Other Two-parameter Equations

Several two-parameter equations have been suggested based on the additivity of I and R effects, each being the combination of various components as described earlier. Taft originally introduced this approach for the electronic effects of benzene and defined Or values employing values of Oj (from the aliphatic series - see Chapter 2) and Hammett s a for para substituents (Equation 18). 

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Redlich-Kwong Equation. The Redlich-Kwong equation, which was proposed in 1949 [5], has been found to reproduce experimental P-V-T data for gases just as well as several equations that use more than two empirical constants and better than other two-parameter equations [6]. It has the form... 

When these values of a and b are used, the van der Waals equation fits the critical point and the slope and curvature of the critical isotherm. By continuity, the equation should also be a good fit to experimental data at temperatures slightly above the critical point. Other values of the constants may provide a better fit to data at conditions far from the critical point. Because it is an analytical function, the van der Waals equation cannot reproduce the discontinuities characteristic of vaporization shown at the two-phase regions in Figs. 7-9. Equation (21) (as well as other two-parameter equations, such as those of Berthelot, and Redlich and... 

Table 1.1 displays several additional equations of state, and values of parameters for several gases are found in Table A.3. The parameters for a given gas do not necessarily have the same values in different equations even if the same letters are used. The accuracy of several of the equations of state has been evaluated. The Redlich-Kwong equation of state seemed to perform better than the other two-parameter equations, with the van der Waals equation coming in second best. The Gibbons-Laughton modification of the Redlich-Kwong equation (with four parameters) is more accurate than the two-parameter equations. 

In Eq. (1.4-15), the parameters a and b have canceled out. The van der Waals equation of state thus conforms to the law of corresponding states. The same equation of state without adjustable parameters applies to every substance that obeys the van der Waals equation of state if the reduced variables are used instead of P, Vm, and T. The other two-parameter equations of state also conform to the law of corresponding states. 

Several assumptions were made in using the broad MWD standard approach for calibration. With some justification a two parameter equation was used for calibration however the method did not correct or necessarily account for peak speading and viscosity effects. Also, a uniform chain structure was assumed whereas in reality the polymer may be a mixture of branched and linear chains. To accurately evaluate the MWD the polymer chain structure should be defined and hydrolysis effects must be totally eliminated. Work is currently underway in our laboratory to fractionate a low conversion polydlchlorophosphazene to obtain linear polymer standards. The standards will be used in polymer solution and structure studies and for SEC calibration. Finally, the universal calibration theory will be tested and then applied to estimate the extent of branching in other polydlchlorophosphazenes. 

Complexes of Th4+ with cupferron (as well as oxime) were investigated by determining the distribution of tracer amount of 234Th between the organic (chloroform or isobutyl methyl ketone) and aqueous phases.118 The distribution curves show the presence of complexes other than the tetra ones, but formation constants for the neutral tetrakis products were calculated on the basis of the two-parameter equation method of Dryssen and Sillen. 

Clearly, by measuring any two of the four parameters, PCO2 or cdCOi, pH, ctC02, or cHCOj, and using the Henderson-Hasselbalch equation with the above appropriate values for pK and a, the other two parameters may be calculated. Although used as constants, these values must... 

With more than two variables the L terms are simply expanded to include more terms. For a two-parameter equation the size of the matrix remains 2. For a six-parameter equation with two variables, the size of the matrix is a symmetrical 6x6. Thus, only 27 sums need be calculated. The 6x6 square matrix is inverted and multiplied by the 1x6 matrix to obtain corrections in the six parameters. These are then adjusted and the process iterated to convergence. The iteration is controlled with a Visual Basic Macro. The rigorous inclusion of estimates for parameters from other experiments is easily incorporated into this procedure. The parameters and errors must be input. Next the program simply adds terms to the appropriate sums. For example, if the value of a has been determined to be ax with an uncertainty of sa, then the quantity 1 / (sa sa) is added to [a a ] and this quantity is multiplied by (aest —ax) and added to /-oa ] The adjustment is made as before, as are the parameters and uncertainties obtained. This has been demonstrated by Wentworth, Hirsch, and Chen [Chapt. 5, 37],... 

Diffraction experiments at high pressures provide information concerning the compression-induced changes of lattice parameters and, thus, sample volume. In pure phases of constant chemical composition and in the absence of external fields, the thermodynamic parameters volume V, temperature T and pressure P are related by equations of state, i.e. each value of a state variable can be defined as a function of the other two parameters. Some macroscopic quantities are partial differentials of these equations of state, e.g. the frequently used isothermal bulk modulus Bq of a phase at a defined temperature and zero pressure 5q = — Fq (9P/9F) for T= constant and P = 0, with the reciprocal of Bq V) being the isothermal compressibility k. Equations of state can also be formulated as derivatives of thermodynamic functions like the internal energy U or the Helmholtz free-energy F. However, for practical use the macroscopic properties of solids are often described by means of semi-empirical equations, some of which will be discussed in more detail. 

Perhaps a main benefit of converting A and B parameters and sensitivities to them to other scales has been the clarification of how two-parameter equations work as applied to solvent effects. Swains statistical method, which avoids the assignment of solvent parameters based on any one reaction, is an... 

After the separation of components, each individual parameter T, S, or P can be calculated by inversion of the solubility equations Ci,eq (T,S,P) if the other two parameters are known. The inverse techniques discussed above offer the possibility to determine all three parameters at once. However, these parameters are rather strongly correlated, in particular in the presence of excess air (Fig. 1 Aeschbach-Hertig et al. 1999b). The reason for this problem is that the dependence of noble gas concentrations on the parameters follows some systematic trends. The effects of changes of T and S on the concentrations increase with molar mass of the gas, whereas the effect of excess air (Ad) decreases, because the solubilities strongly increase with molar mass. P has a uniform effect relative to equilibrium concentrations, but in the presence of excess air it is relatively more important for the heavy noble gases. As a result, especially the effects of P and S are very similar, and both can be approximated by a combination of T and Ai (see Fig. 1). 

Resuming the discussion of isotropic liquids note that the four basic equations for conservation of mass and momentum include only four material parameters mass density p, compressibility p, viscosity T] and secmid viscosity Other two parameters, namely, thermal conductivity k and specific heat capacity Cp (or Cv) would come about as soon as the energy conservation law is applied to thermal processes. So, the isotropic liquid is completely described by six parameters. 

Nichita et al applied the pseudo-component method to the wax precipitation from hydrocarbon mixtures. To do so a general form of a two-parameter equation of state was used for vapour and liquid phases. The heavy components were assumed to precipitate in a single solid solution. Because lumping in pseudo-components often results in difficulties in solid-liquid equihbrium calculations the authors proposed a delumping procedure (mentioned in section 9.3.1). Lira-Galeana et al calculated wax precipitation in petroleum mixtures by assuming the wax consisted of several solid phases each described as a piue component or pseudo-component immiscible with other solid phases. 

The equation shows its best advantages in any situation involving liquid density calculations and in situations near the critical region, but it is usually better than other two parameter models in all regions. 

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

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