Entropies - differential

Fig. XVII-23. (a) Entropy enthalpy, and free energy of adsorption relative to the liquid state of N2 on Graphon at 78.3 K (From Ref. 89.) b) Differential entropies of adsorption of n-hexane on (1) 1700°C heat-treated Spheron 6, (2) 2800°C heat-treated, (3) 3000°C heat-treated, and (4) Sterling MT-1, 3100°C heat-treated. (From Ref 18.)...

In Fig. 5.21, from Dawson s paper, the uptake at X for the 250°C-outgassed sample is dose to the calculated value for a monolayer of water with a (H20) = 101 A. Point X has therefore been ascribed to a close-packed monolayer of water on a hydroxylated surface of rutile. The fact that the differential entropy of adsorption relative to the liquid state (calculated from the isosteric heat of adsorption) changes sharply from negative to positive values in this region with A s 0 at X was regarded as supporting evidence. ... 

Figure 13.13 Compensation relation between differential heat and differential entropy of adsorption.

Differential Entropies in H-Zeolites. The differential entropies, Ss, of krypton in the three H-zeolites were derived for each uptake from the relation... 

The diastereofacial selectivity of the addition of lithioacetonitrile to 2-phenylpro-panol has been studied over a wide range of temperatures, solvents, and bases.256 Eyring plots [In(dr) vs 1 IT], activation parameters, and inversion temperatures have been characterized. In some cases, the differential entropy of activation, AAS, plays an exclusive role in determining anti -selectivity. 

We will illustrate the method by outlining the calculation of the (differential) entropy of adsorption. For S, we use the Sackur-Tetrode expression, Eq. (76) of Chapter 5, with a term added for the internal degrees of freedom ... 

Figure 4. Net differential entropy of adsorption of krypton on ground muscovite...

The integral change in entropy, AS, is determined from the temperature coefficient of AG. The change in differential entropy of the adsorbent, AS, can be determined either from the difference between the changes in entropy of the integral... 

The differential entropy of adsorption can be readily calculated from the differential enthalpy of adsorption since from Equations (2.46), (2.51) and (2.54) we obtain ... 

The integration of the differential entropy of adsorption between 0 and T is conveniently carried out with the variables T, V and A held constant ... 

In the general case, the integral molar entropy of adsorption is not equal to the mean differential entropy of adsorption over the range of surface excess concentration from 0 to r, because of the extra term of the right-hand side of Equation (2.67). 

We can verify with Equation (2.68) that Aads/i<0 since, for physical adsorption, the equilibrium pressure necessary to obtain the surface excess concentration r (or n) increases with the adsorption temperature. It follows that A is necessarily negative. However, since the differential entropy of adsorption, at constant T, given by Equation (2.56) is directly proportional to the differential enthalpy of adsorption, its calculation is not of great value. 

Many systems give linear plots of pjn against p over a limited ranges of pressure, but such linearity does not by itself imply conformity with the Langmuir model. As already indicated, a second condition is that the energy of adsorption should be independent of surface coverage. Thirdly, the differential entropy of adsorption should vary in accordance with the ideal localized model (Everett, 1950). That no real system has been found to satisfy all these requirements is not surprising in view of the complexities noted here and in subsequent chapters. 

Equations (5.2.20) furnish four useful interrelations between Sg and the differential entropies (3Ss/dns). Note that only at constant P, 4>, T, i.e., when all intensive variables are held... 

Now eliminate the differential entropies between (5.2.20) and (5.2.21). This yields... 

In summary, an attempt has been made in sections 5.2 and 5.3 to provide a number of formulations for the thermodynamic properties of surface phases. Special emphasis has been placed on the key role of the Gibbs adsorption equation as a link between the experimentally available isotherm data, and the determination of molar or differential entropies, enthalpies,... 

Therefore, -S is a state property or an exact differential. Entropy cannot be easily defined but can be described in terms of entropy increase accompanying a particular process. 

Substituting into Equations 6 and 7 the values of the derivative and A calculated from Equation 1, we obtained the following expressions for differential entropy and differential heat of adsorption (2) ... 

Thus, do may be interpreted as representing the infinitesimal differential entropy increase in the universe arising from irreversible interactions between the system and its surroundings. Alternative interpretations are furnished below. Moreover, on subtracting (1.12.2a) from (1.12.3b) one obtains... 

We have thus obtained three equivalent formulations for the differential entropy of an ideal gas. We may now carry out an integration between two specific limits selecting Eq. (2.4.6) we obtain... 

The above expressions furnish four interrelations between the molar entropy Ss and the various differential entropies (dSs/dris). Note that it is only when the three intensive quantities T, P,(j) are held fixed that the molar entropy of the adsorbed phase is equal to its partial molal counterpart. The terms involving 1 are usually small and are generally neglected. 

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