Rectilinear diameters, law

Randall, M. 1. 264. 265 Raoult s law 268-73 and phase equilbria 419. 423 standard state 289 Rectilinear diameters, law of 393 Redlich-Kister equation 220, 362 Redlick-K wong equation 256 relative apparent molar enthalpy of solutions 356-7... 

FIG. 10 The coexistence curve for a two-dimensional Lennard-Jones fluid, showing the estimated critical point as the filled circle ( ). The points are simulation results, and the solid lines are fits to the 2D scaling law and the rectilinear diameter law the dashed line is from an earlier equation of state. (From Ref, 75a.)... 

As in tire one-fluid case, the experimental sums are in good agreement with the law of the rectilinear diameter, but the experimental differences fail to give a parabolic shape to tlie coexistence curve. 

The average densities of the liquid and gas will plot as a straight line which passes through the critical point. This property is known as the Law of Rectilinear Diameters. The dashed line on Figure 2-11 shows these average densities. 

At its critical point, the liquid and vapor densities of a fluid become equal. Since this point is very difficult to determine directly, an extrapolation procedure is often used. This procedure is based on the law of rectilinear diameters, which states that... 

Using the resulting pairs of density values, construct a Cailletet-Mathias plot (liquid and vapor densities versus temperature). From this plot, obtain the critical temperature T, test the law of rectilinear diameters [see Eq. (1)], and obtain the critical density p. ... 

Walden found Mendel flf s formula approximately valid for normal liquids between 50° and 125°. Since the density of the vapour is negligible in this interval, the formula is in agreement with Mathias s law of rectilinear diameter < 5.VII B) since the density at 0° is approximately three times the critical density, this is a consequence of the law of corresponding states ( 16.VIIC). For liquefied gases, Walden reduced the density to that at absolute 2 ro, which according to Guldljerg ( lO.VII B) is four times the critical density ... 

If 1 jvg is neglected in comparison with 1 jvi, the law of rectilinear diameter of Cailletet and Mathias ( 5.VIIB) gives llvi=A—BTy where A and B are constants, and r=abs. temp. Hence, from Bakker s equation ( 4.VIIIL), l. =alvi, it foUows that l —a A—BT), where a is van der Waals s constant. From Stefan s equation ( 6.VIII G),... 

It has been found that, if the average density of the liquid and vapor are plotted as a function of temperature, a straight line results. This is known as the Law of Rectilinear Diameters and may be stated mathematically as... 

If the small terms in and higher are ignored, equation (A2.5.41 is the law of the rectilinear diameter as evidenced by the straight line that extends to the critical point in figure A2.5.10 this prediction is in good qualitative agreement with most experiments. However, equation (A2.5.5T which predicts a parabolic shape for the top of the coexistence curve, is unsatisfactory as we shall see in subsequent sections. 

The basic Equation 3 becomes explicit as soon as the parameters P(PT) and A(Pr) are specified. This specification is the result of a lengthy trial procedure, based on the following factors (a) consistency with known near-critical power laws, (b) approximate consistency with the law of the rectilinear diameter (c) the tendency of the low-pressure vapor curve to form a straight line in logarithmic coordinates, as predicted by Equation 5 (d) imposition of a definite form of the corresponding-states principle and (e) consistency with a large collection of experimental data. 

An experimental test of the Guggenheim form (Equation 20) is shown in Figure 4, where the numerical value of (x) required to exactly reproduce the power law (omitting the linear, rectilinear diameter term) is compared with experimental data for ethylene. 

Kudchadker et al. have pointed out that experimental methods which enable visual observation of the critical temperature, the determination of the critical density by the law of rectilinear diameters (see below), and subsequently the direct determination of the corresponding critical pressure, should be preferred over other methods. 

Critical Volume of Pure Components.—The critical volume is the most difficult of the three critical constants to measure and consequently critical volumes are rarely measured to an accuracy of better than 0.5 per cent. The best method is to measure orthobaric liquid and gas densities up to within a kelvin or so of the critical temperature. The critical density is obtained using the law of rectilinear diameters, i.e. by extrapolating the mean of these densities to the critical temperature. This technique was originally proposed by Cailletet and Mathias and has been investigated carefully by Schneider and co-workers and discussed in detail by Rowlinson. ... 

Because the law of rectilinear diameters does not hold for mixtures, the critical volumes of mixtures are very difficult to determine accurately. However, if the maxcondentherm and maxcondenbar are fairly similar (not more than 1 K different in their temperatures) use of the law appears to lead to reasonable values of the critical volume. The law cannot be recast by substituting pressure in place of temperature as suggested by Kaminiski and Toriumi. The only reliable method of obtaining the critical volume of a mixture is to extrapolate the bubble-point volume and dew-point volume to the critical point. 

Does the lack of complete symmetry in the binary mixture lead to a failure of the law of the rectilinear diameter in the critical region ... 

Quite the opposite situation applies when we examine the observed (1 - a) in an attempt to test the validity of the law of the rectilinear diameter.f According to equation (34), unless g is exactly zero, the limiting value of (1 — a) as / approaches zero is not (1 - a), but rather 2jS, since 2j8 < (1 — a). (This spurious 2j8 exponent arises quite generally whenever any variable without the correct symmetry properties is chosen it does not depend on the particular transformation of equation (29). The effect appears to have been pointed out explicitly first by Buckingham for the transformation from p, p or T, p to r, Vja co-ordinates for the lattice-gas coexistence curve.)... 

Fig. 1 Vapor-liquid coexistence densities for pentadecanoic acid monolayers. The results of the isotherm measurements of Kim and Cannell [9] and Pallas and Pethica [11] and the fluorescence data of Moore et al. [15] are shown as open triangles, squares, and diamonds, respectively. The simulation data are depicted by open and filled circles for the original Karaborni and Toxvaerd model [29] and an improved force field (this work). The solid and dashed lines are fits of the simulation data using the scaling law and the law of rectilinear diameters...

The critical temperature of a substance can be measured quite accurately by observing the appearance or disappearance of a liquid-gas meniscus, and the critical pressure can be measured at this temperature with a high-pressure manometer. To evaluate the density at the critical point, it is best to extrapolate tbe mean density of the coexisting liquid and gas phases, (p + p )/2, to the critical temperature as illustrated in Fig. 8.8 on page 207. The observation that the mean density closely approximates a linear function of temperature, as shown in the figure, is known as the law of rectilinear diameters, or the law of Cailletet and Matthias. This law is an approximation, as can be seen by the small deviation of tbe mean density of SFe from a linear relation very close to the critical point in Fig. 8.8(b). This failure of the law of rectilinear diameters is predicted by recent theoretical treatments. ... 

Thus we must view with caution critical data derived from experiments that have not actually approached the immediate critical region. This includes much of the information in the lower pxjrtion of Table 1.1. Note that even some of the data for the lighter alkali metals lithium and sodium involve extrapolation from experiment. Extrapolation of experimental results using rules and methods obeyed for insulating liquids, such as the law of corresponding states or the law of rectilinear diameters, may lead to conclusions that are seriously in error. The reasons for the failure of these empirical rules in electronically conducting fluids forms a central theme of this book. 

The coexistence curves of cesium, rubidium, and mercury violate a century-old empirical rule known as the Law of Rectilinear Diameters (Cailletet and Mathias, 1886). According to this rule, a plot of the diameter, versus T, should be linear right up to the critical point. In contrast, renormalization group theory predicts that the temperature derivative of the diameter, dp /dT, should diverge at least as fast as the constant-volume specific heat c . Specifically, as the reduced temperature T = (T — T)/Tc goes to zero, the diameter varies as... 

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