Desorption rate equations

The Langmuir isotherm equation can also be derived from the formal adsorption and desorption rate equations derived from chemical reaction kinetics. In Section 3.2.2, we see that the mass of molecules that strikes 1 m2 in one second can be calculated using Equation (186), by applying the kinetic theory of gases as [dmldt = P2 (MJ2nRT)m], where P2 is the vapor pressure of the gas in (Pa), Mw is the molecular mass in (kg mol ), T is the absolute temperature in Kelvin, R is the gas constant 8.3144 (nT3 Pa mol-K-1). If we consider the mass of a single molecule, mw (kg molecule-1), (m = Nmw), where N is the number of molecules, by considering the fact that (R = kNA), where k is the Boltzmann constant, and (Mw = NAmw), we can calculate the molecular collision rate per unit area (lm2) from Equation (186) so that... 

By combining the net desorption rate, equation 5.41, with the expression for the equilibrium constant one gets ... 

Desorption rate equation for each desorption site is ... 

From all previously stated, it can be inferred that the starting point for extraction of kinetic parameters from thermodesorpion profiles is desorption rate equation proposed by Polanyi and Wigner (Eqs. (4.7) and (4.14)) [6]. However, it has to be kept in mind that the term 9" is just one particular case of one general function/(a), where a denotes the reacted (desorbed) fraction (the degree of surface event) and a) is the... 

There are several modified forms of effusion sources that are used in certain process tools, especially in MBE machines, such as shown in Figure 11.6. For example, one may include a gas cracker which is simply a series of heated baffles or plates configured such that before a gas molecule can escape the source it must strike at least one of these surfaces. Group V and VI elements usually evaporate from then-own soHd/hquid surfaces as small clusters of four to eight atoms. The desorption rate equations tell us that the rate for such a process will depend upon a power of the species concentration. On the surface this concentration is constant and high. Therefore, evaporation of clusters is relatively easy. However, reaction of these large... 

If the desorption rate is second-order, as is often the case for hydrogen on a metal surface, so that appears in Eq. XVIII-1, an equation analogous to Eq. XVIII-3 can be derived by the Redhead procedure. Derive this equation. In a particular case, H2 on Cu3Pt(III) surface, A was taken to be 1 x 10 cm /atom, the maximum desorption rate was at 225 K, 6 at the maximum was 0.5. Monolayer coverage was 4.2 x 10 atoms/cm, and = 5.5 K/sec. Calculate the desorption enthalpy (from Ref. 110). 

Dj IE, ratio of a crack is held constant but the dimensions approach molecular dimensions, the crack becomes more retentive. At room temperature, gaseous molecules can enter such a crack direcdy and by two-dimensional diffusion processes. The amount of work necessary to remove completely the water from the pores of an artificial 2eohte can be as high as 400 kj/mol (95.6 kcal/mol). The reason is that the water molecule can make up to six H-bond attachments to the walls of a pore when the pore size is only slightly larger. In comparison, the heat of vaporization of bulk water is 42 kJ /mol (10 kcal/mol), and the heat of desorption of submonolayer water molecules on a plane, soHd substrate is up to 59 kJ/mol (14.1 kcal/mol). The heat of desorption appears as a exponential in the equation correlating desorption rate and temperature (see Molecularsieves). 

The latter kind of formulation is described at length in Sec. 7. The assumed mechanism is comprised of adsorption and desorption rates of the several participants and of the reaction rates of adsorbed species. In order to minimize the complexity of the resulting rate equation, one of the several rates in series may be assumed controlling. With several controlling steps the rate equation usually is not exphcit but can be used with some extra effort. 

The adsorption of carbon monoxide retards the reduction reaction with the rate constant k, followed by the desorption reaction with a rate constant k in the overall rate equation... 

Kinetic theories of adsorption, desorption, surface diffusion, and surface reactions can be grouped into three categories. (/) At the macroscopic level one proceeds to write down kinetic equations for macroscopic variables, in particular rate equations for the (local) coverage or for partial coverages. This can be done in a heuristic manner, much akin to procedures in gas-phase kinetics or, in a rigorous approach, using the framework of nonequihbrium thermodynamics. Such an approach can be used as long as... 

This review is structured as follows. In the next section we present the theory for adsorbates that remain in quasi-equilibrium throughout the desorption process, in which case a few macroscopic variables, namely the partial coverages 0, and their rate equations are needed. We introduce the lattice gas model and discuss results ranging from non-interacting adsorbates to systems with multiple interactions, treated essentially exactly with the transfer matrix method, in Sec. II. Examples of the accuracy possible in the modehng of experimental data using this theory, from our own work, are presented for such diverse systems as multilayers of alkali metals on metals, competitive desorption of tellurium from tungsten, and dissociative... 

To get the equilibrium sticking coefficient we assume that at an ambient pressure Pq the adsorbate is in equilibrium at a temperature T with partial coverages Hq, m, and Iq. We then increase the pressure slightly to p = Pq- - AP and linearize the rate equations in the increase in the precursor coverages Am = (m) —m and Al = (/) — Iq. If adsorption into and desorption from the precursors is much faster than transitions from the precursors into the adsorbed state, we can ignore terms proportional to An = n) -6 on the right-hand side of Eqs. (70-72) and also assume that the precursors will be in a steady state. It has been shown that the sticking... 

In the first one, the desorption rates and the corresponding desorbed amounts at a set of particular temperatures are extracted from the output data. These pairs of values are then substituted into the Arrhenius equation, and from their temperature dependence its parameters are estimated. This is the most general treatment, for which a more empirical knowledge of the time-temperature dependence is sufficient, and which in principle does not presume a constancy of the parameters in the Arrhenius equation. It requires, however, a graphical or numerical integration of experimental data and in some cases their differentiation as well, which inherently brings about some loss of information and accuracy, The reliability of the temperature estimate throughout the whole experiment with this... 

An analysis of the rate of release of adsorbed atoms from sites with a continuous energy spectrum for the case of an arbitrary distribution function of initial site populations was given by Carter (32). The rate equation for the t th desorption process with x = 1 and negligible readsorption is... 

The required distribution of initial populations ntu can be obtained in the following manner (32). Let us consider a system with Ed mi = 20 kcal/ mole and Ed max = 45 kcal/mole. Assuming that kd = 1013 sec-1 and x = 1, we can calculate theoretical desorption rates dnai/dt for Ed = 20, 21, 22,..., 45 kcal/mole as a function of nBOi. With increasing temperature, 25 values of dnjdt are measured at temperatures corresponding to Ed of 20, 21, 22,. . ., 45 kcal/mole. Since the total desorption rate at any moment must be equal to the sum of the individual desorption processes, we obtain 25 linear equations. Their solution permits the computation of the initial populations of the surface sites in the energy spectrum considered, i.e. the function n,oi(Edi). From the form of this function, desorption processes can be determined which exhibit a substantial effect on the experimental desorption curve. 

The reaction of Si02 with SiC [1229] approximately obeyed the zero-order rate equation with E = 548—405 kJ mole 1 between 1543 and 1703 K. The proposed mechanism involved volatilized SiO and CO and the rate-limiting step was identified as product desorption from the SiC surface. The interaction of U02 + SiC above 1650 K [1230] obeyed the contracting area rate equation [eqn. (7), n = 2] with E = 525 and 350 kJ mole 1 for the evolution of CO and SiO, respectively. Kinetic control is identified as gas phase diffusion from the reaction site but E values were largely determined by equilibrium thermodynamics rather than by diffusion coefficients. 

Adsorption equilibrium of CPA and 2,4-D onto GAC could be represented by Sips equation. Adsorption equilibrium capacity increased with decreasing pH of the solution. The internal diffusion coefficients were determined by comparing the experimental concentration curves with those predicted from the surface diffusion model (SDM) and pore diffusion model (PDM). The breakthrough curve for packed bed is steeper than that for the fluidized bed and the breakthrough curves obtained from semi-fluidized beds lie between those obtained from the packed and fluidized beds. Desorption rate of 2,4-D was about 90 % using distilled water. 

In particular, reactions in heterogeneous catalysis are always a series of steps, including adsorption on the surface, reaction, and desorption back into the gas phase. In the course of this chapter we will see how the rate equations of overall reactions can be constructed from those of the elementary steps. 

Writing the rate equations for the adsorption desorption equilibrium for each layer, we obtain... 

This model comprises three steps to describe the addition of each alkyne, followed by desorption of the final product. The resulting rate equation is ... 

Equation (1) can be ignored. Assuming further that the desorption rate is fast compared to the Ra half lives, then the ( Ra/ Ra) ratios in the groundwater and adsorbed on surfaces (and so in the mobile Ra pool) are equal. In this case, the measured groundwater ( Ra/ Ra) ratio reflects the ratio of the supply rates of Ra and Ra, which is equal to the ratio, adjusted for any differences in the distributions of and... 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

It is interesting to note that, although the intrinsic rate of desorption is slower than that of adsorption, both rates were found to be sufficiently fast under our experimental conditions so that the adsorption-desorption process on the Pt surface can be assumed to rapidly equilibrate at all times that is, even a ten-fold increase in both the adsorption and desorption rate constants (while keeping their ratio constant) did not significantly change the predicted step responses. With the assumption of chemisorption equilibrium, Equations (1) and (4) can be combined into the form (35)... 

Such processes assume that molecules from a fluid phase in contact with a solid catalytic surface combine chemically with catalyst surface molecules and reaction subsequently proceeds between chemisorbed molecules followed by desorption of the products. A large number of different rate equations with varying numbers of constants can be derived by making various auxiliary assumptions and tested against experimental rate data. Since a more or less plausible mechanism is postulated, the feeling is that a chosen rate equation is somewhat extrapolatable outside an experimental range with greater... 

A triatomic molecule undergoes the reaction, A3 B + C, in contact with a catalytic surface. It dissociates completely on adsorption. Write rate equations for the two cases (a) Surface reaction rate controlling, adsorptive equilibrium of all participants maintained (b) Rate of desorption of substance B controlling, surface reaction equilibrium maintained. 

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