Pressure two-dimensional

The surface viscosity can be measured in a manner entirely analogous to the Poiseuille method for liquids, by determining the rate of flow of a film through a narrow canal under a two-dimensional pressure difference Ay. The apparatus is illustrated schematically in Fig. IV-7, and the corresponding equation for calculating rj is analogous to the Poiseuille equation [99,100]... 

The influence of two-dimensional pressure on the tangential permeability of the layers for various cations was examined. The current in the channel formed by the monomolecular film and the substrate is carried by hydrated cations. The permeability of the channel to various cations diminishes in the order... 

In the case of soluble substances of low capillary activity the evaluation of = surface tension between that of the liquid without the Gibbs film and one in which the Gibbs film is established only as a first approximation may the ideal dynamic surface tension of a solution, i.e. of a solution without a Gibbs film, be regarded as equal to the surface tension of the solvent. 

The apparatus shown in Figure 6.2a resembles a two-dimensional cylinder/piston arrangement. With this similarity in mind, the suggestion that surface tension is analogous to a two-dimensional pressure seems plausible. With certain refinements, this notion will prove very useful in Chapter 7. 

The surface layer does not have zero thickness, of course, even though it is conceptually convenient to think of it as two-dimensional matter. If we assume that the film pressure ir extends over the entire thickness of the film, then it is an easy problem to convert the two-dimensional pressure to its three-dimensional equivalent. Taking 10 mN m 1 as a typical value for it and 1.0 nm as a typical value for t enables us to write... 

FIG. 7.6 Composite two-dimensional pressure ir versus area a isotherm, which includes a wide assortment of monolayer phenomena. Note that the scale of the figure is not uniform so that all features may be included on one set of coordinates. The sketches of the surfactants show the orientations of the molecules in each phase at various stages of compression. 

This force is converted to two-dimensional pressure by dividing it by the length of the edge to which the force is applied. Assuming the accessible surface area to be a square means that the length of this edge is also f the pressure contribution of this one collision equals... 

In the two-dimensional gaseous state, surface viscosities can be as low as 10 8 kg s l, while for condensed states values range from 10 "7 to 10 5 kg s While it makes sense that the surface viscosity increases as we move from G to L to S states, the numbers themselves mean little to us. To remedy this we repeat the sort of calculation done for two-dimensional pressure and examine what Equation (25) tells us about the equivalent bulk viscosity. As with Equation (3), we assume r = 1.0 nm therefore a surface viscosity of 10 7 kg s 1 is equivalent to a bulk viscosity of... 

If the experimental isotherm (n/w as a function of p) is known, then Equation (7) may be integrated either analytically or graphically to give the two-dimensional pressure as a function of coverage. This relationship therefore establishes the connection between the two-and three-dimensional pressures that characterize the surface and bulk phases. This is how adsorption data could be used to determine the film pressure in equilibrium with a drop of bulk liquid on a solid surface as discussed in Section 6.6b. 

Assuming that only the radial velocity is nonzero, but retaining two-dimensional pressure variation, we can reduce the continuity and momentum equations to the following system ... 

Two-dimensional Pressure Distribution in Solids Filling a Rectangular Channel... 

The equilibrium thermodynamic functions describing the retention process are essentially related to the net retention volume. For example, the intermolecular adsorption of Gibbs free energy, —AGa, for one mole of solute vapor from a reference gaseous state with the partial pressure, Po, to a reference adsorbed state with the equilibrium spread pressure (or two-dimensional pressure), no, is given as [115]... 

In the case of adsorbed layers, by analogy with the two-dimensional pressure of spread films, a surface pressure (n) is introduced defined as the change of - interfacial tension (y) caused by the addition of a given species to a base solution. At constant cell potential (E) this is... 

The monolayer of amphiphile spread on water is then compressed into a well-packed state. The profile of monolayer compression is recorded as surface pressure-molecular area (n-A) isotherm. Typical examples of n-A isotherms are shown in Fig. 4.33. The transverse axis of the isotherm represents the molecular area, which can be obtained by dividing the total surface area by the number of amphiphile molecules. The surface pressure is derived by subtracting the surface tension of the monolayer-covered water surface from that of pure water. This has dimensions corresponding to two-dimensional pressure. Surface pressure cannot be regarded as a normal three-dimensional pressure. For convenience, when discussing the phase of two-dimensional monolayer here, the surface pressure is treated as a two-dimensional pressure. 

The phase behavior of multicomponent hydrocarbon systems in the liquid-vapor region is very similar to that of binary systems. However, it is obvious that two-dimensional pressure-composition and temperature-composition diagrams no longer suffice to describe the behavior of multicomponent systems. For a multicomponent system with a given overall composition, the characteristics of the P-T and P-V diagrams are very similar to those of a two-component system. For systems involving crude oils which usually contain appreciable amounts of relatively r on-volatile constituents, the dew points may occur at such low pressures that they are practically unattainable. This fact will modify the behavior of these systems to some extent. 

Fig. 26. Profiles of the density pjv (atOms/A ), two-dimensional pressure P (eV/A ), kinetic temperature T (K), and particle velocity in the direction of the shock detonation Vx (km/s), for a simulation begun with an impact plate velocity of 6.0 km/s. The results shown are at an elapsed time of 65 ps.

This relation shows how the two-dimensional pressure may be determined through measurements of F at a sequence of equilibrium gas pressures F at a fixed temperature, either by graphical integration, or analytically, by curve fitting procedures. 

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... 

FIGURE 10.2 Surface tension and surface pressure, (a) Measurement of surface or interfacial tension by means of a Wilhelmy plate (width L, thickness (5). The plate is attached to a sensitive balance, (b) Illustration of surface pressure (/7) caused by surfactant molecules (depicted by vertical dashes). Between the barriers the surface tension is lowered and a net two-dimensional pressure acts on the barriers. 

Fig. 11-5. Displacement of the barrier due to the action of two-dimensional pressure of the adsorption layers of insoluble surfactants...

If a pure solvent is present on one side of the barrier, and the solvent with the adsorption layer on the other, the forces acting on the barrier from each of these sides differ. Considerations, similar to those used in the description of the Dupre experiment (see Chapter 1,1), suggest that a force per unit barrier length, g0, is acting in the direction of the pure solvent, and a force o(T) < g0 is acting in the opposite direction along the surface covered with the adsorption layer. The net force per unit of barrier length is directed towards the pure solvent (Fig. II-5), and equals the difference in the surface tension between the surface of pure solvent and that surface covered with adsorption layer. This force, equal to g0 - a (T), is referred to as the two-dimensional pressure, n, of the adsorption layer ... 

This principle was utilized by Langmuir in his instrument designed to measure two-dimensional pressure, i.e. to obtain the 7r(T) dependence. 

Equation (II.8) is also valid for soluble surfactants as well. However, in the latter case the value of n can not be directly measured using Langmuir s method, but can be established from surface tension measurements as the drop in the surface tension n - -Aa = o0 - a(c) (refer to Chapter II, 2 regarding the identity between n and -Aa). If the T(p) or T(c) dependence is known, the two-dimensional pressure can be obtained by integrating the Gibbs equation, i.e. ... 

When adsorption takes place at the surface of a highly porous solid adsorbent, the surface excess can be readily measured, e.g. by measuring the increase in the adsorbent weight in the case of adsorption from vapor, or by following the decrease in the adsorbate concentration during adsorption from solutions. Studies of the adsorption dependence on vapor pressure (or solution concentration) reveal T(p) (or T(c)) adsorption isotherms. In both cases the two-dimensional pressure isotherm can be established from the Gibbs equation (see Chapter II, 2, and Chapter VII, 4). Therefore, it is as a rule possible to establish the dependence between the two of three variables present in the Gibbs equation the surface tension isotherm, a(c), for mobile interfaces and soluble surfactants, the two-dimensional pressure, tt(c), isotherm for insoluble... 

Let us next consider the characteristic properties of interface and adsorption layers, comparing the behavior of water soluble surfactants to that of insoluble ones. We will gradually move from the simplest cases to more complex ones, revealing the nature of intermolecular interactions in the adsorption layers. In doing so, we will analyze the typical relationships that describe the properties of adsorption layers, namely the surface tension isotherm, a(c), the adsorption isotherm, F(c), the two-dimensional pressure isotherm, 7r(r), etc. 

Fig. II-6. The surface tension, a(c), and two-dimensional pressure, tt(.vm), isotherms of three surfactants, the consecutive members of a homologous series. The concentration range corresponds to the Henry region...

It is more correct to regard the two-dimensional pressure as analogous to the osmotic pressure. In surfactant solutions, in which T > 0 and c(s) > c, the osmotic pressure is greater at the surface than in the bulk. This causes pumping of water into the surface layer, and thus eases the formation of a new surface, which is in fact equivalent to the decrease in the surface tension. The opposite situation, where T < 0 and c(s) < c, is typical with surface inactive substances, when one can say that the energy required for a new surface formation (determined by a) increases due to the need to perform additional work against the osmotic pressure forces. This work is directed towards the displacement of electrolyte from the surface layer into the bulk. 

Equation (II. 16) is the state equation of the ideal two-dimensional gas, which is represented by the two-dimensional pressure isotherm, 7t(sm), shown in Fig. II-8. To analyze the behavior of real systems, and to study the origin of deviations of such systems from ideal two-dimensional state, the two-dimensional pressure isotherm is plotted in nsM - 7r coordinates. The isotherm of an ideal two-dimensional gas in these coordinates is a straight line, parallel to the x axis (Fig. II-9). If the two-dimensional pressure and area per molecule... 

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