Temperature Langmuir isotherms

Isotherms. When a fibei is immersed, in a dyebath, dye moves fiom the external phase into the fibei. Initially the late is quick but with time this slows and eventually an equiUbrium is reached between the concentration of dye in the fiber and the concentration of dye in the dyebath. For a given initial dyebath concentration of a dye under given dyebath conditions, eg, temperature, pH, and conductivity, there is an equiUbrium concentration of dye in fiber, Dj and dye in the dyebath external solution, D. Three models describe this relationship simple partition isotherm, Freundhch isotherm, and Langmuir isotherm. 

The saturation of the SHG response at high cation concentrations suggests that the process of complex formation at the membrane surface may be treated by a Langmuir-isotherm type analysis [24,27]. At constant temperature, the Langmuir equation is given by... 

This three-parameter equation behaves linearly in the Henry s law region and reduces to the Langmuir isotherm for m = 1. Other well-known isotherms include the Langmuir-Freundlich isotherm or Sips isotherm [Sips,/. Chem. Phys., 16, 490 (1948) Koble and Corrigan, Ind. Eng. Chem., 44,383 (1952)] or loading ratio correlation with prescribed temperature dependence [Yon and Turnock, AlChE Symp. Sen, 67(117), 75 (1971)]... 

Langmuir isotherm equation phys chem An equation, useful chiefly for gaseous systems, for the amount of material adsorbed on a surface as a function of pressure, while the temperature Is held constant, assuming that a single layer of molecules is adsorbed it is / = ap/ 1 + ap). where / is the fraction of surface covered, p is the pressure, and a is a constant. lar),myur Ts-3,th3rm i,kwa zh3n ) lanthana See lanthanum oxide. lan tha-na ... 

At a given temperature adsorption isotherms measure the number of adsorbed molecules as a function of pressure for the fluid that is in contact with the zeolite. The simplest form is the Langmuir isotherm which treats the zeolite as a collection of equivalent adsorption sites in the absence of adsorbate-adsorbate... 

Symbol (0) for characteristic temperature. 2. Symbol (0) for degree of saturation of binding sites as defined in the Langmuir isotherm treatment for adsorption of a ligand onto a surface. See Langmuir Isotherm. 3. Symbol (0) for plane angle. 4. Symbol (0) for one of the space coordinates in the three-dimensional, spherical polar coordinate system. 5. Symbol (0) for Celsius temperature. 

It might also be noted that K (Equation (62)) may be related to AG° for the adsorption process if the model applies to the experimental system. Therefore, from studies of adsorption at different temperatures, values of AH° and AS0 may be determined for the process described by Equation (61). It must be emphasized that compliance with the form predicted by the Langmuir isotherm is not a sensitive test of the model therefore interpretations of this kind must be used cautiously. 

The exponent 1/n of isotherm (1) is linearly dependent on the temperature. At sufficiently high temperatures when, by (18), 1/n > 1, the theory gives simply a linear course of the isotherm at the beginning an isotherm of the form q = Cp1/", 1/n > 1, cannot be obtained by superposition of the Langmuir isotherm. Indeed, in this case a(b) still has the form of equation (9) with 1/n > 1, but we may no longer seek the asymptotic form of the beginning of the isotherm by formulas (11) and (12) since... 

The acidic sites on iron oxides are believed to be FeOH sites (32), much like the well-known SiOH sites on silica. Heats of adsorption on iron oxide of bases of known Cg and Eg, having appreciably different ratios of Cg to Eg ("hardness" or "softness"), allow estimation of the and for the acidic sites of iron oxide. Our initial studies were done by measuring adsorption isotherms at two or more temperatures (Figure 7) and from the temperature coefficient of the equilibrium constant K the enthalpy of adsorption was calculated. In Figure 7 the adsorption data is plotted as a Langmuir isotherm ... 

At low pressures the Langmuir isotherm equation reduces to V— Vmap - i.e. the volume of gas adsorbed varies linearly with pressure. At high pressures a limiting monolayer coverage, V=Vm, is reached. The curvature of the isotherm at intermediate pressures depends on the value of the constant a and, hence, on the temperature. 

There is also another topological limit on C H ratio. Physisorption proceeds only in the monolayer above the boiling point of hydrogen, and the adsorption must follow the Langmuir isotherm. Hence, the storage capacity depends on the pressure of H2 gas above the carbon surface at a fixed temperature. It is also greatly limited by available SSA of carbon phase as provided by porous structure. 

Many different adsorption isotherms may be derived where all or some of the basic assumptions going into the derivation have been released or relaxed [101-105], It should be stated, however, that the general form of the Langmuir isotherm, which is shown for two temperatures in Fig. 13, may be used for a phenomenological description of many processes. It is clear, from the adsorption isotherm, the sticking probability. v may also be determined given that all other parameters are known [1,7]. 

Equilibrium between solution and adsorbed or sorbed phases is a condition commonly used to evaluate adsorption or sorption processes in soils or soil-clay minerals. As previously stated, equilibrium is defined as the point at which the rate of the forward reaction equals the rate of the reverse reaction. Two major techniques commonly used to model soil adsorption or sorption equilibrium processes are (1) the Freundlich approach and (2) the Langmuir approach. Both involve adsorption or sorption isotherms. A sorption isotherm describes the relationship between the dissolved concentration of a given chemical species (adsorbate) in units of micrograms per liter (pg L 1), milligrams per liter (mg L-1), microequivalents per liter (pequiv L-1), or millimoles per liter (mmol L-1), and the sorbed quantity of the same species by the solid phase (adsorbent) in units of adsorbate per unit mass of adsorbent (solid) (e.g., pg kg-1, mg kg-1, peq kg-1, or mmol kg 1) at equilibrium under constant pressure and temperature. Sorption isotherms have been classified into four types, depending on their general shape (Fig. 4.13) ... 

Lewin and coworkers [255-260] developed an accessibility system based on equilibrium sorption of bromine, from its water solution at pH below 2 and at room temperature, on the glycosidic oxygens of the cellulose. The size of the bromine molecule, its simple structure, hydrophobicity, nonswelling, and very slow reactivity with cellulose in acidic solutions, contribute to the accuracy and reproducibility of the data obtained. The cellulose (10 g/1) is suspended in aqueous bromine solutions of 0.01-0.02 mol/1 for 1-3 h, depending on the nature of the cellulose, to reach sorption equilibrium. The diffusion coefficients of bromine in cotton and rayon are 4.6 and 0.37 x 10 cm /min, respectively. The sorption was found to strictly obey the Langmuir isotherm, which enables the calculation of the accessibility of the cellulose as follows ... 

The problem of what constitutes the active surface at any given temperature is a diflBcult one which cannot be resolved by assumption of the simple Langmuir isotherm. As we have noted in the case of Hj, large variations in heat of sorption will imply severe changes in active surface temperature which may thus give rise to anomalous pressure dependence of the rate of reaction. These important details can be resolved only by direct isotherm measurement. 

At the plateau the maximum uptake value of the adsorbent is therefore reached. The type I or Langmuir isotherm describes the formation of a single monolayer on the surface of the porous material. No multilayers of hydrogen are formed at 77 K, because the interaction strength between single layers is too weak at temperatures higher than the critical temperature of H2. Therefore a type I isotherm is typically obtained for porous materials at adsorption temperatures higher than the critical temperature of the gas. 

The adsorption capacity of activated carbon may be determined by the use of an adsorption isotherm. The adsorption isotherm is an equation relating the amount of solute adsorbed onto the solid and the equilibrium concentration of the solute in solution at a given temperature. The following are isotherms that have been developed Freundlich Langmuir and Brunauer, Emmet, and Teller (BET). The most commonly used isotherm for the application of activated carbon in water and wastewater treatment are the Ereundlich and Langmuir isotherms. The Freundlich isotherm is an empirical equation the Langmuir isotherm has a rational basis as will be shown below. The respective isotherms are ... 

The adsorption of methylene blue by coir pith carbon was carried out by varying the parameters such as agitation time, dye concentration, adsorbent dose, pH and temperature. Equilibrium adsorption data obeyed Langmuir isotherm. Adsorption kinetics followed a second order rate kinetic model. The adsorption capacity was found to be 5.87 mg dye per g of the adsorbent. There was no significant change in the per cent removal with pH. The pH effect and desorption studies suggest that chemisorption might be the major mode of the adsorption process. 

Adsorption equilibrium was represented by extended Langmuir isotherm with the same saturation capacity (qs) for both adsorbates and constants (bj) following Arrhenius temperature dependence ... 

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