Langmuir constants

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. 

Table 12. Langmuir Constants for the Absorption of Alkaline-Earth Cations on Manganese Dioxide ...

The curves were determined from Eqs. 24"—26" in order to apply these in a numerical calculation one first has to know the values of the following functions at — 3°C Afz, the difference in chemical potential between the "empty Structure II lattice and ice Cpg. the Langmuir constant for propane in the larger cavities of Structure II (Cpi = 0 for geometrical reasons) Cmi> Cm2> the Langmuir constants for methane in the two types of cavities of Structure II. 

Langmuir, constants, 48 isotherm, 39 theorem, 15 Lattice model, 123... 

Langmuir constants, 47 molecule, ionization energy, 71 magnetic susceptibility, 70, 71... 

Projections, linearly independent, 293 Propagation, of polymerization, 158 Propane, hydrate, 10, 33, 43, 46, 47 hydrate thermodynamic data and lattice constants, 8 + iodoform system, 99 Langmuir constant, 47 water-hydrogen sulfide ternary system, 53... 

The Langmuir equation is derived here from application of the mass law, in a similar way as the surface complex formation equilibria were derived in Chapter 2. In principle at a constant pH there is no difference between a Langmuir constant and a surface complex formation constant. 

Since this step is fast in comparison to the subsequent ones, this reaction can be considered as a pre-equilibrium, is the surface complex formation (equivalent to the Langmuir) constant. 

K Langmuir constant (m moL ) k reaction rate constant (m moL s ) k overall adsorption rate parameter (s ) k intra-particle mass transfer coefficient (s )... 

In Figure 4.23, the model results for solid diffusion control (eq. (4.141)) and two different values of the Langmuir constant La) are presented. In Figure 4.24, the model results for solid diffusion and liquid-film diffusion control (eq. (4.140)) for La = 0.5 are presented. 

Relating the Langmuir Constant to Cell Potential Parameters... 

For an ideal gas Equation 5.22a may be considered as elementary probability of cavity i occupation by molecule J. This is one of the most useful equations in the method of hydrate prediction, and it may also be recognized as the Langmuir isotherm. If the equation were written for one guest component J, it would contain the Langmuir constant Cjj as the only unknown for a given pressure and fraction of the cavities filled (or fraction of monolayer coverage). 

Equation 5.21 shows that the Langmuir constant is a direct function of the particle partition function within the cavity qj/, in particular Cjj contains the nonideal gas translation term. When the fluid in equilibrium with the hydrate is a nonideal gas, the pressure of component J in Equation 5.22a is replaced with its fugacity,//. 

With such corrections, Equation 5.22a finds many uses in the calculation of hydrate properties. The equation relies on the fitting of the Langmuir constant Cjj to experimental hydrate conditions. The method of relating the Langmuir constant to experimental conditions is given in Section 5.1.4. 

Now consider the logarithmic term in Equation 5.18a. It may be simplified through the use of Equation 5.22a, which relates 0j-t, the fractional occupation of a cavity of type i by a molecule of type J, to the Langmuir constant... 

However, it should be remembered that the fractional filling is a function of the product Cjjfj, rather than either factor in the product. Finally, in the original van der Waals and Platteeuw approach the Langmuir constants for both adsorption and enclathration were only functions of temperature for each molecule type retained at the individual site or cavity. In the modified approach below, the Langmuir constants are also a function of cage size, or the unit cell volume, which is a function of the hydrate guests, temperature, and pressure. 

The cavities are assumed to be spherically symmetric, which enables the elimination of the two angular portions of the triple integral, resulting in 47t. Substitution of the resulting equation into Equation 5.21 yields the final expression for the Langmuir constant in terms of the particle potential within the cavity. 

The evaluation of the Langmuir constant may then be determined from a minimum of experimentally fitted Kihara parameters via an integration over the cavity radius. Equation 5.27a shows the Langmuir constant to be only a function of temperature for a given component within a given cavity. 

Equation 5.27a shows that the main contribution to the Langmuir constant comes from integrating the guest-cavity pair potential in the interior of the cells. This is a partial explanation of the reason why the final assumption (2) at the beginning of this subsection for smeared water molecules is a good approximation for the cavity shell in addition the cavities are spherical, to a first approximation. 

The value of fitting the Langmuir constants to simple hydrate formation data is in the prediction of mixture hydrate formation. When the formation data for the simple hydrates are adequately fitted, then mixtures of those guest components can be predicted with no adjustable parameters. Since there are only eight simple hydrate formers of natural gas which form si and sll, but an infinite variety of mixtures, such an advantage represents a substantial savings of time and effort. 

Using the above equations, the Langmuir constant for the large cavity occupant can be determined explicitly from the chemical potential difference and the fugacity. However, for systems in which both cavities are occupied, a second method must be used to supplement Equation 5.27c and d. 

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