Isosteric method

The differential enthalpy of adsorption may be obtained indirectly by the isosteric method (cf. Section 2.6.1). In the past, it was often referred to as the isosteric heat and denoted by -qa. This term is now discouraged, and should be replaced by the isosteric enthalpy of adsorption. The differential energy and differential enthalpy of adsorption are related by the expression ... 

Indirect Derivation of the Quantities of Adsorption from a Series of Experimental Physisorption Isotherms The Isosteric Method... 

The usual method of calculating A from a series of adsorption isotherms obtained at different temperatures is to plot In [p]r for a given T (or n) as a function of l/T. This method of calculation of the differential enthalpy based on the use of Equations (2.68) and (2.69) is called the isosteric method. By applying Equation (2.68) at different values of r (or n), we can determine the variation of with T. 

Equation (2.72) is not easy to use in the general case in which the spreading pressure is unknown. But in the particular case of stepwise isotherms where there are two adsorbed phases in equilibrium with the gaseous adsorptive (i.e. in the case of a univariant adsorption system), Larher (1968, 1970) showed that the isosteric method may be used with the transition pressure p" to give integral molar energies un and entropies sn of the quasi-layer ... 

Provided that sufficient care is taken in the experimentation, the results are more accurate than can be obtained by the isosteric method, especially at low surface coverage (Harkins, 1952 Zettlemoyer and Narayan, 1967). 

As explained in Section 2.6.1, the application of the isosteric method relies on the principles embodied in Equation (2.68). In practice, the isosteric procedure is applied to at least two isotherms at different temperatures, which must not be too far apart. A temperature range of, say, 10 K is often considered to be a good compromise. Whenever possible, more than two isotherms should be measured and the plot of In [p] versus 1 /T checked for linearity. As noted earlier, the isosteric method is very sensitive to any error in the measurement of the equilibrium pressure. Systematic comparisons between the values of differential enthalpy determined by the calorimetric and isosteric methods have revealed serious inaccuracies in the isosteric values at low pressures or low surface coverage (Rouquerol et al. 1972 Grillet et al., 1976). It turns out that one must be particularly careful when applying the isosteric method at surface coverage <0.5. A further constraint is that for each constant n°. there should be no 2-D phase change over the temperature studied. 

In spite of these limitations, two valuable features of the isosteric method should be recognized ... 

The isosteric method , based on the temperature dependence of adsorption. 

The isosteric method can be considered, for adsorption from solution, in a similar manner as for gas adsorption (see Section 2.6.1). For example, by equating the chemical potentials of component 2 (the solute) in its adsorbed state and the liquid phase, by keeping the specific amounts adsorbed constant and by considering a dilute solution, so that the activity can be replaced by a molality (b2 = 1000 n2/mi), we obtain ... 

For all these reasons, one should be cautious in the application of the isosteric method (see Lyklema, 1995). 

The simplest way of obtaining the differential enthalpy of adsorption, A is by the isosteric method, which involves the measurement of at least two isotherms for the given system at different temperatures and application of the Clausius-Clapeyion equation. The method is based on the assumption that the adsorbed phase remains unchanged and in thermodynamic equilibrium over the temperature range studied. In practice, the method is very sensitive to any errors in the measurement of the equilibrium pressures. For these reasons the derived isosteric enthalpies are often unreliable, particularly in the region of low surface coverage. 

Indirect derivation of the quantities of adsorption from a series of experimental physisorption isotherms the isosteric method 43... 

According to equation (2), InC is directly related to the heat of adsorption in the first layer. Accordingly, the values of the C parameter were compared with the differential enthalpies of adsorption. These were obtained using the isosteric method from the adsorption data of N2 at two different but close temperatures 77.3 and 87.3 K. Figure 5 shows that InC and the differential enthalpies of adsorption vary with the surface coverage in the same way, confirming the energetic nature of the C parameter. 

Figure 5. Comparison of the C parameters of the BET equation with the differential enthalpies obtained using the isosteric method.

Now, these calorimetric results allow, in principle, to recalculate adsorption isotherms at other temperatures than 30°C by simple use of the isosteric method derived from the Clausius-Clapeyron equation, for instance in the following form ... 

Adsorption microcalorimetry, finally, allows to check the differential enthalpies of adsorption derived from various adsorption isotherms through the isosteric method. What is more, it is much safer and meaningful in the first, raising, part of the isotherm, specially when it is close to the ordinate, such as for instance for carbon dioxide adsorption. This part is probably the most interesting from the viewpoint of specific interactions and gas separation or storage. 

Sorption thermodynamic functions of nitrous oxide, N2O, are described for zeolites NaLSX and CaLSX in shapes cA clay-bound beads. They were determined by the Sorption Isosteric Method (SIM) over complete ranges of sorption-phase concentration and compared with those for carbon dioxide, C(>2. nepo earlier for the same NaLSX sorbent... 

The Sorption Isosteric Method (SIM) was used to determine thermoifynamic data of N2O sorption by zeolites NaLSX and CaLSX. This technique and its utilization for an investigation of sorption-thermodynamic properties of CO2 on identical NaLSX beads were described in detail in refs. [4-8]. 

Ni( 111) by the isostere method gave slightly lower values ( 90 kJ mol" With cobalt, rhodium, platinum (which is also structure insensitive) and iridium, values are somewhat lower (75—90 kJ mol ) and for copper and manganese lower still. The heat of adsorption (—A//ads) is converted into an M—H bond strength Dmh by the relation... 

When one wishes to measure the adsorption of binary or more complicated mixtures of adsorbates foen the above simple volumetric/gravimetric systems are not suitable and one has to resort to more complicated systems. Such a system has been developed at Imperial College and elsewhere and is referred to as the isosteric method [1,2]. 

Although the isosteric apparatus is primarily designed to study the sorption of mixtures as in the case of single component studies the isosteric principle is only strictly obeyed when both components of the mixture are reasonably strongly sorbed. When one component of the mixture is somewhat weakly sorbed errors will arise in the mixture isosteres generated. The size of these errors can be seen in Figures 9(a) and (b) when a 52 mol% N2/48 mol % CO2 gas mixture is admitted to the silicalite from Ae doser section. If the whole of the admitted dose was sorbed by the silicalite as assumed by the isosteric method then the composition of the sorbed phase would be that of the admitted dose. However, the sorbed phase composition of the... 

Sorption of a 48 mol% C02/52 mol%N2 gas mixture in silicalite-1 a) mole fraction, X(N2), of N2 in the sorbed phase vs temperature b) total amount of mixture soibed, n, vs temperature. The dashed lines represents the values of these quantities if the whole of the admitted dose had been soibed as assumed in the isosteric method. 

Ranke, W. and Josephy, Y. (2002) Determination of Adsorption Energies and Kinetic Parameters by Isosteric Methods. Phys. Chem. Chem. Phys. 4,2483-2498. 

Initially, heat of wetting was used to measure total surface areas of solids including micro-porous materials. Later, this heat of wetting method was used to obtain differential heats of adsorption, as can be done from the isosteric method and by flow calorimetry. 

For such ambitious applications, it is first necessary to understand more deeply the interactions between methane and the microporous adsorbent surface. The enthalpy of CH4 adsorption has been evaluated in various zeolite systems by using isosteric methods via the Clapeyron equation or performed using microcalorimetry measurements which allow direct... 

This paper deals with the principles, advantages and limitations of measurement of sorption equilibria under isosteric conditions. It further assesses the sorption-isosteric method (SIM) as an effective tool for providing complete sets of sorption-thermodynamic functions, viz., enthalpy, standard entropy and standard Gibbs free energy of sorption, for nanoporous solids, i.e., micro- and mesoporous ones, as functions of sorption-phase concentration, n, over its entire range, and to approach such data for mixtures. The usefulness of SIM is exemplified by sorption systems that comprise atmospheric gases on zeolites and carbon dioxide, CO2, on carbonaceous sorbents, as well as several of their mixtures. 

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