equations ... 

The partial and integral thermodynamic values are presented by the terms AO and AO, respectively. 

The work carried out in the workpackage Toolboxes of the OPTICORR project has been based on the availability of three software tools and two major databases. The software tools are FactSage, the integrated thermodynamic databank system [3] ChemApp, the programmer s library for thermo chemical calculations [4] and ChemSheet, the thermochemistry add-in [5] for Microsoft EXCEL. ... 

This picture suggests that the more size-symmetric ion pairs such as KCl or NaCl should exhibit stronger attraction in solution than the size-asymmetric salt LiCl. This local argument should also have a bearing on integral thermodynamic properties such as osmotic coefficients, activity coefficients, maximal solubilities or heats of solution (compare Fig. 1). Most of these properties are difficult to obtain from simulations. The osmotic coefficient (p is comparably straightforward to calculate. It is related to the osmotic pressure O via... 

Figure 1. The integral thermodynamic functions H, C, and JS = H — C for PE. The subsoripts a refer to the amorphous, "o to the orystalline samples the equilibrium melting occurs at 1 H° is set arbitrarily to zero, it oan also be adjusted to the heat of formation AHf( J) by adding the integral of Cp from 298.15 to 0 K to AHf (298.15) (Ref. 34).

In general, integral thermodynamic functions are more easily interpreted than differential functions. For example. Hill et al [1951] emphasize that the usual entropy discussed qualitatively or quantitatively (statistical mechanics) in terms of order-disorder, randomness of motion, etc., of the adsorbed molecules is the integral entropy. .. and not the differential entropy... Hill s treatment was applied to kaolinite-water systems by Martin [I960]. 

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