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Cubic Equations of State in Software
35 Cubic Equations of State QR Code JIS X 0510 Creator In None Using Barcode maker for Software Control to generate, create Denso QR Bar Code image in Software applications. Quick Response Code Scanner In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. most suitable is elimination of Vc to yield expressions relating a(Tc) and b to PCand T, The reason is that PCand Tc are usually more accurately known than V, An equivalent, but more straightforward, procedure is illustrated for the van der Waals equation Since V = Vcfor each of the three roots at the critical point, QRCode Encoder In Visual C# Using Barcode maker for VS .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications. Drawing QR Code In Visual Studio .NET Using Barcode generation for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications. Equation (340) expanded in polynomial form becomes: QR Code 2d Barcode Encoder In VS .NET Using Barcode creator for .NET Control to generate, create Denso QR Bar Code image in VS .NET applications. Creating QR Code 2d Barcode In Visual Basic .NET Using Barcode generation for .NET framework Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications. Recall that for a particular substance parameter a in the van der Waals equation is a constant, independent of temperature Termbyterm comparison of Eqs (A) and (B) provides three equations: Printing Bar Code In None Using Barcode creation for Software Control to generate, create bar code image in Software applications. Data Matrix ECC200 Generator In None Using Barcode creation for Software Control to generate, create DataMatrix image in Software applications. Solving Eq (D) for a , combining the result with Eq (E), and solving for b gives: Painting EAN13 Supplement 5 In None Using Barcode drawer for Software Control to generate, create GTIN  13 image in Software applications. Generate EAN / UCC  14 In None Using Barcode printer for Software Control to generate, create GTIN  128 image in Software applications. Substitution for b in Eq (C) allows solution for V,, which can then be eliminated from the equations for a and b: Bar Code Maker In None Using Barcode creation for Software Control to generate, create barcode image in Software applications. Code 128 Encoder In None Using Barcode encoder for Software Control to generate, create Code 128A image in Software applications. Although these equations may not yield the best possible results, they provide reasonable values which can almost always be determined, because critical temperatures and pressures (in contrast to extensive PV T data) are often known, or can be reliably estimated Substitution for V, in the equation for the critical compressibility factor reduces it immediately to: 3 z = pcvc  "  RT, 8 A single value for Z,, applicable alike to all substances, results whenever the parameters of a twoparameter equation of state are found by imposition of the critical constraints Different values are found for different equations of state, as indicated in Table 31, p 93 Unfortunately, the values so obtained do not in general agree with those calculated from experimental values of Tc, PC,and Vc;each chemical species in fact has its own value of Z, Moreover, the values Leitcode Drawer In None Using Barcode encoder for Software Control to generate, create Leitcode image in Software applications. Drawing ECC200 In Java Using Barcode creator for Java Control to generate, create Data Matrix 2d barcode image in Java applications. CHAPTER 3 Volumetric Properties of Pure Fluids
Generating UCC.EAN  128 In Java Using Barcode creator for Java Control to generate, create GTIN  128 image in Java applications. Bar Code Generator In Java Using Barcode creator for Android Control to generate, create barcode image in Android applications. given in Table Bl of App B for various substances are almost all smaller than any of the equation values given in Table 31 An analogous procedure may be applied to the generic cubic, Eq (341), yielding expressions for parameters a(Tc) and b For the former, GS1 DataBar Truncated Encoder In .NET Using Barcode generation for .NET Control to generate, create GS1 DataBar Expanded image in Visual Studio .NET applications. Code 39 Reader In Visual Basic .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications. This result may be extended to temperatures other than the critical by introduction of a dimensionless function a(T,) that becomes unity at the critical temperature Thus Code128 Creator In Java Using Barcode encoder for Java Control to generate, create Code 128 Code Set A image in Java applications. Create ECC200 In .NET Framework Using Barcode creation for .NET framework Control to generate, create DataMatrix image in Visual Studio .NET applications. Function a(Tr) is an empirical expression, specific to a particular equation of state Parameter b is given by: In these equations C2 and Q are pure numbers, independent of substance and determined for a particular equation of state from the values assigned to t and a The modern development of cubic equations of state was initiated in 1949 by publication of the RedlicWKwong (RK) equation:' where, in Eq (342), a(T,) = T,''~
Theorem of Corresponding States; Acentric Factor
Experimental observation shows that compressibility factors Z for different fluids exhibit similar behavior when correlated as a function of reduced temperature T, and reducedpressure P,; by dejinition, T'I =  T Tc and This is the basis for the twoparameter theorem of corresponding states: All fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor, and all deviate from idealgas behavior to about the same degree Although this theorem is very nearly exact for the simple fluids (argon, krypton, and xenon) systematic deviations are observed for more complex fluids Appreciable improvement 80tto Redlich and J N S Kwong, Chem Rev,vol 44, pp 233244, 1949 35 Cubic Eauations o f State
results from introduction of a third correspondingstates parameter, characteristic of molecular structure; the most popular such parameter is the acentric factor w , introduced by K S Pitzer and coworker^^ Figure 313 Approximate temperature dependence of the reduced vapor pressure
The acentric factor for a pure chemical species is defined with reference to its vapor pressure Since the logarithm of the vapor pressure of a pure fluid is approximately linear in the reciprocal of absolute temperature, d log P,Sat d(llTr) where P T t is the reduced vapor pressure, T, is the reduced temperature, and S is the slope of a plot of log P,Sat vs 1/T, Note that "log" denotes a logarithm to the base 10 If the twoparameter theorem of corresponding states were generally valid, the slope S would be the same for all pure fluids This is observed not to be true; each fluid has its own characteristic value of S, which could in principle serve as a third correspondingstates parameter However, Pitzer noted that all vaporpressure data for the simple fluids (Ar, Kr, Xe) lie on the same line when plotted as log P,Sat vs 1/T, and that the line passes through log P T t = 10 at T, = 07 This is illustrated in Fig 313 Data for other fluids define other lines whose locations can be fixed in relation to the line for the simple fluids (SF) by the difference: log p , S a t ( s ~ ) log p,sat The acentric factor is defined as this difference evaluated at T, = 07:

