Equation Hill-de Boer

Finally, the Hill-de Boer equation, which is equivalent to the 2D Van der Waals equation has been derived in sec. II.1.5e. 

The Langmuir and Volmer equations are special cases of the Fowler-Guggenheim and Hill-de Boer equations, respectively, in which lateral interactions are allowed to vanish the Brunauer-Emmett-Teller equation is a special case of the Broekhoff-van Dongen equation with n = oo and null lateral interactions the model in which all layers are mobile is a special case of Broekhoff-van Dongen model with n = 0. 

In later work, Ross and Morrison [7, 8] were able to make several advances. The van der Waals equation of state for real gases, which is the basis of the Hill-de Boer equation, is known to be rather inaccurate. Ross and Morrison based their kernel function on a two-dimensional form of the much better virial equation of state. But more importantly, advances in computing resources made it possible to solve Eqn (7.10) for the unknown distribution function using a nonnegative least squares method, rather than assuming a form a priori [9]. 

The pore pressure is directly responsible for the molecular layering and the pore filling. Having described the pore pressure, we now address the molecular layering process. This process can be described by any appropriate equation. If there is no or weak fluid—fluid interaction, we can use the BET-type equation, while if the fluid-fluid interaction is strong we can use the modified Hill-de Boer equation as suggested by Do and Do [63] to calculate the adsorbed film thickness t. In these equations the affinity constant is a function of pore size and the pressure involved in those equations is the pore pressure. 

The equation as given in eq. (2.3-27) is known as the Hill-de Boer equation, which describes the case where we have mobile adsorption and lateral interaction among adsorbed molecules. When there is no interaction between adsorbed molecules (that is w = 0), this Hill-de Boer equation will reduce to the Volmer equation obtained in Section 2.3.3. 

Similarly for the Hill-de Boer equation, we obtain the same isosteric heat of adsorption as that for the case of Fowler-Guggenheim equation. This is so as we have discussed in the section 2.3.3 for the case of Volmer equation that the mobility of adsorbed molecule does not influence the way in which solid interacts with adsorbate. 

Similar analysis of the Hill-deBoer equation (2.3-27) shows that the two dimensional condensation occurs when the attraction between adsorbed molecules is strong and this critical value of c is 27/4. The fractional loading at the phase transition point is 1/3, compared to 1/2 in the case of Fowler-Guggenheim equation. A computer code Hill.m is provided with this book for the calculation of the fractional loading versus pressure for the case of Hill-de Boer equation. Figure 2.3-4 shows plots of the fractional loading versus nondimensional pressure (bP) for various values of c= 5, 7, 10, 15. ... 

Figure 2.3-4 Plots of the fractional loading versus bP for the Hill-de Boer equation...

Since there are many fundamental equations which can be derived from various equations of state, we will limit ourselves to a few basic equations such as the Henry law equation, the Volmer, the Fowler-Guggenheim, and the Hill-de Boer equation. Usage of more complex fundamental equations other than those just mentioned needs justification for doing so. 

The first such solutions were carried out by Ross and Olivier [1, p. 129 6,7]. Using Gaussian distributions of adsorptive potential of varying width, they computed tables of model isotherms using kernel functions based on the Hill-de Boer equation for a mobile, nonideal two-dimensional gas and on the Fowler-Guggenheim equation [Eq. (14)] for localized adsorption with lateral interaction. The fact that these functions are implicit for quantity adsorbed was no longer a problem since they could be solved iteratively in the numerical integration. 

The Hill-de Boer Equation. This local isotherm has been discussed above [see equation (56)]. The value of K is given by ... 

Unless the convergence limit of equation (82) is known and the requisite coefficients determinable, some approximation must be made for the infinite series. Ross and Morrison have taken the reduced virial eoefficients, D2o/r, E2olr and higher terms (where r is the Lennard-Jones parameter for the distance of maximum attraction), to be equal to 2, a value supported by the studies of Ree and Hoover. The K used by Ross and Morrison (Krm) may be written as Krm = AJKnit is not clear what advantages this local isotherm offers over other approximate equations derived for a two-dimensional gas, in particular the simpler Hill-de Boer equation. 

One of the most frequently used local adsorption isotherm is the Hill-de Boer equation [34,35]. It should be pointed out that for mobile adsorption, even when lateral interactions are neglected, the additive assumptions about surface topography are necessary [6-8]. 

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