Isosteric heats of adsorption

The isosteric heat, q, is the heat of adsorption at a constant adsorbate amount. In terms of thermodynamics it is related to the pressure and temperature by 

Attempts have been made to determine this from the isotherms (for example see Joyner and Emmett [5]). To do this one measures two or more isotherms at different temperatures that are fairly close. One then fits the isotherms either manually, for example with a spline lit, or mathematically. Unfortunately, errors accumulate very heavily in this case and the choice of fit can greatly distort that answer. Use of the analytical form of the standard curves [6]may aid in this attempt and appears to be successful in some cases but porosity and multiple heats of adsorption make this unreliable as well. 

The differential heat of adsorption is also called the isosteric heat of adsorption, DH, and can be measured either by using a calorimeter or by using a family of isotherms measured at two or more temperatures. 

When a gas or vapor is in adsorption equilibrium with a solid surface, the chemical potential of the adsorbate in the adsorbed phase can be expressed as a function of internal energy as 

The quantity -A// is the differential molar enthalpy of adsorption which is also termed as the isosteric enthalpy of adsorption. Thus, the isosteric heat of adsorption Aff can be written with the opposite sign as 

AH being positive, the heat is evolved when adsorption takes place. The integrated form of Equation 2.23 can be written as 

It may be mentioned that the heat of adsorption calculated by using Equation 2.24 and Equation 2.25 are applicable to a particular value of amount adsorbed. However, these values of heat of adsorption can be modified by energetically nonuniform nature of the carbon surface or by the interactions between the adsorbed molecules in the adsorption layer. It is, therefore, better that the calculations of AH be repeated for a series of values of and then obtain a curve between AH and n . 

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The general type of approach, that is, the comparison of an experimental heat of immersion with the expected value per square centimeter, has been discussed and implemented by numerous authors [21,22]. It is possible, for example, to estimate sv - sl from adsorption data or from the so-called isosteric heat of adsorption (see Section XVII-12B). In many cases where approximate relative areas only are desired, as with coals or other natural products, the heat of immersion method has much to recommend it. In the case of microporous adsorbents surface areas from heats of immersion can be larger than those from adsorption studies [23], but the former are the more correct [24]. 

The constants in Eqs. XVII-88-XVI1-90 may be calculated fiom theory to give the Henry s law constant K from Eq. XVII-87, the experimental n /P dien gives the surface area. Alternatively, the constants may be arrived at from an experimental K (assuming that A is known) and either the isosteric heat of adsorption... 

Fig. XVII-20. Isosteric heat of adsorption of Xe on a stepped Pd surface [8(100) x (110)]. (From Ref. 111.)...

Fig. XVII-21. Continued) (c) Isosteric heats of adsorption of n-hexane on ice powder Vm = 0.073 cm STP. (From Ref. 125). (d) Isosteric heats of adsorption of Ar on graphitized carbon black having the indicated number of preadsorbed layers of ethylene. (From Ref. 126.)...

Fig. XVII-22. Isosteric heats of adsorption for Kr on graphitized carbon black. Solid line calculated from isotherms at 110.14, 114.14, and 117.14 K dashed line calculated from isotherms at 122.02, 125.05, and 129.00 K. Point A reflects the transition from a fluid to an in-registry solid phase points B and C relate to the transition from the in-registry to and out-of-registry solid phase. The normal monolayer point is about 124 mol/g. [Reprinted with permission from T. P. Vo and T. Fort, Jr., J. Phys. Chem., 91, 6638 (1987) (Ref. 131). Copyright 1987, American Chemical Society.]...

The nitrogen adsorption isotherm is determined for a finely divided, nonporous solid. It is found that at = 0.5, P/P is 0.05 at 77 K, gnd P/F is 0.2 at 90 K. Calculate the isosteric heat of adsorption, and AS and AC for adsorption at 77 K. Write the statement of the process to which your calculated quantities correspond. Explain whether the state of the adsorbed N2 appears to be more nearly gaslike or liquidlike. The normal boiling point of N2 is 77 K, and its heat of vaporization is 1.35 kcal/mol. 

Isosteric heat of adsorption (at half coverage) q of nitrogen and argont... 

Now A. the diflerential molar enthalpy of adsorption, is often termed the isosteric enthalpy of adsorption (or alternatively the isosteric heat of adsorption) and is given the opposite sign. Thus... 

If the fraction of sites occupied is 0, and the fraction of bare sites is 0q (so that 00 + 1 = 0 then the rate of condensation on unit area of surface is OikOo where p is the pressure and k is a constant given by the kinetic theory of gases (k = jL/(MRT) ) a, is the condensation coefficient, i.e. the fraction of incident molecules which actually condense on a surface. The evaporation of an adsorbed molecule from the surface is essentially an activated process in which the energy of activation may be equated to the isosteric heat of adsorption 4,. The rate of evaporation from unit area of surface is therefore equal to... 

Fig. 2.14 The isosteric heat of adsorption ( ) of argon, nitrogen and oxygen of rutile at 95 K, plotted as a function of the amount adsorbed (expressed in cm (stp). The uptake of each gas corresponding to the completion of a monolayer is marked. Note the more rapid decrease in as the amount adsorbed approaches monolayer completion. (After Drain.)...

Fig. 2.15 Isosteric heat of adsorption of nitrogen on molecular (low-evergy) solids and on carbons (high-energy solids), plotted as a function of i/n . (A) Diamond (B) gruphitized carbon black. P.33 (D) Benzene (E) Teflon. The curve for amorphous carbon was very close to Curve (A). (Redrawn from a Figure of Adamson . )...

Fig. 4.7 Isosteric heat of adsorption q of nitrogen adsorbed at 77 K on non-porous and porous silica.(I) 4. Fransil", non-porous (2) gel A, mesoporous (. ) A gel B, mesoporous and microporous (4) gel D,...

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. ... 

Fig. 5. Variation of isosteric heat of adsorption with adsorbed phase concentration. Reprinted from ref. 10, courtesy of Marcel Dekker, Inc. To convert kj...

H2O/100 kg of adsorbent. At equilibrium and at a given adsorbed water content, the dew point that can be obtained in the treated fluid is a function only of the adsorbent temperature. The slopes of the isosteres indicate that the capacity of molecular sieves is less temperature sensitive than that of siUca gel or activated alumina. In another type of isostere plot, the natural logarithm of the vapor pressure of water in equiUbrium with the desiccant is plotted against the reciprocal of absolute temperature. The slopes of these isosteres are proportional to the isosteric heats of adsorption of water on the desiccant (see... 

FIG. 16-1 Isotherms (left) and isosteres (right). Isosteres plotted using these coordinates are nearly straight parallel hnes, with deviations caused by the dependence of the isosteric heat of adsorption on temperature and loading. 

Isosteric Heat of Adsorption The most useful heat of adsorption for fixed-bed calculations is the Isosteric heat of adsorption, which is given by the Clausius-Clapeyron type relation... 

The isoteric heat of adsorption qf is composition-dependent, and the sum of integrals Eq. (16-60) is difficult to evaluate for multicomponent adsorption if the isosteric heats indeed depend on loading. Because each isosteric heat depends on the loadings of all components, the sum must be evaluated for a path beginning with clean adsorbent and ending with the proper loadings of all components, if the isosteric heat of adsorption is constant, as is commonly assumed, then the energy balance (Eq. 16-55) becomes... 

From the above argument and Eq. (16) we instantaneously find that the isosteric heat of adsorption cannot be constant within the two-phase region but must also show changes with the surface coverage. In the case of heat capacity we also observe important effects due to the surface heterogeneity. 

Inserted in Eq. (3), this gives the Langmuir isotherm P = Zi ,exp( u ABr)= - exp(-KoABr,, from which we get the isosteric heat of adsorption... 

Using equation (6.34) and the definition of the isosteric heat of adsorption Had=T2(d(jIl ad) /T)pj,6 j one can show easily that the isosteric heat of adsorption, AHadj, is given by ... 

To the extent that Pj may be assumed constant one can then show that the variation in the isosteric heat of adsorption AHadj, with coverage 0j is given by ... 

Similarly, Eq. (16-9) is used to calculate the isosteric heat of adsorption from two isotherms. 

The differential heat of adsorption, Qj, for n-butane over FER framework type was 59kJ/mol until a coverage of about one n-butane per acid site was reached, beyond which there was l-2kJ/mol lower heat, as illustrated in Figure 13.8. The differential heat of adsorption is related to the isosteric heat of adsorption according to the following relation. 

Additionally, the heat of adsorption has been investigated. For example, the amount of Kr adsorbed on Agl increases when temperature is decreased from 79 K (0.13 cc/g) to 77 K (0.16 cc/g). These data allow one to estimate the isosteric heat of adsorption (Jaycock and Parfitt, 1981) ... 

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