Barker equation

The temperature dependence of a can be estimated by different methods. At the first approximation, the a value can be accepted as constant, and direct experimental measurements (which is naturally the most precise method) or the known Barker equation can be applied [77] ... 

Barker J and Henderson D 1967 Perturbation theory and equation of state for a fluids II. A successful theory of liquids J. Chem. Phys. 47 4714... 

Hoover W G, Ross M, Johnson K W, Henderson D, Barker J A and Brown B C 1970 Soft sphere equation of state J. Chem. Phys. 52 4931-41... 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

Worth noting is the fact that Barkers method does not require experimental yf values. Thus the correlating parameters Ot, b, and so on, can be ev uated from a P-X data subset. Common practice now is, in fact, to measure just such data. They are, of course, not subject to a test for consistency by the Gibbs/Duhem equation. The worlds store of X T.E data has been compiled by Gmehling et al. (Vapor-Liquid Lquilibiium Data Collection, Chemistiy Data Series, vol. I, parts 1-8, DECHEMA, Frankfurt am Main, 1979-1990). 

For simplicity, we have shown an expansion wave in which the pressure is linearly decreasing with time. This, in general, is not the case. The release behavior depends on the equation of state of the material, and its structure can be quite complicated. There are even conditions under which a rarefaction shock can form (see Problems, Section 2.20 Barker and Hollenbach, 1970). In practice, there are many circumstances where the expansion wave does not propagate far enough to fan out significantly, and can be drawn as a single line in the x t diagram. 

Integral equations have been developed for inhomogeneous fluids. One such integral equation is that of Henderson, Abraham and Barker (HAB) [88] who assumed the OZ equation for a mixture and regarded the surface as a giant particle. For planar geometry they obtained... 

The vapour-pressure equation (22) has been verified for brom-and iodo-naphthalene by L. Rollo(7) (1909), and for toluene, naphthalene, and benzene by J. T. Barker (1910). The latter finds that the solid and liquid states give the same chemical constant, which is in agreement with Nemst s theory. 

As mentioned earlier, the concepts involved in the two-fermion case are essentially the same and also the expansion in terms of o have the similar problems as in the one-fermion case. Consequently, we have chosen to give only the final form of the Flamiltonian and refer to Barker and Glover [49] for mathematical details. The NRF of the two-fermion Flamiltonian can be written in terms of the operators defined in equation (73) as... 

The data of Levitan and Barker ( ) on the ability of carboxylic acids to promote potassium ion conductance in mollusk neurons (Table III) was reexamined. One can write Equations 15 and 16 for simple benzoic acids, but salicylic acids do not give a good correlation in log alone (r = 0.785). 

The Levitan and Barker series has also been examined by Hansch (18), who reported that the benzoic acids, salicylic acids and four miscellaneous acids could all be correlated using log Pi, Equation 17. The difference is in the calculation of log Pi values which were obtained by subtracting constants from the log P values, 3.69 for salicylic acids and 4.36 for benzoic acids. 

Equation (4) explains immediately the empirical observations made (see e.g. Barker et al. ) of the dependence on relative atomic sizes of the parameter x. 

An additional example of a functional group undergoing reaction at the alpha atom is provided by the reactions of coordinated oximes. The earliest observation on this class of system appears to have been reported by Barker who reacted bis (dimethylglyoxime) nickel with methyl iodide and dimethyl sulfate (3). The formulations suggested for the products are archaic however, the experiments have been repeated and found to be substantially correct (31). The reactions are exemplified in Equation 47. 

A method for interpolation of calculated vapor compositions obtained from U-T-x data is described. Barkers method and the Wilson equation, which requires a fit of raw T-x data, are used. This fit is achieved by dividing the T-x data into three groups by means of the miscibility gap. After the mean of the middle group has been determined, the other two groups are subjected to a modified cubic spline procedure. Input is the estimated errors in temperature and a smoothing parameter. The procedure is tested on two ethanol- and five 1-propanol-water systems saturated with salt and found to be satisfactory for six systems. A comparison of the use of raw and smoothed data revealed no significant difference in calculated vapor composition. 

There are two basic approaches to the calculation of vapor compositions from boiling point-liquid composition data or vapor pressure-liquid composition data (a) the coexistence equation (i) which requires the smoothing of experimental T-x or H-x data first, or (b) a correlating equation which relates the excess free energy with liquid composition. Various equations have been proposed, but Barker (2), who pioneered this method, employed Scatchard s equation (3). Raw or smoothed data are used, but the smoothing process may introduce unwarranted errors. 

The present author (4) has previously preferred to use raw isobaric data coupled with Barker s method (2) and the Wilson equation (5), but interpolation of the calculated discrete vapor composition values requires smoothing of boiling point-liquid composition data at some stage. 

Comparisons of calculated and measured quenching rates provide a useful measure of the accuracy of the wave function used for the system. As an example, the value of Ze for helium calculated from the zero energy static-exchange wave function of Barker and Bransden (1968) is 0.0347, or 0.0445 when the van der Waals potential is added to the static-exchange equation however, the experimental value obtained by Coleman et al. (1975b) at room temperature is 0.125 0.002 (see section 7.3). This rather large discrepancy, a factor of three, shows that the static-exchange wave function provides a poor representation of the electron-positron correlations in this system. 

Plot P-x,y diagram for Margules Equation with parameters from Barker s Method. 

A graphical form of the above equations is shown in Fig. 7.4. The eddy diffusivity DE is calculated from the correlation of Barker and... 

MATRICES AND LINEAR ALGEBRA. Hans Schneider and George Phillip Barker. Basic textbook covers theory of matrices and its applications to systems of linear equations and related topics such as determinants, eigenvalues and differential equations. Numerous exercises. 432pp. 5X x 8X. 66014-1 Pa. 8.95... 

Equation (27) is valid at low densities. The HNC approximation consists in assuming that Eq. (27) is valid at all densities. The HNC approximation was developed by many investigators. A full set of references can be found in the review of Barker and Henderson [4]. 

The PX HNC and MSA equations have been applied to a wide variety of systems. Some applications have been considered by Barker and Henderson [4]. An application that is of particular interest for colloidal systems is an electrolyte we will consider this application now. 

A well-known approximate molecular theory of a fluid at a planar interface is originally due to Helfand, Frisch and Lebowitz [76] and later to Henderson, Abraham and Barker [77] and Perram and White [78]. Consider a binary mixture (A,B) in which one of the species (A) becomes extremely dilute and infinitely large. S.E. [52] show that if the size of species A tends to infinity while the concentration of A tends toward zero, then a consequence of the OZ equation, coupled with the PY equation, is the relation... 

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