Three-constant equation

Calculated from the best three-constant equation recommended by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970). Trt refers to the lower curve, and T,-u, to the upper curve. 

Table II gives the temperature interval, the number of laboratories that Battino judges have published reliable solubility data, the number of experimental values used in the linear regression, the linear regression standard deviation at the midpoint temperature, and the temperature of minimum mole fraction solubility (maximum value of Henry s constant) at one atmosphere partial pressure of the gas. For all of the gases except oxygen only a three constant equation was used.

The temperature of minimum mole fraction solubility at one atmosphere gas partial pressure was calculated from the three constant equation. The differentiation of equation (2) with respect to temperature gives Tm n = 100 A2/A3. The values that fall outside the temperature range of the experimental data used in the regression must be looked on as only tentative values of the temperature of minimum solubility. 

Neon + water. The only solubility data above 350 K are the data of Potter and Clynne (6). These were combined with Battino s selected data (1) for the linear regression. Figure 3 shows the extrapolation of Battino s equation, which is much too high, and the curves for both the three and four constant fits to the entire data set. Values of the parameters for the four constant equation are given in Table V. The four constant equation gives a better fit to the data at the low temperature than does the three constant equation. Of the five noble gas + water systems, the neon + water system is the only one for which the Potter and Clynne values are lower than Battino s selected values near 350 K temperature where the data sets overlap. 

Xenon + water. The solubility data of Potter and Clynne ((5) and of Stephan, Hatfield, Peoples and Pray (10 ) were used to estimate the mole fraction solubility at one atmosphere xenon pressure at the higher temperatures. The two sets of data were combined with the 20 solubilities selected from five papers by Battino (2) in a linear regression. Figure 6 shows the data, and Battino s equation and the three constant equation. The three constants for the tentative equation for use between the temperatures of 350 and 600 K are in Table V. The Stephan et al. solubility value at 574 K was not included in the regression. 

The coefficient of expansion of liquids is a more complicated function of the temperature than that of sohd bodies. The linear equation is practically never sufficient to represent the temperature variation. In general it is necessary to resort to a three-constant equation ... 

The temperature dependence of the second virial coefficient can be fitted by two-constant equations, e.g. that of Lennard-Jones, but these have not a simple algebraic form. It can be fitted by several alternative three-constant equations. The form here used is the simplest derivable from a well defined model, namely a square-well potential (see Guggenheim, Australian Rev. Pure and App. Chem. 1953, 3,1). 

Broul, Nyvlt and Sohnel (1981) came to the conclusion that, when tested against solubility data from 70 inorganic salts in water, the accuracy of the two-constant equations was consistently lower than that of the three-constant equations. However, they found that there was Kttle to choose between individual equations in these two groups, but they did select equation 3.9 as being the most reliable. 

Equation (16) is a three-constant equation of the form nK = A/T + B nT + C, and the three constants can be evaluated if K is known with sufficient accuracy over a range of temperature, thus giving values of AHq, ASq, and ACp, and hence also the values of AH and at some standard temperature such as 25 C. The best method of treating the experimental data has been discussed by many authors, and several equations differing in form from (16) have been suggested to represent the variation of K with temperature. Most of these are empirical, and the... 

Another frequently used three-constant equation is that used by Rankine and by Kirchhoff... 

We use Eq. 12.AT for all the following examples. In Table A.9 all of the entries have only 3 constants, setting either c ox d=Q. The tables in [1, 2] use more complex equations with 5 constants these are presumably slightly more accurate than the three-constant equations in Table A.9. 

As a single equation with three variables, equation 6.26 does not have a unique solution for the concentrations of CHaCOOH, CHaCOQ-, and HaO+. At constant temperature, different solutions of acetic acid may have different values for [HaO+], [CHaCOQ-] and [CHaCOOH], but will always have the same value ofiQ. 

In 1893, it was shown that corresponding states are not unique to van der Waals equation of state (73). Rather, for any equation of state having not more than three constants, corresponding states are only a mathematical consequence. 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

This equation contains three constants (3/2, R, NA) and only one variable, the temperature T. It follows that—... 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles, follows ... 

The nature of the assumptions entertained in the deduction of Kirchhoff s equation ensure the theoretical validity of the latter only at low temperatures. Notwithstanding this, Juliusburger (1900), from a review of existing data, found the equation to give results deviating by 3 per cent, at most from the observed values for 74 substances out of 80 examined. He considers that it may be regarded as an empirical formula with three constants, up to the critical point, and gives the values of A, B, C for a number of substances. 

By means of these equations we can eliminate three constants from (2). But, if the equation (2) is now reduced, as is required by the law of corresponding states, it must not contain any constants characteristic of the substance, hence (2) can contain only three independent characteristic constants. In this case (2) can always be written in the form ... 

Suppose we attempt to evaluate all three constants, k, m, and n. Then the first three components of Equations (7.51) are needed. Evaluating the various... 

In summary, the simple Michaelis-Menten form of Equation (12.1) is usually sufficient for first-order reactions. It has two adjustable constants. Equation (12.4) is available for special cases where the reaction rate has an interior maximum or an inflection point. It has three adjustable constants after setting either 2 = 0 (inhibition) or k = 0 (activation). These forms are consistent with two adsorptions of the reactant species. They each require three constants. The general form of Equation (12.4) has four constants, which is a little excessive for a... 

Consider the case of three, constant-volume tanks in series, as represented in Fig. 2.11, in which the tanks have differing volumes V], V2, V3, respectively.. Assuming well-mixed tanks, the component balance equations are... 

The two-constant equation and the simple Elovich equation efifect-tively described the change of Co in the main source, the ERO fraction in both soils (Fig. 6.30). This suggests that the diffusion processes controlled the rate of Co transformation. It is noted above that the kinetics of Mn transformation in the three main fractions (EXC, CARB, and ERO), in the... 

Note that there are 11 dependent variables, or unknowns in these equations (three u s, six r,y s, P, and p), all of which may depend on space and time. (For an incompressible fluid, p is constant so there are only 10 unknowns. ) There are four conservation equations involving these unknowns (the three momentum equations plus the conservation of mass or continuity equation), which means that we still need six more equations (seven, if the fluid is compressible). These additional equations are the con-... 

Given data of (C, t) or of (r, C) and a correlation for kda, the other constants can be found as for single reactions. All three constants also can be found by nonlinear regression or by the solution of three simultaneous equations. 

P3.06.10. RATE EQUATION WITH THREE CONSTANTS A rate equation has the form... 

This is a cubic equation in r with three constants and the partial pressure of the reactant in the gas phase, pg, as the variable. 

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