Differential quantities of adsorption and experimental results

We have introduced integral molar quantities, which indicates that there are corresponding differential quantities. Integral refers to the fact that the total amount of adsorbed gas is involved. In contrast, the differential molar energy of adsorption is determined only by the last infinitesimal amount adsorbed. It is defined as 

U9 is the total internal energy of the free gas. Since usually the amount adsorbed is small compared to the total amount of gas in the reservoir the properties of the free gas do not change significantly during adsorption. Thus, dU9/dNa = U which leads to 

It involves the change of the internal surface energy upon adsorption of an infinitesimal amount of gas at constant temperature and total surface area. 

By analogy, the differential molar enthalpy of adsorption and the differential molar entropy of adsorption are defined as  

Physical adsorption of gases on solids is virtually always enthalpically driven (Att 0). Entropically driven adsorption can exist but usually the entropy of molecules on a surface is much lower than in the gas phase. Vibrational, rotational, and also translational degrees of freedom are restricted on surfaces. 

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