Isotherm statistical

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

Thus from an adsorption isotherm and its temperature variation, one can calculate either the differential or the integral entropy of adsorption as a function of surface coverage. The former probably has the greater direct physical meaning, but the latter is the quantity usually first obtained in a statistical thermodynamic adsorption model. 

The problem of the theoretical description of biopolymer water adsorption isotherms has drawn the attention of researchers for a long time. In the works [19], [20] a rigorous statistical basis for equations describing the isotherms for the case of homogeneous adsorption surfaces and noninteracting adsorption sites of N different types has been suggested. The general equation is ... 

The results of a comparison between values of n estimated by the DRK and BET methods present a con. used picture. In a number of investigations linear DRK plots have been obtained over restricted ranges of the isotherm, and in some cases reasonable agreement has been reported between the DRK and BET values. Kiselev and his co-workers have pointed out, however, that since the DR and the DRK equations do not reduce to Henry s Law n = const x p) as n - 0, they are not readily susceptible of statistical-thermodynamic treatment. Moreover, it is not easy to see how exactly the same form of equation can apply to two quite diverse processes involving entirely diiferent mechanisms. We are obliged to conclude that the significance of the DRK plot is obscure, and its validity for surface area estimation very doubtful. 

Thermodynamically Consistent Isotherm Models. These models include both the statistical thermodynamic models and the models that can be derived from an assumed equation of state for the adsorbed phase plus the thermodynamics of the adsorbed phase, ie, the Gibbs adsorption isotherm,... 

Statistical Thermodynamic Isotherm Models. These approaches were pioneered by Fowler and Guggenheim (21) and Hill (22). Examples of the appHcation of this approach to modeling of adsorption in microporous adsorbents are given in references 3, 23—27. Excellent reviews have been written (4,28). 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

We can show that the thermodynamic and statistical entropies are equivalent by examining the isothermal expansion of an ideal gas. We have seen that the thermodynamic entropy of an ideal gas increases when it expands isothermally (Eq. 3). If we suppose that the number of microstates available to a single molecule is proportional to the volume available to it, we can write W = constant X V. For N molecules, the number of microstates is proportional to the Nth power of the volume ... 

Therefore, the change in the statistical entropy when a sample expands isothermally from volume Vt to a volume V2 is... 

The number of CrB representatives is highly increased by ternary compounds observed in the isothermal sections M -Mo-B and My-W-B just below the transformation T of the CrB-type high-T modifications of MoB and WB into low-T a-MoB or -WB. These ternary compounds therefore represent a stabilization of the high-T form of MoB and WB, respectively, toward lower T by statistical Mo(W)/metal substitution. Transition T (T ) range for WB from 2100°C (W rich) to 2180°C (B... 

Despite the occurrence of binary AIB2 borides (see also Fig. 2), no ternary representatives are known (Mn, Mo)B2 found from isothermal sections is a stabilized high-T phase by conversion to lower T by a statistical ( ) metal-metal substitution. Both MnB2 and M0B2 are high-T compounds stable above 1075°C and 1517°C respectively WB2 is claimed but is either metastable or impurity stabilized. Similar examples are observed with (W, Pd>2B5 (M02B5 type) as well as (Mo, Rh),, (B3 and (W, Ni), B3 (Mo,., 83 type). The phase relations in the B-rich section of the Mo(W)-B binaries, however, are not known precisely. 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

An approximate analysis of polymer adsorption as a set of sequential reactions leads to a simple equation for the adsorption isotherm expressed in terms of three parameters. Comparison of the model with recently published statistical theories reveals remarkable agreement in both the general shape of the isotherms and the predicted effects of molecular weight. The problems of applying such models to experimental data are discussed. 

It is of considerable interest to compare the present model with the predictions of the statistical theories. In particular, the results will be compared with the theories of Scheutjens and Fleer (9,10,13). These authors have pointed out that their results are in general agreement with other published models. Comparison of the adsorption isotherms shown in Figure 1 with those given by Scheutjens and Fleer (see, for example, Figure 5 of Reference 13) shows excellent qualitative agreement. In each case, the isotherms consist of three regions as described above. 

Figure 2. Comparison of adsorption isotherms based on the present model (solid lines) with the statistical theory of Scheutjens and Fleer (symbols). The following values of the parameters were used...

Adsorption isotherms obtained from the model have been shown to agree very closely with the predictions of recently published statistical theories (9,13). While there can be no doubt that the more sophisticated, statistical models provide more information on the nature of the adsorption process and the structure of the adsorbed film, because of its simple form, the macroscopic model can offer a powerful tool for the analysis, interpretation and utilization of adsorption data. 

Cellulose pyrolysis kinetics, as measured by isothermal TGA mass loss, were statistically best fit using 1st- or 2nd-order for the untreated (control) samples and 2nd-order for the cellulose samples treated with three additives. Activation parameters obtained from the TGA data of the untreated samples suggest that the reaction mechanism proceeded through an ordered transition state. Sample crystallinity affected the rate constants, activation parameters, and char yields of the untreated cellulose samples. Various additives had different effects on the mass loss. For example, phosphoric acid and aluminum chloride probably increased the rate of dehydration, while boric acid may have inhibited levoglucosan... 

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