Reciprocal linear form

The monolayer coverage (Vmon) is hardly determined experimentally with accuracy. So, for practical purposes the Langmuir equation is suitably transformed in the so-called reciprocal linear form, as illustrated by the Eq. 1.6 ... 

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... 

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. 

The linear form of this equation is denoted hy the Lineweaver-Burk or double reciprocal plot, which is derived from the Michaelis-Menten and Hill equation and is denoted as ... 

Linear forms for rate equations. To obtain Km and Vmax from experimental rate data, Eq. 9-15 can be transformed by algebraic rearrangement into one of several linear forms. The popular double-reciprocal or Lineweaver-Burk plot of 1/ v against 1 / [S] (Fig. 9-3) is described by Eq. 9-20. The values of Km/ Vmax and 1 / Vmax can be evaluated from the slope and intercept, respectively, of this straight line plot. 

It is very useful to transform the Michaelis-Menten equation into a linear form for analyzing data graphically and detecting deviations from the ideal behavior. One of the best known methods is the double-reciprocal or Lineweaver-Burk plot. Inverting both sides of equation 3.1 and substituting equation 3.2 gives the Lineweaver-Burk plot 4... 

A double-reciprocal linear plot form can be easily derived ... 

Equations 8.29 and 8.30 are identical in form with the isotherm and double-reciprocal linear plot forms for optical spectroscopy (Equations 8.15 and 8.16). 

Simple, graphical methods for testing the fit of rate data to Equation 9 and for estimating the kinetic parameters k2 and KR involve using linearized forms of Equation 9. By far the most widely used linear form is that of Kitz and Wilson (14). Taking reciprocals of both sides of Equation 9, we obtain... 

Application of a least-squares method to the linearized plots (e.g., Scatchard and Hames) is not reasonable for analysis of drug-protein binding or other similar cases (e.g., adsorption) to obtain the parameters because the experimental errors are not parallel to the y-axis. In other words, because the original data have been transformed into the linear form, the experimental errors appear on both axes (i.e., independent and dependent variables). The errors are parallel to the y-axis at low levels of saturation and to the x-axis at high levels of saturation. The use of a double reciprocal plot to determine the binding parameters is recommended because the experimental errors are parallel to the y-axis. The best approach to this type of experimental data is to carry out nonlinear regression analysis on the original equation and untransformed data. 

Another method to obtain estimates for Km and is the rearrangement of the Michaelis-Menten equation to a linear form. The estimation for the initial velocities, Vo, from progress curves is not a particularly reliable method. A better way to estimate Vn is by the integrated Michaelis-Menten equation (Cornish-Bowden, 1975). Nevertheless, the graphical methods are popular among enzymolo-gists. The three most common linear transformations of the Michaelis-Menten equation are the Lineweaver-Burk plot of 1/Vo vs. 1/[S] (sometimes called the double-reciprocal plot), the Eadie-Hofstee plot, i.e. v vs. vo/[S], and the Hanes plot, i.e., [SJ/vo vs. [S] (Fig. 9.3). 

The Xj s are variations from equilibrium position of interatomic distances or angles. In this harmonic approximation we have terms in xj, but also crossed terms in XjXj, that represent harmonic interactions between two different vibrations j and / that are often related to nearly lying atoms. The presence of such terms means that one cannot excite the vibration defined by Xj without also exciting that defined by Xj,. One can, however, eliminate such cross terms in the harmonic approximation by performing a linear transform on the x s, which depends on the various coefficients found in the harmonic development of V, which are called force constants. The newly defined coordinates are those of normal modes of the molecule. They are linear combinations of the x s, which implies that the x s may also be expressed in the form of the reciprocal linear combination of these normal modes. The... 

Equation 15 may be rearranged in a linear form as the reciprocal equation,... 

A convenient method for testing a tentative one-plus equation and obtaining values of its coefficients is nonlinear regression (see Section 3.5). Alternatively, the equation can be brought into linear form and tested by linear regression or plotting. The rate is a nonlinear function of the concentrations because of the additive terms in the denominator, but the concentration dependence becomes linear when the reciprocal is taken. For example, if the rate is... 

I quation 14-10 is the basis for several computer-based methods for determining the formation constant K,. In the usual econstant concentration of metal is used and the total ligand concentration C , is varied. The change in absorbance i/1 is then measured as a function of total ligand concentration and the results statistically analyzed to obtain Ki. Unfortunately, the relationship shown in Equation 14-1 (Its nonlinear, and thus nonlinear regression must be used unless the eqitation is transformed to a linear form.t We can linearize the equation by taking the reciprocal of both sides to obtain... 

Figure 9 is a plot of the logarithm of the initial rate of carbon dioxide formation versus the reciprocal of the temperature. It is of the usual approximately linear form, which would not be expected if the total catalytic surface area were changing with temperature. 

If the adsorbate is in the gas phase and tends to form many layers on the surfaces of soil clay particles, as do water vapor and the inert gases listed in Table 1.6, then can be calculated as the reciprocal of the sum of slope and y intercept of the line obtained by fitting adsorption data to a linear form of the Brunauer-Emmett-Teller equation- ... 

In order to estimate the initial rate, the entire course of this reversible reaction was integrated according to Eq. (18.73), and redrawn in a linear form in Fig. 8B. Since the reaction was started with a saturating concentration of one of the substrates and, in this case, iCp /Ca and [Bo] [Ao], it is admissible to treat reaction (18.74) as an irreversible monosubstrate reaction and fit the entire reaction progress curve to Eq. (18.73). From the linear plot, one can calculate the maximal velocity V (equd to a reciprocal intercept on ordinate) and V/K (equal to a reciprocal slope). 

A graphical representation of concentration, the square root of the concentration, the logarithm, or the reciprocal concentration versus time leads to a linear form from which the slope, k, may be determined. 

The effective bandwidth (also called the spectral bandpass or spectral slit width, which is one half the bandwidth when the two slit widths arc identical, is seen to be the range of wavelengths that exit the monochromator at a given wavelength setting. The effective bandwidth can be related to the reciprocal linear dispersion by writing Equation 7-9 in the form... 

The calculation of the thermal conductivity of gas hydrate using EMD and the Green-Kubo linear response theory was repeated recently. In that work, convergences of the relevant quantities were monitored carefully as a function of the model size. Subtleties in the numerical procedures were also carefully considered. The thermal conductivity of methane hydrate was found to converge within numerical accuracy for 3 x 3 x 3 and 4x4x4 supercells. In the calculation of the heat flux vector there is an interactive term that is a pairwise summation over the forces exerted by atomic sites on one another. The species (i.e., water and methane) enthalpy correction term requires that the total enthalpy of the system is decomposed into contributions from each species. Because of the partial transformation from pairwise, real-space treatment to a reciprocal space form in Ewald electrostatics, it is necessary to recast the diffusive and interactive terms in this expression in a form amenable for use with the Ewald method using the formulation of Petravic. ... 

It is reasonable to assume that the glass transition temperature Tg drops linearly with increase in chain-end concentration, that is with an increase in the reciprocal of the molecular weight. This will give an equation of the form... 

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