A,-plot

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

The low-pressure extrapolated data points were generated by linear extrapolation of the lowest 4-6 points on a plot of... 

Figure 4.9 shows a plot of Eq. (4.12). As the purge fraction a is increased, the flow rate of purge increases, but the concentration of methane in the purge and recycle decreases. This variation (along with reactor conversion) is an important degree of freedom in the optimization of reaction and separation systems, as we shall see later. 

Rgure 8.2 A plot of economic potential versus conversion shows where the optimum is, but why does the optimum occur where it is ... 

Now using a hydrocarbon component, say ethane, as an example, let us consider the other parameter, volume, using a plot of pressure versus specific volume (i.e. volume per unit mass of the component, the inverse of the density). The process to be described could be performed physically by placing the liquid sample into a closed cell (PVT cell), and then reducing the pressure of the sample by withdrawing the piston of the cell and increasing the volume contained by the sample. 

The primary drive mechanism for gas field production is the expansion of the gas contained in the reservoir. Relative to oil reservoirs, the material balance calculations for gas reservoirs is rather simple the recovery factor is linked to the drop in reservoir pressure in an almost linear manner. The non-linearity is due to the changing z-factor (introduced in Section 5.2.4) as the pressure drops. A plot of (P/ z) against the recovery factor is linear if aquifer influx and pore compaction are negligible. The material balance may therefore be represented by the following plot (often called the P over z plot). 

Routine production tests are performed, approximately once per month on each producing well, by diverting the production through the test separator on surface to measure the liquid flowrate, water cut, and gas production rate. The wellhead pressure (also called the flowing tubing head pressure, FTHP) is recorded at the time of the production test, and a plot of production rate against FTHP is made. The FTHP is also recorded continuously and used to estimate the well s production rate on a daily basis by reference to the FTHP vs production rate plot for the well. 

Fig. Ill-13. (a) Plots of molecular density versus distance normal to the interface a is molecular diameter. Upper plot a dielectric liquid. Lower plot as calculated for liquid mercury. (From Ref. 122.) (b) Equilibrium density profiles for atoms A and B in a rare-gas-like mixmre for which o,bb/ o,aa = 0.4 and q,ab is given by Eq. III-56. Atoms A and B have the same a (of Eq. m-46) and the same molecular weight of SO g/mol the solution mole fraction is jcb = 0.047. Note the strong adsorption of B at the interface. [Reprinted with permission from D. J. Lee, M. M. Telo de Gama, and K. E. Gubbins, J. Phys. Chem., 89, 1514 (1985) (Ref. 88). Copyright 1985, American Chemical Society.]...

Figure IV-10 illustrates how F may vary with film pressure in a very complicated way although the v-a plots are relatively unstructured. The results correlated more with variations in film elasticity than with its viscosity and were explained qualitatively in terms of successive film structures with varying degrees of hydrogen bonding to the water substrate and varying degrees of structural regularity. Note the sensitivity of k to frequency a detailed study of the dispersion of k should give information about the characteristic relaxation times of various film structures.

On compression, a gaseous phase may condense to a liquid-expanded, L phase via a first-order transition. This transition is difficult to study experimentally because of the small film pressures involved and the need to avoid any impurities [76,193]. There is ample evidence that the transition is clearly first-order there are discontinuities in v-a plots, a latent heat of vaporization associated with the transition and two coexisting phases can be seen. Also, fluctuations in the surface potential [194] in the two phase region indicate two-phase coexistence. The general situation is reminiscent of three-dimensional vapor-liquid condensation and can be treated by the two-dimensional van der Waals equation (Eq. Ill-104) [195] or statistical mechanical models [191]. 

At lower temperatures a gaseous film may compress indefinitely to a liquid-condensed phase without a discemable discontinuity in the v-a plot. 

Fig. IV-20. Film pressure-area plots for cerebronic acid (a long-chain a-hydroxy carboxylic acid) and cholesterol (see insert) and for an equimolar mixture. At low pressures the r-a plot is close to that of the average (dashed line), an unanticipated kink then appears, and finally, the horizontal portion probably represents ejection of the cholesterol. (From Ref. 239.)...

A plot of G x versus composition is shown in Fig. IV-22 for condensed films of octadecanol with docosyl sulfate. Gaines [241] and Cadenhead and Demchak [242] have extended the above approach, and the subject has been extended and reviewed by Barnes and co-workers (see Ref. 243). 

Barnes and Hunter [290] have measured the evaporation resistance across octadecanol monolayers as a function of temperature to test the appropriateness of several models. The experimental results agreed with three theories the energy barrier theory, the density fluctuation theory, and the accessible area theory. A plot of the resistance times the square root of the temperature against the area per molecule should collapse the data for all temperatures and pressures as shown in Fig. IV-25. A similar temperature study on octadecylurea monolayers showed agreement with only the accessible area model [291]. 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

Thus a plot of Ci/n versus Ci should give a straight line of slope 1/n and intercept /n b. 

The CMC for sodium dodecylbenzenesulfonate is about lO Af at 25°C. Calculate K for the preceding reaction, assuming that it is the only process that occurs in micelle formation. Calculate enough points to make your own quantitative plot corresponding to Fig. Xni-13. Include in your graph a plot of (Na )(R ). Note It is worthwhile to invest the time for a little reflection on how to proceed before launching into your calculation ... 

A surfactant solution is a mixture of DTAC (dodecyltrimethylammonium chloride) and CPC (cetyl pyridinium chloride) the respective CMCs of the pure surfactants are 2 X lO M and 9 x IO M (Ref. 140). Make a plot of the CMC for mixtures of these surfactants versus the mole fraction of DTAC. 

Emulsion A has a droplet size distribution that obeys the ordinary Gaussian error curve. The most probable droplet size is 5 iim. Make a plot of p/p(max), where p(max) is the maximum probability, versus size if the width at p/p(max) = j corresponds to... 

Fig. XV-7. Fluorescence micrographs showing morphology of crystalline L-a-dimyris-tolphosphatidylethanolamine domains following a t jump to the plateau region of the v-a plot (a) after 2 sec b) after 1 min (c) after 20 min (d) following a second pressure jump after condition (c). (From Ref. 40.)...

While the v-a plots for ionized monolayers often show no distinguishing features, it is entirely possible for such to be present and, in fact, for actual phase transitions to be observed. This was the case for films of poly(4-vinylpyri-dinium) bromide at the air-aqueous electrolyte interface [118]. In addition, electrostatic interactions play a large role in the stabilization of solid-supported lipid monolayers [119] as well as in the interactions between bilayers [120]. 

A plot of P/n versus P should give a straight line, and the two constants and b may be evaluated from the slope and intercept. In turn, n may be related to the area of the solid ... 

The characteristic isotherm concept was elaborated by de Boer and coworkers [90]. By accepting a reference from a BET fit to a standard system and assuming a density for the adsorbed film, one may convert n/rim to film thickness t. The characteristic isotherm for a given adsorbate may then be plotted as t versus P/P. For any new system, one reads t from the standard r-curve and n from the new isotherm, for various P/P values. De Boer and co-work-ers t values are given in Table XVII-4. A plot of t versus n should be linear if the experimental isotherm has the same shape as the reference characteristic isotherm, and the slope gives E ... 

Make a plot of Eq. XVII-69 as 6 versus P, and, for comparison, one of a Langmuir adsorption isotherm of same limiting or Henry s law slope. Comment on the comparison. 

Convert the upper curve of Fig. XVII-31a to an adsorption isotherm, that is, to a plot of V versus P/I. ... 

Show whether a plot of d versus log t should be linear (a form of the Elovich equation). 

If Langmuir adsorption occurs, then a plot of 9 versus p for a particular isothenn will display the fonn of equation (Al.7.3). Measurements of isothenns are routinely employed in this manner in order to detennine adsorption kinetics. 

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