Excess isotherm

Fig. XI-10. Isotherm of composition change or surface excess isotherm for the adsorption of (1) benzene and (2) n-heptane on Graphon. (From Ref. 141.)...

Figure 6.11 Composite (surface excess) isotherms for the adsorption of (a) benzene from solution in methanol on to charcoal177 and (b) chloroform from solution in carbon tetrachloride on to charcoal178 (By courtesy of (a) American Chemical Society, (b) Journal of the Chemical Society)...

Figure 5.14. Typical shapes of surface excess isotherms from dilute solutions (Langmuir or L-shape, S-shape, stepwise (SW)).

Since the reduced and relative surface excess isotherms convey composite information on the adsorption of the two components, there is a strong incentive to determine the individual (or separate ) isotherms, i.e. the adsorbed amount n (or ) versus concentration, mole fraction or mass fraction. It will be recalled that this implies some assumptions about the thickness, composition and structure of the adsorbed layer, and therefore is not to be recommended for reporting adsorption from solution data in a standard form. Indeed, this second step is already part of the theoretical interpretation of the adsorption mechanisms. 

Buszewski, B., Bocian, S., Felinger, A. Excess isotherms as a new way for characterization of the columns for reversed-phase liquid chromatography. J. Chromatogr. A 1191, 72-77 (2008)... 

Equation (2-52) is the total adsorption isotherm derived from experimentally measurable excess isotherm using the model of adsorption process obtained from the analysis of the experimental isotherm profile. 

A second, more analytical, consequence of 12.1.1) is that the change in bulk composition is not only the consequence of the disappearance of. say A from a mixture of A and B, but is also due to the desorption of B. When an isotherm is measured on the basis of depletion of component A in solution, l.e. when is measured, the resulting Isotherm is not an individual isotherm, relating to the Interfaclal properties of A only, but a surface excess (formally called composite) isotherm, relating to the Interfacial properties of A and B. There is no thermodynamic way to decompose such surface excess isotherms into the two individual ones, although there are situations where this can be done with reasonable model assumptions. 

Before further discussing the thermodynamics and model analyses of surface excess isotherms, let us look at some typical shapes, as shown in fig. 2.8. TYpes (a) and (b) are the most common. Schay and Nagy ) distinguish more types by... 

Figure 2.8. Classification of excess isotherms for binary solutions, (a) (inverse) U-shape (b) (toppled) S-shape (c) linear.

Figure 2.9. Types of partial (or individual) isotherms leading to excess isotherms of the U-type (a) or S-type (b).

The more complex the individual isotherms, the more complicated is the surface excess Isotherm. The simple situation of n = 0 and n° = const, leads to... 

Models, methods and examples of decomposing surface excess Isotherms into their constituent parts will be discussed in sec. 2.4. 

Still another variant consists of predicting the shape of, say, the (1 +3) excess isotherm from those for (1+2) and (2+3), and comparing the prediction with experiment. 

In which the l.h.s, is the operational surface excess, telling us by how much the mole fraction of 2 In the adsorbate exceeds that in the liquid. According to (2.3.81 this is (one of) the measured surface excess Isotherms, if plotted as a function of x. 

Figure 2.11 gives an Illustration of both the Individual, fl(x), and the surface excess, (9-x)(x), Isotherms. Systems obeying the present premises give rise to type (a) curves (fig. 2.8). With Increasing K the maximum Increases and moves to the left. At the same time the individual and excess isotherms remain Identical till higher 0, which is in line with expectation. 

Figure 2.11. Individual (-- - ) and excess (---isotherms for adsorption from a binary...

Illustrations of this behaviour are given in figs 2.12 and 13. The curves may be compared with those in fig. 2.11 for K = 5. The individual isotherms are now much steeper. In turn this leads to the near-linear drop of the excess isotherms beyond x = 0.2. For sufficiently high x°< surface phase separation takes place, as... 

These two figures exhibit another feature that is recurrently found, namely that sizable parts of the excess isotherms are linear. Writing the Ostwald-Klpling isotherm (2.3.6] per vmit of mass m of the adsorbent as... 

Figure 2.23. Extrapolation of linear parts of surface excess isotherms. Dotted curve sketch of the individual isotherm for substance 2 in type a.

Excess isotherms are shown on Figure 4 for binary mixtures of ethylene and ethane adsorbed on NaX. Both curves are for 20°C and a vapor composition of 10 mole percent ethylene. The individual excess isotherm for ethylene is the difference between two isotherms. 

Basically the Issue is that [4.2.1] reflects a surface excess isotherm, as already encountered in the adsorption of binary mixtures on solid surfaces (secs. II.2.3 and 4). Adsorption of the one component cannot be effected without depletion from the surface of the other, and because splitting these two processes is Inoperational, thermodynamics tells us that only the two adsorptions together are measurable ). 

The solution of the dividing plane problem is that the adsorptions are not referred to a zero value of one of the two components, but that Tj " cmd are introduced where is the surface excess of 1 per unit cirea over the amoimt of 1 in a reference system of uniform composition with Xj = (1 - x) and x =x. This is the excess that is measured in surface excess isotherms and which determines the surface tension. The definitions are... 

Figure 4.3. Surface excess Isotherms according to (4.2.12]. For conditions see the text. K Is the exchange constant. The curves correspond to the drawn ones in fig. 11.2.11, after normalization.

Non-ideality in the regular solution model yields the following excess isotherm equation (see II.2.4.21) for a monolayer... 

Figure 4.6. Surface excess isotherms (top) ind surface tension (bottom) of a binary mixture of an r-mer and a monomer. z°K(r]l z = 5.

Figure 4.10. Surface excess isotherms of linear alkanes (he), with different chain length n, in mixtures with benzene. Temperature 30°C. (Redrawn from Schmidt and Clever, loc. cit.)...

The set of surface excess isotherms cannot be used to derive individual isotherms eq. [4.2.3b) contains two unknowns. So an additional assumption is needed. The authors took + r A = 1 (results not shown). Equation [4.2.26] might be an alternative. Anedyses like the one leading to fig. 4.10 can of course be extended to all literature data available. 

The specific surface excess isotherms for binary liquid mixtures benzene + n-heptane and benzene + 2-propanol were measured by static immersion method [8] The concentrations of equilibrium solutions were determined using HP 5890 gas chromatograph from Hewlett-Packard. The initial mixtures over the whole concentration range served for detector calibration, The surface excess of a given component was calculated from the relation ... 

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