Single component identity

Ho and Aris (1987) argued that any formulation of reaction in continuous mixtures must satisfy the single-component identity (SCI), namely that it should reduce to the kinetics of a single component when the mixture is pure. This is true of Eq. 29, for with/(x) = S(x - x0), U(t) = V(x0t). The corresponding H(x, y) = discrete component each satisfying the kinetic law given by G. We see that this is... 

The meaning of the single-component identity is of interest, for if we take... 

Equation (98) should be discussed in some detail. When a approaches < (which is the case where only reactant x = 1 is present in the mixture), it correctly predicts the single-reactant result dCldt = —C. This is worded by saying that the alias satisfies the single-component identity (SCI) When the initial concentration distribution approaches a delta function, one recovers the result for a single component. It is clear that the SCI requirement must be satisfied for every kinetic equation, not only the linear one. In fact, one can generalize the requirement to that of the discrete component identity—when the initial concentration distribution approaches the sum of N distinct delta functions, one must recover the corresponding discrete description for A components (Aris, 1991a). 

Aris ° was the first to address the theoretical aspects of total lumping of first-order reactions. Luss and Hutchinson later noticed that serious problems arise if one extends the continuum approach to nonlinear kinetics. Ho and Aris ° discussed the origin of the difficulties in liunping nonlinear kinetics in continuous mixtures. They proposed a single-component-identity that must be satisfied by any continuum treatment in order to overcome the difficulties. Other aspects of the mathematical and conceptual difficulties have also been examined." " Krambeck" addressed thermodjmamic issues. Ocone and Astarita"" reviewed many aspects of continuous mixtures. 

PCA = principal components analysis SCI = single component identity. 

Many processes of interest, especially in catalysis, are known to have nonlinear intrinsic kinetics, for which the above lumping approach leads to a physically invalid model. The lumped model must satisfy an important limiting case called the single component identity (SCI) if all components have the same value for their kinetic parameters, a valid lumped model must reduce to the kinetics of a single component. 

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

For any single-component system such as a pure gas the molar Gibbs energy is identical to the chemical potential, and the chemical potential for an ideal gas is thus expressed as... 

An alternative way of looking at monolayers is to consider them as two-dimensional binary solutions rather than two-dimensional phases of a single component. The advantage of this approach is that it does acknowledge the presence of the substrate and the fact that it plays a role in the overall properties of the monolayer. Although quite an extensive body of thermodynamics applied to two-dimensional solutions has been developed, we consider only one aspect of this. We examine the film pressure as the two-dimensional equivalent of osmotic pressure. It will be recalled that, at least for low osmotic pressures, the relationship among uosm, V, n, and Tis identical to the ideal gas law (Equation (3.25)). Perhaps the interpretation of film pressure in these terms is not too farfetched after all ... 

An unequivocal identification of an unknown compound is unlikely by chromatographic processes alone. Not the least of the reasons for this is the need for comparison to standards thereby assuming reasonable prior assurance of the possible identity of the unknown. It should be noted that in addition to retention time measurements obtained on two or more column systems, if reasonable care has been exercised, quantitative measures of the suspect compound should also correspond, thus providing an additional secondary identification. In other words, whatever the unknown compound may be, it cannot be a mixture of two components on one column and a single component on the second column without... 

Trend outputs consist of a continuous electrical signal [0 to 10 volts or 4 to 20 mini (inA)] from Ihe processor. As many as 30 to 40 such outputs may be available from a single processor. Each output represents Ihe concentration of a particular component in one of the sample streams on a given analyzer sealed lo some convenient range. Component identity and scale factors for each output channel are user-assigned from the processor keyboard. 

Other combinations of rate constants in the two cases might produce identical profiles for either loss of reactant (R) or formation of product (P) in both cases, profiles for the other two components will differ. It is worth repeating that monitoring the concentration of a single component cannot distinguish between intermediate and cul de sac transient species. 

Next consider the triple point of the single-component system at which the solid, liquid, and vapor phases are at equilibrium. The description of the surfaces and tangent planes at this point are applicable to any triple point of the system. At the triple point we have three surfaces, one for each phase. For each surface there is a plane tangent to the surface at the point where the entire system exists in that phase but at the temperature and pressure of the triple point. There would thus seem to be three tangent planes. The principal slopes of these planes are identical, because the temperatures of the three phases and the pressures of the three phases must be the same at equilibrium. The three planes are then parallel. The last condition of equilibrium requires that the chemical potential of the component must be the same in all three phases. At each point of tangency all of the component must be in that phase. Consequently, the condition... 

The thermodynamic equations for the Gibbs energy, enthalpy, entropy, and chemical potential of pure liquids and solids, and for liquid and solid solutions, are developed in this chapter. The methods used and the equations developed are identical for both pure liquids and solids, and for liquid and solid solutions therefore, no distinction between these two states of aggregation is made. The basic concepts are the same as those for gases, but somewhat different methods are used between no single or common equation of state that is applicable to most liquids and solids has so far been developed. The thermodynamic relations for both single-component and multicomponent systems are developed. 

A Scatchard plot obtained for the interaction of M2 and "intact" con A, Figure 6, was found to be linear as would be expected for single component systems and a value of 1.9 x 105 M-1 was obtained for the association constant. Also, identical difference spectra were obtained when the concentration of ligand and protein were symmetrically interchanged. Both of these results demonstrate the homogeneous behavior of the "intact" con A with respect to M2. Interestingly, the "nicked" con A that was obtained... 

Figure 7.17 Gel electrophoresis of haptoglobins. Haptoglobin 1-1 moves as a single component, whereas the other two types show genetically derived polymorphisms. Hemoglobin-binding activity is identical in the three types.

In some cases, single component representation is unsuitable. Figure 14.4 shows a simple sequence of structure standardization executed by a compound registration system including structure correction of the benzothiadiazole, salt/addend stripping and neutralization to depict the canonical parent structure representation. Any alternative salt form or structure representation of this compound will be standardized into the same parent structure and thus recognized as identical (at that level). 

Irradiations of Pure Substrates. Irradiation of one-component systems is a desirable prerequisite for the study of multi-component systems. The irradiation of a pure compound provides data which may indicate the identity of active intermediates which may then be considered for use as reactants in mixed systems. In addition, products which may be interesting in themselves may result from such treatment. Also, it is necessary to obtain as much product identification data as possible in single-component systems in order to simplify the analytical problems encountered when mixtures are irradiated. 

The upper limit of the vapor pressure line is the point A. This is known as the critical point and the temperature and pressure represented by this point are the critical temperature To and the critical pressure Pc, respectively. At this point the intensive properties of the liquid phase and the vapor phase become identical and they are no longer distinguishable. For a single-component system the critical temperature may also be defined as the temperature above which a vapor cannot be liquefied, regardless of the applied pressure. Similarly, the critical pressme of a single-component system may be... 

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