Corresponding states theorem

In this subsection, a brief overview is given of thermodynamic methods for evaluation of the surface tensions. Most of these methods are empirical or semiempirical, like power correlations for specific substances [43] or correlations between surface tensions and viscosities [44—46]. Many methods have been developed only for single-component fluids (at least, to the best of our knowledge). Among them are the methods based on the corresponding states theorem [47—49]. 

The volumetric properties of fluids are represented not only by equations of state but also by generalized correlations. The most popular generalized correlations are based on a three-parameter theorem of corresponding states which asserts that the compressibiHty factor is a universal function of reduced temperature, reduced pressure, and a parameter CO, called the acentric factor ... 

Critical Temperature The critical temperature of a compound is the temperature above which a hquid phase cannot be formed, no matter what the pressure on the system. The critical temperature is important in determining the phase boundaries of any compound and is a required input parameter for most phase equilibrium thermal property or volumetric property calculations using analytic equations of state or the theorem of corresponding states. Critical temperatures are predicted by various empirical methods according to the type of compound or mixture being considered. 

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. 

Some consequences of the theorem of corresponding states have already been considered with reference to experimental results it has been shown that there is a good general agreement, but this is not strict. The question arises as to whether the deviations observed are due to the errors of experiment, or are indications of an inherent fault in the equation itself. 

The results of Amagat s and Raveau s work may be summed up in the statement that, whereas the theorem of corresponding states holds good very approximately, the equation of van der Waals gives results quite inconsistent with the experimental values, especially near the critical point. 

There still remains for consideration the question whether the theorem of corresponding states, which we have seen is at least approximately true, is in fact rigorously exact, or is only a more or less close approximation. This problem is, thanks to the now classical investigations of S. Young and his students, quite satisfactorily solved. Very careful measurements have shown that there are small deviations, the magnitude of which is much greater than the experimental errors, and the theorem of corresponding states, in the form previously employed ... 

The results of Young, and others, have shown that substances may be divided into two large groups according as they do or do not agree closely with the theorem of corresponding states. These may be called normal and abnormal substances,... 

The normal substances, however, really exhibit small deviations which are all the greater the more complex is the molecule of the substance. The theory of van der Waals, or in fact any hypothesis from which a theorem of corresponding states could be derived, assumes however that the transition from the gaseous to the liquid state, as well as the changes of density in either state, result from alterations in the propinquity of molecules which otherwise remain unaltered. Any association or dissociation of the substance would therefore give rise to abnormalities, and in fact the substances which deviate most from the normal relations (e.g.l water, acetic acid) are those which appear, on other grounds, to be associated in the liquid state. In the case of acetic acid the commencement of polymerisation, even in the state of vapour, is evident from the abnormal densities. 

A similar explanation may account for the slight deviations exhibited by normal substances, but fails to explain the anomalous behavior of the monatomic gases. A mechanical interpretation of the theorem of corresponding states has, how ever, been advanced by Earnerlingh Onnes ( Principle of Uniformity ) which appears to embrace all known cases. 

Two of them, 6 and p, can be deduced from the properties of pure A and B by applying the theorem of corresponding states which is implied in assumption (1). The two other parameters, 0 and a, were eliminated by assuming the combination rules ... 

Concerning point (b), a generalized theory was developed independently by Prigogine and his co-workers10 11 and by Scott.18 The main idea was to combine the concept of average potential involved in the cell model with the theorem of corresponding states for pure compounds, in such a way that ... 

The APM relies essentially on two bases. The first one is the theorem of corresponding states which says that the configurational partition function of a classical assembly of N molecules may be written in the form ... 

All these expressions clearly reduce to the theorem of corresponding states for a one-component system (cf. Eqs. (8) and (10)). The problem is now to attribute values to the reduced volumes and for A and B molecules in their respective mean fields in other words how is the available volume V shared between the molecules A and B We recover here a typical problem of the cell model. Three different assumptions on , (vBy have been proposed11 leading to slightly different versions of the APM ... 

The functions rj0(T) and experimental data of selected substances which closely follow the theorem of corresponding states.20 Six substances were retained argon, krypton, xenon, methane, carbon monoxide, and nitrogen (neon was discarded on account of quantum translational effects). 

The two parameters 8 and p for a given pair of substances can be obtained by application of the theorem of corresponding states to any suitable thermodynamic property. For example, in Section III we used the critical temperatures and pressures to determine the values of e and r for a series of substances, Kr being taken as reference. Of course all ratios of e and r for two substances, obtained from different thermodynamic properties, should agree closely the contrary would mean that the theorem of corresponding states is badly violated. 

The estimated uncertainties in the average values bb/ aa and bbAaa °f Tables V and VI are respectively 0-02 and 0.01 (resulting from both experimental errors and deviations from the theorem of corresponding states). This unavoidably leads to rather high inaccuracies in 8 and p (20% in the case of CH4-Kr considered above). 

SOL.35. I. Prigogine, A. Bellemans, and C. Naar-Colin, Theorem of corresponding states for polymers, J. Chem. Phys. 26, 710 (1957). 

Thompson. William H.. "A Molecular Association Factor for Use in the Extended Theorem of Corresponding States." Ph D. dissertation, Pennsylvania State University. University Park. 1966. 

For use of the generalized Redlich/Kwong equation one needs only the critical temperature and critical pressure of the gas. This is the basis for the two-parameter theorem of corresponding states All gases, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor, and all deviate from ideal-gas behavior to about the same degree. 

Although use of an equation based on the two-parameter theorem of corresponding states provides far better results in general than the ideal-gas equation, significant deviations from experiment still exist for all but the simple fluids argon, krypton, and xenon. Appreciable improvement results from the introduction of a third corresponding-states parameter, characteristic of molecular structure the most popular such parameter is the acentric factor , introduced by K. S. Pitzer and coworkers.t... 

The definition of w makes its value zero for argon, krypton, and xenon, and experimental data yield compressibility factors for all three fluids that are correlated by the same curves when Z is represented as a function of Tr and Pr. Thus the basic premise of the three-parameter theorem of corresponding states is that all fluids having the same value of w have the same value of Z when compared at the same Tr and Pr. 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

This is the basis for tlie two-parameter theorem of corresponding states ... 

If the two-parameter theorem of corresponding states were generally valid, the slope S would be the same for all pure fluids. This is observed not to be true each fluid has its own characteristic value of S, which could in principle serve as a third corresponding-states parameter. However, Pitzer noted that all vapor-pressure data for the simple fluids (Ar, Kr, Xe) lie on tlie same line when plotted as log vs. 1/71 and that the line passes tlirough logi j. = —1.0 at Tr = 0.7. This is illustrated in Fig. 3.13. Data for other fluids define other lines whose locations can be fixed in relation to the line for the simple fluids (SF) by the... 

Recently Nernst has proposed a formula (based partly on the theorem of corresponding states), which is also a particular case of the general formula, viz. 

We shall show in the next paragraph that the vapour pressure constants play an important part in the calculation of chemical equilibria in gases. The first problem which Nernst had to solve after the discovery of his theorem was therefore the calculation of at least the approximate value of C for as many simple substances as possible. For this purpose he made use of the theorem of corresponding states, and assumed further that the specific heat of solid and liquid bodies diminishes to a small but finite value (viz. nx 1 5, where n is the number of atoms in the molecule) as the temperature is lowered. On the evidence of the measurements published up to that time he also assumed that the molecular specific heat of gases and vapours is a linear function of the temperature which approaches the value 3-5-l-7ixl-5 at very low temperatures. In this way he arrived at the vapour pressure formula... 

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