Thermodynamics numerical calculations

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

Figure 5.55 Mutual dependence of Q i and Q d order parameters. In the upper part of the figure is outlined the T dependence of substitutional disorder Qod for different values of Qdi and, in the lower part, the T dependence of the displacive disorder parameter Qdt for different values of The heavy lines on the surface of local curves represent the solution for thermal equilibrium. From E. Salje and B. Kuscholke, Thermodynamics of sodium feldspar II experimental results and numerical calculations. Physics and Chemistry of Minerals, 12, 99-107, figures 5-8, copyright 1985 by Springer Verlag. Reprinted with the permission of Springer-Verlag GmbH Co. KG.

Salje E., Kuscholke B., Wruck B., and Kroll H. (1985). Thermodynamics of sodinm feldspar, II Experimental results and numerical calculations. Phys. Chem. Minerals, 12 99-107. 

The partition function provides the bridge to calculating thermodynamic quantities of interest. Using the molecular partition function and formulas derived in this section, we will be able to calculate the internal energy E, the heat capacity Cp, and the entropy S of a gas from fundamental properties of the molecule, such as its mass, moments of inertia, and vibrational frequencies. Thus, if thermodynamic data are lacking for a species of interest, we usually know, or can estimate, these molecular constants, and we can calculate reasonably accurate thermodynamic quantities. In Section 8.6 we illustrate the practical application of the formulas derived here with a numerical example of the thermodynamic properties for the species CH3. 

Principles of Thermodynamics should be accessible to scientifically literate persons who are either learning the subject on their own or reviewing the material. At Emory University, this volume forms the basis of the first semester of a one-year sequence in physical chemistry. Problems and questions are included at the end of each chapter. Essentially, the questions test whether the students understand the material, and the problems test whether they can use the derived results. More difficult problems are indicated by an asterisk. Some problems, marked with an M, involve numerical calculations that are most easily performed with the use of a computer program such as Mathcad or Mathematica. A brief survey of some of these numerical methods is included in Appendix B, for cases in which the programs are unavailable or cumbersome to use. 

In figures 7-13 the results of numerical calculation using formulae (6)-(12) are given with the following values of the coefficients of the incomplete thermodynamic potential (5) ... 

A predictive molecular thermodynamics approach is developed for microemulsions, to determine their structural and compositional characteristics [3.7]. The theory is built upon a molecular level model for the free energy change. For illustrative purposes, numerical calculations are performed for the system water, cyclohexane, sodium dodecyl sulfate as surfactant, pentanol as cosurfactant and NaCl as electrolyte. The droplet radius, the thickness of the surfactant layer at the interface, the number of molecules of various species in the droplets, and the distribution of the components between droplets and the continuous phase are calculated. The theory also predicts the transition from a mi-... 

Figure 5. Numerical calculation of the Mn between cross-links, Afc, os a function of the thermodynamic parameter x and the coal volume fraction v2,8. Curves are calculated for v = 0.769 cm3/g, V, = 80.56 cm3/mol and cluster size M0 = 150.

When the surface activity is stronger, the surface coverage is high throughout the values of Aq calculated (curve 2). The peak becomes dull and there is no big difference between the surface coverages in the forward and reverse scans. Nevertheless, by and large, the results of numerical calculation in Figure 7.5 indicate that the properties of the electrochemical instability under the current flow are essentially the same as those in the thermodynamic equilibrium. 

Salje E, Devarajan V, Bismayer U, Guimaraes DMC (1983) Phase transitions in Pb3(Pi xAS c04)2 Influence of the central peak and flip mode on the Raman scattering of hard modes. J Phys C 16 5233-5243 Salje E, Kuscholke B, Wrack B, Kroll H (1985) Thermodynamics of sodium feldspar 11 Experimental results and numerical calculations. Phys Chem Minerals 12 99-107 Salje E, Wrack B (1983) Specific-heat measurements and critical exponents of the ferroelastic phase transition in Pb3(P04)2 and Pb3(Pi.j,AS c04)2. Phys Rev B28 6510-6518 Schofield PF, Chamock JM, Cressey G, Henderson CMB (1994) An EXAFS study of cation site distortions through the E2/c-ET phase transition in the synthetic cuproscheelite-sanmartinite solid solution. Mineral Mag 58 185-199... 

The replica method can be also applied to express some of the thermodynamic properties of a confined fluid in terms of the correlation functions of the system. The details of the procedure can be found elsewhere [6, 26], In our system, we need to rearrange some expressions so as to avoid the difficulties in the numerical calculations due to the long range behavior of some correlation functions. 

Concentration-dependent activity coefficients can be accommodated with relative ease by an added term (e.g., [see Helfferich, 1962a Brooke and Rees, 1968] and variations in diffusivities are easily included in numerical calculations (Helfferich and Petruzzelli, 1985 Hwang and Helfferich, 1986). In both instances, however, a fair amount of additional experimental information is required to establish the dependence on composition. Electro-osmotic solvent transfer and particle-size variations are more difficult to deal with, and no readily manageable models have been developed to date. A subtle difficulty here is that, as a rule, there is not only a variation in equilibrium solvent content with conversion to another ionic form, but that the transient local solvent content is a result of dynamics (electro-osmosis) and so not accessible by thermodynamic considerations (Helfferich, 1962b). Theories based on the Stefan-Maxwell equations or other forms of (hcrniodyiiainics of ir-... 

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. 

Most of the numerical calculations have been based on the use of (i) single-step irreversible reaction models and (li) the assumption of a poly tropic gas mixture. In order to catch the essential features of real detonations and detonation initiation phenomena, Lee and Higgins [7] suggested that one should use (i) chemical mechanisms with at least two or three reaction steps and (li) thermodynamics relations for gas mixtures at high temperatures. 

Numerical calculations of phase equilibria require thermodynamic data or cotestations of data. For pure componeats, the requisite data may include saturation pressures (or temperatures), hem capacities, latent hems, and volumetric properties. For mixtures, one requires a PVTx equation of state (for determitiation of 4>j), and/or en expression for the molar excess Gibbs eenrgy g (for deiermiention of y,). We have disoussed in Sections 1.3 and 1.4 the correlating capabilities of selected equations or mite and expressions for gR, and the behavior of the fogacily cnefficients and activity coefficients derived from them. 

Molecular orbital computations are currently used extensively for calculation of a range of molecular properties. The energy minimization process can provide detailed information about the most stable stmcture of the molecule. The total binding energy can be related to thermodynamic definitions of molecular energy. The calculations also provide the total electron density distribution, and properties that depend on electron distribution, such as dipole moments, can be obtained. The spatial distribution of orbitals, especially the HOMO and LUMO, provides the basis for reactivity assessment. We illustrate some of these applications below. In Chapter 3 we show how MO calculations can be applied to intermediates and transitions structures and thus help define reaction mechanisms. Numerical calculation of spectroscopic features including electronic, vibrational, and rotational energy levels, as well as NMR spectra is also possible. 

In view of the numerous oxidation states, an extensive oxidation-reduction chemistry of technetium is expected. Polarographic reductions of pcrtechnetate in aqueous and in non-aqueous solutions, supplemented by coulometric and cyclic voltammetric measurements, were conducted to study the electrochemical behavior of technetium, to identify some oxidation stales and to synthesize new technetium compounds. Electrode reactions frequently proved to be irreversible and therefore not adequate for calculating thermodynamic data. The electrochemistry of technetium is reported in detail in several review articles [11-13]. 

The simplified-kinetic-theory treatment of reaction rates must be regarded as relatively crude for several reasons. Numerical calculations are usually made in terms of either elastic hard spheres or hard spheres with superposed central attractions or repulsions, although such models of molecular interaction are better known for their mathematical tractability than for their realism. No account is taken of the internal motions of the reactants. The fact that every combination of initial and final states must be characterized by a different reaction cross section is not considered. In fact, the simplified-kinetic-theory treatment is based entirely on classical mechanics. Finally, although reaction cross sections are complicated averages of many inelastic cross sections associated with all possible processes by which reactants in a wide variety of initial states are converted to products in a wide variety of final states, the simplified kinetic theory approximates such cross sections by elastic cross sections appropriate to various transport properties, by cross sections deduced from crystal spacings or thermodynamic properties, or by order-of-magnitude estimates based on scientific experience and intuition. It is apparent, therefore, that the usual collision theory of reaction rates must be considered at best an order-of-magnitude approximation at worst it is an oversimplification that may be in error in principle as well as in detail. 

A somewhat more interesting problem is the implicit solvent potential of mean force (PMF) between two Bom ions. Unlike the single ion solvation, no closed-form analytic solution is available for this system. Instead, it must be either modeled with one of the numerous published series solutions or with numerical calculations. Figure 5 presents a thermodynamic cycle illustrating a typical PB calculation of an ion—ion PMF. The energy of bringing the ion to a distance R in solution is... 

A numerical calculation needs knowledge of the solvent activity of die corresponding homopolymer solution at the same equilibrium concentration (here characterized by the value of the Flory-Huggins %-function) and the assumption of a deformation model that provides values of the factors A and B. There is an extensive literature for statistical thermodynamic models which provide, for example, Flory A = 1 and B = 0.5 Hermans A = 1 and B = 1 James and Guttf or Edwards and Freed A = 0.5 and B = 0. A detailed explanation was given recently by Heinrich et al. ... 

The outline of the paper is as follows. In Sect. 2 we describe the basic RISM and PRISM formalisms, and the fundamental approximations invoked that render the polymer problem tractable. The predicticms of PRISM theory for the structure of polymer melts are described in Sect. 3 for a variety of single chain models, including a comparison of atomistic calculations for polyethylene melt with diffraction experiments. The general problem of calculating thermodynamic properties, and particularly the equation-of-state, within the PRISM formalism is described in Sect. 4. A detailed application to polyethylene fluids is summarized and compared with experiment. The develojanent of a density functional theory to treat polymer crystallization is briefly discussed in Sect. 5, and numerical predictions for polyethylene and polytetrafluoroethylene are summarized. 

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