Thermodynamic potential parameter

The molecular and liquid properties of water have been subjects of intensive research in the field of molecular science. Most theoretical approaches, including molecular simulation and integral equation methods, have relied on the effective potential, which was determined empirically or semiempirically with the aid of ab initio MO calculations for isolated molecules. The potential parameters so determined from the ab initio MO in vacuum should have been readjusted so as to reproduce experimental observables in solutions. An obvious problem in such a way of determining molecular parameters is that it requires the reevaluation of the parameters whenever the thermodynamic conditions such as temperature and pressure are changed, because the effective potentials are state properties. 

Here the nucleation barrier AO is the excess thermodynamic potential needed to form the critical embryo within the uniform metastable state, while the prefactor Jq is determined by the kinetic characteristics for the embryo diffusion in the space of its size a. Expressions for both AO and Jo given by Zeldovich include a number of phenomenological parameters. 

The inverse of H determines the geometric compliance matrix (Nalewajski, 1993, 1995, 1997, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008) describing the open system in the Qi,F)-representation. The relevant thermodynamic potential is defined by the total Legendre transform of the system BO potential, which replaces the state-parameters (N, Q) with their energy conjugates (/a, F), respectively ... 

Let us now turn to the mixed, partly inverted (N, F)-representation describing the geometrically relaxed, but externally closed molecular system. The relevant thermodynamic potential is now defined by the partial Legendre transformation of W(N, Q) which replaces Q by F in the list of the system parameters of state ... 

Here, we want to emphasize that The correct way to find the ground state of the homogeneous neutral u, d quark matter is to minimize the thermodynamical potential along the neutrality line Q nQ=o = Qu,d,e nQ=o> not like in the flavor asymmetric quark system, where (3-equilibrium is required but pe is a free parameter, and the ground state is determined by minimizing the thermodynamical potential klu,d,e-... 

The Greek indices a,j3= II, B,G) count colors, the Latin indices i = u,d,s count flavors. The expansion is presented up to the fourth order in the diquark field operators (related to the gap) assuming the second order phase transition, although at zero temperature the transition might be of the first order, cf. [17], iln is the density of the thermodynamic potential of the normal state. The order parameter squared is D = d s 2 = dn 2 + dG 2 + de 2, dR dc dB for the isoscalar phase (IS), and D = 3 g cfl 2,... 

The gaps have been calculated by minimizing the thermodynamic potential Q, Eq. (2), in the space of order parameters ((/), A) and the results are shown in Figs. 1 and 2. 

Figure 1. Dependence of the thermodynamic potential on the light flavor gap q = 4>u = 4>d (order parameter) for different values of the chemical potential, 4>s = 682 MeV.

For a homogeneous system in equilibrium, the minimum of the thermodynamic potential ilq with respect to the order parameters negative pressure therefore the constant Qvac = Ec Wvac IS chosen such that the pressure of the physical vacuum vanishes. 

At the present stage, we include only contributions of the first family of leptons in the thermodynamic potential. The conditions for the local extremum of ilq correspond to coupled gap equations for the two order parameters [Pg.387]

Although the potential energy functions can be made to reproduce thermodynamic solvation data quite well, they are not without problems. In some cases, the structure of the ion solvation shell, and in particular the coordination number, deviates from experimental data. The marked sensitivity of calculated thermodynamic data for ion pairs on the potential parameters is also a problem. Attempts to alleviate these problems by introducing polarizable ion-water potentials (which take into account the induced dipole on the water caused by the ion strong electric field) have been made, and this is still an active area of research. 

Auerbach et al. (101) used a variant of the TST model of diffusion to characterize the motion of benzene in NaY zeolite. The computational efficiency of this method, as already discussed for the diffusion of Xe in NaY zeolite (72), means that long-time-scale motions such as intercage jumps can be investigated. Auerbach et al. used a zeolite-hydrocarbon potential energy surface that they recently developed themselves. A Si/Al ratio of 3.0 was assumed and the potential parameters were fitted to reproduce crystallographic and thermodynamic data for the benzene-NaY zeolite system. The functional form of the potential was similar to all others, including a Lennard-Jones function to describe the short-range interactions and a Coulombic repulsion term calculated by Ewald summation. 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

Physical measurements are directly input to the statistical thermodynamics theory. For example three-phase hydrate formation data, and spectroscopic (Raman, NMR, and diffraction) data were used to determine optimum molecular potential parameters (e,o,a) for each guest, which could be used in all cavities. By fitting only a eight pure components, the theory enables predictions of engineering accuracy for an infinite number of mixtures in all regions of the phase diagram. This facility enables a substantial savings in experimental effort. 

The parameters rp and 7J2 are linear combinations of the magnetic modes converted on the representation T5. The fact that below TN the magnetic subsystem of copper metaborate forms an easy-plane weak ferromagnet, twisted below 7] in a spiral, permits to compose rp as a combination of the ferromagnetic modes (1) and (3), and tj2 as a combination of the antiferromagnetic modes (2) and (4). Accordingly II = (Hn, Hn) = (Hx, -Hy). It is necessary to note that in the thermodynamic potential given by Eq. (5) the order parameter responsible for the transition at 7] is not chosen in an explicit form as it was done in our previous paper [8],... 

ATP = [Tj - T2 - (Dn/an - D22/a22)q2]2 + 4(A122 + C122)/(a a22) 1/2/2. Such a distinction corresponds to the properties of copper metaborate as described above. Indeed, at particular values of parameters of the thermodynamic potential (5) it is possible to obtain Tv > Tql with a stable homogeneous state (q = 0) at intermediate temperatures. Then for copper metaborate the temperature TN can be connected with Tv (12), and 7) - with... 

In conclusion, field dependent single-crystal magnetization, specific-heat and neutron diffraction results are presented. They are compared with theoretical calculations based on the use of symmetry analysis and a phenomenological thermodynamic potential. For the description of the incommensurate magnetic structure of copper metaborate we introduced the modified Lifshits invariant for the case of two two-component order parameters. This invariant is the antisymmetric product of the different order parameters and their spatial derivatives. Our theory describes satisfactorily the main features of the behavior of the copper metaborate spin system under applied external magnetic field for the temperature range 2+20 K. The definition of the nature of the low-temperature magnetic state anomalies observed at temperatures near 1.8 K and 1 K requires further consideration. 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

The thermodynamic potential of the redox reaction can also be obtained by averaging the anodic peak potential and the cathodic peak potential. However, for a reversible system, kinetic parameters such as reaction rate cannot be obtained because the reaction rates for both forward and backward reactions are extremely fast. 

Thermodynamics is the basis of all chemical transformations [1], which include dissolution of chemical components in aqueous solutions, reactions between two dissolved species, and precipitation of new products formed by the reactions. The laws of thermodynamics provide conditions in which these reactions occur. One way of determining such conditions is to use thermodynamic potentials (i.e., enthalpy, entropy, and Gibbs free energy of individual components that participate in a chemical reaction) and then apply the laws of thermodynamics. In the case of CBPCs, this approach requires relating measurable parameters, such as solubility of individual components of the reaction, to the thermodynamic parameters. Thermodynamic models not only predict whether a particular reaction is likely to occur, but also provide conditions (measurable parameters such as temperature and pressure) in which ceramics are formed out of these reactions. The basic thermodynamic potentials of most constituents of the CBPC products have been measured at room temperature (and often at elevated temperatures) and recorded in standard data books. Thus, it is possible to compile these data on the starter components, relate them to their dissolution characteristics, and predict their dissolution behavior in an aqueous solution by using a thermodynamic model. The thermodynamic potentials themselves can be expressed in terms of the molecular behavior of individual components forming the ceramics, as determined by a statistical-mechanical approach. Such a detailed study is beyond the scope of this book. 

A thermodynamic model of dissolution is presented in this chapter, which relates the solubility product constant to the thermodynamic potentials and measurable parameters, such as temperature and pressure of the solution. The resulting relations allow us to develop conditions in which CBPCs are likely to form by reactions of various oxides (or minerals) with phosphate solutions. Thus, the model predicts formation of CBPCs. 

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