Rectilinear diameter

If the small temis in p- and higher are ignored, equation (A2.5.4) is the Taw 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 (A2.5.5). which predicts a parabolic shape for the top of the coexistence curve, is unsatisfactory as we shall see in subsequent sections. 

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. 

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... 

Figure 3.3 Illustration of the calculation of the phase diagram of a mixed biopolymer solution from the experimentally determined osmotic second virial coefficients. The phase diagram of the ternary system glycinin + pectinate + water (pH = 8.0, 0.3 mol/dm3 NaCl, 0.01 mol/dm3 mercaptoethanol, 25 °C) —, experimental binodal curve —, calculated spinodal curve O, experimental critical point A, calculated critical point O—O, binodal tielines AD, rectilinear diameter,, the threshold of phase separation (defined as the point on the binodal curve corresponding to minimal total concentration of biopolymer components). Reproduced from Semenova et al. (1990) with permission.

The analogy between gas-liquid and metal-hydrogen systems has been mentioned elsewhere in this chapter, and therefore the rule of rectilinear diameter, that holds for gas-liquid systems, might apply to metal-hydrogen systems. This rule suggests that the following relations should hold at any temperature. 

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. 

Zcoii/y is the sum of two Gaussians, the first centered at y = 0 to ensure the correct behavior in the limit of zero density, and the second centered at the rectilinear diameter yT to provide the correct value of residual Helmholtz energy at high density. The temperature dependence of yu is given universal [2], The parameters b and c, which determine the second and third virial coefficients, are given universally for non-polar substances [3]. The pre-exponential factor w has a dominating influence on the vapor pressure. Its temperature dependence requires two substance-specific correction parameters wi and w2. 

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. ... 

Nernst, Theoretische Chemie, 7 Aufl., 1913, 234 from rectilinear diameter. 

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... 

Figure 9. The coexistence densities of the full (solid lines and filled symbols) and short-range models of polar and associating fluids. The straight lines correspond to the rectilinear diameter rule.

Certainly the most important models for the development of modem scaling theory of critical phenomena have been the discrete Ising model of ferromagnetism and its antipode - the continuum van der Waals model of fluid. The widespread belief is that real fluids and the lattice-gas 3D-model belong to the same universality class but the absence of any particle-hole-type symmetry in fluids requires the revised scaling EOS. The mixed variables were introduced to modify the original Widom EOS and account the possible singularity of the rectilinear diameter. 

Figure 1. Comparison of the reduced CXC-densities pi,g/pc for real substances Ar C2H, (m/u), CO2 ( /<>), H2O c/o) with the van der Waals-Maxwell-Gibbs model s predictions ) based on the disorder parameter x the respective rectilinear diameter (pi+pg)/2pc as a function of x (—).

One may note from eq. (3) that the rectilinear diameter pa and its derivative dpa/dT can be explicitly represented by equations in terms of As. ... 

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