Andrews’ curve

Above the critical temperature, the isotherms show two curvatures on the right, pv increases as p decreases, on the left, pv and p increase together. This change of curvature, suggesting vapour and liquid states respectively, is clearly seen in Andrews curves for 32° 5 and 35° 5. It is also observed in the isotherms of permanent gases, and will be discussed later. 

AMSP = Autocorrelation of Molecular Surface Properties autocorrelation descriptors > Andrews curves molecular descriptors (0 transformations of molecular descriptors)... 

Another interesting tool to obtain linear combinations of descriptors is that based on the Andrews curves [Andrews, 1972]. The Andrews curves are a pictorial way to represent and compare multivariate objects. In this representation of ap-dimensional space, each ith object is represented by a mathematical function as... 

Chernoff Faces, Andrews Curves, Star Diagrams, and Parallel Coordinate Plots... 

This is an intriguing idea, as it allows each original multivariate point to be recovered from the curve. Since the Andrews curve is dominated by the low-frequency terms, some arbitrary decision must be made regarding the relative importance of the original variables. 

Conditions (30) and (31) are sufficient to discuss the principal properties of the critical state of a one-component system. We observe that the existence of a critical state for such a system cannot be inferred from a j)riori considerations, because it is not necessary that the two branches of the connodal curve should ultimately coalesce that such is the case must be regarded as established for systems containing liquid and vapour by the experiments of Andrews ( 86), and the following discussion is limited to such systems (cf. 103). 

FIGURE 28.2 Standard oscillating disc curemeter curve. (Redrawn from Andrew, C., Introduction to Rubber Technology, Knovel e-book publishers, 1999.)... 

Kies141 made an extensive study of oxidimetric titrations of iodide with iodate according to Andrews in the presence of hydrochloric acid. The shape (in 1N HC1) of the two hyperbolic sections followed by a flat part of the curve is explained by the occurrence of two (nearly) reversible systems, I2C1 /I and IC12 /I2C1 , according to the titration reactions... 

Figure 5.5. Isotherms for CO2 taken from the work of Andrews [3]. The shaded area indicates the region of stability of a one-phase liquid system. Liquid and vapor exist together at equilibrium under the dashed curve. Above the critical isotherm, 31.1°C, no distinction exists between liquid and gas. Andrews suggested that the term vapor be used only to represent the region to the right of the dashed curve below the critical temperature. The dashed curve ABCD represents the van der Waals equation. (By permission, from J. R. Partington, An Advanced Treatise on Physical Chemistry, Vol. 1, Longman Group, Ltd., London, 1949,...

Figure 5. Calibration curve for 6% agarose, prepared with unlabeled polypeptides (open circles), plotted according to the method of Andrews. The log M is plotted vs. Kd. Super-imposed on the curve are data points obtained with fluorescein (filled triangles) and rhodamine (filled squares) labeled proteins.

There are a number of instances in which (with the aid of sensitive measurements) well-defined transitions between gaseous and coherent states are observed as the film is compressed. The tt-A curves show a marked resemblance to Andrews p-V curves for the three-dimensional condensation of vapours to liquids. The tt-A curve for myristic acid, given as an example, has been drawn schematically to accentuate its main features (Figure 4.26). Above 8 nm2 molecule-1 the film is gaseous and a liquid-expanded film is obtained on compression to 0.5 nm2 molecule-1. Fluctuating surface film potentials verify the heterogeneous, transitional nature of the surface between 0.5 nm2 molecule-1 and 8 nm2 molecule-1. 

The curves which are obtained by plotting pressure against volume at various constant temperatures are known as isotherms (isos = equal therm = heat). Andrews obtained isotherms of carbon dioxide at different temperatures which are shown in figure (7). 

As the area is diminished below some thousands of sq. A., where the molecules cover only a small fraction of the surface, the surface pressure rapidly becomes much smaller than that of a perfect gas, and in the four acids with the longest chains becomes constant over a considerable region. The curves are indeed a very faithful reproduction of Andrews s curves for the relation between pressure and volume, for carbon dioxide, at temperatures near the critical. The horizontal regions in the curves correspond to the vapour pressure of liquids, and indicate the presence of an equilibrium between two surface phases, the vapour film, and islands of liquid, coherent film. 

Andrew and Bersohn (2) have shown that for rigid methyl groups the intensity of the spectrum in the central part of the absorption curve is twice as high as that of the lateral parts—i.e.9... 

FIG. 13.65 Stress-strain behaviour of PMMA at various temperatures. Curve a shows rubberlike behaviour of a cross-linked thermoplastic above Tg. According to Andrews (1968). 

The P-V Isotherms of Carbon Dioxide The importance of critical temperature of a gas was first discovered by T. Andrews in his experiments on pressure-volume relationships (isotherms) of carbon dioxide gas at a series of temperatures. The isotherms of carbon dioxide determined by him at different temperatures is shown in the figure given above. Consider first the isothermal at the lowest temperature, viz., 13.1 C. The point A represents carbon dioxide in gaseous state occupying a certain volume under a certain pressure. On increasing the pressure, its volume diminishes as is indicated by the curve AB. At B which represents a pressure of 49.8 atm, liquefaction of the gas commences and thereafter a rapid decrease in volume takes place at the same pressure, as more and more of the gas is converted into the liquid state. At C, the gas has been completely liquefied. Now, as the liquid is only slightly compressible, further increase of pressure produces only a very small decrease in volume. A steep line CD that is almost vertical shows this. 

The isotherm EFGH at 21.5 C shows a similar behaviour except that now the liquefication commences at a higher pressure and the horizontal portion FG, representing decrease in volume, becomes smaller. At still higher temperatures, the horizontal portion of the curve becomes shorter and shorter until at 31.1 C it reduces just to a point (represented by X). At this temperature, therefore, the gas passes into liquid state imperceptibly. Above 31.1 C, the isotherm is continuous. There is no evidence of liquefaction at all. Andrews concluded that if the temperature of carbon dioxide is above 31.1 C, it cannot be- liquefied, no matter how high the pressure may be. He called 31.1 C as the critical temperature of carbon dioxide. Since then, other gases have been... 

It may be pointed out that the isotherms plotted in the figure given above are based on theoretical calcula tions of Vcorresponding to different values of P obtained by using the van der Waals equation. The isotherms for carbon dioxide, obtained by Andrews experimentally, were in close resemblance with these curves, with the difference that the wavelike portion LMNOQ was replaced by a horizontal line. Since then more careful experiments have shown that small portions corresponding to curves LM and OQ can be realised in practice also. These represent supersaturated vapour and superheated liquid, respectively. 

Fracture energy master curves were determind as a function of nearly equivalent ranges of reduced crack velocity (King and Andrews), and extension rate (Swetlin). In both cases, T was used as the reference temperature. King and Andrews master curves were obtained using the WLF Equation and the universal constants, while Swetlin s master curves were determined via numerical best-fit shifting. 

We have already seen that there are three chief types of freezing-point curves in systems of two components, viz. those obtained when (i) the pure components crystallise out from the molten mass (2) the components form one or more compounds (3) the components form solid solutions. In the case of the metals, representatives of these three classes are also found. A mathematical analysis of the freezing-point curves on the basis of the formulae for ideal solutions (p. 104) has been carried out by Andrews and Johnston. ... 

The treatment of temperature-heat content curves from a theoretical viewpoint has been carried out by a number of investigators, starting with White (6) in 1920. Other early papers on the subject are by Andrews et al. (7), Skau (8), Mair et al. (9), Glasgow et al. (10), Malotaux and Straub (11), and Thomas and Parks (12). More recent treatments have been given by Rossini (13), Mastrangelo and Dorntei 14). Badley (15), and Smit (1). The reader is referred to these references for a more comprehensive presentation than that given here. 

If we take the experimental curves obtained by Andrews for the relation between the pressure, p, and volume, v, of carbon dioxide at different temperatures T, t0, tv r . .. we get a set of curves resembling Fig. 90. At any temperature T above the... 

Average curve from results of AUuard Miczynski — Monatsh. Ch. 58, 86 Henry — Compt rend. 99 1X57 84 Lamouroux —/Wd. 1281 998, 99 at as roote and Andrew —... 

The intensity of the morphine abstinence syndrome is related to the dosage of morphine required to prevent signs of abstinence from appearing. Himmelsbach and Andrews found that, if the stabilization dose were known, the intensity of abstinence in a given individual could be reasonably accurately predicted. The curve relating abstinence syndrome intensity... 

In other words, just as was shown by Andrews et al. (1948) for unfilled polymers, the ratio of two relaxation moduli for a given system at a common time but at different strains was found to be essentially independent of time, at least within a certain range of time and strain. Thus stress relaxation curves for a filled polymer at different strains 8 could be shifted along the modulus axis to give a master curve at a reference strain (Table 12.1). The corresponding modulus function becomes... 

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