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Extended Corresponding States Expressions for the Changes in Enthalpy, Compressibility Factor and Constant-Volume Heat Capacity at Vaporization ⇑ S

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Extended Corresponding States Expressions for the Changes in Enthalpy, Compressibility Factor and Constant-Volume Heat Capacity at Vaporization ⇑ S J. Chem. Thermodynamics 85 (2015) 68–76 Contents lists available at ScienceDirect J. Chem. Thermodynamics journal homepage: www.elsevier.com/locate/jct Extended corresponding states expressions for the changes in enthalpy, compressibility factor and constant-volume heat capacity at vaporization ⇑ S. Velasco a,b, M.J. Santos a, J.A. White a,b, a Departamento de Física Aplicada, Universidad de Salamanca, 37008 Salamanca, Spain b IUFFyM, Universidad de Salamanca, 37008 Salamanca, Spain article info abstract Article history: By analyzing data for the vapor pressure curve of 121 fluids considered by the National Institute of Received 2 October 2014 Standards and Technology (NIST) program RefProp 9.1, we find that the first and the second derivatives Received in revised form 19 December 2014 with respect to reduced temperature (Tr) of the natural logarithm of the reduced vapor pressure at the Accepted 16 January 2015 acentric point (T ¼ 0:7) show a well defined behavior with the acentric factor x. This fact is used for Available online 31 January 2015 r checking some well-known vapor pressure equations in the extended Pitzer corresponding states scheme. In this scheme, we then obtain analytical expressions for the temperature dependence of the Keywords: enthalpy of vaporization and the changes in the compressibility factor and the constant-volume heat Vapor pressure curve capacity along the liquid–vapor coexistence curve. Comparisons with RefProp 9.1 results are presented Saturation properties Acentric factor for argon, propane and water. Furthermore, very good agreement is obtained when comparing with Enthalpy of vaporization experimental data of perfluorobenzene and perfluoro-n-heptane. Compressibility factor Ó 2015 Elsevier Ltd. All rights reserved. Constant-volume heat capacity 1. Introduction are the two-phase vapor and liquid molar heat capacities at con- stant volume. The vapor pressure equation for a pure fluid gives the relation Equations (1) and (2) are exact thermodynamic relations, and between saturation pressure pr and temperature T from the triple they can be rewritten in different convenient forms that involve to the critical point along the liquid–vapor phase boundary r. The the natural logarithm of the vapor pressure as a function of the form of this equation is one of the most fascinating unsolved prob- temperature. One of these forms for equation (1) is lems in physics. Perhaps, it is the equation for which a larger num- ber of proposals have been made. The only two informations that dlnpr DvH ¼ 2 ; ð3Þ thermodynamics provides about it concern to its first temperature dT RT DvZ derivative, the so-called Clapeyron–Clausius equation, where Z is the compressibility factor defined as prV=RT; R being the dp D H g l 0 r ÀÁv gas constant, and therefore DvZ ¼ Z À Z is the change in the com- prðTÞ ¼ ; ð1Þ dT T V g À V l pressibility factor associated to vaporization. On the other hand, taking into account the relationship and to its second temperature derivative, the so-called Yang–Yang 2 2 2 equation [1], d ln pr dlnpr 1 d pr 2 ¼ þ 2 ; ð4Þ dT dT pr dT d2p Cg Cl 00 r ÀÁv2 À v2 prðTÞ ¼ ; ð2Þ dT2 T V g À V l and using equations (2) and (3) one obtains ! 2 g l 2 where D H is the molar enthalpy of vaporization, V and V are the d ln p DvH DvCv v r ¼ þ ; ð5Þ g l 2 2 2 vapor (g) and liquid (l) molar volumes at saturation, and Cv2 and Cv2 dT RT DvZ RT DvZ g l ⇑ Corresponding author at: Departamento de Física Aplicada, Universidad de where DvCv ¼ Cv2 À Cv2. In terms of the reduced variables Tr T=Tc Salamanca, 37008 Salamanca, Spain. Tel.: +34 923294436; fax: +34 923294584. and pr pr=pc, with Tc and pc being the temperature and the E-mail address: [email protected] (J.A. White). pressure of the critical point, equation (3) becomes http://dx.doi.org/10.1016/j.jct.2015.01.011 0021-9614/Ó 2015 Elsevier Ltd. All rights reserved. S. Velasco et al. / J. Chem. Thermodynamics 85 (2015) 68–76 69 dlnp D H f 0ð0:7; xÞ and f 00ð0:7; xÞ show a well defined behavior with the r ¼ v r ; ð6Þ dT 2 acentric factor x. This information can be used for testing vapor r Tr DvZ pressure equations or for obtaining adjustable coefficients appear- where DvHr ¼ DvH=RTc is the reduced enthalpy of vaporization, ing in them. On the other hand, in this work we also analyze the while equation (5) becomes behavior of the reduced enthalpy of vaporization ! 2 2 d ln p 1 D H 1 D Cà DvHrx DvHrðTr ¼ 0:7Þ; ð14Þ r ¼ v r þ v v ; ð7Þ 2 4 D Z 2 D Z dTr Tr v Tr v the change in the compressibility factor à g l DvZx DvZðTr ¼ 0:7Þ; ð15Þ where DvCv ¼ Cv2 À Cv2 =R. If one knows the temperature depen- dence of the right-hand side of equations (6) or (7), integrations and the change in the heat capacity at constant volume of these equations yield the vapor pressure curve. For example, by à à DvCvx DvCvðTr ¼ 0:7Þ; ð16Þ assuming a constant ratio DvHr=DvZ and imposing that pr ¼ 1at Tr ¼ 1, integration of equation (6) leads to the so-called Clausius– at the acentric point. We find that DvHrx is almost linearly corre- Clapeyron (CC) equation, à lated with x while the x-dependence of DvZx and DvCvx is well 0 00 1 explained from DvHrx; f ð0:7; xÞ and f ð0:7; xÞ. Finally, the obtained ln pr ¼ A 1 À ; ð8Þ equations are used for predicting the effect of temperature on the Tr reduced enthalpy of vaporization and on the changes in the com- where the parameter A can be obtained from a known point in the pressibility factor and the constant-volume heat capacity during vapor-pressure curve. vaporization. We compare the obtained equations with NIST data The two-parameter corresponding-states principle (CSP) states for argon, propane and water and also with experimental data of that fluids at equal reduced pressure and temperature should perfluorobenzene and perfluoro-n-heptane, with very good behave identically [2]. If the CSP were true, the parameter A in agreement. equation (8) should take the same value for all fluids. Guggenheim [3] proposed a value of A 5:4 by fitting experimental vapor pres- 2. Checking vapor-pressure equations at the acentric point sure data for a small number (seven) of nonpolar fluids. In order to extend the CSP to a wider range of fluids more parameters must be The first temperature derivative of the natural logarithm of the introduced in the corresponding-states correlations. In particular, vapor pressure at the acentric point, equation (12), can be obtained in order to quantify the deviation of fluids with respect the from equation (6), two-parameter CSP predictions, Pitzer proposed in 1955 a third parameter x, named acentric factor and defined as [4,5] 0 DvHrx f ð0:7; xÞ¼ 2 ; ð17Þ x À1:0 À log10pr at Tr ¼ 0:7; ð9Þ 0:7 DvZx Although the acentric factor was originally introduced to represent with DvHrx and DvZx defined by equations (14) and (15), respec- the nonsphericity (acentricity) of a molecule, nowadays it is used as tively. Figure 1(a) shows a plot of f 0ð0:7; xÞ vs. x. Symbols corre- a measure of the complexity of a molecule with respect to its size/ spond to data obtained from RefProp 9.1 and equation (17) for shape and polarity and it is tabulated for many fluids [6]. Fluids the 121 substances listed in table 1. One can see that f 0 0:7; with nearly x ¼ 0 are called simple fluids (e.g., the seven Guggen- ð xÞ heim’s fluids). The Pitzer three-parameter CSP states that fluids increases with x in a practically linear way. with the same value of x should behave identically at equal At this point it is important to remark that RefProp 9.1 is a com- reduced pressure and temperature. We note that the acentric factor puter program that uses the most accurate equations of state and can be used to introduce the so-called acentric point given by models available for calculating a large number of properties for the 121 pure fluids included in the program [8]. Therefore, the T ¼ 0:7T and p ¼ 10ð1þxÞp . c r c results obtained from RefProp are not experimental data and, in In the Pitzer CSP theory, a vapor-pressure equation has the form some particular cases, are subjected to deviations that can be ln pr ¼ f ðTr; xÞ; ð10Þ ascribed to the equation of state considered for the fluid. Nonethe- less, in most cases RefProp 9.1 gives very accurate results that can with f ð1; xÞ¼0. Furthermore, taking into account definition (9), be used to test the extended corresponding states expressions the function f ðT ; xÞ must satisfy the condition r derived in this work. f ð0:7; xÞ¼ð1 þ xÞ ln 10; ð11Þ The second temperature derivative of the natural logarithm of the vapor pressure at the acentric point, equation (13), can be in order to be self-consistent at the acentric point. Recently [7] we obtained from equation (7) have checked condition (11) for testing some vapor-pressure equa- ! tions often used for estimating the vapor pressure of a large variety 2 à of fluids. In this work, we analyze the behavior of the first and 00 1 DvHrx 1 DvCvx f ð0:7; xÞ¼ 4 þ 2 : ð18Þ DvZ DvZ second derivatives with respect the reduced temperature of ln pr 0:7 x 0:7 x at the acentric point, i.e., the functions Figure 1(b) shows a plot of f 00ð0:7; xÞ vs.
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