Differential molar energy of adsorption

Thus by measuring the small amount of heat 5Q which is evolved when the adsorption increases by the small amount 6n mole at constant temperature, the differential molar energy of adsorption can be evaluated calorimetri-... 

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... 

Figure 11.4. Differential molar energy of adsorption of Ar and N2 on samples of kaolinite as a function of surface coverage, (a) On sample GB3 (b) On sample FU7 (reproduced courtesy of Cases et al., 1986).

The differential molar energy of adsorption can be measured by means of a closed isothermal calorimeter. This system consists of two compartments contained in a closed isothermal calorimeter. Initially one compartment is evacuated and contains a given amount of adsorbent but no adsorbate, and the other compartment contains n moles of gas at pressure p. The two compartments are then connected physically until the pressure equilibrates. 

The differential heat of adsorption is related to the integral heat and to the differential molar energy of adsorption according to... 

The integral molar energy of adsorption is therefore obtained by integrating the differential energy of adsorption between the limits 0 and n°. It equals the mean of the differential energy of adsorption over the range of surface excess concentration from 0 to r. 

The characteristic energy is a measure of the adsorption strength between adsorbate and adsorbent. The function f is regarded as the distribution function of filling of micropores 0 over the differential molar work of adsorption, and n is the parameter associated with the distribution function. 

With any form of the distribution and a particular choice of the local isotherm, eq. (4.4-2) can be in general integrated numerically to yield the overall adsorption isotherm equation. The local fractional loading 0 can take the form of either the DR or DA equation. As discussed earlier the DA equation with n = 3 describes well solids having narrow micropore size distribution, and hence this makes this equation a better candidate for a local isotherm equation rather than the DR equation. However, since the selection of a distribution function is arbitrary, this does not strictly enforce the local isotherm to reflect the intrinsic local isotherm for a specific characteristic energy. Moreover, the DR or DA itself stems from a Weibull s distribution function of filling of micropore over the differential molar work of adsorption. Thus, the choice of the local isotherm is empirical, and in this sense the procedure of eq. (4.4-2) is completely empirical. The overall result, however, provides a useful means to describe the equilibrium data in microporous solids. 

Let AB be the corresponding differential molar heat of adsorption (and, according to the previous assumptions, this is independent of 0 and also independent of gas pressure, if the gas is perfect). Let /i be the chemical potential of the adsorbed layer and let/e be the chemical potential of the gas at unit pressure. Then the differential Gibbs free energy of adsorption, relative to the gas at unit pressure, is... 

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... 

Integration of the differential energy of adsorption is quite straightforward from Equation (2.50). Since the gas is ideal, its molar internal energy does not vary with pressure so that ... 

The heat evolved will now be a differential heat of adsorption, equal at constant volume to Qd or per mole, to qd - AI2, where Ae2 is the change in partial molar energy. It follows that... 

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. 

By substituting the averaged molar enthalpy Ha(na) of the adsorbate for the averaged molar internal energy Ea(na) of the adsorption phase, the differential adsorption molar enthalpy Ha (na) is obtained. The isosteric heat of adsorption QjS0 is given as the difference between Hg the molar enthalpy of the gaseous phase and Ha (na) the differential molar enthalpy. 

Here uG — is the difference in the partial molar internal energy of the gas G and the adsorbed layer 1, and —p (iV/tym)s is the adiabatic differential heat of compression. Reversible adsorption can only occur at temperatures such that diffusion of the adsorbed gas is fast enough to establish an equilibrium distribution at the surface during the time of the measurements. This criterion holds for chemisorption as well, but implies that observations are made at correspondingly higher temperatures. 

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 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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