barcode generator vb.net code Residual Properties in Software

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62 Residual Properties
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Since V = Z R T I P , the residual volume and the compressibility factor are related:
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The definition for the generic residual property is:
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where M is the molar value of any extensive thermodynamic property, eg, V, U , H , S, or G Note that M and Mig, the actual and ideal-gas properties, are at the same temperature and pressure Equation (637), written for the special case of an ideal gas, becomes:
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Subtracting this equation from Eq (637) itself gives:
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This fundamental property relation for residual properties applies to fluids of constant composition Useful restricted forms are:
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and In addition, the defining equation for the Gibbs energy, G = H - T S , may also be written for ~ ; the special case of an ideal gas, Gig = Hig - T S ~by difference,
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GR= f f R The residual entropy is therefore:
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Thus the residual Gibbs energy serves as a generating function for the other residual properties, and here a direct link with experiment does exist It is provided by Eq (643), written: (const T) Integration from zero pressure to arbitrary pressure P yields:
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CHAPTER 6 Thermodynamic Properties of Fluids
where at the lower limit G ~ / R is equal to zero because the zero-pressure state is an ideal-gas T state In view of Eq (640):
Differentiation of Eq (646) with respect to temperature in accord with Eq (644) gives:
I%=-T~
(E),P
d~ (const T )
The residual entropy is found by combination of Eqs (645) through (647):
The compressibility factor is defined as Z = P V / R T ;values of Z and of (a Z/a T ) p therefore come from experimental P V T data, and the two integrals in Eqs (646) through (648) are evaluated by numerical or graphical methods Alternatively, the two integrals are evaluated analytically when Z is expressed as a function of T and P by a volume-explicit equation of state Thus, given P V T data or an appropriate equation of state, we can evaluate H R and sR and hence all other residual properties It is this direct connection with experiment that makes residual properties essential to the practical application of thermodynamics Applied to the enthalpy and entropy, Eq (641) is written: H = fyig
+ffR
s = s i p + sR
Thus, H and S follow from the corresponding ideal-gas and residual properties by simple addition General expressions for Hig and s i g are found by integration of Eqs (623) and (624) from an ideal-gas state at reference conditions To and Po to the ideal-gas state at T and P : ~ H i p = H$
C$ d~
sip= S +
lo c"-
dT T
P R ln Po
Substitution into the preceding equations gives:
3~hermodynamic properties for organic compounds in the ideal-gas state are given by M Frenkel, G J Kabo,
K N Marsh, G N Roganov, and R C Wilhoit, Thermodynamics of Organic Compounds in the Gas State, Thermodynamics Research Center, Texas A & M Univ System, College Station, Texas, 1994
62 Residual Properties
Recall (Secs 41 and 55) that for purposes of computation the integrals in Eqs (649) and (650) are represented by:
and Equations (649) and (650) may be expressed alternatively to include the mean heat capacities introduced in Secs 41 and 55:
where H~ and S R are given by Eqs (647) and (648) Again, for computational purposes, the mean heat capacities are represented by:
Since the equations of thermodynamics which derive from the first and second laws do not permit calculation of absolute values for enthalpy and entropy, and since in practice only relative values are needed, the reference-state conditions To and Po are selected for convenience, and values are assigned to H and S: arbitrarily The only data needed for application of Eqs (651) and (652) are ideal-gas heat capacities and P V T data Once V , H, and S are known at given conditions of T and P , the other thermodynamic properties follow from defining equations
The true worth of the equations for ideal gases is now evident They are important because they provide a convenient base for the calculation of real-gas properties
Residual properties have validity for both gases and liquids However, the advantage of Eqs (649) and (650) in application to gases is that H~ and S R , the terms which contain all the complex calculations, are residuals that generally are quite small They have the nature of corrections to the major terms, Hig and S'g For liquids, this advantage is largely lost, because H~ and sR must include the large enthalpy and entropy changes of vaporization Property changes of liquids are usually calculated by integrated forms of Eqs (628) and (629), as illustrated in Ex 61
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