11007
J. Phys. Chem. 1995, 99, 11007-11012
Corresponding States of Compressed Fluids Based on Their Published Equations of State Eugene M. Holleran Chemistry Department, St. John's Universiv, Jamaica, New York 11439 Received: March 8, 1995; In Final Form: May 8, 1999
A correlation of the properties of compressed fluids is presented in terms of a seven-coefficient reduced equation which is common to a group of fluids called corresponding fluids, thirteen of which are investigated here. With three characteristic constants for each fluid, this one equation provides an accurate equation of state and allows the calculation of the thermodynamic properties derivable therefrom. The equation is valid in the compressed fluid region for densities ranging roughly between two and three times the critical density, as determined for each fluid by the range in which its unit compressibility line (UCL) is linear within a few parts per ten thousand. The form of the equation is determined by the fact that isotherms of the residual Helmholtz energy, A', have a minimum on the UCL. The coefficients are evaluated by a simultaneous leastsquares fitting of the P@Tbehavior of the published modified Benedict-Webb-Rubin (MBWR) or SchmidtWagner (SW) equations of state for five fluids: oxygen, argon, ethane, ethylene, and propane. The reduced equation is used to calculate the density, e, as a function of T and P , agreeing within experimental error with the MBWR or SW equations for these five fluids. It is applied with similar success to n-butane, isobutane, nitrogen trifluoride, methane, and carbon monoxide. The equation also succeeds in representing accurately @(T,P)for sets of original experimental data for nitrogen, krypton, and xenon. Thermodynamic properties that can be calculated from the reduced equation include the difference between the heat capacities, CP- CV, at any point in the region and isothermal changes in the thermodynamic properties E, H,S,A , G , CP,and CV between any two densities in this range. A table is given showing the favorable comparison between these changes as calculated from the reduced equation and from the SW equations for ethane.
Introduction In recent years, equations of state covering the entire P@T range of experimental data have been published for many fluids. Most of these equations can be written in the form
Z = 1+ g K
(1)
where Z is the compressibility factor, defined as P/@RT. An expression for K(T,@)is found from experimental data in different ways. In the modified Benedict-Webb-Rubin (MBWR or MB) form the coefficients in a multiterm expression for K(T,@)are found by least squares directly from the data. The Schmidt-Wagner (SW) treatment is based on the reduced Helmholtz energy, 4 = A/RT, so that Z = e&. (Here we adopt the notation in which the subscript indicates the partial derivative with respect to that variable.) 4 is written as the sum of two terms: an "ideal" part, and a "residual" part, Q. 4' is given as In(&) plus a function of T only, and Q is found as a function of T and e by fitting to experimental data. Thus Z = 1 @Qe, and
e,
+
K = d',
(2)
Treatments such as the MB and SW ones provide the best available overall description of the volumetric and thermodynamic properties of these fluids. One of the uses often proposed for these equations is for corresponding states studies. That is the purpose to which they are put in this paper, for compressed fluids in the range of density roughly between two and three times the critical density. Behavior of the Residual Helmholtz Free Energy The residual Helmholtz energy, Q = A'/RT, is a fundamental quantity in the SW treatment. For temperatures below the Boyle @Abstractpublished in Advance ACS Absrructs, June 15, 1995.
temperature, each isotherm of Q versus g as calculated from the SW expression (e.g. for ethylene') passes through a minimum. According to eqs 1 and 2, the compressibility factor Z equals unity when the derivative, Q,, equals zero. Therefore the minima of these isotherms lie on the unit compressibility line (UCL), along which P = gRT. In the compressed-fluid density region under consideration here, T versus e on the UCL is linear according to various individual sets of experimental data for many fluids and also according to their SW and MBWR equations of state, as will be confirmed below. If the zerodensity and zero-temperature intercepts, TOand eo, of this linear segment are used to reduce T and @ (0 = T/To and 6 = @/eo), then the UCL in reduced form is
e+d=i
(3)
Figure 1 shows the Q isotherm for T = 260 K for ethylene in the compressed-fluid region according to its SW expression.' The figure shows (Q - QM),where QMis the minimum value, plotted against (4 - @M), where @M is the density at the minimum. It can be seen that (Q - QM)is nearly symmetrical about ( g - @M) = 0, and in fact near @M it is nearly proportional to (8 - @M)2. Adding cubic and quartic terms permits a highly accurate representation of (Q - QM)on this isotherm. Other isotherms in this region behave similarly, as do the isotherms of other fluids. If we reduce the density to 6 by eo and note that, according to eq 3, 6 at the minimum is 1 - 0, where 0 is the reduced temperature of the isotherm, we see that (@ - @M)/@O is (0 $. 6 - 1). Thus, defining u = 8 f 6 - 1, we can write
4' = #IM
+ au2 + bu3 + cu4
(4)
For a given fluid a, b, c, and QMare functions of 0 only. To illustrate the accuracy of eq 4 for the full compressedfluid P@Tregion under consideration here, we consider the fluid
0022-3654/95/2099-11007$09.00/0 0 1995 American Chemical Society
Holleran
11008 J. Phys. Chem., Vol. 99, No. 27, 1995
'. O0B9
8 I40
,07 120
.06 100
P MPa
.05
9'- 9b4
80
D
.03
60
.02 40
.oi 20
-
0.00 0
-2.5
.2
.i.5
-1
..5
,5
0
1
1.5
2
2.5
P-PM
. 2 0 4 , 50
Figure 1. Isotherm at 260 K of Q versus density for ethylene. The points are for pressures from about 10 to 110 MPa. The curve is for a polynomial in (e - e M ) , powers 2, 3, and 4.
ethylene. The first step is to determine values for TOand eo and at the same time to set the upper and lower limits to the density range to be considered. This was done by finding T, e points at which the SW equation of state (EOS)' gives @e = 0 (UCL points) and then fitting them by multiple least squares to the equation 1 = c,T+
(5)
C2@
with T and as independent variables and unity as the dependent variable. This gives TOas llcl and eo as 1/c2 by eq 3. The density limits were taken as those within which the deviations of the dependent variable from exactly unity were equal to or less than 0.0003, thus assuring the practical constancy of the reducing constants. TOcan be determined to within a few parts per thousand, and eo even more precisely (because the compressed-fluid region is closer to the eo intercept of the UCL). As noted in earlier p~blications,2-~ the UCL changes to a slightly different straight line at lower densities. Values of the constants for our region are included in Table 1. Next, additional @, T, 8 points were derived from the SW EOS within these density limits (15.0 and 22.3 mom) along isobars and isotherms for pressures up to 100 MPa and for temperatures (130-350 K) within these P and limits. This data set is listed in Table 1, and is similar in appearance to the set for propane shown in Figure 2. These 486 points were then
I
100
,
I
,
150
8
2W
,
I
250
.
8
300
.
6
350
.
1
400
.
i
450
,
9
.
500
tc 550
T (K)
Figure 2. Data set for propane. The circles locate the points derived from the MB EOS,' including the curved line of P,TUCL points. The T versus e UCL is closely linear within the isochores shown. The squares outline the region in which eq 16 agrees with the MB EOS within 0.1 % in density. fitted to eq 4 by multiple least squares using polynomials in powers of x =1/8 to express a, b, c, and @M. Selected integer powers of x were good for a, b, and c, and half-integer powers for @M. The fit was excellent, with R2 = 0.999 999 998 and the average absolute deviation of @ of 0.002%. In other words eq 4 provides an accurate reproduction of the SW residual Helmholtz free energy for ethylene in this region. Other fluids behave similarly, and the formulation provided by eq 4 will be useful in our corresponding states study. Equation of State Equations 1,2, and 4 provide a simple form for the equation of state in this region in terms of K which equals (Z - l)/@or @e :
e& = 2au + 3bu2+ 4cu3
(6)
This equation can be tested by its fit of individual sets of PeTdata. With TOand eo found as above, values of K(8,6)are fitted by multiple least squares to eq 6 with terms in a, b, and c chosen from powers of x up to 5, with a minimum power of 1 for a, 2 for b, and 3 for c (because, empirically, isochores of
TABLE 1: Characteristic Constants, P and e Ranges, and the Average Absolute Deviation (AAD) in % Density between the Source Listed and Ea 16 fluid source ref TO,K KO,*mL/mol no. pts max. MPa e range,' moVL 6 range AAD, %e eo, m o l L argon oxygen ethane ethylene propane isobutane n-butane NF3 methane
co
nitrogen krypton xenon
MB
sw sw sw
MB MB MB MB MB EOS data data data
8 5 6 1 7 7 7 8 7 12 14 15 16
413.4 419.5 787.7 729.0 926.0 997.8 1044 591.8 530.1 346.7 331.6 575.5 788.6
46.334 41.520 24.654 27.550 18.315 14.454 14.608 29.867 34.932 38.912 39.120 38.028 29.601
1.87 1.95 2.465 2.38 2.95 3.24 3.24 2.75 1.84 2.21 2.06 1.85 1.89
40.4 41.0 100.0 86.4 161 224 222 92.0 52.1 56.9 52.7 48.6 64.0
49 1 314 277 486 439 160 180 292 227 153 98 174 254
24.2-35.8 29.4-39.3 15.4-21.2 15.0-22.3 11.7-15.3 9.1 - 12.2 9.0-11.9 18.5-25.7 22.9-29.0 22.1-32.2 22.8-32.4 21.2-30.2 15.7-23.8
100 80
10 100 100 35 70 50 100 100 69 100 100
0.52-0.77 0.62-0.83 0.62-0.86 0.54-0.81 0.64-0.84 0.63-0.85 0.60-0.81 0.62-0.86 0.66-0.83 0.57-0.83 0.58-0.83 0.56-0.79 0.53-0.80
0.04 0.05 0.02 0.03 0.04 0.02 0.03 0.07 0.07 0.06 0.06 0.04 0.05
a Values of ko and KO are relative based on 100.0 mL/mol for KOfor ethane. They can be multiplied by a common factor provided the coefficients in Table 4 are all divided by this same factor. The maximum P used in the fits is100 MPa unless restricted by the data or the MB or SW EOS. Generally, eq 16 and the source equation agree to higher pressures. This is the range in which t9 6 on the UCL is within f0.0003 of exactly one. In most cases, the range in which the deviation in density is within 0.1% is appreciably wider.
+
Corresponding States of Compressed Fluids
K are nearly linear in 1IT near the UCL). Direct fits of various sets of original experimental data (e.g. those listed in Table 1 for nitrogen, krypton, and xenon, among others) were found to agree within experimental error. More importantly, eq 6 also agrees very closely with the P@T behavior of the SW and MBWR equations, which incorporate and represent well multiple sets of experimental data for each fluid. For example, eq 6 reproduced the density at the 486 T, P points of the SW ethylene data set described above with an average absolute deviation (AAD) of 0.01%. (More detail on finding the density at each T, P point is provided later.) Similar data sets were derived from the SW equations for oxygen5 and ethane6 and from the MB equations for propane7 and argon.8 Table 1 lists the numbers of points in these data sets, along with the density ranges (determined by the linearity of the UCL) and the pressure ranges (taken to the limit covered by the EOS or 100 MPa, whichever is lower). Figure 2 shows the 439 T, P points of the propane data set, including UCL points. Also shown are the isochores at 11.7 and 15.3 m o m , within which the UCL meets the strict linear requirement. As with ethylene, fits of these four data sets individually to eq 6 gave very good results, with AAD's in density ranging from 0.01% to 0.02%. These five fluids were used to establish a common reduced equation, as described in a later section. Other published equations of state for these fluids were not used in this study. For ethylene, an MB EOS8 covers P only to 40 MPa and does not agree well with the SW EOS' beyond that. For oxygen, an MB EOS8 could have been used, since it agrees at the 314 T, P points from the SW EOS5 with an AAD in density of less than 0.01%. For argon, an SW EOS9 shows an uncommon pattern of deviations for the UCL (eq 5 ) . This may be due to the fact that it stretches over a much wider range of variables and so may not be able to devote adequate attention, so to speak, to our region of interest. For ethane, an MB EOS7 also exhibits an atypical deviation pattern for the UCL. For the fluids with SW equations, the coefficients of a($, b(x), and c ( x ) in eq 6 can be found from those of eq 4. In general, however, it is better to start with eq 6 for K because this equation can be fitted as noted above directly to experimental P@Tdata and also to data sets from the MBWR or other equations of state in addition to the SW equations. (@ - @M) can then be found from eq 4 using the same a(x), b(x), and c(x), and this is the direction that will be taken in what follows. Note that, in reversing the isothermal differentiationthat gives eq 6 from eq 4, we have
dqY = K ~ = Q (2au
+ 3bu2 + 4cu3)du
J. Phys. Chem., Vol. 99, No. 27, 1995 11009 TABLE 2: Dependence of Isothermal Changes in Various Properties on the Reduced Function U and Its Derivative
AZ = koA(6 U,j) AEIRT = kGAU, AHIRT= koA(xU, dud) ASIR = kOA(xU, - CJ) - A In 6 AAIRT = koAU A In 6 AGIRT = koA(U dud) A In 6 ACvIR = -&2AU.
+
+ +
The subscripts indicate partial derivatives with respect to 6 and x.
division by ko, such a property becomes a corresponding property in the traditional sense of having the same value at the same 8, 6 for all corresponding fluids. The basic corresponding property in this sense is taken to be @alb, which is directly related to the nonideality of the P@T behavior of fluids. This assumed correspondence allows us to write eq 6 as
e& =
The system of corresponding states adopted here uses TOand T and e to 8 and 6. Corresponding states of different fluids are defined as those states that have the same 8, 6 values. In addition to TO and eo, the system requires a third, dimensionless, characteristic constant, h,called the nonideality factor. The determination of relative values of ICO is described below. In this system a correlated property is one which, after having been rendered dimensionless by the proper combination of R, TO,and eo, obeys the same reduced equation for different fluids, called corresponding fluids. That is, a correlated property is the same function of 8, 6, and k~ for all corresponding fluids. An important subclass of correlated properties is directly proportional to ko times a function of 8 and 6, so that, upon
eo to reduce
@Id
= k0(2au i3Pu2
+ 4yu3)
(7)
in which the terms and coefficients for a ( x ) , ,&), and y ( x ) are the same for all corresponding fluids. Letting
U(x,u) = au2
+ pu3 + yu4
(8)
eq 6 can be written as
where KO is a characteristic volume defined as kd@o. The following P@T properties, closely related to KIKo, are also corresponding properties:
where n is a reduced pressure: n = Pl@oRTo. From these equations we see that what coresponds is the difference of @OKand Z and n from their ideal-gas (UCL) values. In this sense, states along the UCL are standard states. The relative deviation of these properties from ideality for different fluids at the same 8, 6 equals the ratio of their b values. This is the reason for calling ko the nonideality factor. From eq 4, written as @ = @M h U , it follows that (@ @M)& = (A' - A ' M ) / ~ isT also a corresponding property, with
+
which when integrated between the limits @M and @, or zero and u, gives (@ - @M). Thus the temperature dependence of @M is not available from P@T data. Corresponding States
+
A#'lko = AA'Ik,,RT = AU
(12)
where A indicates an isothermal change, in general from one density to another in our range and in particular from the UCL density, 6 = (1 - e), where U = 0, to another density. This in tum leads to the correspondence of (ff - GIucL)/ m T , where G is the Gibbs free energy, A PV. Thus
+
Changes in the full thermodynamic properties, rather than just the residual parts, depend on changes in the full 4 = 4' @ and/or its derivatives. From the definition of AIRT = 4, it follows that EIRT = x&, HIRT = ~4~ 648, SIR = x& - 4, GIRT = 4 646, and CVIR = -x24=. Table 2 lists the expressions for isothermal changes of these properties in terms of changes in U and its derivatives. It is seen that changes in EIRT, HIRT, and CVIR are proportional to k~ (and so yield
+
+
+
Holleran
11010 J. Phys. Chem., Vol. 99, No. 27, 1995 corresponding properties), while changes in SIR, AIRT, and GIRT are not, because of the In 6 term in 4’. These latter three are still correlated properties, because of their dependence on common reduced equations. And of course, corresponding properties can be contrived from them by subtracting In 6. This is equivalent to considering changes in the residual parts only. Thus, for example, A(SIR - In 6)lb = APIM = A(xU, - V) corresponds, as do A @ I M T and AA‘flbRT as noted above. The difference between the two heat capacities, (Cp - CV), is interesting because actual values rather than just isothermal changes can be calculated from PeT data via derivatives of U . Thus,
TABLE 3: Summary of the Reduced Expressions for U and Its Derivative
+
U = au2 pu3 f yu4 Uh = 2au 3pu2 4yu3 U66 = 2 a 6pu 12yu2 U, = a,u2 pXu3 y,u4 - Ud/x2 Udr = 2a,u 3p,u2 4y,u3 - Ud~lx2 U, = a,u2 pUus yxxu4- U& 2U6/x3 - 2Ud,/x2 = e 6 - 1 = iix 6 - 1 a = a l x ag3 ad a d p = PSx3+P& y = YQ’
+ + + +
+ + + + + + + + + + +
+
6 = eleo x = TdT a On the UCL, u = U = UO= U v = 0, and Udd = 2 a , Udx = - 2 d x2, and U,, = 2aJP.
TABLE 4: Coefficients for Reduced Equations 0.255 300 25 0.285 184 86 -0.058 175 2 0.003 810 31 0.462 462 37 -0.053 305 1 0.342 236 97
al
Reduced Equations for Corresponding Fluids As described above, eq 6 (and therefore eqs 9 and 10) agrees excellently with the EOS’s of fluids in our range. It is flexible enough to follow the individual equations of state even if the latter have small systematic errors due to insufficient or inaccurate primary experimental data in part of our range. For a corresponding states treatment, however, it is necessary to fit two or more fluids simultaneously (SIMFIT) in order to find a common reduced equation. We must expect that SIMFITs will yield not quite such excellent agreement (AAD) if any systematic errors in the EOS’s happen to go in different directions. In fact, we should not be surprised to find deviations as large as the usual experimental errors, which in our range are often 0.1% or 0.2%in density. Still, the results for SIMFITs of the MB and/or SW data sets for many pairs of fluids remain excellent, with AAD’s of about 0.02%. To obtain these results, the To’s and eo’s from the individual fits were used, and c& was taken as the dependent variable in eq 9 (in place of KIKo), with the constant co taken as unity for one of the fluids and as the ratio of the unknown KO’sfor the other. With a trial value for CO, the best powers of x for a, /3, and y were selected by repeated trial fits of COK= Us = 2au 3/3u2 4yu3. Then co was varied until the best least-squaresfit of the combined data sets was obtained. Finally, the To’s and eo’s were adjusted slightly if helpful. Note that only relative values of KO(and hence b)are determined in this way. Groups of three and four fluids were SIMFIT, with the results only moderately worsening. Finally, one single reduced equation for coK(u,x) was obtained from a simultaneous fitting of the five fluids: argon, oxygen, ethane, ethylene, and propane. This yielded the following equation for U ( U J ) :
+
+
u = [alx+ a3x3+ a4x4+ a& + The seven coefficients of this equation, arbitrarily adjusted by setting KO for ethane to 100 mLImol, are listed in Table 4.It should be noted that somewhat different terms and coefficients can be found that would serve about as well. Expressions for the necessary derivatives of U are listed in Table 3. Results obtained from eq 15 for thirteen corresponding fluids are discussed below. Accuracy of the Reduced Equations
PpT. As we have seen, Z and P can be calculated at any e, T in our range for any corresponding fluid from its three
a3 a4 a 5 P3
P4 Y3
characteristic constants and the one common reduced equation of state. From eq 10 this equation of state is
P = @ R q1
+ kodU,]
(16)
with U, given as a function of u and x (Le. e and 6 ) as in Table 3 and with the coefficients as in Table 4. Note that only the second term, the deviation from ideality, needs to be expressed in reduced form. The procedure for checking agreement of a fluid with eq 16 can be summarized as follows. 1. Find TO,eo, and the density limits. (a) If an EOS such as MBWR or SW has been fitted to the experimental data, it can be solved for T, points at which K = 0. These are then fitted to eq 5 , giving TOas llcl and eo asllc;! and the range of e as that within which the UCL is straight, e.g. with clT cze = 1 k 0.0003. (b) Alternatively, K, T, e data points throughout the entire range, either from experiment directly or via an EOS, can be fitted to
+
1 = c,T
+ + KT(C,@ + c4e2+ c,T + c& + C2@
C,Q2>
(17) to give TO,eo, and the limits. (Equation 17 is a good EOS in its own right, but it is less convenient for deriving thermodynamic properties.) 2. Find the relative l~ and the goodness of fit (AAD %e)of the reduced EOS. (a) Select a trial value for b. At each P, T point, find the density, e, by iterating until eq 16 gives the correct P. (b) Vary b to find the best AAD of %e. TO,eo, and the e limits may be adjusted slightly if helpful. As might be expected, the five fluids used in the SIMFIT yield very good results, with AAD’s in density ranging from 0.02% for ethane to 0.05% for oxygen, as listed in Table 1. Figure 2 shows the 439 points of the MB data set for propane, which are reproduced by eq 16 with an AAD of 0.04% in density. The figure includes the isochores within which the UCL is strictly linear (in the T, 0 plane) and also outlines the P,T area within which the density deviation is within 0.1%. It is seen that some of the low-pressure points show deviations greater than 0.1% in e. This is true for some of the other fluids as well. Otherwise, the region of agreement within 0.1% generally extends to higher pressures and densities than those on which the equations were based, suggesting that both eq 16 and the MB or SW equations may extrapolate accurately beyond
J. Phys. Chem., Vol. 99, No. 27, 1995 11011
Corresponding States of Compressed Fluids the ranges of the primary experimental data. The area of agreement within 0.2% is considerably more extensive. Our reduced EOS was then tested for other fluids, as listed in Table 1. The MB equations for butane7 and isobutane7 gave excellent results (AAD's of 0.02 and 0.03% in e) even though they were not used in determining the coefficients. For methane, the MB EOS7 was used, yielding an AAD of 0.07% in e, as seen in Table 1. At these same 220 T, P points, two SW EOS'S'~-"show average absolute density differences of 0.05% and 0.12% from the MB EOS and 0.13% from each other. The MB EOS covers pressures up to 200 MPa and shows a typical pattern of UCL deviations. The SW EOS's cover pressures up to 1000 MPa and show atypical UCL deviations. Still, with slightly different constants, they can be fitted by our reduced EOS with AAD's of 0.08% and 0.07%. For carbon monoxide, an EOS given by GoodwinI2was used, giving an AAD of 0.06% in e. For NF3, an MB EOS8 gives an AAD of 0.07%. For nitrogen, an MB8 and an SWI3EOS both cover P to 1000 MPa (and T to about 2000 K). Their UCL's show atypical deviation patterns and disagree with each other. Rather than use these, it was decided instead to test eq 16 on an individual set of raw experimental data due to Streett and Staveley,I4who report measurements of 107 points for pressures up to 69 MPa on eight isotherms from 77 to 120 K. Equation 16 fits well through the highest experimental density of 32.46 m o m but indicates a lower limit of about 22.7 m o m . With nine points at lower densities omitted, the remaining 98 points show an AAD of 0.06% and a maximum deviation of 0.18% in e. Streett and Staveley represented all 107 points by an EOS with an AAD of 0.11%. Equation 16 can fit all 107 points with this same AAD. Equation 16 was also applied to individual experimental data sets for krypton and xenon. For krypton, Streett and Staveley" measured 385 points at 11 temperatures from 120 to 220 K. They fitted these points to a Strobridge EOS (which is similar to the MBWR one) with an AAD of 0.03%. This EOS gives a UCL which is linear within one part in 104 but which is limited by the T range to densities from about 23.5 to 30.0 mom. Equation 16 fits a somewhat wider range (21.2-30.2 m o m ) with an AAD of 0.04% when applied to a data set derived from the Strobridge EOS for pressures up to 100 MPa at the 11 experimental T's. For xenon the measurements reported by Streett, Sagan, and Stave1eyl6were used. These authors fitted their raw data to a Strobridge EOS which gives a linear UCL with good estimates of TOand eo but which again is not extensive enough to set the density limits. These limits, however, are readily established at about 15.7 and 23.8 m o m by fitting eq 16 to a set of 254 points at temperatures from 170 to 280 K derived from the Strobridge equation. Within these limits and for pressures up to 100 MPa, the AAD is 0.04% in e. Thus we see that in our region the P@Tbehavior of 13 corresponding fluids can be represented within the usual experimental error by a single reduced equation. Part of the reason for this success is the fact that the nonideal term in eq 16 contributes only a part (and near the UCL only a relatively small part) of the pressure. Therefore K/Ko does not have to be known to greater than, say, 1% accuracy to give a good representation of P@Tdata. The reverse of this is that even good data do not give KIKO to really high accuracy. This is reflected in Table 1, where KO and ko are reported to fewer significant figures than TO and eo. This in turn limits the accuracy with which the nonideal part of the thermodynamic properties can be derived from PQT data.
Thermodynamic Properties The isothermal changes in the thermodynamic properties depend on U(x,6)and its derivatives, as shown in Table 2. As we have seen, these are all evaluated by way of Ua (Le. WKo),which is our link to P@Tdata. Because the reduced expressions represent P@Twithin experimental error, they must also represent the thermodynamic changes as well as can be expected from P e T data. Table 5 shows the results of thermodynamic calculations for ethane for three isotherms at pressures from 10 to 100 MPa. At each T, P the SW density was found from eqs 1 and 2 using the SW expression for @a in ref 6. This density was then used with the given T i n the SW expressions for @ and its derivatives to find the other SW entries. The density labeled red. was found from eq 16 and used with T to find the other red. entries from the equations in Tables 2 and 3, with the coefficients in Table 4, and the reducing constants in Table 1. As expected, the SW and red. densities show excellent agreement, all within 0.04% except for the point at 10 MPA and 250 K. This illustrates the trend, mentioned in connection with Figure 2, of points in the low-P, high-T comer of the data sets to exceed 0.1% deviation in e. At high pressures, results are shown up to 100 MPa, and the agreement remains good even though both the SW and the reduced equations are extrapolated beyond the 70 MPa limit of the experimental data. The difference in the heat capacities, CP- CV,is listed next, because absolute values rather than just isothermal changes can be calculated for this quantity from the reduced expressions via eq 14. The agreement is within a few tenths of a J/(mol K ) , which compares favorably with the accuracy with which SW treatment represents the heat capacities themselves. Next the table lists isothermal changes in H , S, G , and CV. These changes could have been shown between any pressure pairs; in Table 5 they are listed as the changes from the UCL pressure, P = edT(1 - 0). Except for the poorest point mentioned above, the agreement for AH is within a few Jlmol, and for AS it is within 0.02 J/(mol K). AG is included in the table, even though it is readily available as AH - TAS, because the remarkable agreement continues even through the worst point, where the errors in AH and TAS apparently cancel. The results for ACV are less good. Two reasons can be suggested for this. First, changes in CV with pressure are relatively small compared to the experimental error and the magnitude of CV. Second, the changes calculated from the reduced equation are given by changes in U,, which is removed from our fit of USto PeT by one integration (with respect to 6) and two derivatives (with respect to x), leading to low expectations for the accuracy of ACV. Still, the results are surprisingly good for the 200 K isotherm. Ethane is one of the best-fitting fluids (note the AAD's for %e in Table 1). However, the results for the thermodynamic properties of the other fluids are also very good, usually within the uncertainties with which these properties are represented by the SW and MB equations. Discussion
A seven-coefficient reduced equation (eq 15) has been derived and applied to thirteen compressed fluids. With three characteristic constants for each fluid, this one common equation provides an accurate representation of their PeT behavior. It also provides a good representation of the thermodynamic properties that can be found from the equation of state. Equation 15 covers a range of reduced density, 6 = @/eo, from about 0.52 to 0.86. The range for an individual fluid (in which its UCL is essentially linear) usually includes about 60
11012 J. Phys. Chem., Vol. 99, No. 27, 1995
Holleran
TABLE 5: Comparison of Property Values for Ethane Calculated from the Reduced Equations and from the SW Equations P, MPa @W), mom
10 20 24.893 30 40 50 60 70 80 90 100
19.677 19.868 19.955 20.044 20.208 20.362 20.507 20.644 20.775 20.899 21.018
@(red.), (Cp-Cv)(SW), (Cp-Cv)(red.), AHfSWk AHhed.). ASfSW). AShd.). AGfSW). AGfred.). ACdSW). ACdred.). J/mol J/mol ’ moa J/(mol/K) J/(molK) J/mol ’ J/mol ‘’ J/(mol Kj’ J/(mol K j 150 K 25.4 -553 -553 1.32 1.32 -751 -751 -0.6 -0.7 19.677 25.6 0.42 0.42 -246 -246 -183 -183 24.3 -0.2 -0.2 19.870 24.5 0 0 0 0 0.0 0.0 23.8 0.00 0.00 24.1 19.959 192 192 -0.42 -0.42 23.4 255 255 0.2 0.2 20.048 23.7 570 570 -1.22 -1.21 752 752 0.6 22.6 20.214 22.9 0.5 950 950 -1.97 -1.96 1245 1245 22.0 0.9 0.7 20.369 22.2 1333 1333 -2.68 -2.67 1735 1734 1.2 0.9 21.5 20.5 14 21.6 1717 1717 -3.36 -3.35 2221 2220 1.6 21.0 21.1 1.o 20.652 -4.00 2102 2103 1.2 -4.01 2703 2702 1.9 20.6 20.782 20.6 2488 2489 -4.63 -4.62 2.1 20.2 1.3 3183 3182 20.905 20.2 2875 2876 1.4 -5.23 -5.22 366 1 3659 2.4 19.9 21.023 19.8
10 20 30 30.588 40 50 60 70 80 90 100
17.788 18.104 18.381 18.396 18.628 18.852 19.057 19.248 19.425 19.591 19.747
17.782 18.100 18.379 18.394 18.628 18.853 19.059 19.249 19.426 19.591 19.747
29.6 27.6 26.1 26.1 25.0 24.0 23.2 22.5 21.9 21.3 20.9
29.4 27.6 26.2 26.1 25.1 24.1 23.4 22.7 22.1 21.5 21.0
-677 -357 -20 0 328 685 1049 1417 1789 2164 2542
200 K -676 -356 -20 0 327 683 1046 1413 1785 2160 2537
2.30 1.12 0.06 0.00 -0.90 -1.78 -2.61 -3.37 -4.10 -4.79 -5.44
2.31 1.12 0.06 0.00 -0.91 -1.79 -2.62 -3.39 -4.12 -4.81 -5.46
-1137 -580 -32 0 508 1042 1570 2092 2609 3121 3630
-1137 -580 -32 0 508 1042 1569 2092 2609 3121 3630
-0.9 -0.4 0.0 0.0 0.3 0.7 1.o 1.3 1.5 1.8 2.0
-0.7 -0.3 0.0 0.0 0.3 0.6 0.9 1.1 1.4 1.7 2.0
10 20 30 34.982 40 50 60 70 80 90 100
15.609 16.187 16.638 16.832 17.013 17.337 17.623 17.880 18.114 18.329 18.529
15.630 16.190 16.636 16.829 17.010 17.334 17.621 17.879 18.115 18.332 18.533
35.8 31.3 28.6 27.6 26.7 25.3 24.3 23.4 22.7 22.0 21.5
35.2 31.0 28.5 27.5 26.6 25.3 24.1 23.2 22.4 21.7 21.0
-590 -398 - 142 0 150 465 796 1139 1491 1849 2213
250 K -603 -401 - 143 0 150 466 797 1140 1492 1851 2216
3.78 2.04 0.62 0.00 -0.59 -1.65 -2.62 -3.50 -4.32 -5.08 -5.79
3.73 2.02 0.62 0.00 -0.58 -1.65 -2.62 -3.50 -4.31 -5.07 -5.78
-1535 -907 -298 0 297 879 1451 2014 2570 3118 3661
-1535 -907 -298 0 297 879 1451 2014 2570 3119 366 1
-0.9 -0.6 -0.2 0.0 0.2 0.5 0.8 1.1 1.3 1.5 1.7
-1.1 -0.6 -0.2 0.0 0.2 0.6 0.9 1.3 1.6 2.0 2.3
”
or 70% of the total range of eq 15. Some fluids, like ethane and propane, are closer to the higher density end. Others, like the noble gas fluids, are nearer the lower end. For these latter, 6 = 0.86 is beyond their triple points and (at low r ) their melting lines. Some comments regarding corresponding states and reduced equations are in order. (a) It is clear from our results that the traditional use of the critical constants as the basis for reduction cannot serve to provide the sort of correspondence of properties obtained here. The reason is that in general the ratio of the Tc’s of two fluids is different from the ratio of their TO’S,and the ratio of their et's is different from that of their eo’s. Thus, reduction by the critical constants cannot yield the congruence of their UCL’s and the resultant correlations described above. (b) A third characteristic constant is definitely needed for accurate corresponding states treatments. Two-constant systems can apply only to very limited groups of fluids whose third constant has about the same value. A fourth constant is apparently not necessary, at least for the (mostly nonpolar) fluids investigated here. (c) In our system, the difference of P from its ideal value, when properly reduced, is a corresponding property. Reduced pressure itself is not a corresponding property (except on the UCL),and attempts to make it one cannot succeed with any great accuracy in any system, except for fluids with similar values of ko. A good example of such a case is provided by the two butanes, which both have = 3.24 as seen in Table 1. All that is needed to match these two fluids is a T ratio (1.0467 from the TO’S)and a ratio (1.0105). With these and the resultant P ratio of 1.OS77 the density of is0 butane at given T,
P points can be calculated from the EOS of normal butane at the corresponding T, P. This procedure has two small disadvantages compared to a SIMFIT of the two: (1) it is valid only in the overlap region of reduced T, P, and (2) it puts all the error on one fluid, whereas S M F I ” G splits the errors. Still, for the 132 points in the overlap region, the e of isobutane calculated from the EOS of n-butane differs from that calculated from its own EOS by an AAD of only 0.02%. References and Notes (1) Jahangiri, M.; Jacobsen, R. T.; Stewart, R. B.; McCarty, R. D. J. Phys. Chem. Re$ Data 1986, 15, 593. (2) Holleran, E. M. Ind. Eng. Chem. Res. 1990, 29, 632. (3) Holleran, E. M. J. Phys. Chem. 1992, 96, 8568. (4) Holleran, E. M. Ind. Eng. Chem. Res. 1993, 32, 3143. (5) Stewart, R. B.; Jacobsen, R. T.; Wagner, W. J. Phys. Chem. Re$ Data 1991, 20, 917. (6) Friend, D. G.; Ingham, H.; Ely, J. F. J. Phys. Chem. Re$ Data 1991, 20, 275. (7) Younglove, B. A,; Ely, J. F. J. Phys. Chem. Ref: Data 1987, 16, 577. (8) Younglove, B. A. J. Phys. Chem. Ref: Data, Suppl. 1982, 11, 1. (9) Stewart, R. B.; Jacobsen. R. T. J. Phys. Chem. Ref: Data 1989,18, 639. (10) Friend, D. G . ; Ely, J. F.; Ingham, H. J. Phys. Chem. Re$ Data 1989, 18, 583. (11) Setzmann, U.; Wagner, W. J. Phys. Chem. Ref: Data 1991, 20, 1061. (12) Goodwin, R. D. J. Phys. Chem. Re$ Data 1985, 14, 849. (13) Jacobsen, R. T.; Stewart, R. B.; Jahangiri, M. J. Phys. Chem. Ref: Data 1986, 15, 735. (14) Streett, W. B.; Staveley, L. A. K. Adv. in Cryog. Eng. 1968, 13, 363. (15) Streett, W. B.; Staveley, L. A. K. J. Chem. Phys. 1971, 55, 2495. (16) Streett, W. B.; Sagan, L. S.;Staveley,L. A. K. J. Chem. Thermodyn. 1973, 5, 633. JP9506595
