Adsorption on Equilibrium Surfaces

Function (28) inserted into integral (4) allows the computation of the overall adsorption isotherm on equilibrium surfaces. Two cases are conveniently distinguished weak heterogeneity (qM qm keTf/ai) and strong 

Weak Heterogeneity.—In this case (qM qm kaTf/a,) the upper bound of integration can be ignored, r=l, and distribution (28) can reasonably be supposed to hold in the whole range (qm,+°°)- Putting Tp= xo = i/ 2, equation (28) becomes 

Distribution (30) is known to be the distribution function computed, in the condensation approximation, from the Freundlich isotherm,  

Since the work of Zeldovich it has been established that the above behaviour is due to an exponential distribution of adsorption energy. Various expressions of the Freundlich isotherm are known for instance, Sips modified the original expression to show saturation at high pressure and to permit the exact evaluation of the distribution function equation (32) is suggested by the use of the condensation approximationand is in good agreement with rigorous 

Distribution (31), compared with distribution (10), shows that in the high-energy range the distribution function of equilibrium surfaces strictly resembles the distribution function explaining, in the condensation approximation, the modified DR isotherm if B is identified with l/ksT xo- 

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Each of features (i) to (iv) involved a remarkable amount of work, and the solutions to these problems led to the theory of adsorption on equilibrium surfaces . 

That the Temkin isotherm could be ascribed to adsorption on equilibrium surfaces was first guessed by lonescu, but his derivation was based on rather weak arguments. The above consideration, on the contrary, is already satisfactory and can be made more precise by inserting distribution function (28) in equation (4) and by computing this integral numerically. The results in a log-log plot are shown in Figure 5, and confirm the Henry behaviour at low pressures and the saturation at high pressures. 

A simplified model of equilibrium surface suggests that the DR behaviour is observed in low-pressure adsorption on patchwise, weakly heterogeneous surfaces which were grown in equilibrium conditions and hence were quenched at the adsorption temperature. At higher pressures, these surfaces should exhibit the Freundlich behaviour, while in the case of strong heterogeneity adsorption should be described by the Temkin isotherm. The three classic empirical isotherms, Freundlich, Dubinin-Radushkevich, Temkin, seem therefore to be related to adsorption on equilibrium surfaces, and the explanation of these experimental behaviours can be seen as a new chapter of the theory of adsorption the theory of physical adsorption on equilibrium surfaces. 

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