Model systems linear additive models

Once a system has been dissected into parts, attempts at reassembly can be made to achieve some degree of understanding. This is done by building models made of two ingredients, namely parts and linear interactions between them. In this context, we can define a model as an effort to decouple and to discard events between ingredients that have no perceivable effect on the system under study. A model mimics a system in a simplified manner because it calls upon fewer states. A model is thus an abbreviation or a subset of the system it addresses (see also Section 3.3.1, and the enlightening review on models by Boxenbaum, 1992). In such linear additive models, small causes can only have small effects, while big causes necessarily elicit big effects. 

In the resolution of any multicomponent system, the main goal is to transform the raw experimental measurements into useful information. By doing so, we aim to obtain a clear description of the contribution of each of the components present in the mixture or the process from the overall measured variation in our chemical data. Despite the diverse nature of multicomponent systems, the variation in then-related experimental measurements can, in many cases, be expressed as a simple composition-weighted linear additive model of pure responses, with a single term per component contribution. Although such a model is often known to be followed because of the nature of the instrumental responses measured (e.g., in the case of spectroscopic measurements), the information related to the individual contributions involved cannot be derived in a straightforward way from the raw measurements. The common purpose of all multivariate resolution methods is to fill in this gap and provide a linear model of individual component contributions using solely the raw experimental measurements. Resolution methods are powerful approaches that do not require a lot of prior information because neither the number nor the nature of the pure components in a system need to be known beforehand. Any information available about the system may be used, but it is not required. Actually, the only mandatory prerequisite is the inner linear structure of the data set. The mild requirements needed have promoted the use of resolution methods to tackle many chemical problems that could not be solved otherwise. 

Polyesters derived from maleic anhydride and 2,2-di(4-hydroxyphenyl)pro-pane were copolymerised with styrene and then studied by CP/MAS NMR [39] spectroscopy. The three dimensional-crosslinked network formed by the polymerisation was examined using spin-lattice relaxation times in the rotating frame. A correlation between reaction conditions and the structure of the resulting material was found. The degree of residual unsaturation was determined by subtraction of two relaxation times from a linear additivity model used for erosslinked polymer systems. 

Adipic (aa) and succinic (sa) acid surface tension in 2 wt% aqueous NaCl solutions was studied by Heiming et al. [166]. Three different mixtures (in mass%) of the organics and salt were tested between 273 and 306 K 93%aa/5%sa, 80% aa/18%sa, and 5%aa/93%sa. The concentrations of organics were chosen to correspond to those at the moment of droplet activation for dry particles with d = 50, 100 nm (mixtures 1 and 2), and d = 40, 50, and 100 nm (mixmre 3). All mixtures showed a linear dependence of surface tension on temperature, and pure adipic acid was fotmd to cause more surface tension depression than pure succinic acid. Surface tension depression in this non-reactive system was described very well by a linearly additive model based on the S-L equation. 

Figure 6.2. Binding isotherms and the average correlation, g(C) - 1 for the tetrahedral (T), square (S), and linear (L) models. The sites are identical and all correlations are due to direct ligand-ligand pairwise additive interactions, (a) Curves for positive cooperativity, S(2) = 10 (b) curves for negative coopera-tivity, S(2) = 0.1. Note that in these systems the cooperativity increases in absolute magnitude from L to S to T.

This same data set was analyzed using the DCLS method with unsuccessful results (see Section 5.2.1.2). Analyzing the data with PLS reveals wh the classical approach failed. The number of factors required for the PLS models ranges from four to six when the system only contains three independent sources of variability (four components whose concentrations add to unity). This indicates that the system does not obey the assumptions of the DCLS model (e.g., linear additivity). The PLS technique has been able to model this even though the source and form of the violation is not known. 

The advantage of utilizing the standardized form of the variable is that quantities of different types can be included in the analysis including elemental concentrations, wind speed and direction, or particle size information. With the standardized variables, the analysis is examining the linear additivity of the variance rather than the additivity of the variable itself. The disadvantage is that the resolution is of the deviation from the mean value rather than the resolution of the variables themselves. There is, however, a method to be described later for performing the analysis so that equation 16 applies. Then, only variables that are linearly additive properties of the system can be included and other variables such as those noted above must be excluded. Equation 17 is the model for principal components analysis. The major difference between factor analysis and components analysis is the requirement that common factors have the significant values of a for more than one variable and an extra factor unique to the particular variable is added. The factor model can be rewritten as... 

Figure 28 shows comparisons of the transient gas and solid axial temperature profiles for a step-input change with the full model and the reduced models. The figure shows negligible differences between the profiles at times as short as 10 sec. Concentration results (not shown) show even smaller discrepancies between the profiles. Additional simulations are not shown since all showed minimal differences between the solutions using the different linear models. Thus for the methanation system, Marshall s model reduction provides an accurate 2Nth-order reduced state-space representation of the original 5/Vth-order linear model. 

Note that the applicahon of representation theory to quantum mechanics depends heavily on the linear nature of quantum mechanics, that is. on the fact that we can successfully model states of quantum systems by vector spaces. (By contrast, note that the states of many classical systems cannot be modeled with a linear space consider for example a pendulum, whose motion is limited to a sphere on which one cannot dehne a natural addition.) The linearity of quantum mechanics is miraculous enough to beg the ques-hon is quantum mechanics truly linear There has been some inveshgation of nonlinear quantum mechanical models but by and large the success of linear models has been enormous and long-lived. 

There are numerous other examples of two-box models. For instance, a two-box epilimnion/hypolimnion model was discussed in Chapter 21, and additional examples are given as problems at the end of this chapter. We must remember that as long as these models are linear, their solutions can be constructed with the help of Box 21.6. They always consist of the sum of not more than two exponential functions and are thus fairly simple. This situation changes drastically if we allow the differential equations to become nonlinear. A system of two or more nonlinear differential equations rarely can be solved analytically, yet the available computer tools (such as MATLAB) make their solution easy. 

The monomolecular reaction systems of chemical kinetics are examples of linear coupled systems. Since linear coupled systems are the simplest systems with many degrees of freedom, their importance extends far beyond chemical kinetics. The linear coupled systems in which we are interested may be characterized, in general terms, as arising from stochastic or Markov processes that are continuous in time and discrete in an appropriate space. In addition, the principle of detailed balancing is observed and the total amount of material in the system is conserved. The system is characterized by discrete compartments or states and material passes between these compartments by first order processes. Such linear systems are good models for a large number of processes. 

The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deemter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibrium-dispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. 

Additional data of Arnold and Toor are compared to the predictions of the linearized equations and of the effective diffusivity models in the triangular diagram in Figure 6.4. Clearly, the agreement with the data is very bad indeed. Thus, we have our second demonstration of the inability of the effective diffusivity method to model systems that exhibit strong diffusional interactions. ... 

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