Equation of state surface

The Gibbs adsorption isotherm shows the dependence of the extent of adsorption of an adsorbent on its bulk concentration or pressure. However, we also need to know the state of the adsorbate at the surface. These are interrelated because the extent of material adsorb-tion on a surface depends on the state of the surface. The behavior of the molecules in the surface film is expressed by a surface equation of state which relates the spreading pressure, n, which is the difference between the solvent and solution surface tensions, %= % - y to the surface concentration of the adsorbent. This equation is concerned with the lateral motions and interactions of the molecules present in an adsorbed film. In general, the surface equation of state is a two-dimensional analogue of the three-dimensional equation of state of fluids, and since this is related to monomolecular films, it will be described in Sections 5.5 and 5.6. It should be remembered that on liquid surfaces, usually monolayers form, but with adsorption on solid surfaces, usually multilayers form (see Section 8.3). 

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The adsorption isotherm —Equation (8) —associated with this surface equation of state is called the Henry law limit, in analogy with the equation that describes the vapor pressure of dilute solutions. The constant m, then, is the adsorption equivalent of the Henry law constant. When adsorption is described by the Henry law limit, the adsorbed state behaves like a two-dimensional ideal gas. 

The second type of interaction possible for adsorbed molecules is direct adsorbate-adsorbate interaction. Interactions of this sort are expected to lead to deviations from ideality in the two-dimensional phase just as they lead to deviations from ideal behavior for bulk gases. In this case surface equations of state, which are analogous to those applied to nonideal bulk gases, are suggested for the adsorbed molecules. The simplest of these allows for an excluded area correction (see Equation (7.23)) ... 

In this section, the derived surface equation of state [Eq. (33)] is compared with the experimental 7t - A isotherms involving the LE/LC phase transition for the following six phospholipid surfactants 1,2-dimyris-toylphosphatidylcholine (DMPC), 1,2-dipalmitoylpho-sphatidylcholine (1,2-DPPC), 1,3-dipalmitoylphosph-atidylcholine (1,3-DPPC), 1,2-dimyristoylphosphatidic acid (DMPA), 1,2-dilauroylcephalin (DLPE), and 1,2-dimyristoylphosphatidylethanolamine (DMPE). The general formula of the phospholipid surfactants can be written as R-C02- (hJ-CbC—R, where R stands... 

Ruckenstein, E. Li, B. Surface equation of state for insoluble surfactant monolayers at the air/water interface. J. Phys. Chem. 1998,102, 981. 

Ruckenstein," E. Li, B. A surface equation of state based on clustering of surfactant molecules of insoluble monolayers. Langmuir 1995,11, 3510. 

Equation (2.34) is often referred to as the Gibbs adsorption equation where the interdependence of r and p is given by the adsorption isotherm. TTie Gibbs adsorption equation is a surface equation of state which indicates that, for any equilibrium pressure and temperature, the spreading pressure II is dependent on the surface excess concentration r. The value of spreading pressure, for any surface excess concentration, may be calculated from the adsorption isotherm drawn with the coordinates n/p and p, by integration between the initial state (n = 0, p = 0) and an equilibrium state represented by one point on the isotherm. 

These considerations have their consequences regarding the interpretation of experimental data should it be done in terms of isotherms or in equations of state Preferably both should be considered, but when specific features are under study a choice may have to be made. For instance, surface heterogeneity shows up very strongly in the shapes of the Isotherms (sec. 1.7) but very little in the equation of state in the model case of local Langmuir isotherms without lateral interaction heterogeneity is not seen at all in the equation of state (because the energy is not considered and the entropy not affected) whereas the isotherm shape is dramatically Influenced. On the other hand, for homogeneous model surfaces equations of state may be more suited to observe subtle distinctions in lateral mobility or lateral interaction. 

Langmuir trough or film balance, to be described in sec. 3.3.1). With such compression the surface pressure k increases. Surface equations of state, relating n to the area A and the temperature T can be formulated, entirely analogous to the three-dimensional equivalent. For instance, for a very dilute, gaseous, monolayer the two-dimensional equation of state is... 

With regard to the surface pressure, for ideeilized Langmuir monolayers 7t is an independent, externally applied veirlable, whereas for idealized Gibbs monolayers n is determined by adsorption of molecules. The obvious question is whether surface equations of state are identical between Langmuir and Gibbs monolayers. The answer is, in principle, yes. Relations between n, A and r are completely determined by the numbers of, and Interactions between, the molecules in the monolayer, irrespective of whether or not equilibrium with an adjacent phase has been established. The statistical derivation underlines this. The two-dimensional pressure can be obtained canonically from... 

With this in mind, this chapter will be set up system-wise, working from the more simple towards the more complex monolayers and inserting a section on surface rheology as soon as it is needed. In order to conUiin the treatment within reasonable limits, we shall mostly restrict ourselves to binary systems. Each section starts, where possible and appropriate, with the required (thermodynamic, statistical, electrostatic,. ..) background. Recall from sec. 3.1 that, at equilibrium, surface equations of state are the same as for Langmuir monolayers. 

In Figure 14.5b we show that a common trend of the tt dependence of for LMWE monolayers is that E increased with increasing tt up to the collapse point. This increase is a result of an increase in the interactions between the monolayer molecules, as deduced from monolayer reflectivity. However, for the more condensed monolayer (saturated-LMWE), this increase is higher than for the more expanded imsaturated-LMWE monolayer. In summary, I-TT and -tt curves (Figure 14.5) could reflect the surface equation of state... 

Surface Equation of State for Very Dilute Charged Monolayers at Aqueous Interfaces... 

Using surface pressure (u)-area (A) isotherms for sodium octadecyl sulfate (C18 sulfate) and octadecyl trimethylam-monium bromide (C18 TAB) monolayers spread at the air/water (A/W) and n-heptane/water (O/W) interfaces, we show that the Davies-Guastalla and Davies surface equations of state are not followed over a wide range of temperature and salt concentration. We attribute the discrepancies to Davies incorrect estimate of the chain-cohesion term at the A/W interface, the inadequacy of the electrical contribution to n at all but the highest A studied, and a possible chain-cohesion term at the O/W interface. 

However, there are conceptual differences between the surface equation of state and the adsorption isotherm, so that the surface equation of state is only concerned with the lateral motions of the monolayer molecules and their lateral cohesive and adhesive interactions with the solvent molecules present in the monolayer, whereas an adsorption isotherm is also concerned with the interactions normal to the surface, between the monolayer molecules (as adsorbate) and solvent molecules (as adsorbent). 

For ideal point molecules, the surface equation of state is as given in Equation (435) and can also be written as... 

Equation (608) is exactly equal to Equation (433), given in Section 5.5.3 for the two-dimensional perfect gas for liquid solution surfaces. Equation (608) relates 7Tto the surface excess and is called the surface equation of state. Similarly to Equation (436), we can write [ Amoiecuie = kT] for gas-solid adsorption, where A oWe is the area available per adsorbate molecule in the monolayer, and k is the Boltzmann constant (R = kNA). The adsorption isotherm given by Equations (607) and (608) corresponds to the so-called Henry s law limit, in analogy with the Henry s law equations that describe the vapor pressures of dilute solutions. Equation (606) predicts a linear relation between m (or fractional surface coverage, 0f, and adsorbate gas pressure, P2, as shown in the linear plot in Figure 8.1. 

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