Adjustable constants

There are many simple two-parameter equations for Hquid mixture constituents, including the Wilson (25), Margules (2,3,18), van Laar (3,26), nonrandom two-Hquid (NRTI.v) (27), and universal quasichemical (UNIQUAC) (28) equations. In the case of the NRTL model, one of the three adjustable parameters has been found to be relatively constant within some homologous series, so NRTL is essentially a two-parameter equation. The third parameter is usually treated as a constant which is set according to the type of chemical system (27). A third parameter for Wilson s equation has also been suggested for use with partially miscible systems (29,30,31). These equations all require experimental data to fit the adjustable constants. Simple equations of this type have the additional attraction of being useful for hand calculations. 

Cullinan presented an extension of Cussler s cluster diffusion the-oiy. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally vahd at higher concentrations). 

With two adjustable constants, you can fit a straight line. With five, you can fit an elephant. With eight, you can fit a running elephant or a cosmological model of the universe. ... 

Equation (7.35) is used to find Kthemo as a function of temperature. Since Kkinetic was given, and K,hermo can be calculated from it. Equation 7.38 contains only n and kf as adjustable constants, although kf can be divided between k and Tact if measurements are made at several temperatures. 

It appears that the complete model for both mass and heat transfer contains four adjustable constants, Dr, Er, K and Xr, but Er and Xr are constrained by the usual relationship between thermal diffusivity and thermal conductivity... 

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... 

Here b is an adjustable constant. This criterion can be written in a more general form, which will also include effects of regions with high curvature ... 

The unweighted least squares analysis is based on the assumption that the best value of the rate constant k is the one t,hat minimizes the sum of the squares of the residuals. In the general case one should regard the zero time point as an adjustable constant in order to avoid undue weighting of the initial point. An analysis of this type gives the following expressions for first-and second-order rate constants... 

Figure 3. Comparison of experimental data of network D 0.4 and calculated master curves, using any of the curves of Fig. 2 and Eqs. 5 and 6 with appropriate adjustable constant and integration limits.

Note that the useful empirical functional form discussed in Ref. 7 differs from the unsatisfactory Equation (29) in having the additional adjustable constant a, which can be adjusted to make this a useful equation. 

The Lennard-Jones (6-12) potential has served very well as an inter-molecular potential and has been widely used for statistical mechanics and kinetic-theory calculations. It suffers, however, from having only two adjustable constants, and there is no reason why it should not gradually be replaced by more flexible and more realistic functions. Recently a number of applications have been made of the Buckingham (6-exp) potential [Eq. (82)], which has three adjustable parameters. For this potential the first approximation to the coefficient of diffusion is written by Mason (M3) in the form... 

In these equations, T ), C, and represent the empirically adjusted constants associated with changes in bond lengths, angles, dihedrals, and... 

However c2 and C3 are arbitrary constants. Mathematically, Equation (6) may fit a given set of experimental data well since it has 3 adjustable constants, but the values of c2 and c3 are physically meaningless. 

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