Entropy of the adsorbed phase

Adsorbed Phase Entropy. Since Equations 7 and 8 can accurately describe the relationship between q, T, and p, we may use them to calculate the integral molar entropy of the adsorbed phase. At temperatures significantly lower than critical for the adsorbate, the entropy of the adsorbed phase is usually compared with the entropy of the liquid at same temperature in order to compare the freedom of each phase. Because our experimental domain was higher, we shall make this comparison with the gaseous phase compressed to the same density p as determined by Equation 8. 

Numerical values of Ss were obtained after replacing ps by fs according to Lewis method. Figure 5 shows that the entropy of the adsorbed phase always lies below that for gaseous phase compressed to the same density. Thus the adsorbed phase is more localized and has less freedom than the compressed phase. 

Figure 5. Comparison of the integral entropy of the adsorbed phase (solid lines) with the entropy of the gaseous phase of same density (dashed lines)...

If the integral molar entropy of the adsorbed phase is to be compared with the molar entropy of the immersion liquid, it is necessary to express the molar entropy of the gas as a function of the relative pressure and the enthalpy of vaporization. Thus,... 

This shows how one may obtain the molar entropy of the adsorbed phase from an observation of the variation of in P with T, where P is the pressure of the gas in equilibrium with the adsorbate. The problem here is that one cannot readily devise an experiment in which the variation of in P with T is carried out at constant 4 It is much simpler to conduct experiments under conditions where As or T remains constant. 

In principle, knowing the molar entropy of the perfect gas (Section 1.17), and by measuring the change of equilibrium gas pressure as a function of temperature, one can determine the molar entropy of the adsorbed phase. The problem here is that the experiment has to be carried out at constant 0, a problematic task. Methods for circumventing this difficulty are shown below. Meanwhile, for completeness, we observe that at equilibrium the chemical potentials of the gas and adsorbate must match then Hg — Hs — T(Sg — Ss), so that we obtain the alternative formulation... 

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. 

Riccardo and coworkers [50, 51] reported the results of a statistical thermodynamic approach to study linear adsorbates on heterogeneous surfaces based on Eqns (3.33)—(3.35). In the first paper, they dealt with low dimensional systems (e.g., carbon nanotubes, pores of molecular dimensions, comers in steps found on flat surfaces). In the second paper, they presented an improved solution for multilayer adsorption they compared their results with the standard BET formalism and found that monolayer capacities could be up to 1.5 times larger than the one from the BET model. They argued that their model is simple and easy to apply in practice and leads to new values of surface area and adsorption heats. These advantages are a consequence of correctly assessing the configurational entropy of the adsorbed phase. Rzysko et al. [52] presented a theoretical description of adsorption in a templated porous material. Their method of solution uses expansions of size-dependent correlation functions into Fourier series. They tested... 

The adsorbed water hcis a lower enthalpy and a lower entropy than free water. The standard entropy of the adsorbed phase is very close to the valne found for ice H20(s), viz. S 69.9 — 22.0 50 J/molK (see the example in section 4.6). This suggests that the adsorbed water has a more organized molecular structure and a lower potential energy than found for free water. 

To characterize the state of the adsorbed phase, it is useful to evaluate its molar entropy, s , defined as the mean molar value for all the molecules adsorbed over the complete range of surface coverage up to the given amount adsorbed. The molar integral entropy of adsorption. As, is then defined as... 

A summary of developments in physical adsorption during the period from 1943 to 1955 has been given recently by Everett 94). The chief difference between the approach used by Brunauer in his book published in 1943 and that in vogue in 1955 is in the great development of the thermodynamic aspects of the subject. Prior to 1943, the main effort was in developing theories to predict the shape of adsorption isotherms. Since then, emphasis has shifted towards the thermodynamic properties of the adsorbed phase, particularly its entropy. 

Thermodynamic methods, which have been those most widely used in the past, utilize isotherms and heats of adsorption as their foundations. Entropy changes calculated from such data are not easy to transform unambiguously into specific descriptions of the adsorbed phase. 

Let Sg° be the standard entropy of the gas at the standard pressure (p0 = 1 atm) and at the experimental temperature. The partial molar entropy Ss and the integral entropy Ss of the adsorbed phase are given by ... 

We may assume that both argon and krypton adsorbed on mica are essentially two-dimensional liquids at the completion of the first monolayer (2, 4). The rise in the entropy functions for argon adsorption on potassium and barium mica as the monolayer point is approached may then reflect the transition from substantially localized adsorption at lower coverages to a mobile film. No such phenomenon is observed with krypton, suggesting that there is no change in the behavior of the adsorbed phase during formation of most of the first monolayer. 

The Langmuir adsorption isotherm is represented hy P — K C/(Cm — U), where K is a function of T only, and Cm represents the maximum concentration of adsorbed species that can be accommodated as a monolayer on the surface. Derive the corresponding two-dimensional equation of state. Determine the molar enthalpy and entropy of the adsorbate, using the perfect gas approximation for the gaseous phase. 

Summarizing, an attempt has been made to provide a systematic account of the thermodynamic properties of the adsorbed phase. The Gibbs adsorption equation, as an extension of the Clausius-Clapeyron equation, has played a key role in linking experimental isotherm data to the determination of molar or differential entropies and enthalpies. Similarly, calorimetric measurements can be systematically applied to obtain the same type of information. 

The two-dimensional gas model assumes no mutual interaction of the adsorbed molecules. It is believed that the adsorbent creates a constant (across the surface) adsorption potential. Thus, in the framework of statistical thermodynamics, the model describes adsorption as the transition of a gas with three translational degrees of freedom into an adsorbed state with one vibrational and two translational degrees. Assuming ideal behavior and using molar quantities, one obtains the standard entropy in the adsorbed phase as the sum of the translational and vibrational entropies from Eqs. 5.28 and 5.29 ... 

A second approach attempts to relate the entropy of adsorption to the loss of mobility when a gas phase species, with three degrees of freedom, forms a two dimensional fluid on the surface. This line of reasoning leads to the conclusion that the entropy of adsorption must be negative and no larger than the total entropy of the adsorbate in the gas phase. The authors of this idea (M. Boudart et al. (1967)) go so far as to propose that... 

Here U is the internal energy, S is the entropy, m is the mass of adsorbent, ( ) is the surface potential of the adsorbed phase per unit mass of the adsorbent, and Uj is the chemical potential, related to the fligacity as follows ... 

The measurement and interpretation of the thermodynamic function of the adsorbed phase, such as the free energy of adsorption, the enthalpy of adsorption, and the entropy of adsorption have been the subject matter of large number of investigations. These functions have been evaluated using adsorption isotherms and are compared with those obtained from theoretical considerations. In all... 

---