Kinetic Derivation

The derivation that follows is essentially that given by Langmuir [9] in 1918, in which one writes separately the rates of evaporation and of condensation. The surface is assumed to consist of a certain number of sites S of which S are occupied and Sq = S - S arc free. The rate of evaporation is taken to be proportional to 5, or equal tokiSi, and the rate of condensation proportional to the bare surface So and to the gas pressure, or equal to k PSo. At equilibrium. 

Since S /S equals 6, the fraction of surface covered, Eq. XVII-4 can be written in the form 

Alternatively, 6 can be replaced by n/n , where denotes the moles per gram adsorbed at the monolayer point. Thus 

It is of interest to examine the algebraic behavior of Eq. XVII-7. At low pressure, the amount adsorbed becomes proportional to the pressure 

A plot of P/n versus P should give a straight line, and the two constants and b may be evaluated from the slope and intercept. In turn, n may be related to the area of the solid  

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The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

This difference looks large enough to be diagnostic of the state of the adsorbed film. However, to be consistent with the kinetic derivation of the Langmuir equation, it was necessary to suppose that the site acted as a potential box and, furthermore, that a weak adsorption bond of ifi corresponding to 1 /tq was present. With these provisions we obtain... 

The rate of physical adsorption may be determined by the gas kinetic surface collision frequency as modified by the variation of sticking probability with surface coverage—as in the kinetic derivation of the Langmuir equation (Section XVII-3A)—and should then be very large unless the gas pressure is small. Alternatively, the rate may be governed by boundary layer diffusion, a slower process in general. Such aspects are mentioned in Ref. 146. 

Since in chemisorption systems it is reasonable to suppose that the strong adsorbent-adsorbate interaction is associated with specific adsorption sites, a situation that may arise is that the adsorbate molecule occupies or blocks the occupancy of a second adjacent site. This means that each molecule effectively requires two adjacent sites. An analysis [106] suggests that in terms of the kinetic derivation of the Langmuir equation, the rate of adsorption should now be... 

Langmuir-Type Relations For systems composed of solutes that individually follow Langmuir isotherms, the traditional iTuilti-component Langmuir equation, obtained via a kinetic derivation, is... 

Here a four-step mechanism is described on the framework of methanol synthesis without any claim to represent the real methanol mechanism. The aim here was to create a mechanism, and the kinetics derived from it, that has an exact mathematical solution. This was needed to perform kinetic studies with the true, or exact solution and compare the results with various kinetic model predictions developed by statistical or other mehods. The final aim was to find out how good or approximate our modeling skill was. 

A reactive gas is slowly bubbled into a column of liquid. The bubbles are small, approximately spherical, and are well separated from each other. Assume Stokes law and ignore the change in gas density due to elevation. The gas is pure and reacts in the liquid phase with first-order kinetics. Derive an expression for the size of the bubbles as a function of height in the column. Carefully specify any additional assumptions you need to make. 

From the measurements published in the paper it cannot be inferred that the concentration-time curves can safely be extrapolated to zero time. The authors do not communicate details as to how well the beginning of the reaction could be defined. It should be pointed out that reaction kinetics derived by these means are subject to the uncertainties inherent in the extrapolation method. This holds particularly for a rapidly accelerated reaction. Furthermore, the possibility that the decomposition was initiated heterogeneously could not be excluded with certainty. These objections have to be considered when regarding the following kinetic results. 

Figure 9.3 pictures the oligomerisation reaction Ni is an abbreviation for the nickel-ligand moiety, kg stands for the rate of the growth reaction, and kt for the rate of the termination reaction. These rate constants are the same for all intermediate nickel alkyls, except perhaps for the first two or three members of the sequence owing to electronic and steric effects. Interestingly, a simple kinetic derivation leads to an expression for the product distribution. One can... 

The Nernst equation defines the equilibrium potential of an electrode. A simplified thermodynamic derivation of this equation is given in the Sections 5.3 to 5.5. Here we will give the kinetic derivation of this equation. 

A method for deriving enzyme-rate expressions combining both rapid equilibrium and steady-state procedures first illustrated by Chak With this method, demonstrated by Fromm and Huang, a different rate expression will be obtained depending on which steps are chosen to be in rapid equilibrium and which steps are not. See Enzyme Kinetic Derivations Turnover Number S. Cha (1988) J. Biol. Chem. 243, 820. 

Fromm and Cleland provide valuable discussions of the utility of Haldane relations in excluding certain kinetic reaction mechanisms based on a numerical evaluation of the constants on each side of the equal sign in the Haldane relation. If the equality is maintained, the candidate mechanism is consistent with the observed rate parameter data. Obviously, one must be concerned about the quality of experimentally derived estimates of rate parameters, because chemists have frequently observed that thermodynamic data (such as equilibrium constants) are often more accurate and precise than kinetically derived parameters. See Haldane Relations for Multisubstrate Enzymes... 

This kinetic-theory-based view of the Langmuir result provides no new information, but it does draw attention to the common starting assumptions of the Langmuir derivation and the BET derivation (Section 9.5a). This kinetic derivation of the Langmuir equation is especially convenient for obtaining an isotherm for the adsorption of two gases. This is illustrated in Example 9.4. 

Within a limited group of hydrocarbons, cycloalkenes, the kinetically derived association constants on Pt/Al203 correlate with both model reactions and the strain in the double bond which suggests that the relief of strain is a principal factor in determining relative reactivity in this series, Table 1. 

Although alkenes appear to be less tightly bound to Pd than to Pt, the relative individual rates in the two series differ little. The competitive rates on Pd were not determined so a comparison with the kinetically derived relative adsorption constants on Pt is unavailable. The zero order rate of norbornene relative to cyclohexene on 1-5% Pd/Alz03 at 30 °C is 3.4 while the relative competitive rate is 4.7 which increases to 7.6 in the presence of triphenylphosphine (ref. 24). 

An explanation for this difference in selectivity of the Ni catalysts is suggested by the studies of Okamoto et al. who correlated the difference in the X-ray photoelectron spectra of various nickel catalysts with their activity and selectivity in hydrogenations (ref. 28,29). They find that in individual as well as competitive hydrogenations of cyclohexene and cyclooctene on Ni-B, cyclooctene is the more reactive while the reverse situation occurs on nickel prepared by the decomposition of nickel formate (D-Ni). On all the nickel catalysts the kinetically derived relative association constant favors cyclooctene (ref. 29). The boron of Brown s P-2 nickel donates electrons to the nickel metal relative to the metal in D-Ni. The association of the alkene with the metal is diminished which indicates that, in these hydrocarbons, the electron donation from the HOMO of the alkene to an empty orbital of the metal is more important than the reverse transfer of electron density from an occupied d-orbital of the metal into the alkene s pi orbital. 

The kinetic derivation has the disadvantage that it refers to a certain model. The Langmuir adsorption isotherm, however, applies under more general conditions and it is possible to derive it with the help of statistical thermodynamics [8,373], Necessary and sufficient conditions for the validity of the Langmuir equation (9.21) are ... 

Kinetics Derived from Tracer Signal Dispersion in a Channel Reactor... 

The Langmuir adsorption isotherm is based on the characteristic assumptions that (a) only monomolecular adsorption takes place, (b) adsorption is localised and (c) the heat of adsorption is independent of surface coverage. A kinetic derivation follows in which the velocities of adsorption and desorption are equated with each other to give an expression representing adsorption equilibrium. 

Binding of NO occurs very rapidly, and the kon and kot t values of 105 dm3/(mol s) and 4 s-1, respectively, were reported for 14.7 °C. An improvement in precision of the /i ii values was secured by using the NO trapping method. While there is some concern for the lack of match of the rate constant from the direct and indirect methods, the kinetically derived value of KNO corresponds very well to that obtained from equilibrium measurements (ZfNO = 1.7 x 105mol/dm3 at 25 °C). Markedly positive entropies and volumes of activation for both on and off reactions were obtained A/Uon = + 138, AS off = + 161 J/(molK), AV on I 10.8, AV off = 16.9 cm3/mol. 

Example 7.6 Residence Time Distribution in a CST Kinetic Derivation Consider a CST of volume V and volumetric flow rate Q (depicted in the figure below). At time t — 0 we increase the inlet concentration, in volume fraction, from zero to X(0). Unlike in the previous Example, in a CST the concentration of the exiting stream at time t is identical to that in the tank and equals X(t). A simple mass balance gives... 

The presence of the cofactor xi serves as a hint of a need for the presence of a correlation cofactor fc which is bound to appear during a more detailed kinetic derivation of the expression for the diffusion coefficient due to changes in the functional relation between the local density and the probabilities of multibody configurations, the relation being controlled by the adspecies concentration gradient. This factor has been considered in the case of binary alloys [155,165] (see Subsection 5.2) ... 

If two different substrates bind simultaneously to the active site, then the standard Michaelis-Menten equations and competitive inhibition kinetics do not apply. Instead it is necessary to base the kinetic analyses on a more complex kinetic scheme. The scheme in Figure 6 is a simplified representation of a substrate and an effector binding to an enzyme, with the assumption that product release is fast. In Figure 6, S is the substrate and B is the effector molecule. Product can be formed from both the ES and ESB complexes. If the rates of product formation are slow relative to the binding equilibrium, we can consider each substrate independently (i.e., we do not include the formation of the effector metabolites from EB and ESB in the kinetic derivations). This results in the following relatively simple equation for the velocity ... 

In this situation cH/icD still has the significance discussed above (Section IIA3), as does the isotopic composition of the unreacted starting material, but all kinetically derived quantities will contain contributions from the equilibrium parameters of the dissociating substrate. 

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