Equation 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. 

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... 

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. ... 

Examination of equation 5 shows that if there are no chemical reactions, (R = 0), or if R is linear in and uncoupled, then a set of linear, uncoupled differential equations are formed for determining poUutant concentrations. This is the basis of transport models which may be transport only or transport with linear chemistry. Transport models are suitable for studying the effects of sources of CO and primary particulates on air quaUty, but not for studying reactive pollutants such as O, NO2, HNO, and secondary organic species. 

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 ... 

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... 

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... 

First degree equations linear equations) have the form... 

In most cases the authors prefer the second way of treatment of the desorption data, which is analytic in its nature the Arrhenius equation, whose parameters are assumed to be constant, is solved either in a closed form or numerically. The resulting quantities determining the location, height, and shape of a maximum on the desorption curve are analyzed and expressed whenever possible, in at least approximately linear form, and then compared with the experimental results. A simple analytical expression of the time-temperature function is essential for this kind of treatment. 

This equation is given in a linearized form. The slopes and intercepts allow one to calculate the desired ratios of rate constants. 

Equation (29) can be represented in a linear form, which allows the constant, b, and its evolution to be obtained as a function of the different electrochemical variables. From experimental data... 

A Linear Form of the Michaelis-Menten Equation Is Used to Determine... 

The direct measurement of the numeric value of and therefore the calculation of often requires im-practically high concentrations of substrate to achieve saturating conditions. A linear form of the Michaelis-Menten equation circumvents this difficulty and permits and to be extrapolated from initial velocity data obtained at less than saturating concentrations of substrate. Starting with equation (29),... 

A linear form of the Michaelis-Menten equation simplifies determination of and V. ... 

A linear form of the Hill equation is used to evaluate the cooperative substrate-binding kinetics exhibited by some multimeric enzymes. The slope n, the Hill coefficient, reflects the number, nature, and strength of the interactions of the substrate-binding sites. A... 

Equations 26 form a linear system whieh ean be solved without any difficulty. Let us first of all divide eqn 26 by An, and pass to the limit, so as to work directly in terms of partial derivatives. Let us then define the matrix ... 

The A2 parameter ean be ealeulated from chromatographic data by transfomfing the fundamental equation (Equation 4.22) and substituting values obtained from ehromatograms for eoneentration 0.3, 0.5, and 0.7 of one of the components, or by transformation of Equation 4.22 to the linear form assuming that... 

From scheme I, together with the experimentally observed first-order dependence on the total ester concentration, the rate relationship illustrated in Eq. (1) may be derived. In applying this equation, the cycloamylose concentration must be at least tenfold greater than the initial substrate concentration to ensure first-order conditions. Equation (1) may be rearranged in two ways to yield linear forms which permit graphical evaluation of fa, the maximal rate constant for release of phenol from the fully com-plexed ester and Kd, the cycloamylose-substrate dissociation constant (defined in Scheme I as A i/fa). These two methods are illustrated in Eqs. (2) and (3) and may be attributed to Lineweaver and Burk (1934) and to Eadie (1942), respectively. Although in theory both methods should give... 

In this equation, we used the fact that the potential energy difference between the states Aj, or A + AX, and state A + A /2 is equal to AUiyi+1/2, which is a consequence of the linear form of (2.42). 

Brunauer, Emmett, and Teller extended the Langmuir theory to multimolecular layer adsorption [8]. They related the condensation rate of gas molecules onto an adsorbed layer and the evaporation rate from that layer for an infinite number of layers. The linear form of the relationship is called the BET equation ... 

This may be used as a test to establish the value of n, by trial, from a series of experiments carried out to measure tV2 for different values of cAo. The value of kA can then be calculated from the value of n obtained, from equation 3.4-14 or -15. Alternatively, equation 3.4-15 can be used in linear form (In ty2 versus In cAo) for testing similar to that described in the previous section. 

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