Laws Langmuir isotherm

Extrathermodynamic approach to selectivity Theoretical analysis Thermodynamics of adsorption Henry s law Langmuir isotherm... 

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 Radke-Prausnitz isotherm [Radke and Prausnitz, Ind. Eng. Chem. Fundam., 11, 445 (1972) AIChE J., 18, 761 (1972)]... 

With the adsorbate concentration low enough, the Langmuir isotherm transforms into the linear equation and becomes the simplest isotherm of adsorption, as described by Henry s law. 

Earlier work on the application of the concept of dual mode sorption and diffusion to glassy polymer-gas systems has been reviewed in detail 6) and important aspects of more recent work have been dealt with in more recent reviews 7 10). Eq. (5) was first applied by Michaels et al U). Sorption in the polymer matrix and in the specific sorption sites was represented by linear (Henry s law) and Langmuir isotherms respectively so that Sj in Eq. (5) is given by... 

The concept of unrelaxed volume in glassy polymers is used to interpret sorption and transport data for pure and mixed penetrants A review of recent sorption and permeation data for mixed penetrants indicates that competition for sorption sites associated with unrelaxed gaps between chain segments is a general feature of gas/glassy polymer systems This observation provides convincing support for the use of the Langmuir isotherm to describe deviations from simple Henry s law sorption behavior. 

This simple physical model can be described analytically up to reasonably high pressues (generally for pressures less than or equal to the maximum conditioning pressure employed (4) ) in terms of the sum of Henryfs law for and a Langmuir isotherm for Cjj. 

At sufficiently low pressures (8 1), the Langmuir isotherm reduces to Henry s law with... 

This means that the decomposition occurs uniformly across the surface, where the products are weakly bound and rapidly desorbed consequently, the rate-determining step is the surface decomposition step. This type of reaction shows two rate-limiting laws corresponding to the two extreme behaviors of the Langmuir isotherm. That is, at low pressure, 0A is small and proportional to the pressure, and the rate becomes first order in A(g) ... 

In some cases, adsorption of analyte can be followed by a chemical reaction. The Langmuir-Hinshelwood (LH) and power-law models have been used successfully in describing the kinetics of a broad range of gas-solid reaction systems [105,106]. The LH model, developed to describe interactions between dissimilar adsorbates in the context of heterogeneous catalysis [107], assumes that gas adsorption follows a Langmuir isotherm and that the adsorbates are sufficiently mobile so that they equilibrate with one another on the surface on a time scale that is rapid compared to desorpticm. The power-law model assumes a Fre-undlich adsorption isotherm. Bodi models assume that the surface reaction is first-order with respect to the reactant gas, and that surface coverage asymptotically approaches a mmiolayer widi increasing gas concentration. 

Note that, if H2 followed a Langmuir isotherm, 1/(8) = Xg (Hj), we would obtain on substitution in Eq. (XVII.6.1) a zero-order rate law, which is in fact observed. The true isotherms, however, follow 1/(S) =ks(H2) , and it is only vrhen allowance is... 

Solution On the basis of the reaction model (equations (10.81) and (10.82)), the impedance can be derived under the assumption that the adsorbate Mg obeys a Langmuir isotherm and that the rate constants of electrochemical reactions are exponentially dependent on potential (e.g., following Tafel s law). Each reaction with index i has a normalized rate constant K, corresponding to its rate constant hi by... 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

Isotherms depend significantly on the coliunn temperatme. We have shown earlier (see Figmes 3.33a [38] and 3.33b [54]) isotherms determined at different temperatures for the same systems. From these results, it seems that the amotmt of low-molecular-weight components adsorbed at equilibrium with a constant mobile phase concentration increases by about 1% when the temperatme decreases by about 1°C. For more complex isotherms, Van t Hoff law may not apply. For example, with the bi- or tri-Langmuir isotherms, the equilibrium constant of the... 

When A"Ce 1, the Langmuir isotherm reduces to a linear form analogous to Henry s Law. Sorbate-sorbent pairs that display a Type 1 isotherm can use the Langmuir equation. 

The dual-mode isotherm is merely a combination of Hemy s law and the Langmuir isotherm. Other isotherms, such as the Redlich-Peterson, Langmuir-Ereundlich, Sips, and Toth [g, h, i, and j, respectively, in Table 14.3] versions extend the Langmuir isotherm by accounting for subtle... 

Equation (9-34) is the Langmuir isotherm equation. At low partial pressures, if Kp 1, it reduces to the linear Henry s law form of equation (9-31), while at high partial pressures, if Kp 1, q = qm. 

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