Linear form

A variety of experimental data has been found to fit the Langmuir equation reasonably well. Data are generally plotted according to the linear form, Eq. XVn-9, to obtain the constants b and n from the best fitting straight line. The specific surface area, E, can then be obtained from Eq. XVII-10. A widely used practice is to take to be the molecular area of the adsorbate, estimated from liquid or solid adsorbate densities. On the other hand, the Langmuir model is cast around the concept of adsorption sites, whose spacing one would suppose to be characteristic of the adsorbent. See Section XVII-5B for an additional discussion of the problem. 

When plotted according to the linear form of the BET equation, data for the adsorption of N2 on Graphon at 77 K give an intercept of 0.004 and a slope of 1.7 (both in cubic centimeters STP per gram). Calculate E assuming a molecular area of 16 for N2. Calculate also the heat of adsorption for the first layer (the heat of condensation of N2 is 1.3 kcal/mol). Would your answer for Vm be much different if the intercept were taken to be zero (and the slope the same) Comment briefly on the practical significance of your conclusion. 

The second application of the CFTI approach described here involves calculations of the free energy differences between conformers of the linear form of the opioid pentapeptide DPDPE in aqueous solution [9, 10]. DPDPE (Tyr-D-Pen-Gly-Phe-D-Pen, where D-Pen is the D isomer of /3,/3-dimethylcysteine) and other opioids are an interesting class of biologically active peptides which exhibit a strong correlation between conformation and affinity and selectivity for different receptors. The cyclic form of DPDPE contains a disulfide bond constraint, and is a highly specific S opioid [llj. Our simulations provide information on the cost of pre-organizing the linear peptide from its stable solution structure to a cyclic-like precursor for disulfide bond formation. Such... 

The second application of the CFTI protocol is the evaluation of the free energy differences between four states of the linear form of the opioid peptide DPDPE in solution. Our primary result is the determination of the free energy differences between the representative stable structures j3c and Pe and the cyclic-like conformer Cyc of linear DPDPE in aqueous solution. These free energy differences, 4.0 kcal/mol between pc and Cyc, and 6.3 kcal/mol between pE and Cyc, reflect the cost of pre-organizing the linear peptide into a conformation conducive for disulfide bond formation. Such a conformational change is a pre-requisite for the chemical reaction of S-S bond formation to proceed. The predicted low population of the cyclic-like structure, which is presumably the biologically active conformer, agrees qualitatively with observed lower potency and different receptor specificity of the linear form relative to the cyclic peptide. 

Therefore, in the cases of both additives, the kinetic law for the catalysis will assume a linear form when the concentration of the added species, or, in the case of sulphuric acid, the nitronium ion generated by its action, is comparable with the concentration of the species already present. This effect was observed to occur when the concentration of additive was about o-2 mol 1, a value in fair agreement with the estimated degree of dissociation of nitric acid ( 2.2.1). 

The following paper discusses some of the problems that maybe encountered when using linear regression to model data that have been mathematically transformed into a linear form. 

A second approach for determining the piQ of an acid is to replot the titration curve in a linear form as a Gran plot. For example, earlier we learned that the titration of a weak acid with a strong base can be plotted in a linear form using the following equation... 

Values of Vmax and Km for reactions obeying the mechanism shown in reaction 13.15 can be determined using equation 13.18 by measuring the rate of reaction as a function of the substrate s concentration. The curved nature of the relationship between rate and the concentration of substrate (see Figure 13.10), however, is inconvenient for this purpose. Equation 13.18 can be rewritten in a linear form by taking its reciprocal... 

Gran plot a linearized form of a titration curve, (p. 293) graphite furnace an electrothermal atomizer that relies on resistive heating to atomize samples, (p. 414) gravimetry any method in which the signal is a mass or change in mass. (p. 233)... 

Equation 6 shows that the adsorption of component 1 at a partial pressureis reduced in the presence of component 2 as a result of competition for the available surface sites. There ate only a few systems for which this expression (with 5 1 = q 2 = 5 ) provides an accurate quantitative representation, but it provides useful quaUtative or semiquantitative guidance for many systems. In particular, it has the correct asymptotic behavior and provides expHcit recognition of the effect of competitive adsorption. For example, if component 2 is either strongly adsorbed or present at much higher concentration than component 1, the isotherm for component 1 is reduced to a simple linear form in which the apparent Henry s law constant depends onp. ... 

The nonlinear constant in these equations cannot be evaluated dkecdy by the methods previously described. Even forms such as these can be handled, however. For example, subtracting a trial value of a fromjy and taking logarithms transforms equation 97 into the linear form ... 

A value of q is assumed and values of k are calculated for each data point. The correct value of q has been chosen when the values of/c are nearly constant or show no drift. This procedure is applicable for a rate equation of any complexity if it can be integrated. Eqs. (7-28) and (7-29) can also be put into linear form ... 

The use of CO is complicated by the fact that two forms of adsorption—linear and bridged—have been shown by infrared (IR) spectroscopy to occur on most metal surfaces. For both forms, the molecule usually remains intact (i.e., no dissociation occurs). In the linear form the carbon end is attached to one metal atom, while in the bridged form it is attached to two metal atoms. Hence, if independent IR studies on an identical catalyst, identically reduced, show that all of the CO is either in the linear or the bricked form, then the measurement of CO isotherms can be used to determine metal dispersions. A metal for which CO cannot be used is nickel, due to the rapid formation of nickel carbonyl on clean nickel surfaces. Although CO has a relatively low boiling point, at vet) low metal concentrations (e.g., 0.1% Rh) the amount of CO adsorbed on the support can be as much as 25% of that on the metal a procedure has been developed to accurately correct for this. Also, CO dissociates on some metal surfaces (e.g., W and Mo), on which the method cannot be used. 

LINEARIZED FORM OF THE INTEGRATED MICHAELIS-MENTEN (MM) EQUATION... 

A limitation of the linearized forms of the MM equation is that no aeeurate estimates of and ean be established. Using the... 

Lineweaver-Burk plot Method of analyzing kinetic data (growth rates of enzyme catalyzed reactions) in linear form using a double reciprocal plot of rate versus substrate concentration. 

Various types of fluids, known as plastic fluids, may be encountered, which do not start to flow until a certain minimum shear stress is reached. The relationship between shear stress and the rate of shear strain may or may not take a linear form. 

The Arrhenius equation is best viewed as an empirical relationship that describes kinetic data very well. It is commonly applied in the linearized form... 

Usually the Arrhenius equation is placed in the linear form... 

We wish to apply weighted linear least-squares regression to Eq. (6-2), the linearized form of the Arrhenius equation. Let us suppose that our kinetic studies have provided us with data consisting of Tj, and for at least three temperatures, where o, is the experimental standard deviation of fc,. We will assume that the error in T is negligible relative to that in k. For convenience we write Eq. (6-2) as... 

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