Integral molar quantities of adsorption

The difference between a molar surface excess thermodynamic quantity xar r and the corresponding molar quantity x p for the gaseous adsorptive at the same equilibrium T and p is usually called the integral molar quantity of adsorption, and is denoted Aads r,r ... 

We may derive the relation between these integral molar quantities of adsorption from Equations (2.20) using the expression of surface excess chemical potential p° given by Equation (2.41) and assuming the gas to be ideal ... 

Advantages and limitations of differential and integral molar quantities of adsorption... 

In any investigation of the energetics of adsorption, a choice has to be made of whether to determine the differential or the corresponding integral molar quantities of adsorption. The decision will affect all aspects of the work including the experimental procedure and the processing and interpretation of the data. 

Evaluation of integral molar quantities of adsorption 42 Integral molar energy of adsorption 42... 

The differential molar entropies can be plotted as a function of the coverage. Adsorption is always exothermic and takes place with a decrease in both free energy (AG < 0) and entropy (AS < 0). With respect to the adsorbate, the gas-solid interaction results in a decrease in entropy of the system. The cooperative orientation of surface-adsorbate bonds provides a further entropy decrease. The integral molar entropy of adsorption 5 and the differential molar entropy are related by the formula = d(n S )ldn for the particular adsorbed amount n. The quantity can be calculated from... 

It is the energy difference between Na moles of gas adsorbed U"n (per mol) and the same amount free in the gas phase U. The next important quantities are the integral molar enthalpy of adsorption... 

The integral molar entropy of adsorption is the difference between the entropy of an adsorbed mole and the entropy of the adsorptive in the ideal gas state, at given p and T. It is a mean integral quantity taken over the whole amount adsorbed and it is characteristic of a given state of equilibrium. This is distinguished by the standard integral molar entropy of adsorption, which is the entropy of one adsorbed mole with respect to the entropy of the adsorptive in the ideal gas state at the same T, but under standard pressure. 

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... 

When the main purpose of the gas adsorption measurements is to characterize the adsorbent surface or its pore structure, the preferred approach must be to follow the change in the thermodynamic quantity (e.g. the adsorption energy) with the highest available resolution. This immediately leads to a preference for the differential option, simply because the integral molar quantity is equivalent to the mean value of the corresponding differential quantity taken up to a recorded amount adsorbed. Their relationship is indicated by the mathematical form of Equation (2.64), which is explained in the following section. 

The integral molar quantities are of importance for modelling adsorption systems or in the statistical mechanical treatment of physisorption. For example, they are required for comparing the properties of the adsorbed phase with those of the bulk... 

Some experimental techniques are to be preferred for the accurate determination of integral quantities (e.g. from energy of immersion data or a calorimetric experiment in which the adsorptive is introduced in one step to give the required coverage), while others are more suitable for providing high-resolution differential quantities (e.g. a continuous manometric procedure). It is always preferable experimentally to determine the differential quantity directly, since its derivation from the integral molar quantity often results in the loss of information. 

Experimentally, q is very difficult to measure directly. Attempts to find the partial of ln(P/Pj) with respect to l/Tby measuring the isotherm at two or more temperatures have not been very accurate. This is due to the uncertainty in the shape of the isotherm compared to the precision that is acceptable. Direct calorimetric measurements have been more successful. Calorimetric measurements are more precise but they measure the integral heat of adsorption, Q, and the molar heat of adsorption, Q, as defined by Morrison et al. [17]. Another quantity, the integral energy of adsorption, Q, was defined by Hill [18,19] for constant volume conditions. These quantities can be obtained with more accuracy and precision than the isosteric heat. Nevertheless, the isosteric heat is often reported. 

In flow microcalorimetry a small amount of filler is put into the cell of the calorimeter and the probe molecule passes through it in an appropriate solvent. Adsorption of the probe results in an increase of temperature and the integration of the area under the signal gives the heat of adsorption [98]. This quantity can be used for the calculation of the reversible work of adhesion according to Eq. (16). The capabilities of the technique can be further increased if a HPLC detector is attached to the microcalorimeter. The molar heat of adsorption can also be determined with this setup. 

Let us now consider how these quantities are related to experimentally determined heats of adsorption. An essential factor is the condition under which the calorimetric experiment is carried out. Under constant volume conditions, AadU 1 is equal to the total heat of adsorption. In such an experiment a gas reservoir of constant volume is connected to a constant volume adsorbent reservoir (Fig. 9.3). Both are immersed in the same calorimetric cell. The total volume remains constant and there is no volume work. The heat exchanged equals the integral molar energy times the amount of gas adsorbed ... 

The aim of this chapter is simply to introduce a selection of the most appropriate thermodynamic quantities for the processing and interpretation of adsorption isotherm and calorimetric data, which are obtained by the methods described in Chapter 3. We do not consider here the thermodynamic implications of capillary condensation, since these are dealt with in Chapter 7. Special attention is given to the terminology and the definition of certain key thermodynamic quantities, for example, the difference between corresponding molar integral quantities and differential quantities. 

The principles of phase equilibrium do not apply to excess adsorption variables at high pressure where the excess adsorption passes throu a maximum. Under these conditions, the pressure is no longer a single-valued function of excess adsorption so that n cannot serve as an independent variable for the determination of partial molar quantities such as activity coefficients. Additional complications which arise at high pressure are (1) the selectivity for excess adsorption (S12 = (nf/j/i)/(n2/y2)) approaches infinity as nj — 0 and (2) the differential enthalpy of the ith component has a singularity at the pressure corresponding to maximum nf. For excess variables, the diffierential functions are undefined but the integral functions for enthalpy and entropy are smooth and well-behaved (1). 

The quantities of interest are (i) n, moles of adsorbate (ii) m, mass of adsorbent (iii) V, volume (iv) p, pressure (v) T, absolute temperature (vi) R, molar ideal gas constant (vii) A, surface area of the adsorbent (viii) Q heat (ix) U, internal energy (x) H, enthalpy (xi) 5, entropy and (xii) G, Gibbs free energy. Superscripts refer to differential quantities (d) experimentally measured quantities (exp) integral quantities (int) gas phase (g), adsorbed phase (s) and solid adsorbent (sol) quantities standard state quantities (°). Subscript (a) refers to adsorption phenomena (e.g. AaH) [13, 91]. 

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