Affine linear relationship

Since X4, >5 << 4.2, and both are close to the threshold min = 0.006, we expect to find two exact linear dependences in the data. From an exemination ofthe original paper (ref. 20) Box at al. (ref. 21) found that 4 had been not measured because of experimental difficulties, but rather had been assumed to constitute . of the total conversion of a-pinene (y ). That is, it was assumed that 4 = 0.031100-y ), which gives the exact affine linear relationship... 

Unfortunately, there are some technical difficulties associated with the determinant criterion (ref. 28). Minimizing the determinant (3.66) is not a trivial task. In addition, the method obviously does not apply if det[ V(p) ] is zero or nearly zero for all parameter values. This is the case if there exist affine linear relationships among the responses y, y2,. .., yny, as we discussed in Section 1.8.7. To overcome this problem the principal component analysis of the observations is applied before the estimation step. 

Kawabata, Tsuruta, and Furukawa (121) have reported a linear relationship between the logarithms of their Q values and the logarithms of the methyl affinities of Szwarc and co-workers (111, 123, 124). James and MacCallum (125) have found a linear relationship between the logarithms of the Qo values calculated from the definition of Zutty and Burkhart (122) and the logarithms of the rates of addition of ethyl radicals to various substituted ethylenes. Similar... 

A linear relationship between calculated An0+ values and experimental values of proton affinity (PA) for N-heteroaromatic compounds was also established (Figure 3). the points 1-4 (compounds 32h, 46, 32d and 31d respectively) have been excluded from the statistical treatment, r = 0.969]. 

An ab initio MO study has been carried out on a linear relationship between free energies of activation of Menshutkin reactions and the proton affinities of the nitrogen bases used as nucleophiles. The relationship had been discovered some... 

Other important cases are those in which the hydrogen affinities of Bt and the proton affinities of B are linearly related. Then, a linear relationship between PA(B ) and IP(B ) follows. Such situations are well documented (74JA6252 80OR457 830R45). 

In both cases [that is, AH(B ) = constant and AH(B ) oc IP(B )] one can rationalize the linear relationships found between solution pK and IP values (74HCA546 74JA3314 79JOC2093 84JHC269). They appear as a consequence of the corresponding linear relationships between solution pK values and gas phase proton affinities. 

There is a linear relationship between dose and plasma drug levels (i.e., linear or first-order pharmacokinetics) in normal and ultrarapid metabolizers. In these individuals, the earlier equation can be used to predict the daily dose needed to produce a specific plasma drug level once TDM has been done to estimate the patient s elimination rate. In poor metabolizers, TCAs follow nonlinear pharmacokinetics (i.e., disproportionate increases in plasma drug levels with dose increases) because they lack the CYP 2D6 and must use lower affinity enzymes to metabolize these drugs. 

The two oxides require no correction for CFSE, and the Me-O bondings are predominantly ionic. A linear relationship between PZC + 1/2 log (2 - v/v) and (v/L) confirms the plausibility of the Eqs. 25 and 26 for predicting PZC s in a series of oxides. A certain deviation behaviour of hydrogen-bonded hydroxides and oxyhydroxides from the linear relationship is explained by the fact that structure-forming OH- have a specific affinity toward the broken H-bonds on the surface layer and may be classified as non-dissocia-tive adsorption . 

Clearly, in the thermodynamic cycle above for the estimation of hydrogen affinities there must indeed exist, for a family of homologous compounds, a linear relationship between the basicity, defined as PA or as AG° (because the entropic correction would be constant within the family), and the IPV for the lone electron pair that undergoes the protonation. 

If we expand the expression in brackets, and consider the case of a near-equilibrium state, which may be specified by the inequality Aj RT 1, then we have a linear relationship between the reaction rate and the chemical affinity... 

Near thermodynamic equilibrium, similar linear relationships are also valid for elementary chemical processes, as well as for stepwise processes where the rates are proportional to the difference between the thermodynamic mshes of the initial and final reaction groups (see Section 1.4.2). Here, the criterion of proximity to thermodynamic equilibrium is relationship jA jl < RT, where Arij is the affinity for the transformation of reaction group i to reaction group j. In fact, while... 

The relationship between the concentration of an element in "whole" coal and the ash content can be used as a guide to the affinity of that element for, or incorporation in, the mineral matter or the carbonaceous material. If the concentration of an element increases with increasing ash content that element is presumed to be associated with the inorganic species that form ash, or in other words may be said to have an inorganic affinity. If the concentration shows no correlation with ash content, that element would be said to have an organic affinity. Linear least squares correlation coefficients were calculated for the concentrations of 39 elements... 

Tests show that the horsepower consumption of a seal does not follow the affinity laws with change of speed it is actually much closer to a linear relationship. 

For a chemical reaction, for example, non-equilibrium thermodynamics formulates a linear relationship between the reaction rate and the affinity, which constitute only the first term in the development of the law of mass action. To obtain the full law, one has to take into account not only the initial and final states of the kinetics but all intermediate configurations, i.e. one has to introduce a mesoscopic degree of freedom accounting for the different molecular configurations. When this is done, in the framework of mesoscopic non-equilibrium thermodynamics, one arrives at the law of mass action governing the kinetics for arbitrary values of the thermodynamic... 

---