States, corresponding

Essentially, the theorem states that if the properties are scaled properly, then the scaled properties of all substances should be the same. Most applications of the theory begin with the critical point. Thus, we define the reduced temperature as  

In its simplest form, the theory of corresponding states says that if two substances are at the same reduced temperature and reduced pressure, then the other reduced properties should be equal. 

According to the principle, the properties of any fluid were dependent on only the reduced temperature and pressure. Therefore, the properties of a fluid depend only on its temperature and pressure relative to its critical point  

The z-factor thus obtained is used in equation (2.7) in order to calculate the density of the gas. 

The observation that the properties could be expressed in terms of the reduced quantities had many important ramifications. These including the possibility that if you plotted the reduced vapor pressure as a function of reduced temperature, all substances would fall onto a single curve. Furthermore, if you plotted the compressibility factor versus the reduced pressure with the reduced temperature as a parameter, all fluids would lie on the same plot. 

The theorem of corresponding states goes back to van der Waals, who formulated the principle of corresponding states based on his equation of state. In addition, van der Waals deduced straightforwardly that in reduced coordinates, the vapor pressure curve and the coexistence curve must be the same for all fluids [3]. An extensive treatment of the corresponding state principle has been published by Xiang [4]. 

The theorem of corresponding states usually starts with the van der Waals equation. Temperature, pressure, and volume are replaced with the corresponding reduced quantities, e.g., T = Tct. We arrive finally at an equation with a complete similar form than like that we started before. But all the constants, which are material specific went away, even the gas constant. 

The properties of various equations of state have been reviewed in much more detail as presented here [5, pp. 37-53]. In Table 4.2, various gas equations are summarized. 

Now we are going the reverse way. We are starting with a reduced equation and we are asking whether there are other forms possible than the common form 

Resolving to p and forming the first and second derivative with respect to V, and equating to zero dp/dV and d pjdV, respectively, we obtain for these three equations at = 1, y = 1, and f = l,just andonly a = 3, jS = 1/3, p = 8/3. So there is no other possible solution for this set. 

One of the useful consequences of the vdW EOS is that, if it were indeed a correct representation of reality, then it could be written in the form 

The most widely used corresponding states approach is that due to Pitzer and his co-workers [8], often called Pitzer-type equations. The common form of their approach is 

Lee and Kessler [9] developed their own tables of those functions, which differ slightly from those of Pitzer et. al. The Lee and Kessler tables are probably the most widely used tables of this type. In addition to tables of z° and z they present similar tables for other thermodynamic functions, all in the Pitzer-type format a base function for d = 0.00 and a second function to be multiplied by w and added to the base function. 

While the tables, with proper interpolation are probably the best corresponding states estimates of fluid properties, they are inconvenient for computer use. Many attempts have been made to replace those tables with EOSs. The following simple, totally empirical estimating EOS called the little EOS [10, p. 89], is reasonably accurate at low pressures  

Based on the steam table (which may be considered as reliable as the experimental data), the value of z is 0.804.  

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Using corresponding-states arguments, it is possible to derive a generalized version of Equation (2) which has the form... 

Appendix C-1 gives corresponding-state parameters and UNI-QUAC surface and volume parameters. 

Corresponding-State Parameters and UNIQUAC Surface and Volume Parameters... 

Using the principle of corresponding states for the following characteristics avoids the use of the contributing groups method ... 

Using the principle of corresponding states requires knowledge of pseudo-critical constants of petroleum fractions these should be estimated starting from characteristic properties which are the normal boiling temperature and the standard specific gravity. 

The correction for pressure is often determined by applying the principle of corresponding states. 

Principle of Corresponding States 4.2.3.1 Concept of Corresponding States... 

Hexane, for example, is a component whose properties are well known and follow the principle of corresponding states very closely. The acentric factor recommended by the DIPPR is 0.3046 and is considered by convention not to vary with temperature. 

The COSTALD (Corresponding states liquid density) method was originally developed for calculating the densities of liquefied gases its use has become generally widespread. 

This method utilizes essentially the concept developed by Fitzer in 1955. According to the principle of three-parameter corresponding states, the compressibility factor z, for a fluid of acentric factor w, is obtained by interpolating between the compressibilities Zj and Z2 for the two fluids having acentric factors w, and (p -... 

The other method is to employ the principle of corresponding states and calculate the Cp/ of the mixture in the liquid phase starting from the mixture in the ideal gas state and applying an appropriate correction ... 

The principle of corresponding states enables the enthalpy of a liquid mixture to be expressed starting from that of an ideal gas mixture and a reduced correction for enthalpy ... 

The specific heat of gases at constant pressure is calculated using the principle of corresponding states. The for a mixture in the gaseous state is equal to the sum of the C g of the ideal gas and a pressure correction term ... 

Calculation of thermophysical properties of gases relies on the principle of corresponding states. Viscosity and conductivity are expressed as the sum of the ideal gas property and a function of the reduced density ... 

Returning to multilayer adsorption, the potential model appears to be fundamentally correct. It accounts for the empirical fact that systems at the same value of / rin P/F ) are in essentially corresponding states, and that the multilayer approaches bulk liquid in properties as P approaches F. However, the specific treatments must be regarded as still somewhat primitive. The various proposed functions for U r) can only be rather approximate. Even the general-appearing Eq. XVn-79 cannot be correct, since it does not allow for structural perturbations that make the film different from bulk liquid. Such perturbations should in general be present and must be present in the case of liquids that do not spread on the adsorbent (Section X-7). The last term of Eq. XVII-80, while reasonable, represents at best a semiempirical attempt to take structural perturbation into account. 

Keller J B and Zumino B 1959 Determination of intermolecular potentials from thermodynamic data and the law of corresponding states J. Chem. Phys. 30 1351... 

A theoretical basis for the law of corresponding states can be demonstrated for substances with the same intemiolecular potential energy fimction but with different parameters for each substance. Conversely, the experimental verification of the law implies that the underlying intemiolecular potentials are essentially similar in fomi and can be transfomied from substance to substance by scaling the potential energy parameters. The potentials are then said to be confomial. There are two main assumptions in the derivation ... 

The equation of state detemiined by Z N, V, T ) is not known in the sense that it cannot be written down as a simple expression. However, the critical parameters depend on e and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 bl illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenlieim [19], the curvature near the critical pomt is consistent with a critical exponent (3 closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the fomi associated with van der Waals equation. Figure A2.3.5 (b) shows tliat PIpkT is approximately the same fiinction of the reduced variables and... 

Figure A2.3.6 illustrates the corresponding states principle for the reduced vapour pressure P and the second virial coefficient as fiinctions of the reduced temperature showmg that the law of corresponding states is obeyed approximately by the substances indicated in the figures. The useflilness of the law also lies in its predictive value.

This is Widom s scaling assumption. It predicts a scaled equation of state, like the law of corresponding states, that has been verified for fluids and magnets [102]. 

Cook and Rowlinson J S 1953 Deviations form the principles of corresponding states Proc. R. Soc. A 219 405... 

The generalized Prony analysis of END trajectories for this system yield total and state resolved differential cross-sections. In Figure 5, we show the results. The theoretical analysis, which has no problem distinguishing between the symmetric and asymmetric str etch, shows that the asymmetric mode is only excited to a minor extent. The corresponding state resolved cross-section is about two orders of magnitude less than that of the symmetric stretch. 

A single calculation of the discrete path integral with a fixed length of time t can be employed to compute the state conditional probability at many other times. It is possible to use segments of the path of time length At, 2At,..., NAt sampled in trajectories of total length of NAt and to compute the corresponding state conditional probabilities. The result of the calculations will make it possible to explore the exponential relaxation of P Ao B,t) for times between 0 and t. 

The physical properties of argon, krypton, and xenon are frequendy selected as standard substances to which the properties of other substances are compared. Examples are the dipole moments, nonspherical shapes, quantum mechanical effects, etc. The principle of corresponding states asserts that the reduced properties of all substances are similar. The reduced properties are dimensionless ratios such as the ratio of a material s temperature to its critical... 

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