Infinite pressure, limit

Figure 23. Temperature dependence of the reduced configurational entropy per unit site (defined by Eq. (70)) for the free association equilibrium polymerization model in the infinite pressure limit. The values of the enthalpy Afip = -35 KJ/mol and entropy Aip = -105 J/(mol K) of polymerization are identical to those used iu our extensive studies of equilibrium polymeriza-tiou, and the initial monomer concentration 4)° is taken as 4)° = 0.1. The crossover...

A number of related efforts have studied the impact of applying the infinite pressure limit, or the zero pressure limit. 1 The zero pressure limit would appear to be closer to the basis applied in developing activity models. Once again, the more complex model is technically more correct, but the improvement in accuracy is small. A reasonable compromise in accuracy and simplicity is offered by the PRWS mixing rule. 

The infinite pressure limit of EOS theories can be obtained from Equation (2.100) as... 

Obtain the infinite pressure limit for the nine EOS models listed in Problem 29. 

This equation has the expected behavior that AG< becomes more positive with decreasing solubility of the solute. However, free energies of solvation for different solutes cannot be related to their relative solubilities unless the vapor pressures of the different solutes are similar or one takes account of this via Equation 76. Furthermore, if the solubility is high enough that Henry s law does not hold, then one must consider finite-concentration activity coefficients, not just the infinite-dilution limit. 

The site entropy is thus a sensible candidate for describing fluid relaxation outside the immediate vicinity of the glass transition. In a more precise language, is actually an entropy density, and the maximum in Sc T) derives from an interplay between changes in the entropy and fluid density as the temperature is varied. Explicit calculations demonstrate that the maximum in Sc T) disappears in the limit of an incompressible fluid, which is physically achieved in the limit of infinite pressure. The pressure dependence of Sc T) is described in Section X, where it is found that the maximum in Sc T) becomes progressively shallower and 7a becomes larger with increasing pressure. 

At infinite pressure, where 1/[M] = 0, the rate constant should have its high-pressure limiting value. It is seen that this high-pressure limiting value, Aft, is equal to Jta. One would also qualitatively expect Aft = A., from the reaction scheme consisting of (12), ( —12), and (13) thus in the limit of infinite pressure, all the energized adducts formed in (12) will be stabilized in (13) and none will have a chance to decompose back to reactants via (—12). In this case, the rate constant will just be that for formation of HOSO, that is, Aa. 

Because QRRK theory was developed long before computing became readily available, it had to employ significant physical approximations to obtain a tractable result. The most significant assumption was that the molecule is composed of s vibrational modes with identical frequency i and that other molecular degrees of freedom are completely ignored. RRKM theory relies on neither approximation and thus has a much sounder physical basis. In the limit of infinite pressure, RRKM theory matches the transition state theory discussed in Section 10.3. 

As in the analogous case of gases (Section 2.4), corrections for nonideality can be obtained by measurements of osmotic pressure at different solute concentrations, with extrapolation toward the infinite-dilution limit. For electrolytes, the correction for ionic dissociation is important. 

However, these new mixing rules (based both to infinite- or zero pressure limit) give, for the composition dependence of the second virial coefficient, results that are inconsistent with those obtained from statistical mechanics. 

The original van der Waals idea was that pressure in a fluid is the result of both repulsive forces or excluded volume effects, which increase as the molar volume decreases, and attractive forces which reduce the pressure. Since the molecules have a finite size, there would be a limiting molar volume, b, which could be achieved only at infinite pressure. At large intermolcular separations, London dispersion theory establishes that attractive forces increase as r6, where r is the intermolecular distance. Since volume is proportional to r3, this provides some explanation also for the attractive term in the van der Waals equation of state. 

By relaxing mathematical rigor in establishing the connection between excess free-energy models and EOS, several successful approximate models have been developed in the limit of infinite pressure. One such model that uses excess Helmholtz free energy was introduced by Orbey and Sandler (1995c) and is as follows ... 

The first logarithmic term in eqn. (C.l) is zero in the limit of infinite pressure because for the PR EOS... 

In terms of the above discussion the intercept in Figure 4 corresponds to a limiting (infinite pressure) two body rate constant, ku of 4 X 10"12 cm.3 sec."1. This value is considerably lower than the minimum values for ki which can be estimated from several other studies [2.5 X 10"11 (13), 4 X 10"11 (5), 7 X 10"11 (8,10)]. Because of this inconsistency the value of ki determined from the present results should be considered to be in doubt. Experiments are presently being carried out to clarify this point. 

If the solution is dilute, then yw approaches its infinite dilution limit yw —> 1. Therefore, at high dilution, the vapor pressure of water is given by Raoult s law... 

Henry s law owes its name to William Heniy, the British chemist who reported the linearity between solubility and partial pressure in the early 1800s. The empirical observation of linearity was made independently of thermodynamics and took the force of a physical law. It is not a new physical principle, however, and the constant it introduces is fully accounted for by thermodynamics. To establish the relationship between formal quantities introduced earlier and Henry s law constant, we return to the general expression for the fugacity of a species in a mixture in eq. fio.it I. Applying this equation to the infinite dilution limit, we have... 

For a given system of solute/solvent, Henry s law constant is a function of pressure and temperature but not of composition. This is because it refers to a very specific composition of the solution, the infinite dilution limit. 

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