The Hill-de Boer Equation

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

This local isotherm has been discussed at length by de Boer, Ross and Olivier and also Broekhoff and van Dongen. Phase-transition loops occur when 2a/fcT 3 6.75 and the transition step position (i.e. coverage limits and corresponding pJKi) have been tabulated.  

The Virial Equation for an Imperfect Two-dimensional Gas/ temperature limit of equation (60) may be written as  

B°d and C2D, have been evaluated using the Lennard-Jones 6-12 interatomic potential and for argon using the exp-six potential to describe adsorbate self-interactions. 

When equation (82) is transformed into the generalized form, p = Kg 6, t), the value of K is given by  

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

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 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. 

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. 

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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