Kinetic Henri equation

In the special case of an ideal single catalyst pore, we have to take into account that diffusion is quicker than in a porous particle, where the tortuous nature of the pores has to be considered. Hence, the tortuosity r has to be regarded. Furthermore, the mass-related surface area AmBEX is used to calculate the surface-related rate constant based on the experimentally determined mass-related rate constant. Finally, the gas phase concentrations of the kinetic approach (Equation 12.14) were replaced by the liquid phase concentrations via the Henry coefficient. This yields the following differential equation ... 

The many circumstances leading to the Henri equation for enzyme conversion of soluble substrates are first noted, followed by some kinetic forms for particulate and polymer hydrolysis. Effects common to immobilized enzyme systems are summarized. Illustrative applications discussed Include metabolic kinetics, lipid hydrolysis, enzymatic cell lysis, starch liquefaction, microenvironment influences, colloidal forces, and enzyme deactivation, all topics of interest to the larger themes of kinetics and thermodynamics of microbial systems. 

Substances at high dilution, e.g. a gas at low pressure or a solute in dilute solution, show simple behaviour. The ideal-gas law and Henry s law for dilute solutions antedate the development of the fonualism of classical themiodynamics. Earlier sections in this article have shown how these experimental laws lead to simple dieniiodynamic equations, but these results are added to therniodynaniics they are not part of the fonualism. Simple molecular theories, even if they are not always recognized as statistical mechanics, e.g. the kinetic theory of gases , make the experimental results seem trivially obvious. 

The diffusion coefficient, sometimes called the diffusivity, is the kinetic term that describes the speed of movement. The solubiHty coefficient, which should not be called the solubiHty, is the thermodynamic term that describes the amount of permeant that will dissolve ia the polymer. The solubiHty coefficient is a reciprocal Henry s Law coefficient as shown ia equation 3. 

The kinetics of the general enzyme-catalyzed reaction (equation 10.1-1) may be simple or complex, depending upon the enzyme and substrate concentrations, the presence/absence of inhibitors and/or cofactors, and upon temperature, shear, ionic strength, and pH. The simplest form of the rate law for enzyme reactions was proposed by Henri (1902), and a mechanism was proposed by Michaelis and Menten (1913), which was later extended by Briggs and Haldane (1925). The mechanism is usually referred to as the Michaelis-Menten mechanism or model. It is a two-step mechanism, the first step being a rapid, reversible formation of an enzyme-substrate complex, ES, followed by a slow, rate-determining decomposition step to form the product and reproduce the enzyme ... 

Further experiments by Brown and particularly Henri were made with invertase. At that time the pH of the reactions was not controlled, mutarotation did not proceed to completion, and it is no longer possible to identify how much enzyme was used (Segal, 1959). Nevertheless, in a critical review of kinetic studies with invertase, Henri concluded (1903) that the rate of reaction was proportional to the amount of enzyme. He also stated that the equilibrium of the enzyme-catalyzed reaction was unaffected by the presence of the catalyst, whose concentration remained unchanged even after 10 hours of activity. When the concentration of the substrate [S] was sufficiently high the velocity became independent of [S]. Henri derived an equation relating the observed initial velocity of the reaction, Vq, to the initial concentration of the substrate, [S0], the equilibrium constant for the formation of an enzyme-substrate complex, Ks, and the rate of formation of the products, ky... 

The simplest adsorption equation is Henry s Law, that is, the loading is directly proportional to the sorbate partial pressure. X= KP This linear isotherm equation adequately describes some adsorbents and, in the limit of low coverage, it actually describes most sorbents. For adsorption that is truly described by Henry s Hnear relationship, the loadings are low, the adsorption is bound to be essenhaUy isothermal and there are several published analytical solutions to describe both batch kinetics and column dynamic behavior for such systems. 

In Eqn. 5.3-1, rj is the effectiveness factor of the catalyst with respect to the dissolved gaseous reactant and the temperature of the outer surface. The rate of reaction within the catalyst pores is comprised in rj. R is the reaction rate expressed in moles of gaseous reactant, A, per unit of bubble-free liquid, per unit of time. Reaction is irreversible. In equation (1) it has not been assumed that the gas is pure gas A, its concentration in the bubbles being Cg. Also, Henry s law for the gas is assumed and written as in Eqn. 5,3-4. Using Henry s law, Eqn. 5.3-4, the intermediate concentrations (Cs, CL) can be eliminated using the above system of equations. This provides an expression of the global rate in terms of an apparent constant, ko, that contains the various kinetic and mass transfer steps. Therefore, the observed rate can be written as ... 

The constant Km defined in equation (23) is called the Michaelis constant and is one of the key parameters in enzyme kinetics. It is a simple matter to proceed from this point to an expression comparable to the Henri-Michaelis-Menten equation (18), but with Km in place of Ks. First, rearranging equation (23) gives... 

The kinetics of enzyme reactions was first established by Michaelis and Menten, following the earlier work of Henri [23]. The famous Michaelis-Menten equation for the kinetics of an enzyme reaction with a single substrate is often written [23]... 

The permeation of a gas through a porous polymer is generally described by equations based on the kinetic theory of gases. The sorption isotherm described by Eq. 1 is concave to the pressure axis and is commonly observed for a penetrant gas in a glassy polymer. It is composed of Henry s law and Langmuir-terms [20] ... 

If adsorption equilibrium is established rapidly and the adsorbed and bulk species remain in equilibrium throughout the reaction, cAads can be expressed in terms of a suitable isotherm. This allows the differential kinetic equation to be integrated. For example, if Henry s law adsorption is presumed to apply [43]... 

The flux expression in Equation (4.16) displays the canonical Michaelis-Menten hyperbolic dependence on substrate concentration [S], We have shown that this dependence can be obtained from either rapid pre-equilibration or the assumption that [S] [E]. The rapid pre-equilibrium approximation was the basis of Michaelis and Menten s original 1913 work on the subject [140], In 1925 Briggs and Haldane [24] introduced the quasi-steady approximation, which follows from [S] 2> [E], (In his text on enzyme kinetics [35], Cornish-Bowden provides a brief historical account of the development of this famous equation, including outlines of the contributions of Henri [80, 81], Van Slyke and Cullen [203], and others, as well as those of Michaelis and Menten, and Briggs and Haldane.)... 

On the other hand, kinetics of reactions occiuring on a solid surface, that is, catalysis or photocatalysis, must be significantly different. There may be two representative extreme cases. One is so-called a diffusion controlled process, in which siuface reactions and the following detachment process occur very rapidly to give a negligible surface concentration of adsorbed molecules, and the overall rate coincides with the rate of adsorption of substrate molecules. In this case, the overall rate is proportional to concentration of the substrate in a solution or gas phase (bulk), that is, first-order kinetics is observed IS). The other extreme case is so-called surface-reaction limited, in which surface adsorption is kept in equilibrium during the reaction amd the overall rate coincides with the rate of reaction occurring on the surface, that is, reaction of e and h+ with surface-adsorbed substrate (l9). Under these conditions, the overall rate is not proportional to concentration of the substrate in the bulk unless the adsorption isotherm obeys a Henry-type equation, in which the amount of adsorption is proportional to concentration in the bulk (20). In the former case, the rate... 

The Henri-Michaelis-Menten equation describes the curve obtained when initial velocity is plotted versus substrate concentration. The curve shown in Figure 4-7 is a right rectangular hyperbola with limits of and - K . The curvature is fixed regardless of the values of and V mxx- Consequently, the ratio of substrate concentrations for any two fractions of Vj m is constant for all enzymes that obey Henri-Michaelis-Menten kinetics. For example, the ratio of substrate required for 90% of Vmat to the substrate required for... 

The expression for the simplest form of enzyme kinetics was first given by Henri (Compt. Rend. 135 (1902) 916), but is often named after Michaelis Menten (Biochem. Z. 49 (1913) 333), who investigated the same enzyme and used the same equation. The expression is Svm... 

One additional kinetic theory that deserves to be mentioned is the theory based on the Eyring equation. The essence in this theory is captured by the following phrase credited to Henry Eyring in 1945 "... a molecular system. .. passes. .. from one state of equilibrium to another. .. by means of all possible intermediate paths, but the path that is most economical in energy will be more often traveled." The equation of the chemical reaction (6.19), which was used for the interpretation of the kinetic theory based on the Arrhenius equation, is modified in order to explain the Eyring equation, as shown in equation (6.20). 

Z32 What is Henry s law Define each term in the equation, and give its units. How would you account for the law in terms of the kinetic molecular theory of gases Give two exceptions to Henry s law. 

For the most simple column models under certain simplifying conditions there are analytical solutions of the model equations available. Related to the equilibria this holds for problems vhere all components of interest are characterized by linear isotherm equations, in vhich the Henry constants are not affected by the presence of other component. Then all kinetic effects causing band broadening can be described by a single lumped parameter, for example, the number of theoretical plates. Consequently, the usage of two or more kinetic parameters is not justified. 

Kinetic investigations [32, 33] show dependencies, which are represented by Henry and Moiseev s equation (see Eq. (9.5)). Optimal technical catalysts have a deficit of the chloride ion concentration with respect to the Cu concentration. In a completely oxidized catalyst, the Cu + exists partly as copper oxychloride - its insoluble form has the composition Cu2(OH)3Cl. The chloride ion concentration controls the pH value of the catalyst and has, therefore, a great influence on the reaction rate. Figure 9.1 shows this correlation [6, 34]. 

Equation (2.207), like Henry s isotherm, is applicable for relatively low concentrations. At such kinetics there may be no difference between the adsorption and desorption rates. With increase in concentration the effect of a limited munber of active sites begins to appear. A consequence is that the adsorption rate law becomes nonlinear and equation (2.206) assiunes the following format... 

Henry s coefficient, (mol/m )Hquid/(niol/m )gj5 adsorption equation constant for Langmuir-Hinshelwood kinetics, m mol... 

---