Virial value errors

In the virial methods, therefore, the activity coefficients account implicitly for the reduction in the free ion s activity due to the formation of whatever ion pairs and complex species are not included in the formulation. As such, they describe not only the factors traditionally accounted for by activity coefficient models, such as the effects of electrostatic interaction and ion hydration, but also the distribution of species in solution. There is no provision in the method for separating the traditional part of the coefficients from the portion attributable to speciation. For this reason, the coefficients differ (even in the absence of error) in meaning and value from activity coefficients given by other methods. It might be more accurate and less confusing to refer to the virial methods as activity models rather than as activity coefficient models. 

Despite the importance of mixtures containing steam as a component there is a shortage of thermodynamic data for such systems. At low densities the solubility of water in compressed gases has been used (J, 2 to obtain cross term second virial coefficients Bj2- At high densities the phase boundaries of several water + hydrocarbon systems have been determined (3,4). Data which would be of greatest value, pVT measurements, do not exist. Adsorption on the walls of a pVT apparatus causes such large errors that it has been a difficult task to determine the equation of state of pure steam, particularly at low densities. Flow calorimetric measurements, which are free from adsorption errors, offer an alternative route to thermodynamic information. Flow calorimetric measurements of the isothermal enthalpy-pressure coefficient pressure yield the quantity 4>c = B - TdB/dT where B is the second virial coefficient. From values of obtain values of B without recourse to pVT measurements. 

A very severe test of these virial-coefficient equations for the sea-water-related Na-K-Mg-Ca-Cl-S0,-H 0 system has been made by Harvie and Weare (37) who calculated tne solubility relationships for most of the solids which can arise from this complex system. There are 13 invariant points with four solids present in the system Na-K-Mg-Cl-SO - O and the predicted solution compositions in all 13 cases agree with the experimental values of Braitsch (38) substantially within the estimated error of measurement. In particular, Harvie and Weare found that fourth virial coefficients were not required even in the most concentrated solutions. They did make a few small adjustments in third virial coefficients which had not previously been measured accurately, but otherwise they used the previously published parameters. 

The slope of the lines in Figure 3.10, i.e., the virial constant B, is related to the CED. The value for B would be zero at the theta temperature. Since this slope increases with solvency, it is advantageous to use a dilute solution consisting of a polymer and a poor solvent to minimize extrapolation errors. 

Before comparing theory and experiment let us discuss the convergence of the semiclassical expansion of the dielectric second virial coefficient. In Table 1-15 the classical dielectric virial coefficient the first and second quantum corrections, and the full quantum result are reported. An inspection of this table shows that the quantum effects are small for temperatures larger than 100 K, and /it(/) can be approximated by the classical expression with an error smaller than 2.5%. At lower temperatures the dielectric virial coefficient of 4He starts to deviate from the classical value. Still, for T > 50 K the quantum effects can be efficiently accounted for by the sum of the first and second quantum corrections. Indeed, for T = 50, 75, and 100 K the series (7) + lli 1 (7) + (7) reproduces the exact results with errors... 

Approximately what percentage errors are allowable in the measured variables if the maximum allowable error in calculated values of the second virial coefficient B is I percent Assume that Z — 0.9 and that values of B are calculated by Eq. (332). 

Tables 2—4 (see Appendix, p. 95) list what the author believes are the most accurate values of the dielectric virial coefiBdents obtained to date. The values of the coefiSdents for a particular gas are all taken from a angle paper, except in the case of a few dipolar gases where the authors published values of Ae and B, in separate papers. Emphasis has been placed on accmate values of Be, and for some gases, particularly non-dipolar ones, more accurate values of Ae can be found elsewhere in the literature (e.g. Table II of ref. 61). The uncertainty limits listed are those given by the authors. In most cases involving expansion techniques these reflect deviations of the experimental points from the computed least-squares curve, and contain no estimate of possible systematic errors. The papers listed under other references contain data which will lead to values of the virial coefiSdents considered less accurate than those given in the tables. In many cases these papers contain no actual values of the coefiSdents, but rather data from which values can be obtained.

Fig. I. The differential cross section of neon. The squares represent experimental data and the error bars, average deviations. The solid line represents the values calculated from the scattering potential of Fig. 3. The dashed curve was calculated using the Lennard-Jones potential obtained from virial coefficient data.

Values of the second virial coefficient of ethylene for temperatures between 0° and 175°C have been determined to an estimated accuracy of 0.2 cm3/mol or less from low-pressure Burnett PVT measurements. Our values, from —167 to —52 cm3/mol, agree within an average of 0.2 cm3/mol with those recently obtained by Douslin and Harrison from a distinctly different experiment. This close agreement reflects the current state of the art for the determination of second virial coefficient values. The data and error analysis of the Burnett method are discussed. 

The results themselves have a subtlety associated with their interpretation owing to the presence of the volume-ratio parameter and, optionally, the initial density parameter. The Burnett equations have more flexibility to fit Burnett data than only a density series to PVT data. The statistical uncertainties reflect the quality of the experimental data relative to the particular model used to describe the experiment. The estimation of accuracy for Burnett results is necessarily somewhat subjective since the effect of systematic errors on parameter values is not explicit in nonlinear equations, such as the Burnett equations. Accuracy, however, can be estimated from a study of the effects of systematic errors in computer model calculations and from the magnitude of the change in the volume-ratio value determined with nonideal and nearly ideal gases. For these reasons, we include such information along with our virial coefficient results for ethylene. 

All theses 7 parameters were obtained from squared-error minimizations. The data analysis reveals that the best fits for hydrogen follow a very shallow minimum valleys so that the reported 7 parameters obtained from these fits should be taken with a grain of salt. The value 7j = 2 which was tested here is consistent with - and loosely justified by - the virial ratio for H, V e/E = 2. A reasonable explanation for possible distortions of the 7 value is perhaps linked to the basis used for hydrogen. This point is briefly examined with the help of DFT results (Table 5) and SCF computations using enriched bases. 

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