Phenomenological equations of state

It is well known that the equation of state of Eq. (28) based on the Gaussian statistics is only partially successful in representing experimental relationships tension-extension and fails to fit the experiments over a wide range of strain modes 29-33 34). The deviations from the Gaussian network behaviour may have various sources discussed by Dusek and Prins34). Therefore, phenomenological equations of state are often used. The most often used phenomenological equation of state for rubber elasticity is the Mooney-Rivlin equation 29 ,3-34>... 

Shen 39) has also considered the thermoelastic behaviour of another widely used phenomenological equation of state, the so-called Valanis-Landel equation. Valanis and Landel40) have postulated that the stored energy function W should be expressible as the sum of three independent functions of principle extension ratios. This hypothesis leads to the following equation of state... 

A comprehensive consideration of new phenomenological equations of state for rubber elasticity have been carried out lately by Tschoegl et al.41 45One of their equations of state is given by... 

Because no particular molecular model was assumed, theoretical values cannot be assigned to C, C, and C", nor can any molecular mechanisms be assigned. These phenomenological equations of state, however, accurately express the form of the experimental stress-strain data. 

A straightforward, important application of the statistical ensemble approach is to construct phenomenological equations of state the analogs o( PV = NRT or the van der Waals equations for ideal gases and gas-liquid transitions, respectively. As in thermal systems, equations of state can be used to predict the macroscopic behavior of a system that can have different microsopic interactions. Equations of states capture the relationships between macroscopic variable that emerge from the collective behavior of the microscopic objects. The angoricity-stress equations of state obtained from experiments [54], for example, predict... 

The gas-liquid phase behavior described by the van der Waals theory is considered simple . In this sense it is a reference distinguishing simple from complex . We emphasize that this refers to the qualitative description rather than to the quantitative prediction of fluid properties. Other phenomenological equations of state may be better in this respect, but it is the physical insight which here is important to us. Because of this we compute a number of other thermophysical quantities in terms of t, p, and/or v. 

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

The phenomenological magnetic equation of state, based on the Brillouin formula, has a form related to a scaling law ... 

Purely phenomenological as well as physically based equations of state are used to represent real gases. The deviation from perfect gas behaviour is often small, and the perfect gas law is a natural choice for the first term in a serial expression of the properties of real gases. The most common representation is the virial equation of state ... 

Fet us now confront the EOS predicted by the phenomenological TBF and the microscopic one. In both cases the BHF approximation has been adopted with same two-body force (Argonne uis). In the left panel of Fig. 4 we display the equation of state both for symmetric matter (lower curves) and pure neutron matter (upper curves). We show results obtained for several cases, i.e., i) only two-body forces are included (dotted lines), ii) TBF implemented within the phenomenological Urbana IX model (dashed lines), and iii) TBF treated within... 

The first difficulty derives from the fact that given any values of the macroscopic expected values (restricted only by broad moment inequality conditions), a probability density always exists (mathematically) giving rise to these expected values. This means that as far as the mathematical framework of dynamics and probability goes, the macroscopic variables could have values violating the laws of phenomenological physics (e.g., the equation of state, Newton s law of heat conduction, Stokes law of viscosity, etc.). In other words, there is a macroscopic dependence of macroscopic variables which reflects nothing in the microscopic model. Clearly, there must exist a principle whereby nature restricts the class of probability density functions, SF, so as to ensure the observed phenomenological dependences. 

In this connection, it is very interesting that the volume and intrachain changes obtained by various experimental methods 24,29,85) [Eq. (101)] agree well with Eq. (56) following from the Tobolsky-Shen semiempirical equation of state or the related phenomenological Eq. (76). The values of y determined from the data are rather small (0.1-0.3). As has been mentioned above, according to the semiempirical approach by Tobolsky and Shen one can formally suggest that the front-factor in Eq. (28) is pressure dependent. If it is really so, then the parameter y for rubbers can be considered as an experimental coefficient similar to the coefficient of thermal expansion and compressibility 29). 

First-principle calculations of the thermodynamic properties are more or less hopeless enterprise. One of the most famous phenomenological approaches was suggested by van der Waals [6, 8, 9]. Using the dimensionless pressure it = p/pc, the density v = n/nc and the temperature r = T/Tc, the equation of state for the ideal gas reads it = 8zzr/(3 -u) — 3zA Its r.h.s. as a function of the parameter v has no singularities near u = 1 v = it = t = is the critical point) and could be expanded into a series in the small parameter 77 = [n — nc)/nc with temperature-dependent coefficients. Solving this... 

It has been shown that die BCS theory does lead to die phenomenological equations of London. Pippard and Ginzburg and Landau, and one may therefore state that the basic phenomena of superconductivity are now understood from a microscopic point of view, i.e., in terms of the atomic and electronic structure of solids. It is true, however, that we cannot yet, ub initio, calculate V For a given metal and therefore predict whether it will be superconducting or not. The difficulty here is our ignorance of the exact wave functions to be used in describing the electrons and phonons in a specific metal, and their interactions. However, we believe that the problem is soluble in principle at least. 

In a noteworthy departure from the approach used in the theoretically-based equations of state, Arends L42,43] developed a phenomenologically-based equalion-of-state for polymer melts directly from observations on experimental data for five thermoplastic polymers and then attempted to extract physical meaning from this equation. 

Hart and coworkers developed a phenomenological theory of plastic deformation by using the concept of equation of state [6, 7]. The proposed deformation model consists essentially of two parallel branches (Fig. 6.10). Branch 1 represents... 

The relaxation of certain properties of the system can often be described by simple phenomenological equations called relaxation equations. In chemical kinetics, for example, the constrained state may be a mixture of gases in metastable equilibrium—for example, hydrogen and oxygen. A spark is then introduced and the gas mixture reacts. The concentration of the reactants and products change with time until a new equilibrium state is achieved. The relaxation equations are the familiar phenomenological equations of chemical kinetics and the relaxation times are related to the chemical rate constants. 

The media with which one has to deal when investigating preparation processes of hydrocarbon systems are invariably multi-phase and multi-component mixtures. Section II thus covers the aspects of the hydromechanics of physical and chemical processes necessary for an understanding of the more specialized material contained in following sections. Among these are transfer phenomena of momentum, heat, mass, and electrical charge conservation equations for isothermal and non-isothermal processes for multi-component and multi-phase mixtures equations of state, and basic phenomenological relationships. 

In contrast to Vf, the specific volume V in the glassy state of the polymer exhibits a larger slope than above Tg, which is due to the thermal contraction of Vocc ( occ,g O.Sag). The agreement of the slope dVld vh) fl-om the phenomenological relation Eq. (11.6) with dVf/d vh) from Eq. (11.5) above Tg we consider as evidence that o-Ps detects precisely the free volume calculated fl om S-S hole theory. The larger slope of dVld vh) below Tg supports the conclusion from the S-S equation of state calculation that Vocc shows here, as distinct from above Tg, a certain thermal expansion. 

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