The Two-Dimensional Ideal-Gas Law

For dilute solutions, solute-solute interactions are unimportant (i.e., Henry s law will hold), and the variation of surface tension with concentration will be linear (at least for nonelectrolytes). Thus 

A sample calculation shows how Fig. HI-15c is computed from the data of Fig. HI-15a. Equation III-97 may be put in the form 

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Equation (23) obviously gives the two-dimensional ideal gas law when a > a2 and with the o2 term included represents part of the correction included in Equation (15). This model for surfaces is, of course, no more successful than the one-component gas model used in the kinetic approach however, it does call attention to the role of the substrate as part of the entire picture of monolayers. We saw in Chapter 3 that solution nonideality may also be considered in osmotic equilibrium. Pursuing this approach still further results in the concept of phase separation to form two immiscible surface solutions, which returns us to the phase transitions described above. 

We saw in connection with the discussion of Figure 9.5 that measurable gas adsorption occurs even at gas pressures as low as 10 10 torr. As a matter of fact, the two-dimensional density of the adsorbed molecules is not low enough to conform to the two-dimensional ideal gas law even when the pressure is on the order of 10 10 torr. A question of considerable practical importance, then, is how low the pressure must be for an initially clean surface to remain that way for a reasonable period of time. The above reference to adsorption cites equilibrium data that are not useful for answering questions of rate. 

We saw in Chapter 7 that charged monolayers are likely to obey the two-dimensional ideal gas law, and we also saw that areas per molecule of 10 nm2 or so were also required for this ideal law to apply. Hence we may estimate o for a monovalent ion to be... 

As the barrier moves, the area available to the amphiphilic molecules decreases. This automatically causes the average area per molecule as well as the average distance between the molecules to decrease. Initially, it does not affect the interaction between the molecules (or domains), since they are still far away from each other. In this situation, the pressure is inversely proportional to the area per molecule, according to the two-dimensional ideal gas law. Eventually, however, the intermolecular distances are short enough that the intermolecular interactions can no longer be neglected, and the film... 

Gaseous G-films, which to a good approximation follow either the two-dimensional ideal gas law (II. 16), or the equation describing gas with molecules of finite dimensions (Fig. 11-21). For instance, such films are formed by fatty acids at low two-dimensional pressures or sufficiently high temperatures. Sometimes vapor-type films which exist at temperatures below the condensation point of the adsorption layer are singled out from this class (see below). 

Show that, in this case, the surface pressure ji is given by the so-caUed two-dimensional ideal gas law... 

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 implicit assumption behind the H-K equation is that the adsorbed phase is a two-dimensional ideal gas, i.e., Henry s law region adsorption. Cheng and Yang [112] modified the H-K equation by including the nonlinearity of the isotherm. The modified H-K equation is... 

Assiuning the adsorbed phase to be a two-dimensional ideal gas and assuming the adsorption occurs in the Henry s law region (linear isotherm), the following equation of state can be substituted in Eq. (29) ... 

One of the assumptions made by Horvath and Kawazoe [35] in their original derivation was that the adsorbate behaved as a two-dimensional ideal gas. This implies that the isotherm obeys Henry s law and is therefore linear in nature. The equation of state shown in Eq. (30) was thus substituted in Eq. (29), which caused the term T/d) dli/dT) = RT to cancel out with the other RT term in the expression, resulting in Eq. (31). However, the isotherms for the typical sorbates used in HK analysis, such as N2 at 77 K and Ar at 87 K, clearly show a type 1 adsorption behavior. An example of such an isotherm is shown in Fig. 10. As is obvious from the figure, the assumption of linearity is only valid for the steeply rising portion of the isotherm, whereas the concave portion of the steep rise may also provide useful information. For this reason, Cheng and Yang [40] proposed the use of a Langmuir-type equation of state in place of Eq. (30) because it is known to represent type 1 isotherms in the best manner ... 

The slope is -b as obtained from Equation 4.3.1. Equation 4.3.2 is often called the ideal gas law in surface science and is, in fact, the equivalent in two dimensions of the well-known (three-dimensional) ideal gas law (PV = nRigT). 

In the above example, V, was considered to be a function of only a single variable t Such an equation, V, =/(f), can be represented by a series of points on a two-dimensional Cartesian coordinate system. Physicochemical systems, however, usually depend on more than one variable. Thus, it is necessary to extend the definition of function given above to include functions of more than one variable. For example, we find experimentally that the volume of a gas will vary with temperature according to Equation (2-2) only if the pressure of the gas is held constant. Thus, the volume of a gas is not only a function of temperature, but also is a function of pressure. Careful measurements in the laboratory will show that for most gases at or around room temperature and one atmosphere pressure the law relating the volume of a gas simultaneously to the temperature and the pressure of the gas is the well-known ideal gas law... 

The film balance may be regarded as a two-dimensional piston, and the most commonly studied property is the surface pressure (n) versus area (A) isotherm. The analogy to a PV isotherm is so appropriate that in the gaseous monolayer regime the two-dimensional analogue of the ideal gas law pertains 114 = nRT. It is therefore reasonable to relate discontinuities in n/A isotherms as the monolayer film is compressed in two dimensions to... 

As evident from the above discussion, if measurements can be made at sufficiently low pressures, all monolayers will display gaseous behavior, represented by region G in Figure 7.6. The gaseous region is characterized by an asymptotic limit as n - 0. In the limit of very low film pressures, a two-dimensional equivalent to the ideal gas law applies ... 

An alternative way of looking at monolayers is to consider them as two-dimensional binary solutions rather than two-dimensional phases of a single component. The advantage of this approach is that it does acknowledge the presence of the substrate and the fact that it plays a role in the overall properties of the monolayer. Although quite an extensive body of thermodynamics applied to two-dimensional solutions has been developed, we consider only one aspect of this. We examine the film pressure as the two-dimensional equivalent of osmotic pressure. It will be recalled that, at least for low osmotic pressures, the relationship among uosm, V, n, and Tis identical to the ideal gas law (Equation (3.25)). Perhaps the interpretation of film pressure in these terms is not too farfetched after all ... 

For some purposes, a graphical presentation of information is preferable to an equation. As shown in Eq. (6), the ideal gas law is a relation among three variables. In order to represent this equation in a two-dimensional plot, one variable must be held constant. The plots are called isotherms, isobars, or isochores, depending on whether temperature, pressure, or volume, respectively, is held constant. The plots for an ideal gas are shown in Fig. 1. 

Molecules at low density on a surface, such as surfactants at an interface of air and water, often obey a two-dimensional equivalent of the ideal gas law. The two-dimensional equivalent of p is tt, where rr is a lateral two-dimensional pressure. A is area. Using... 

This remarkable Langmuir s relationship is, in fact, the equation of the state of a rarified (ideal) two-dimensional gas consisting of adsorbed molecules. This equation is analogous to a well-known gas law describing a conventional three-dimensional ideal gas. 

Note the resemblance to the ideal gas law Note also how the dotted line in Figure 22.11 mimics the hyperbolic curve of an inverse relationship between tt and A, that is, Boyle s law. In these regions, each molecule can wiggle around independently of the others, and can be modeled as a sort of two-dimensional gas. 

Of interest is the derivation of the Freundlich isotherm with Vp= from the ideal two-dimensional surface gas. Assuming a surface equation of state similar to the ideal gas law, using n in place of P and in place of V, one has... 

A two-dimensional graph can represent a function of one independent variable. You plot the value of the independent variable on the horizontal axis and represent the value of the dependent variable by the height of a curve in the graph. To make a two-dimensional graph that represents the ideal gas law, we must keep two of the three independent variables fixed. Figure 1.1a shows a set of graphical curves that represent the dependence of P on V for an ideal gas for n = 1.000 mol and for several fixed values of T. 

The deviation of Gibbs monolayers from the ideal two-dimensional gas law may be treated by plotting xA// 7 versus x, as shown in Fig. III-15c. Here, for a series of straight-chain alcohols, one finds deviations from ideality increasing with increasing film pressure at low x values, however, the limiting value of unity for irAfRT is approached. 

It is assumed that the molecules adsorbed on the electrode surface behave as a two-dimensional gas that can be described by an equation of state. For ideal noninteracting molecules, Henry s law, Pa = R 70, is fulfilled. When the mutual interaction between adsorbed molecules is taken into account, the equation with the virial coefficient B can be presented in the form ... 

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