The reduced equation-of-state

Once the partition function of a system is known, the thermodynamic functions are readily found. The total partition function of the system can be written in this case as 

On evaluation of this differential, the following equation-of-state is obtained pv /f = l)-(l/vf) (3.70) 

It so happens that p 0 under normal experimental conditions which means that the equation-of-state defined by equation (3.70) can be simplified to 

Note that these equations-of-states are quite general, applying to both pure macromolecules and pure minimolecules alike. 

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According to the van der Waals equation of state, the value of compressibility at the critical point should be 3/8 = 0.375. When does a real gas depart significantly from an ideal gas We can write equation (4.9) as the reduced equation of state, with the reduced temperatures, pressures, and volumes = TITc, Pi = P/Pc, Vr = V/Vc- Then, all gases would have the same equation of state in the form of reduced parameters ... 

Equation (98) contains the implicit assumption that a for viscosity is the same as a for diffusion. The quantity y may be expressed in terms of the reduced equation of state Z — Z(pr,T,) thus ... 

Since r remains explicit in the reduced equation of state, a simple corresponding-state principle is not, in general, satisfied. For a pol5nneric liquid, however, r oo, and the equation of state is reduced to... 

In the case of small quantum effects (small values of A) and under the conditions given by (1-8) and (1-9), de Boer and Lun-beck have shown that the quantum deviations from the classical theorem of corresponding states of various thermodynamic properties should be proportional to A - Thus, e.g., the reduced equation of state would take the form... 

From (V-20) we obtain the reduced equation of state correct to order... 

Using the theory of Lennard-Jones and Devonshire for the description of the liquid state, De Boer and Lunbeck have calculated Pci and Pj at various values of v and T. Their expression of the reduced equation of state differs from (V-21) only by the fact that Aes is replaced by A. 

For the second illustration of the quantitative failure of the van der Waals equation close to criticality, consider small deviations t=x—l=(T— Tc)/Tc and (p = v — = V — Vc) Vc from the critical point, at which 7r = T = v= l. In terms of these parameters, the reduced equation of state then becomes... 

All variables are now dimensionless. Calculation of the partition function and the equation of state shows that the reduced equation of state is of the form... 

The reduced equation of state (16.5.6) leads us to consider more dosely the possibility of an extension of the theorem of corresponding states to polymers (Prigogine, Beixemans and Colin-naar [1957]). This extension which we shall introduce in tins paragraph opens the way to a treatment of polymer solutions by an appropriate modification of the average potential method (cf. Ch. IX-XII). 

Figure A2.3.4 The equation of state P/pkT- 1, calculated from the virial series and the CS equation of state for hard spheres, as a fimction of q = where pa is the reduced density.

Reduced Equations of State. A simple modification to the cubic van der Waals equation, developed in 1946 (72), uses a term called the ideal or pseudocritical volume, to avoid the uncertainty in the measurement of volume at the critical point. 

Equation (6.30) leads to a final method of obtaining an approximate value for In4> by making use of the law of corresponding states. This law states that all gases obey the same equation of state when expressed in terms of the reduced variables T, — T/Tc, pT - p/pc. and V, — V/Vc, where T., pc. and Vc are the critical temperature, pressure, and volume, respectively. 

The principle of corresponding states will be used as the basis for comparing the different equations of state. This principle states that, to a reasonable approximation, all gases show the same (p. Fni. T) behavior when compared in terms of the reduced variables. The extent to which this principle is followed is... 

From the ideal gas equation, it is found that for 1 mole of gas, PV/KT = 1, which is known as the compressibility factor. For most real gases, there is a large deviation from the ideal value, especially at high pressure where the gas molecules are forced closer together. From the discussions in previous sections, it is apparent that the molecules of the gas do not exist independently from each other because of forces of attraction even between nonpolar molecules. Dipole-dipole, dipole-induced dipole, and London forces are sometimes collectively known as van der Waals forces because all of these types of forces result in deviations from ideal gas behavior. Because forces of attraction between molecules reduce the pressure that the gas exerts on the walls of the container, van der Waals included a correction to the pressure to compensate for the "lost" pressure. That term is written as w2a/V2, where n is the number of moles, a is a constant that depends on the nature of the gas, and V is the volume of the container. The resulting equation of state for a real gas, known as van der Waals equation, is written as... 

The nonclassical equation of state of Levelt Sengers et al. (1983) for the critical region of H2O may be expressed in terms of reduced parameters (cf. section 9.2) as follows ... 

The empirical principle of corresponding states may then be stated in the form All substances obey the same equation of state in terms of the reduced variables ... 

Fickett in "Detonation Properties of Condensed Explosives Calculated with an Equation of State Based on Intermolecular Potentials , Los Alamos Scientific Lab Rept LA-2712 (1962), pp 34-38, discusses perturbation theories as applied to a system of deton products consisting of two phases one, solid carbon in some form, and the other, a fluid mixt of the remaining product species. He divides these theories into two classes conformal solution theory, and what he chooses to call n-fluid theory. Both theories stem from a common approach, namely, perturbation from a pure fluid whose props are assumed known. They differ mainly in the choice of expansion variables. The conformal solution method begins with the assumption that all of the intermolecular interaction potentials have the same functional form. To obtain the equation of state of the mixt, some reference fluid obeying a common reduced equation of state is chosen, and the mixt partition function is expanded about that of the reference fluid... 

The concept of the van der Waals conformational gas yields also a reduced equation of state 52)... 

What we learn from Eqs. (36) and (37) is that both the reduced ir-electron energy and the reduced HOMO-LUMO separation scale (approximately) according to a simple law. The analogy with the gas equation of state is appropriate here. One can... 

The above equation known as vander Waals reduced equation of state does not involve vander Waals constants as well as gas constant, hence it is a general equation applicable to all substances. 

There is a reduced equation of state that is followed by many gases the critical point data are used to define a reduced pressure, volume, and temperature ... 

It is not surprising that attempts have been made to derive equations of state along purely theoretical lines. This was done by Flory, Orwoll and Vrij (1964) using a lattice model, Simha and Somcynsky (1969) (hole model) and Sanchez and Lacombe (1976) (Ising fluid lattice model). These theories have a statistical-mechanical nature they all express the state parameters in a reduced dimensionless form. The reducing parameters contain the molecular characteristics of the system, but these have to be partly adapted in order to be in agreement with the experimental data. The final equations of state are accurate, but their usefulness is limited because of their mathematical complexity. 

The Kumar equation of state (Kumar, 1986 Kumar et al., 1987) is a modification of the Panayiotou-Vera model that was developed to simplify the calculations for multicomponent mixtures. Since the Panayiotou-Vera equation is based on the lattice model with the quasichemical approach for the nonrandomness of the molecules in the mixture, the quasichemical expressions must be solved. For a binary system the quasichemical expressions reduce to one quadratic expression with one unknown, but the number of coupled... 

The advantages of the Kumar equation of state are purely computational. The resulting expressions are approximations to the Panayiotou-Vera equation of state that will reduce to the proper forms for random conditions. Kumar et al. (1987) state that the expressions in Panayiotou and Vera (1982) differ because of errors in the Panayiotou and Vera work. The Vera and Panayiotou expressions have been shown to be correct with the methods described by High (Chapter 5, 1990). Thus, the discrepancies between the Kumar equation of state and the Panayiotou and Vera equation of state must occur in the approximations due to the Taylor series expansion. 

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