Pseudo-equilibrium, 6.33

The microcalorimetric measurements of Della Gatta and his co-workers in their investigation of the interaction of water vapour with highly dehydroxylated y-alumina confirm that in this system also, the nondissocia-tive chemisorption of water is nonactivated, whilst the dissociative chemisorption is always activated. Thus the pseudo-equilibrium between the two chemisorbed states is displaced towards dissociative chemisorption as the temperature is increased above 150 C. 

The application of a 50 percent Murphree vapor-phase efficiency on a y-x magram is illustrated in Fig. 13-40. A pseudo-equilibrium cui ve is drawn halfway (on a vertical line) between the operating hnes and the true-equilibrium cui ve. The true-equilibrium cui ve is used for the first stage (the partial reboiler is assumed to be an equilibrium stage), but for 1 other stages the vapor leaving each stage is assumed to approach the equilibrium value only 50 percent of me way Consequently, the steps in Fig. 13-40 represent actual trays. 

Under these eonditions, the desorption proeess and surfaee reaetions are in a pseudo-equilibrium state ... 

At pseudo equilibrium (-rj = 0, implying tliat tlie steps are very rapid ... 

The left side of Eq. (4-11) is also displayed in Fig. 4-2, and it is characterized by a rate constant of 0.30 s 1. In the case where k < k.2, a pseudo equilibrium state is attained. That is, after a certain time, the rates of loss of A and I become constant, which also can be seen from Eq. (4-7). After a sufficiently long time, exp(- 2t) becomes negligible compared with exp(-itif) then... 

Frequently the rate constants k for the forward reaction A B and 2 for the reverse reaction are much larger than 3, so that the pseudo-equilibrium A B is strongly coupled and the reaction B C serves merely to drain off the product C from the equilibrium. If we define the pseudoequilibrium constant ATab = k lk2, then the rate of formation of C is... 

We attribute the form of this rate law to be due to the pseudoequilibrium 6. We refer to 6 as a pseudo-equilibrium, because it is in fact a steady state rather than a true equilibrium. If... 

ILLUSTRATION 4.2 USE OF STEADY-STATE AND PSEUDO EQUILIBRIUM APPROXIMATIONS FOR INTERMEDIATE CONCENTRATIONS... 

Instead of using the steady-state approximation in the manipulation of the individual rate expressions, the same result may be reached by assuming that a pseudo equilibrium condition is established with respect to reaction B and that reaction C continues to be the rate limiting... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

TP space time for a plug flow reactor PE pseudo equilibrium... 

Unfortunately, the method is based on a fairly large nunber of assumptions. If we want to relate GN to the pseudo-equilibrium rubber plateau modulus, G , and to the effect of chain entangling in ordinary networks produced by cross-linking in the unstrained state, the following assumptions are required ... 

Figure 3. Modulus contributions from chemical cross-links (Cx, filled triangles) and from chain entangling (Gx, unfilled symbols) plotted against the extension ratio during cross-linking, A0, for 1,2-polybutadiene. Key O, GN, equibiaxial extension , G.v, pure shear A, Gx, simple extension Gx°, pseudo-equilibrium rubber plateau modulus for a polybutadiene with a similar microstructure. See Ref. 10.

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

From Eyring, the rate constant of reaction k depends on a pseudo equilibrium constant AT, relating to the formation of a transition-state complex, TS. Clearly, AT will always be virtually infinitesimal. 

Depending on the food item, only pseudo-equilibrium (or a stationary state) may be reached in the time frame of a measurement, as a very long... 

Based on their earlier results of forward and reverse shift, dissociation of H20 or C02 can be the rate limiting step, so that includes steps 3, 4 and 6. The other steps were assumed to be in pseudo-equilibrium, resulting in the following series of equations to describe the kinetics ... 

Dankwerts and McNeil ( 3) have employed the method of Van Krevelen et al. to predict the partial pressure of carbon dioxide over carbonated alkanolamine solutions. The central feature of this model is the use of pseudo-equilibrium constants and their dependence on ionic strength. The ratio of the pseudo-equilibrium constant at a certain ionic strength to that at zero ionic strength has been termed the "ionic characterization factor". However, ionic strength alone is insufficient to determine the ionic characterization factors. As well the ionic characterization factors are sometimes not a simple linear function of ionic strength. 

Kent and Eisenberg (5) also correlated solubility data in the system S+CC +alkanoleimines+ O using pseudo-equilibrium constants based on molarity. Instead of using ionic characterization factors, they accepted published values of all but two pseudoequilibrium constants and found these by fitting data for MEA and DEA solutions. They were able to obtain excellent fits by this approach and also discovered that the fitted pseudo-equilibrium constants showed an Arrhenius dependence on temperature. 

T. Kokugan, A. Trianto, and H. Takeda, Dehydrogenation of pure cyclohexane in the membrane reactor and prediction of conversion by pseudo equilibrium model, J. Chem. Eng. Jpn. 31,596-603 (1998). 

Although the work was in the form of pseudo-equilibrium in the presence of irradiation, and did not take into account the effect of kinetics, it nevertheless gives a clear indication of an additional energy term due to irradiation. Similar studies were done by Kaufman et al. (1978) to show the magnitude of the Gibbs energy necessary to prevent [Pg.417]

When a drug reaches the circulation, it quickly distributes outside the capillary beds into well-perfused tissues but may distribute slowly or not at all to less-accessible tissues protected by barriers, such as the brain. The volume of distribution is the ratio of the amount of drug in the body divided by the drug concentration in plasma once a pseudo-equilibrium is estabhshed between blood and tissues. For small molecules, a low volume of distribution generally signifies extensive plasma protein binding that restricts distribution outside the capillary bed, while a large volume of distribu-... 

Ion-exchange equilibrium can be considered to be analogous to chemical equilibrium. From that point of view, the mass-action law can be used to express the state of equilibrium despite the fact that this law is defined exclusively for homogeneous systems. Derived this way, the so-called pseudo-equilibrium constant Ke is not really a constant, since it depends on the total concentration ... 

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