Applications of Corresponding-States Theory

In 1974, Goodwin published a very high-precision, wide-range equation of state for methane whieh was capable of providing complete thermodynamic data from the triple point p = 70MPa and T=500K. This reference-fluid equation of state, along with the Leach shape factors, was used in a series of studies of liquefied-natural-gas (LNG) properties by Mollerup and co-workers. 

In the first studyMollerup and Rowlinson found that it was possible to reproduce liquefied natural gas densities to within 0.2%, even down to reduced temperatures of 0.3. In 1975, Mollerup continued his study of liquefied natural gas properties and reported results for phase equilibria, densities and enthalpies in both the critical- and normal-fluid regions. The method was also applied to natural gas, liquefied petroleum gas and related mixtures in a 1978 investigation. Mixtures studied included methane through pentane and common inorganics such as N2, CO, CO2 and H2S. The paper reported density predictions to within 0.2%, dew- and bubble-point errors not exceeding those of good experimental data and errors in liquid-phase enthalpies which were less than 2kJ- kg .  

During the mid-1970s, researchers at the National Institute of Standards and Technology (NIST) at Boulder, Colorado undertook a series of projects with the objectives of measuring and predicting the properties of liquefied natural gas and related mixtures. One result from this project was an extended corresponding-states model for liquefied natural gas densities developed by McCarty. That implementation used a 32-term, modified Benedict-Webb-Rubin equation of state for methane as the reference fluid and shape factors which had the same functional form as those proposed by Leach, but which had been re-fit to liquefied natural gas density data. The model reproduced the available liquefied natural gas density data to within 0.1 %. Eaton et alP used McCarty s methane equation to predict critical lines and VLB in (methane ethane). 

Another part of the NIST study focused on the development of predictive extended corresponding-states models for transport properties. That work has been reviewed recently and will not be included here. As mentioned in Section 4, however, the transport property work produced another reference-fluid equation of state for methane that was extrapolated to 7 = 40 K so as to avoid problems with the relatively high triple point of methane. That equation was later used by Romig and Hanley as the reference-fluid equation to predict the l AlnQ phase equilibria of (nitrogen ethane). 

In addition to the studies mentioned here, Mentzer, et a/. summarized extended corresponding-states results for phase equilibrium especially for systems containing hydrogen and common inorganics. They found accurate pure-fluid predictions for non-polar compounds up to about C7H16. For mixtures they found a strong dependence on the binary interaction parameters but once those parameters were optimized for phase equilibrium, they could be used to represent a variety of properties accurately, without further optimization. 

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To illustrate the application of corresponding-states theory to polymer solution calculations, we consider two cases of sol-vent/polymer vapor-liquid equilibria. The first case we consider is that of the chloroform/polystyrene solution. The second is that of benzene/polyethylene oxide. 

The applicability of corresponding-states theories of liquid mixtures has been tested for all the dense gas systems for which the excess functions have been directly determined. 186-139 results are inconclusive. For many of the mixtures studied there is a large difference in the critical temperatures between the components. In such cases, as shown above, the excess functions at moderate pressures depend primarily on the properties of the component of higher critical temperature - they are insensitive to the details of the mixed interactions. At higher pressures, where the test of theory is more significant, the calculated properties depend markedly on the choice of reference fluid. Even simple... 

The effect of an applied pressure on the UCFT has been investigated for polymer particles that are sterically stabilized by polyisobutylene and dispersed in 2-methy1-butane. It was observed that the UCFT was shifted to a higher temperature as the hydrostatic pressure applied to the system increased. There was also a qualitative correlation between the UCFT as a function of applied pressure and the 6 conditions of PIB + 2-methylbutane in (P,T) space. These results can be rationalized by considering the effect of pressure on the free volume dissimilarity contribution to the free energy of close approach of interacting particles. Application of corresponding states concepts to the theory of steric stabilization enables a qualitative prediction of the observed stability behaviour as a function of temperature and pressure. 

An example of the application of transition state theory to atmospheric reactions is the reaction of OH with CO. As discussed earlier, this reaction is now believed to proceed by the formation of a radical adduct HOCO, which can decompose back to reactants or go on to form the products H + COz. For complex reactions such as this, transition state theory can be applied to the individual reaction steps, that is, to the steps shown in reaction (15). Figure 5.3 shows schematically the potential energy surface proposed for this reaction (Mozurkewich et al., 1984). The adduct HOCO, corresponding to a well on the potential energy surface, can either decompose back to reactants via the transition state shown as HOCO./ or form products via transition state HOCO,/. ... 

Note that 4T/h has units of s and that the exponential is dimensionless. Thus, the expression in (3.1.17) is dimensionally correct for a first-order rate constant. For a second-order reaction, the equilibrium corresponding to (3.1.11) would have the concentrations of two reactants in the denominator on the left side and the activity coefficient for each of those species divided by the standard-state concentration, C, in the numerator on the right. Thus, C no longer divides out altogether and is carried to the first power into the denominator of the final expression. Since it normally has a unit value (usually 1 M ), its presence has no effect numerically, but it does dimensionally. The overall result is to create a prefactor having a numeric value equal to 4T/h but having units of M s as required. This point is often omitted in applications of transition state theory to processes more complicated than unimolecular decay. See Section 2.1.5 and reference 5. 

Regardless of the method, applications of transition-state theory for simulating penetrant diffusion are ultimately limited by the physical diffusion mechanism. For sufficiently large penetrant molecules, the separation of time scales that accompanies the jump mechanism could diminish, making it more difficult to distinguish between events that do or don t contribute to diffusive motion. How large a penetrant corresponds to this limit is still a research question. 

It is difficult to point to the basic reason why the average-potential model is not better applicable to metallic solutions. Shimoji60 believes that a Lennard-Jones 6-12 potential is not adequate for metals and that a Morse potential would give better results when incorporated in the same kind of model. On the other hand, it is possible that the main trouble is that metal solutions do not obey a theorem of corresponding states. More specifically, the interaction eAB(r) may not be expressible by the same function as for the pure components because the solute is so strongly modified by the solvent. This point of view is supported by considerations of the electronic models of metal solutions.46 The idea that the solvent strongly modifies the solute metal is reached also through a consideration of the quasi-chemical theory applied to dilute solutions. This is the topic that we consider next. 

It is often of industrial interest to be able to predict the equilibrium sorption of a gas in a molten polymer (e.g., for devolatilization of polyolefins). Unfortunately, the Prigogine-Flory corresponding-states theory is limited to applications involving relatively dense fluids 3,8). An empirical rule of thumb for the range of applicability is that the solvent should be at a temperature less than 0.85 Tp, where Tp is the absolute temperature reduced by the pure solvent critical temperature. 

It is important to point out here, in an early chapter, that the Born-Oppenheimer approximation leads to several of the major applications of isotope effect theory. For example the measurement of isotope effects on vapor pressures of isotopomers leads to an understanding of the differences in the isotope independent force fields of liquids (or solids) and the corresponding vapor molecules with which they are in equilibrium through use of statistical mechanical theories which involve vibrational motions on isotope independent potential functions. Similarly, when one goes on to the consideration of isotope effects on rate constants, one can obtain information about the isotope independent force constants which characterize the transition state, and how they compare with those of the reactants. 

A second interesting new theoretical development of a quite different kind has been the application of an information theory approach to the classification of experimental results [478-488]. The outcome of collisions, for example, the distribution of product translational or vibrational energies, is compared with the statistically predicted result for a closed system and the difference between the two is recorded as the surprisal. The surprisal is defined by an expression of the type found in information theory for example, if the probability of a reaction producing a molecular product in a final vibrational state v, corresponding to a fractional energy yield/,., is P fv.) and the corresponding statistical expression for a closed system is P°(fV ), then the surprisal is given by... 

As for most applications of statistical rate theory, the most difficult aspect is the characterization of the transition state. As discussed above, if the E0 value determined from a TCID experiment corresponds to the bond energy of AB+, then the transition state must be a loose one. In such a case, it is generally appropri-... 

Nucleophilic alkyl radicals, i.e. c-C6Hir or t-Bu% add to activated alkynes 3.0-5.2 times slower than the corresponding substituted alkenes, whereas nucleophiles having lone pairs of electrons attack alkynes markedly faster than alkenes. Application of Frontier Orbital theory indicates early and late transition states for radical nucleophiles and nonradical nucleophiles, respectively. ... 

Calculation of the Chains Unperturbed Dimensions. Inspection of the conformational energy maps (Figures 2 and 3) reveals that adoption of the familiar three-state (i.e., T, G", G") scheme is sufficient for application of the RIS theory to these chains.The corresponding statistical weight matrix U, inclusive of both first-order and second-order interactions (those interactions depending on, respectively, one and two skeletal-bond rotations), is represented by ... 

The possibility of producing such transitions was calculated by Goppert-Mayer 28 at the beginning of quantum mechanics. It was one of the firs applications of the perturbation theory to second order. This calculation requires a summation on all the other levels. But we assume that only one term of this summation, corresponding to the intermediate level Ej., has a predominant role we can then interpret the calculation in the following manner after absorption of the first photon, the atom is in the virtual state E with the energy defect Eq + r... 

This is the entire formal structure of classical statistical mechanical perturbation theory. The reader will note how much simpler it is than quantum perturbation theory. But the devil lies in the details. How does one choose the unperturbed potential, y How does one evaluate the first-order perturbation It is quite difficult to compute the quantities in Equation P5 from first principles. Most progress has been made by some clever application of the law of corresponding states. It is not the aim of this chapter to follow this road to solution theory any further. 

The principle of corresponding states is a two-parameter theory and works well only for simple molecules, which are the noble gases and a few nonpolar or very slightly polar ones. Two main approaches have been used in expanding its range of applicability. One is to introduce a third parameter, the most successful being the acentric factor, and the other is based on manipulating the intermolecular potentials. We will mention only the acentric factor. 

By application of the theory of corresponding states, these authors derived the following equation for the relative viscosity ... 

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