Ideal reduced volume

A modified law of corresponding states suitable for real gases was proposed in 1946 by Gouq-Jen Su of Univ of Peiping, China (Ref 1). A term called the "ideal critical volume was defined and the ratio of volume over the ideal critical volume was called the "ideal reduced volume . It was shown that for 17 gases within the temperature and pressure ranges studied, the over-all deviation was only 1%. The value of the critical ratio was not a restrictioh or a criterion for the applicability of the modified law... 

Calculate the ideal critical volume and ideal reduced volume, and be able to use V - as a parameter in the Nelson and Obert charts. 

Unlike the pressure where p = 0 has physical meaning, the zero of free energy is arbitrary, so, instead of the ideal gas volume, we can use as a reference the molar volume of the real fluid at its critical point. A reduced Helmlioltz free energy in tenns of the reduced variables and F can be obtained by replacing a and b by their values m tenns of the critical constants... 

The reduction of dimensions also reduces volumes which are accessible to the detector. Thus, detection principles related to geometric dimensions of the detector cell ai e not ideally suited for coupling to microsystems, whereas surface sensitive principles, such as electrochemical methods or optical methods utilizing the evanescent field of a waveguide, or methods which can be focussed on a small amount of liquid, such as electrochemiluminescence (ECE), ai e better suited. This is why electrochemiluminescence detectors ai e combined to microsystems. Moreover ECE has found wide applications in biochemistry because of its high sensitivity, relatively simplicity and feasibility under mild conditions. 

This parameter is equal to one when the perfect gas law applies. When the pressure is increased to 10 atm, the predicted perfect gas volume is reduced to 2.24 L and the average distance between the molecules shrinks to 15.5 A when the pressure is increased to 100 atm, the predicted volume is reduced to 0.224 L and the average distance shrinks further to 7.2 A when the pressure is further increased to 1000 atm, the volume predicted by the perfect gas law is 0.0224 L and the average distance shrinks further to 3.3 A. By then, the intermolecular distance is roughly equal to the molecular diameters. As the molecules come together, the attractive forces are first felt, which manifests itself by a drop from the ideal gas volume, which is represented by Z < 1. With further compression, the molecules begin to touch, and the repulsive forces become dominant, which manifests itself as a resistance to further volume reduction and Z > 1. 

Let us examine the value of Z under different conditions. The first term is always greater than one, which represents the repulsion term making the volume greater than the ideal gas volume and the second term reduces the value of Z, which represents the attraction term. At a fixed value of T above the critical temperature, compression will cause V to decrease so that Z will drop below one, and further compression will cause V to decrease even more so that Z will rise above one. When the temperature is at or below the critical temperature, compression will eventually cause the gas to condense into a liquid at or above the critical pressure Pc. The relations between the critical constants and the values of van der Waals a and b are... 

As with single-component gases, if you know V and either T or P you can estimate the ideal pseudoreduced volume = VPj RT and use the other known reduced property to determine the unknown temperature or pressure from the compressibility chart. 

The idea of using the reduced variables to correlate the p-V-T properties of gases, as suggested by van der Waals, is that all substances behave alike in their reduced (i.e., their corrected) states. In particular, any substance would have the same reduced volume at the same reduced temperature and pressure. If a relationship does exist involving the reduced variables that can be used to predict pr, Tr, and K, what would the equation be Certainly, the simplest form of such an equation would be to imitate the ideal gas law, or... 

EXAMPLE 3.17 Use of Pseudo-reduced Ideal Molal Volume... 

In 1873, van der Waals [2] first used these ideas to account for the deviation of real gases from the ideal gas law P V= RT in which P, Tand T are the pressure, molar volume and temperature of the gas and R is the gas constant. Fie argried that the incompressible molecules occupied a volume b leaving only the volume V- b free for the molecules to move in. Fie further argried that the attractive forces between the molecules reduced the pressure they exerted on the container by a/V thus the pressure appropriate for the gas law isP + a/V rather than P. These ideas led him to the van der Waals equation of state ... 

This result should be compared with Eq. (8.28) for the case of the ideal mixture. It is reassuring to note that for n = 1, Eq. (8.36) reduces to Eq. (8.28). Next let us consider whether a change of notation will clarify Eq. (8.36) still more. Recognizing that the solvent, the repeat unit, and the lattice site all have the same volume, we see that Ni/N is the volume fraction occupied by the solvent in the mixture and nN2/N is the volume fraction of the polymer. Letting be the volume fraction of component i, we see that Eq. (8.36) becomes... 

The most significant commercial product is barium titanate, BaTiO, used to produce the ceramic capacitors found in almost all electronic products. As electronic circuitry has been rniniaturized, demand has increased for capacitors that can store a high amount of charge in a relatively small volume. This demand led to the development of highly efficient multilayer ceramic capacitors. In these devices, several layers of ceramic, from 25—50 ]lni in thickness, are separated by even thinner layers of electrode metal. Each layer must be dense, free of pin-holes and flaws, and ideally consist of several uniform grains of fired ceramic. Manufacturers are trying to reduce the layer thickness to 10—12 ]lni. Conventionally prepared ceramic powders cannot meet the rigorous demands of these appHcations, therefore an emphasis has been placed on production of advanced powders by hydrothermal synthesis and other methods. 

Reduced Properties. One of the first attempts at achieving an accurate analytical model to describe fluid behavior was the van der Waals equation, in which corrections to the ideal gas law take the form of constants a and b to account for molecular interactions and the finite volume of gas molecules, respectively. 

Reduced Equations of State. A simple modification to the cubic van der Waals equation, developed in 1946 (72), uses a term called the ideal or pseudocritical volume, to avoid the uncertainty in the measurement of volume at the critical point. 

Because the value of Z for natural gas is significantly less than unity at ambient temperatures and at pressures greater than 1 MPa (145 psia), the compressibility must be taken into account in gas measurement gas purchased at high line pressure will fflve a greater volume when the pressure is reduced than it would if the gas were ideal. Natural gas pipeline operators use a.. supercompre.s.sihility faclor, also called Z, out defined as... 

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