Langmuir vaporization

The L vov sublimation model. The applications (outlined in Sections 2.4.5. and 2.4.6.) of the Hertz-Knudsen-Langmuir vaporization model to decompositions by L vov et al. [94] are based on the assumption that decomposition involves an initial sublimation step, followed by condensation of the less volatile products. Because sublimation is an endothermic process, the condensation process would need to make a significant energetic contribution for the decomposition to be exothermic overall. 

Equations 3.7-3.9 will be referred to in this text as the Langmuir vaporization equations. 

Advantages of the Langmuir Vaporization Equations A comprehensive analysis of these equations lead us to the following conclusions. 

Relation Between the Quantities k and J To compare the Arrhenius equation with the Langmuir vaporization equations, consider first how the rate constant k is related to the absolute decomposition rate J. For the steady-state... 

There is an essential difference between the decomposition rates expressed by the quantities J and k. Unlike J, which does not depend on the particle size, k is inversely proportional to the initial dimensions of the particle. For pro = 1 (e.g., for p = 2,000 kg m and rg = 0.5 mm = 5 x 10 m), the rates J and k are numerically equal. The difference between these rates increases proportionately with increasing size and density of the particles. Equation 3.32 permits conversion from relative values of the rate constants k expressed in per second to the absolute rates J in units of kg m s. This opens up an attractive possibility for the interpretation of data obtained by traditional measurement of the a—t kinetic curves in terms of the Langmuir vaporization equations. 

The Langmuir vaporization equations thus open up broader possibilities for description of decomposition processes than the Arrhenius approach. First, the key physical quantity entering all vaporization equations is the equilibrium pressure of products, which is directly related to the thermodynamic parameters of the process. As a result, the A and E parameters of the Arrhenius equation receive a straightforward physical interpretation. 

Third, these equations permit the calculation of the absolute rates of a process, a possibility that had been believed unrealizable before their first application in 1981 to the kinetics of solid decomposition [25], The interest in theories of the transition state and of the activated complex was primarily stimulated by the possibility of calculating absolute reaction rates, although the attempts to use them in studies of heterogeneous processes met with only limited success [1, 2]. In contrast, the first comparison of theoretical with experimental values of the A parameters performed within the framework of Langmuir vaporization equations was much more successful [25]. 

The common CDV decomposition mechanism for different substances and some well-forgotten or unclaimed ideas (Langmuir vaporization models and third-law method), underlying this approach, appeared to be the necessary and mutually supplementing elements. Without any of them it would be impossible to develop a sufficiently rigorous and consistent theory. The essence of these three aspects of the approach is expressed in a simplified form in Table 17.3. 

Mechanism Kinetics Methodology Congruent dissociative vaporization Langmuir vaporization equations Third-law method R(s/1) A g)i + B(g) feqp = f J) ArHf = - uR nPeqp)... 

Balankin et al. [3] found that only the liquid Th appears under Langmuir vaporization conditions. The vapor partial pressures given as a function of stoichiometry are... 

The case of a vapor adsorbing on its own liquid surface should certainly correspond to mobile adsorption. Here, 6 is unity and P = the vapor pressure. The energy of adsorption is now that of condensation Qu, and it will be convenient to define the Langmuir constant for this case as thus, from Eq. xvn-39. 

Equation XVII-70 bears a strong resemblance to the Langmuir equation (see Ref. 4)—to the point that it is doubtful whether the two could always be distinguished experimentally. An equivalent form obtained by Volmer [53] worked well for data on the adsorption of various organic vapors on mercury [54] (see Problem XVII-40). 

Films or membranes of silkworm silk have been produced by air-drying aqueous solutions prepared from the concentrated salts, followed by dialysis (11,28). The films, which are water soluble, generally contain silk in the silk I conformation with a significant content of random coil. Many different treatments have been used to modify these films to decrease their water solubiUty by converting silk I to silk II in a process found usehil for enzyme entrapment (28). Silk membranes have also been cast from fibroin solutions and characterized for permeation properties. Oxygen and water vapor transmission rates were dependent on the exposure conditions to methanol to faciUtate the conversion to silk II (29). Thin monolayer films have been formed from solubilized silkworm silk using Langmuir techniques to faciUtate stmctural characterization of the protein (30). ResolubiLized silkworm cocoon silk has been spun into fibers (31), as have recombinant silkworm silks (32). 

Langmuir equations The mathematical expressions that describe vapor adsorption equilibria. 

Kinetic analysis based on the Langmuir-Hinshelwood model was performed on the assumption that ethylene and water vapor molecules were adsorbed on the same active site competitively [2]. We assumed then that overall photocatalytic decomposition rate was controlled by the surface reaction of adsorbed ethylene. Under the water vapor concentration from 10,200 to 28,300ppm, and the ethylene concentration from 30 to 100 ppm, the reaction rate equation can be represented by Eq.(l), based on the fitting procedure of 1/r vs. 1/ Ccm ... 

Next, let the example of vanadium, which, in the as-reduced condition, may contain a variety of impurities (including aluminum, calcium, chromium, copper, iron, molybdenum, nickel, lead, titanium, and zinc) be considered. Vanadium melts at 1910 °C, and at this temperature it is considerably less volatile than many of the impurity metals present in it. The vapor pressure of pure vanadium at this temperature is 0.02 torr, whereas those of the impurity elements in their pure states are the following aluminum 22 torr calcium 1 atm, chromium 6 torr copper 23 torr iron 2 torr molybdenum 6 1CT6 torr nickel 1 torr lead 1 torr titanium 0.1 torr and zinc 1 atm. However, since most of these impurities form a dilute solution in vanadium, their actual partial pressures over vanadium are considerably lower than the values indicated. Taking this into account, the vaporization rate, mA, of an element A (the evaporating species) can be approximated by the following free evaporation equation (Langmuir equation) ... 

All the work we describe in this chapter was carried out in UHV on the rutile Ti02(l 1 0)1 x 1 surface. Exposures to vapors and gases are given in Langmuirs (L) where 1 L = 1.333 x 10 s mbar s. Coverages of defects or molecules adsorbed at the surface will be given in monolayers (ML), where 1 ML corresponds to the density of primitive surface unit cells. 

Gao, T. Gao, J. Sailor, M. J., Tuning the response and stability of thin film mesoporous silicon vapor sensors by surface modification, Langmuir. 2002, 18, 9953 9957... 

Langmuir cells (Fig. 8 h) are very similar to Knudsen cells. Here the vaporization of a solid from its uncovered surface can be measured in a similar way by thermo-gravimetric methods. 

Brunauer-Emmett-Teller (BET) adsorption describes multi-layer Langmuir adsorption. Multi-layer adsorption occurs in physical or van der Waals bonding of gases or vapors to solid phases. The BET model, originally used to describe this adsorption, has been applied to the description of adsorption from solid solutions. The adsorption of molecules to the surface of particles forms a new surface layer to which additional molecules can adsorb. If it is assumed that the energy of adsorption on all successive layers is equal, the BET adsorption model [36] is expressed as Eq. (6) ... 

LANGMUIR TROUGH AND BALANCE LAPLACE TRANSFORM LARMOR PRECESSION LASER-FLASH KINETIC ANALYSIS LATENT ACTIVITY LATENT HEAT Latent heat of fusion LATENT HEAT Latent heat of vaporization LATENT HEAT Lateral binding proteins,... 

The next set of examples show an entropy of adsorption roughly equal to the entropy change on losing the degree of translational freedom normal to the surface, i.e., in the adsorbed state the molecules are equivalent to a two-dimensional gas or vapor. The data for a variety of different adsorbates and adsorbents is given in Table V. The isotherms obtained by Armbruster (20) for the adsorption of CO and N2 on silver were not S-shaped, and they could be fitted to equations of the Langmuir type. The amount of adsorbate required to saturate the surface was given for each substance at both temperatures. Armbruster calculated the heats of adsorption by the method of Brunauer, Emmett and Teller (22) and there is some doubt about the validity of such heats. 

When the functional form of the correlation is suggested by theory, there is a great deal more confidence that the correlation can be extrapolated into regions of P that have no experimental data, and can be used for other families of compounds other than the training set S. Examples of theory-suggested functional forms include the van der Waals equation of state for gases, the Langmuir isotherm for adsorption and catalysis, and the Clausius-Clapeyron equation for the vapor pressure of liquids. 

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