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A correlation of critical points
Critical points were correlated for fluids which obey the three-parameter equation o f state according to which the compressibility factor, Z=P/dR T, is given by Z= 1 + kBSt~ (0, 8), in which K (0, 8) is a universal function o f reduced temperature, 0 = T/I"B, and reduced density, 8 = d/d o. The three parameters which distinguish one fluid from another are k B, TB, and d o. It is shown that experimental inflection points o f the pressure versus density isotherms o f gases obeying this equation o f state all fall on the same reduced curve o f 0 versus 8. Critical points also must lie on this curve, at a location dependent
on k B. Critical densities are too uncertain experimentally to permit a definite correlation, but the critical temperatures o f sixteen fluids are shown to fall on a common 0c versus k B curve with an average deviation o f O.4 degrees. The critical pressures o f these same fluids lie within a few tenths o f an atmosphere on a common Irc (Tr = P/doRT B) versus k B line. This correlation o f critical points is accomplished using values o f the three parameters that had previously been determined by fitting supercritical PVT data to the equation o f state.
A correlation of critical points E. M. Holleran, R. E. Walker, and C. M. Ramos
The purpose of this paper is to investigate the correlation of critical points implicit in the three-parameter reduced equation of state according to which the compressibility factor, Z =P/(dRT), is gt'ven by Z = 1 +kB 8 K (0,8)
(1)
In this equation 0 is the reduced temperature, 8 is the reduced density, with T d 0 = --, ~ = -rB ao
(2)
The three constants TB, do, and k B are characteristic of each fluid. The quantity K is a function of 0 and 8 only, and the states of the fluid for which K = O and Z = 1 are represented by the unit-compressibility law (UCL): 0 +6 = 1
(3)
Equation 1 holds accurately for compressed liquids, 1 and recently it has been applied with considerable success to gases at super-critical temperatures and volumes. 2 With a single expression for to, and only the three adjustable parameters, it reproduces within experimental error a large quantity of PVT data in this range. For such gases as methyl fluoride, tetrafluoro-methane, ethylene, propene, isobutane, neopentane, perfluorocyclobutane, carbon dioxide, and hydrogen sulphide, the average deviation in Z is a few parts per thousand. An even more excellent fit, with an average deviation in Z of a few parts per ten thousand is obtained with the very precise literature data for neon, argon, krypton, xenon, nitrogen, oxygen, fluorine, hydrogen at high 0, carbon monoxide, methane, ethane, and many mixtures. Poor fits, with deviations greater than 1%, were found for water, methyl alcohol, and hydrogen at low 0. The authors are with the Chemistry Department, St. John's University, Jamaica, NY 11439, USA. Received23 August1974.
210
Because of the accuracy, simplicity, and wide applicability of this equation of state, it is of interest to investigate its implications for the critical point. The first problem encountered in attempting to do this is the fact that PVT data in the vicinity of the critical point, though often precise, are not usually accurate. 3 This is due to several factors, including the extremely rapid change in density with pressure in this region. In fact, in order to obtain the excellent data fits of (1) it was necessary to avoid the critical region entirely. Thus, for a given fluid, • (0, 6) cannot usually be determined reliably from its own experimental data near the critical point. However, there are several avenues open to us. First, we can investigate, as we do in this paper, the qualitative critical behaviour of fluids which obey an equation of state of the general form of (1). Other possibilities then become apparent and will be discussed below. Behaviour of dissimilar fluids
By dissimilar fluids we mean those having appreciably different values of the third characteristic constant, kB. In Table 1 the fluids mentioned above are listed, together with the values of TB, do, and kB found by Holleran and Hammes 2 by minimizing the rms percentage deviations between experimental and calculated Z. In Fig.1 are shown the isotherms of Z versus 6, as calculated from (1), for two fluids, A and B, at the same reduced temperature, 0 = 0.39. The curves are drawn for two different values ofkB, 1.94 for A, and 2.37 for B, so that A behaves like methane, and B like ethane. The actual temperatures (T = OTB) represented are 198.8 K for methane, which is above its critical temperature of 190.7 K, and 296.1 K for ethane, which is below its critical temperature of 305.3 K. Both curves cross the Z = 1 line at 6 = 0.61, in accordance with (3). The dotted portion of the curve for ethane lies in the two-phase region. We see that Z is always farther from unity for ethane than it is for methane at equal 0s and equal 6s. In fact, since
CRYOGENICS. APRIL 1975
region where this can be verified. It is reasonable to assume that they continue to be identical when one of them penetrates its two-phase region. If this is true, then the subcritical behaviour of some fluids can be calculated from the supercritical behaviour of others (having smaller kB values).
1.0
In Fig.2 are plotted, for the same two fluids A and B, the 0 = 0.39 isotherms o f reduced pressure n, where P -
-
06Z
(4)
d o R TB
N
0.5 \\ k
\
\
x
NX \ N
I
O
I
I
t
OI.
I
I
0.3
t
0.5
5
Fig.1 Isotherms of Z versus 8 for t w o gases at 0 = 0.39. Note t h a t Z = 1 a t 6 = 1--0. The ratio of ( Z - - l ) for the t w o gases is constant. The dashed section of curve B is in t h e t w o phase region
The 7r isotherms of Fig.2 are thus proportional to the Z isotherms of Fig. 1 multiplied by 6, which pulls the curves down to zero n at zero 6. In Fig.2 it is seen that curve A for methane loses the minimum it had in Fig.l, but curve B for ethane, which had a lower minimum in the Z isotherm, still retains a minimum in the pressure isotherm. The pressure then exhibits a van der Waals' loop in the two phase region. We can calculate these sub-critical pressure loops for fluids having large k B values from the general corresponding experimental super-critical behaviour of gases with lower k B values. This then opens the possibility that, by applying the Maxwell integral to these loops, we can calculate the vapour pressures and orthobaric densities of some fluids from the supercritical P V T behaviour of others. This will be attempted in a later report. For the present, we will investigate the implied correlation of the critical points themselves.
0.10
Pressure inflection points It has been shown by Holleran 4 that the reduced critical points of fluids which obey (1) all lie on the same 0 versus 6 curve. This can be seen as follows. At the critical temperature, the isotherm of n versus fi exhibits a flat (n '= O) inflection point (n" = O). (The primes indicate partial derivatives with respect to 6.) From (1) and (4) we have n = 0 (6 + k B 62K)
0.05
(5)
Differentiation gives / r ' = 0 [1 + k B (62 K ' + 2 6a)]
(6)
and n" = Ok B (4 6K '+ 82 K" + 2 K)
V
o
1
I
I
o.I
0.3
I
[
0.5
5
Fig.2 Isotherms of reduced pressure for the same t w o gases as in F i g . l , at the same reduced temperature. Again, the dashed section of curve B is in the t w o phase region, where it exhibits a van der Waals' loop
r (0, 6) is the same for these (as for all the fluids under discussion), Z is always 2.37/1.94 = 1.22 times as far from unity for ethane as it is for methane. However, when ( Z - 1 ) / k B isotherms are plotted for these fluids, the curves for equal 0 are identical within experimental error in the
CRYOGENICS.
APRIL
1975
(7)
Since ¢ and its derivatives are functions of 0 and 6 only, (7) shows that the locus of the 0, 6 points for which n" vanishes is independent of ka. The inflection densities of all pressure isotherms of all gases obeying (1) should therefore lie on a single 0 versus 6 curve. This is tested by experimental data in the next section. In addition, the critical points (0c, 6c) of all these fluids must also lie on this same curve. Setting n ' = O in (6) shows that the location of the critical point on this curve depends on the value of kB. Therefore 0c, 6c, lrc, and Zc all depend on kB. We have already seen in Figs 1 and 2 that 0c is greater than 0.39 for ethane (kB = 2.37), and 0c is less than 0.39 for methane (kB = 1.94). The detailed experimental dependence of 0 c on kB is described later.
211
The common
inflection point curve
In this section we verify from experimental data that isothermal inflection points of rr versus 8 do lie on a common 0 versus 8 curve. Because o f the anticipated difficulty o f finding reliable second derivatives from experimental data, considerable care was taken in evaluating the isothermal inflection densities. At first we found least squares polynomials o f P versus d, Z versus d, and K versus d. (K = (Z- 1)/d = K VB, where VB = kB/do). Where available, we took densities up to or a little beyond critical. In this range, orders of five or more in the pressure polynomial provided satisfactory fits of the data. Orders o f four or more were satisfactory for Z, and three or more for K, which is consistent with the progressive division by d in changing from P to Z to K. At first, inflection densities were calculated from polynomials of orders 5, 6, and 7 for P; 4, 5, and 6 for Z; and 3, 4, and 5 for K. The most self-consistent results were found with the K isotherms, and the densities for the 4th and 5th orders agreed to within a few hundredths of a mole 1-1 in most cases, Larger differences (0.1 or 0.2 mole 1"1 occasionally resulted from adding or subtracting data points. We decided to calculate all inflection densities from the fourth power polynomials o f K. As the lowest-order reliable polynomial this had the advantage of depending the least on curve-fitting, o f avoiding the possible introduction o f extraneous wiggles, and of requiring fewer data points per isotherm. We decided to use only those isotherms for which at least eight data points were available. To avoid the possible distortion o f curvature at the ends of the data range, we required at least two experimental points beyond (above and below) the calculated inflection density. Isotherms above TB have no inflection points (see below), and usually we had to stay below about 0.85 TB to obtain inflection densities in the data range. Finally, no reliable results could be found for isotherms near the critical temperature, where inflection densities are near the critical density. Only isotherms with temperatures at least twenty degrees above the critical temperature could be included in most cases. Table 2 lists the inflection densities calculated in this way for the twelve gases for which the isotherms satisfied all the above conditions. We estimate most o f them to be correct to within about -+ 0.05 mole 1-1. Using the TB and do values in Table 1, we plotted the reduced temperatures, 0, and the reduced inflection densities, 8. The points for all the gases fell very close to the c o m m o n curve shown in Fig.3. The next step was to find an empirical representation of this c o m m o n 0 dependence of the reduced inflection densities. We made use o f the fact that the 0 versus ~ curve begins at zero 6 with 0 = 1 and 30/0/5 = - 3 . This can be seen as follows. The function K (0, 8) can be expressed in the virial series K =/3 +~,6 + A 6 2 + ....
(8)
in which 13= B/VB, 7 = Cdo/VB, A = D doZ /Va ...... and B, C, D, .., are the ordinary virial coefficients, which for a given fluid are functions of temperature only. Equation 7 shows that at zero density the inflection curve begins with K = O, so that/3 = O and 0 = 1. That is, the curve begins at the Boyle temperature, with T-- TB, where the second virial coefficient is zero. At this point d/3/dO = 1, because VB =
212
Table 1. Characteristic constants for the HH equation of state Gas
kB
TB
do
6
1.20
7 8
1.81 1.81
109.5 575.6 795.2
50.75 37.45 29.04
Reference
H2 Kr Xe Ar CH3F CH4
9
1.87
10 11
1.89 1.94
407.6 841 509.7
46.8 25.1 35.58
02
12
1.98
408.7
Ne
13
2.09
115.3
N2
14
2.14
324.3
47.9 90.1 40.67
F2 CO
15 16 17
2.21 2.22 2.24
366 335 715
55.03 41.0 27.8
18 20
2.36 2.36 2.37
926 898 759.2
37.7 19.3 25.48
C3H 8 CO2 neo- C5H12
21 22 23
2.54 2.75 2.75
C4H10 CF 4 cyclo-C4F8
24 25 26
2.80 3.02 4.31
C2H4 H2S C3 H6 C2H 6
iso-
°
19
894 714 1011 955 518.7 792
18.7 40.4 11.95 15.1 28.07 14.08
The gases listed are those for which good (few parts per thousand) to excellent (few parts per ten thousand) fits of super critical P V T data have been found thus far. It should be emphasized that these values are not unique, and rather different sets can still provide satisfactory data fits
T(dB/dT) at TB. 5 Evaluating K 'and t~" from (8), and substituting in (7) gives / 3 + 3 7 8 +6A62 + .... = 0
(9)
as the equation for the 0 versus 6 inflection point curve in terms of the reduced virial coefficients. Differentiating (9) we find that on this curve,
(10)
so that the initial slope, at 6 = O, is given by
('0)
(11)
(It is interesting to note that the inflection point curve follows roughly, but not exactly, the equation 0 = ( 1 - 6 ) . 3) Similarly, it can be shown that the 0 versus/5 curve connecting the isothermal Z versus 6 minima begins with a slope of - 2 , and roughly obeys the equation 0 = ( 1 - 8 ) 2 ; and, of course, the Z = I (UC) line obeys the relation, 0 = ( 1 - 6 ) . Various empirical forms were investigated for the inflection point curve. The following equation, explicit in 6, was finally adopted: 6 = ( 1 - 0 ) / 3 + C2 ( 1 - 0 ) 2 + C3 ( 1 - 0 ) 3
(12)
CRYOGENICS. APRIL 1975
Table 2. Observed and calculated inflection densities
1.0
UCL
Gas
7", K
Kr
298.15 323.15 348.15 373.15 398.15 423.15 448.15 473.15
~
7.20 6.40 5.62 4.86 4.17 3.52 2.90 2.29
0.5180 0.5614 0.6048 0.6483 0.6917 0.7351 0.7786 0.8220
0.1921 0.1710 0.1500 0.1300 0.1113 0.0941 0.0775 0.0610
7.21 6.40 5.63 4.90 4.20 3.54 2.91 2.31
0.01 0,00 0.01 0.04 0.03 0.02 0.01 0.02
Xe
323.15 348.15 373.15 398.15 423,15
7.45 6.86 6.31 5.81 5.34
0.4064 0.4378 0.4693 0.5007 0.5321
0.2567 0.2361 0.2174 0.2001 0.1839
7.40 6.86 6.35 5.86 5.38
-0.05 0.00 0.04 0.05 0.04
Ar
173.15 188.15 203.15 223.15 248.15 273.15 298.15 323.15 348.15
11.37 10.35 9.41 8.22 6.87 5.64 4.49 3.39 2.35
0.4248 0,2428 0.4616 0,2214 0.4984 0,2011 0.5475 0,1756 0.6088 0,1469 0,6701 0,1206 0.7315 0,0960 0.7928 0.0724 0.8541 0.0503
11.42 10.43 9.49 8.32 6.95 5.68 4.50 3.39 2.34
0.05 0,07 0,08 0,10 0,08 0,04 0,01 0,00 -0.01
CH 4 273.15 298.15 323.15 348.15 373.15 398.15 423.15
6.45 5.63 4.84 4.10 3.42 2.72 2.10
0.5359 0.1813 0.5849 0.1581 0.6340 0.1358 0.6830 0.1151 0.7320 0.0960 0,7811 0.0765 0.8301 0,0589
6.53 5.68 4.88 4.12 3.41 2.73 2,09
0.08 0.05 0.04 0.02 -0.01 0,01 -0.01
10.95 10.60 10.26 9.93 9.30 8.68 8.11 7.54 6.98 6.48 6.03 5.50 5.04 4.63
0.4527 0.2286 0.4649 0.2214 0.4771 0.2143 0.4894 0.2074 0.5138 0.1942 0.5383 0.1813 0.5628 0.1692 0.5872 0.1573 0.6117 0.1456 0,6362 0.1353 0.6606 0.1258 0.6851 0.1148 0.7096 0.1053 0.7340 0.0966
10.92 10.59 10.27 9.95 9.33 8.73 8.15 7.59 7.05 6.52 6.01 5.51 5.02 4.55
-0.03 -0.01 0.01 0.02 0.03 0.05 0.04 0.05 0.07 0.04 -0.02 0.01 -0,02 -0.08
8.15 5.71 2.21
0,04 0.01 -0.04
(D
0.6
0.4
O
O.I
6
0,2
0.3
Fig.3 The reduced inflection point curve. The common locus of 0 versus 6 for the isothermal inflection points of ¢r versus ~ for all corresponding gases is shown, along with the unit compressibility line (0 = 1--6) and the two-phase boundaries for gases A and B (methane and ethane). The three marks show the predicted location of the critical point on the inflection point curve for values of k B of 2, 3, and 4. The dotted line shows the approximate location of some reported critical points; others are more scattered
A least squares fit of (12) to the experimental points gave C2 = 0.0324, and 6"3 = 0.2180 for the range of 6 from O to about 0,26. Equation 12 is related in Fig.3. Table 2 includes values of inflection densities calculated from (12) using TB and do values from Table 1. The agreement with the experimental values is very good. The rms deviaition is 0.04 m 1q and the average absolute deviation is 0.03 m 1-1. An even better agreement can be obtained by changes in TB and do which are within their experimental uncertainty. For example, the calculated and experimental inflection densities for CF4 can be made to agree within 0.05 m 1-1 (0.03 average) by using TB = 519 K and do = 28.25 m 1"1 , and these constants still permit a satisfactory P V T data fit by the HolleranHammes (HH) equation. We conclude that the inflection densities of all the gases listed in Table 2 lie, within experimental error, on the same reduced temperature versus reduced density curve, as required by (1). If the more common corresponding-states reduction by the critical constants were employed, then the T/Tc versus did e curves of all the gases would include the common point (1, 1). But at lower densities they would all diverge and intercept the zero-density axis at the various values of TB/Tc = 1/0e, which range from 2.02 to 2.75 for the gases in Table 2. This is another example of the inadequacy of the critical constants in corresponding-states correlations. 4 Reduced inflection densities can also be calculated directly from the HH equation of state with given n (0, 5). When this is done it is found that the HH values of 6 agree to well within 0.001 with those calculated from (12) for 0 from 1.0 down to 0.5. At lower 0 the agreement worsens, with the difference reaching 0.004 at 0 = 0.4. Evidently the HH equation of
CRYOGENICS. APRIL 1975
d (calc), Ad, ml 1 m1-1
0
0.8
dm1-1
02
185 190 195 200 210 220 230 240 250 260 270 280 290 300
N2
163.15 203.15 273.15
8.11 5.70 2.25
F2
175.02 180.03 185.03 190.03 195.03 200.04 210.03 220.03 230.03 240.02 250.01 260.01
11.83 11.40 10.99 10.58 10.19 9.81 9.06 8.32 7.68 7.07 6.38 5.71
0.5031 0.6264 0.8423
0.1993 0.1401 0.0553
0.4782 0.2149 0.4919 0.2072 0.5055 0.1997 0.5192 0.1923 0.5329 0.1852 0.5465 0.1783 0.5739 0.1646 0,6012 0.1512 0.6285 0.1396 0.6558 0.1284 0.6831 0.1160 0.7104 0.1038
"11.76 -0.07 "11.35 -0.05 "10.96 -0.03 "10.57 -0.01 '10.18 -0.01 9.80 -0.01 9.07 0.01 8.36 0.04 7.68 0.00 7.01 -0.06 6.37 -0.01 5.75 0.04
213
Gas
T,K
d m 1-] 0
(5
d (calc), Ad, m 1-1 m 1l
270.00 280.00 289.99 299.99
5.18 4.57 4.03 3.41
0.7377 0.7650 0.7923 0.8197
0.0941 0.0832 0.0733 0.0619
5.15 4.56 3.99 3.44
-0.03 -0.01 -0.04 0.03
C2H4 348.15 373.15 398.15 423.15
5.82 5.29 4.80 4.34
0.4869 0.5219 0.5569 0.5918
0.2092 0.1904 0.1727 0,1560
5.81 5.30 4.81 4.35
-0.01 0.01 0.01 0.01
C2H6 348.15 373.15 398.15 423.15 448.15 473.15 498.15 523.15
5.75 5.26 4.83 4.38 3.98 3.60 3.23 2.89
0.4586 0.4915 0.5244 0.5574 0.5903 0.6232 0.6562 0.6891
0.2255 0.2065 0.1897 0.1719 0.1562 0.1412 0.1269 0.1132
5.72 5.26 4.82 4.40 4.00 3.61 3.24 2.89
-0.03 0.00 -0.01 0.02 0.02 0.01 0.01
iso- 448.15 C4H10473.15 498.,15 523.15 548.15 573.15
3.38 3.14 2.91 2.71 2.50 2.31
0.4693 0.4954 0.5216 0.5478 0.5740 0.6002
0.2238 0.2076 0.1924 0.1793 0.1656 0.1528
3.30 3.09 2.88 2.68 2.49 2.30
-0.08 -0.05 -0.03 -0.03 -0.01 -0.01
CF4 273.15 298.15 303.15 323.15 348.15 373.15 398.15 423.15 448.15
5.37 4.68 4.54 4.03 3.44 2.88 2.35 1.85 1.35
0.5266 0.5748 0.5844 0.6230 0.6712 0.7194 0.7676 0.8158 0.8640
0.1912 0.1666 0.1618 0.1436 0.1224 0.1027 0.0838 0.0657 0.0480
5.28 .4.61 4.49 3.98 3.39 2.83 2.30 1.79 1.30
-0.09 -0.07 -0.05 -0.05 -0.05 -0.05 -0.05 -0.06 -0.05
cyclo-423.15 C4F8 448.15 473.15 498.15 523.15 548.15
2.67 2.44 2.22 2.00 1.80 1.61
0.5343 0.5658 0.5974 0.6290 0.6605 0.6921
0.1898 0.1730 0.1574 0.1422 0.1281 0.1146
2.59 2.38 2.16 1.96 1.77 1.58
-0.08 -0.06 -0.06 -0.04 -0.03 -0.03
0.00
The observed values are f r o m 4th order polynomials of (Z--1)/d versus d. The data references are the same as in Table 1. For Ar, the points at T = 248.15 K and lower are from reference 9b; the others are f r o m 9a. The calculated densities are from (12)
state, though accurate for PVT points down to 0 = 0.4 or less, nevertheless begins to deviate in its second derivative, tf 7r , for 0 < 0.5 and 8 > 0.2.
Critical densities We have already noted that the critical points of all fluids obeying (1) with the same K (0, 8) must lie on this same inflection point curve (or its extension to somewhat higher densities). Thus the curve in Fig.3 is also a plot of 0 c versus 6c. Any attempt to verify this is confronted with the wide variation in reported values o f critical densities. Differences of several percent are common. An accurate correlation is
214
not possible under these circumstances. However, if the des given in reference 27 are used, the 0c, 8c points for Kr, Ar, Xe, 02, N2, CO, and CII 4 fall within 1% on a common curve. This line is nearly straight and lies above the extension of (12) by about 2% at 0 = 0.40 and about 4% at 0 = 0.36, as shown by the dotted curve in Fig.3. It is possible that the extended reduced inflection point curve could bend up to match this line. But points for the other fluids are even higher (except for H2 which falls below our line), and scattered. Perhaps our simple representation is not adequate for the critical region, and r (0, 5), which corresponds so well in the supercritical region and the liquid region, goes its separate way for each fluid around its critical point. But the disagreement among reported critical densities is too great to permit any defmite conclusion.
Critical temperatures Fortunately, critical temperatures are known much more accurately than critical densities. Reported values o f Tc usually agree to within a few tenths of a degree. We can therefore hope to correlate these. In the section on pressure inflection points we saw that the location o f the critical point on the inflection point curve depends on the value of the characteristic constant kB. In this section we investigate the dependence of Oc on k a. The HH equation for r (0, 8) will not be useful here, since 8e < 0.5 for the fluids under consideration. Table 3 lists Tc for sixteen fluids. Fig.4 shows a plot o f 0c versus k a for these fluids, found by using T~ and k B from Table 1. Once again, these points fall close to a common curve. By a least squares fit, omitting CO, the following dependence o f 0 c on ka was derived 0c = 0.1839 + 0.1230 kB -- 0.0128 kB2
(13)
for ka from 1.8 to 3.0. The solid curve in Fig.4 represents this equation. To show the closeness o f the correlation, Table 3 includes values of Tc calculated from (13) and the constants of Table 1. The average difference between calculated and observed critical temperatures for these fluids is less than 0.4 degree. Hydrogen with k B = 1.20 and 0 c = 0.301, and C4F8 with kB --- 4.31 and Oc = 0.490, lie far beyond the limits o f ( 1 3 ) but on reasonable extensions of it. They could be used to extend the curve if necessary, though they are isolated points at the present time. Points for the other fluids o f Table 1, namely methyl fluoride, propene, and neopentane, lie above the curve. Still, they are within 1 or 2% of(13), and so could be brought into agreement by corresponding changes in Ta. Such changes may be possible within the experimental uncertainty of this parameter, as discussed in the next section. Thus the correlation o f reduced critical temperatures predicted by (1) is excellent for many substances, and at least fair for some others.
Critical pressures Table 3 also lists Pc and lre for the sixteen gases that follow the smooth 0 c versus ka curve. It can be seen that 7rc is nearly constant. Iflr c is taken as 0.0308, then
CRYOGENICS. APRIL 1975
Table 3. Observed and calculated critical temperatures and critical pressures Fluid
CP ref
Tc, K
0c(calc)
Tc,K (calc)
ATc, K
Pc, atm
¢rc
Pc, atm (calc)
APc, atm
Kr
27
209.4
0.3646
209.9
+ 0.5
54.3
0.00307
54.5
+ 0.2
Xe
27
289.8
0.3546
289.9
+ 0.1
58.0
0.00306
58.4
+ 0.4
Ar
150.9
0.3691
150.4
-0.5
48.3
0.00309
48.3
0.0
CH 4
27
9b
190.7
0.3743
190.8
+ 0.1
45.8
0.00308
45.9
+ 0.1
02
12
154.6
0.3773
154.2
-0.4
49.8
0.00310
49.5
-0.3
Ne
28
44.4
0.3851
44.4
0.0
26.2
0.00307
26.3
+ 0.1
N2
27
126.2
0.3885
126,0
-0.2
33.5
0.00310
33.4
-0.1
F2
15
144.3
0.3932
143,9
-0.4
51.4
0.00311
51.0
-0.4
CO
27
133.0
0.3939
132,0
-1.0
34.5
0.00306
34.7
+ 0.2
C2H 4
27
283.1
0.3952
282.6
-0.5
50.5
0.00310
50.3
-0.2
H2S
27
373.6
0.4029
373,1
--0.5
88.9
0.00310
88.3
-0.6
C2H 6
20
305.3
0.4035
306,3
+ 1.0
48.1
0.00303
48.9
+ 0.8
C3H 8
27
369.9
0.4138
369,9
0.0
42.0
0.00306
42.3
+ 0.3
CO2
27
304.2
0.4253
303.7
-0.5
72.9
0.00308
73.0
+ 0.1
iso-C4H10
27
408.1
0.4280
408.7
+ 0.6
36.0
0.00304
36.5
+ 0.5
CF 4
29
227.5
0.4386
227.5
0.0
37.0
0.00310
36.8
-0.2
Pc = 0.00253 doT B
(14)
for kB from 1.8 to 3.0. Pc values calculated from (14) with the constants from Table 1 agree with observed values to within a few tenths o f an atmosphere. The average deviation is less than 0.3 arm. Since reported critical pressures often disagree by this amount, the agreement may be considered to be within experimental error. Actually, 7rc versus k B appears to curve slightly, and to have a maximum o f about 0.0309. This impression is enhanced if the 7rc, kB points for Ha (0.285, 1.20)and C4F 8 (0.299, 4.31) are taken into account. Once again, these are isolated points, but they might be used to extend the curve if necessary.
0.44
~o.4o
The three fluids (propene, neopentane, methyl fluoride) which fell one or two percent above the 0 e versus k B curve, now fall from 4 -9% above the 7rc versus k B curve, well beyond the usual experimental error in Pc- Evidently these fluids do not fit with the others, although conceivably the deviations could be due to inaccuracies in the constants TB, do, and kB. Ambiguities
in t h e constants
It is significant t h a t the above correlations o f critical temperatures and critical pressures were achieved with the use of values o f TB, do, and kB found by fitting supercritical PVT data to the Htt equation of state. Data near the critical point are not needed, and in fact must be disregarded in the fit even if available, as already noted. The results are even more remarkable when the uncertainties in the three constants are considered. If we assume a group of fluids that exactly obey (1) with the same ¢ (0, 6), then each has a true set o f values for the three constants. However, it turn~ out to be very difficult to find these true
CRYOGENICS. APRIL 1975
0.36
118
~
21.2
~
216
~
310
kB
Fig.4 Reducedcritical temperaturesversus the characteristic constant, k B. The points are for the sixteen fluids in Table 3. The solid curve is for (13) found by a least squaresfit omitting CO, the point abovethe curve
215
values from P V T data. The reason for this is the general similarity in behaviour o f r (0, 8) in neighbouring regions o f temperature and density. From data o f limited range or accuracy it is therefore difficult to pinpoint the exact O, 8 region, and the exact values o f Ta and do and hence kB. For this reason the values o f the constants listed in Table 1 are not unique. Other sets o f values will still produce satisfactory P V T fits. Changes in one constant can largely be compensated for by related adjustments in the others. Generally speaking if TB i~ increased, do and kB decrease. Fortunately the magnitudes and directions o f these compensating changes are such that, if one set o f constants places an inflection density on the c o m m o n 0 versus 6 curve, another set will usually displace the point to a new location which is still approximately on the curve. If one set locates a point off the curve, other sets tend to move the point nearly parallel to the curve. If the points are not too far off, then acceptable constants can be found that will bring them into line, as was seen with CF4. An example o f two rather different compensating sets of constants is provided by the values o f TB and do found for two different sets o f P V T data for ethane 2o,3o to be 759.2K, 25.48 m 1-1 , and 753.OK, 25.65 m 1-1 . Both sets locate the inflection densities approximately on the common 0 versus 8 curve, though at different positions. A similar compensation occurs with rrc and with the 0 c versus k B curve. For example, two different sets o f P V T data from neon 28,31 gave k~, TB, do values o f 2.09, 115.3K, 90.1 m 1"1 . The critical temperature calculated from (13) is 44.4 K for the first set and 44.3 K for the second. The critical pressure calculated from (14) is 26.3 atm for the first set and 26.1 atm, for the second. Thus it appears that it is not necessary for the present purpose to know the true values o f the three constants o f fluids obeying (1). Even though these constants cannot easily be determined unambiguously, nevertheless, a set o f values that provides a good fit of supercritical P V T data will also correlate the critical points.
References 1
216
Holleran, E. M. Cryogenics 11 (1971) 19
2 3 4 5 6 7 8 98 9b 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3t
Holleran, E. M., Hammes, J. Cryogenics 15 (1975) 95 Engelstaff, P. A., Ring, J.W. [Temperley, H.N.V., Rowlinson, J. S., Rushbrooke, G. S. (eds)] Physics of Simple Liquids (North-Holland, Amsterdam, 1968) Ch 7, p 256 Holleran, E. M. Cryogenics 15 (1975) 137 ltolleran, E. M. JPhys Chem 73 (1969) 167 Goodwin, R. D. JResNBS: A Phys and Chem 71A (1967) 203 Beattie, J. A., Brierly, J. S., Baxriault, R. J. J Chem Phys 20 (1952) 1613 Michels, A., Wassenaar, T., Louwerse, P. Physica 20 (1954) 99 Michels, A., Wijker, H., Wijker, H. K. Physica 15 (1949) 627 Michels, A., Levelt, J. M., DeGraaff, W. Physic¢ 24 (1958) 659 Michels, A., Visser, A., Lunbeck, R. J., Wolkers, G. J. Physica 18 (1952) 114 Douslin, D. R., Harrison, R. H., Moore, R. T. J Chem Eng Data 9 (1964) 358 Weber, L. A. JResNBS: A Phys and Chem 74A (1970) 93 Sullivan, J. A., Sonntag, R. E. Advances in Cryogenic Engineering 12 (1967) 706 Cmin, Jr, R. W. Sonntag, R. E. Advances in Cryogenic Engineering 11 (1966) 379 Prydz, R., Straty, G. C. JResNBS:APhysandChem 74A (1970) 747 Bartlett, E., Hetherington, H., Kvalnes, H., Tremearne, T. JACS 52 (1930) 1374 Miehels, A., Geldermans, M. Physica 9 (1942) 967 Reamer, H. H., Sage, B. H., Lacey, W. N. IndandEngChem 42 (1950) 140 Miehels, A., Wassenaar, T., Louwerse, P., Wolkers, G. J. Physica 19 (1953) 287 Douslin, D. R., Harrison, R. H. J Chem Thermo 5 (1973) 491 Beattie, J. A., Kay, W. C., Kaminsky, J. JACS 59 (1973) 1589 Michels, A., Michels, C. Proc Roy Soc A 153 (1935) 201 Beattie, J. A., Douslin, D. R., Levine, S. W. J Chem Phys 20 (1952) 1619 Beattie, J. A., Marple, Jr, S., Edwards, D. G. J Chem Phys 18 (1950) 127 Douslin, D. R., Harrison, R. H., Moore, R. T. J Phys Chem 71 (1967) 3477 Douslin, D. R., Moore, R. T., Waddington, G. J Phys Chem 63 (1959) 1959 Reid, R., Sherwood, T. K. The Properties of Gases and Liquids, 2nd edn (McGraw-Hill, 1966) MeCarty, R. D., Stewart, R. B. Advances in Thermophysical Properties 84 (ASME, 1965) Martin, J. B., Bhada, R. K. AIChE J 17 (1971) 683 Reamer, H. H., Sage, B. H., Lacey, W. N. Ind & Eng Chem 36 (1944) 956 Gibbons, R. M. Cryogenics 9 (1969) 251
CRYOGENICS. APRIL 1975
on k B. Critical densities are too uncertain experimentally to permit a definite correlation, but the critical temperatures o f sixteen fluids are shown to fall on a common 0c versus k B curve with an average deviation o f O.4 degrees. The critical pressures o f these same fluids lie within a few tenths o f an atmosphere on a common Irc (Tr = P/doRT B) versus k B line. This correlation o f critical points is accomplished using values o f the three parameters that had previously been determined by fitting supercritical PVT data to the equation o f state.
A correlation of critical points E. M. Holleran, R. E. Walker, and C. M. Ramos
The purpose of this paper is to investigate the correlation of critical points implicit in the three-parameter reduced equation of state according to which the compressibility factor, Z =P/(dRT), is gt'ven by Z = 1 +kB 8 K (0,8)
(1)
In this equation 0 is the reduced temperature, 8 is the reduced density, with T d 0 = --, ~ = -rB ao
(2)
The three constants TB, do, and k B are characteristic of each fluid. The quantity K is a function of 0 and 8 only, and the states of the fluid for which K = O and Z = 1 are represented by the unit-compressibility law (UCL): 0 +6 = 1
(3)
Equation 1 holds accurately for compressed liquids, 1 and recently it has been applied with considerable success to gases at super-critical temperatures and volumes. 2 With a single expression for to, and only the three adjustable parameters, it reproduces within experimental error a large quantity of PVT data in this range. For such gases as methyl fluoride, tetrafluoro-methane, ethylene, propene, isobutane, neopentane, perfluorocyclobutane, carbon dioxide, and hydrogen sulphide, the average deviation in Z is a few parts per thousand. An even more excellent fit, with an average deviation in Z of a few parts per ten thousand is obtained with the very precise literature data for neon, argon, krypton, xenon, nitrogen, oxygen, fluorine, hydrogen at high 0, carbon monoxide, methane, ethane, and many mixtures. Poor fits, with deviations greater than 1%, were found for water, methyl alcohol, and hydrogen at low 0. The authors are with the Chemistry Department, St. John's University, Jamaica, NY 11439, USA. Received23 August1974.
210
Because of the accuracy, simplicity, and wide applicability of this equation of state, it is of interest to investigate its implications for the critical point. The first problem encountered in attempting to do this is the fact that PVT data in the vicinity of the critical point, though often precise, are not usually accurate. 3 This is due to several factors, including the extremely rapid change in density with pressure in this region. In fact, in order to obtain the excellent data fits of (1) it was necessary to avoid the critical region entirely. Thus, for a given fluid, • (0, 6) cannot usually be determined reliably from its own experimental data near the critical point. However, there are several avenues open to us. First, we can investigate, as we do in this paper, the qualitative critical behaviour of fluids which obey an equation of state of the general form of (1). Other possibilities then become apparent and will be discussed below. Behaviour of dissimilar fluids
By dissimilar fluids we mean those having appreciably different values of the third characteristic constant, kB. In Table 1 the fluids mentioned above are listed, together with the values of TB, do, and kB found by Holleran and Hammes 2 by minimizing the rms percentage deviations between experimental and calculated Z. In Fig.1 are shown the isotherms of Z versus 6, as calculated from (1), for two fluids, A and B, at the same reduced temperature, 0 = 0.39. The curves are drawn for two different values ofkB, 1.94 for A, and 2.37 for B, so that A behaves like methane, and B like ethane. The actual temperatures (T = OTB) represented are 198.8 K for methane, which is above its critical temperature of 190.7 K, and 296.1 K for ethane, which is below its critical temperature of 305.3 K. Both curves cross the Z = 1 line at 6 = 0.61, in accordance with (3). The dotted portion of the curve for ethane lies in the two-phase region. We see that Z is always farther from unity for ethane than it is for methane at equal 0s and equal 6s. In fact, since
CRYOGENICS. APRIL 1975
region where this can be verified. It is reasonable to assume that they continue to be identical when one of them penetrates its two-phase region. If this is true, then the subcritical behaviour of some fluids can be calculated from the supercritical behaviour of others (having smaller kB values).
1.0
In Fig.2 are plotted, for the same two fluids A and B, the 0 = 0.39 isotherms o f reduced pressure n, where P -
-
06Z
(4)
d o R TB
N
0.5 \\ k
\
\
x
NX \ N
I
O
I
I
t
OI.
I
I
0.3
t
0.5
5
Fig.1 Isotherms of Z versus 8 for t w o gases at 0 = 0.39. Note t h a t Z = 1 a t 6 = 1--0. The ratio of ( Z - - l ) for the t w o gases is constant. The dashed section of curve B is in t h e t w o phase region
The 7r isotherms of Fig.2 are thus proportional to the Z isotherms of Fig. 1 multiplied by 6, which pulls the curves down to zero n at zero 6. In Fig.2 it is seen that curve A for methane loses the minimum it had in Fig.l, but curve B for ethane, which had a lower minimum in the Z isotherm, still retains a minimum in the pressure isotherm. The pressure then exhibits a van der Waals' loop in the two phase region. We can calculate these sub-critical pressure loops for fluids having large k B values from the general corresponding experimental super-critical behaviour of gases with lower k B values. This then opens the possibility that, by applying the Maxwell integral to these loops, we can calculate the vapour pressures and orthobaric densities of some fluids from the supercritical P V T behaviour of others. This will be attempted in a later report. For the present, we will investigate the implied correlation of the critical points themselves.
0.10
Pressure inflection points It has been shown by Holleran 4 that the reduced critical points of fluids which obey (1) all lie on the same 0 versus 6 curve. This can be seen as follows. At the critical temperature, the isotherm of n versus fi exhibits a flat (n '= O) inflection point (n" = O). (The primes indicate partial derivatives with respect to 6.) From (1) and (4) we have n = 0 (6 + k B 62K)
0.05
(5)
Differentiation gives / r ' = 0 [1 + k B (62 K ' + 2 6a)]
(6)
and n" = Ok B (4 6K '+ 82 K" + 2 K)
V
o
1
I
I
o.I
0.3
I
[
0.5
5
Fig.2 Isotherms of reduced pressure for the same t w o gases as in F i g . l , at the same reduced temperature. Again, the dashed section of curve B is in the t w o phase region, where it exhibits a van der Waals' loop
r (0, 6) is the same for these (as for all the fluids under discussion), Z is always 2.37/1.94 = 1.22 times as far from unity for ethane as it is for methane. However, when ( Z - 1 ) / k B isotherms are plotted for these fluids, the curves for equal 0 are identical within experimental error in the
CRYOGENICS.
APRIL
1975
(7)
Since ¢ and its derivatives are functions of 0 and 6 only, (7) shows that the locus of the 0, 6 points for which n" vanishes is independent of ka. The inflection densities of all pressure isotherms of all gases obeying (1) should therefore lie on a single 0 versus 6 curve. This is tested by experimental data in the next section. In addition, the critical points (0c, 6c) of all these fluids must also lie on this same curve. Setting n ' = O in (6) shows that the location of the critical point on this curve depends on the value of kB. Therefore 0c, 6c, lrc, and Zc all depend on kB. We have already seen in Figs 1 and 2 that 0c is greater than 0.39 for ethane (kB = 2.37), and 0c is less than 0.39 for methane (kB = 1.94). The detailed experimental dependence of 0 c on kB is described later.
211
The common
inflection point curve
In this section we verify from experimental data that isothermal inflection points of rr versus 8 do lie on a common 0 versus 8 curve. Because o f the anticipated difficulty o f finding reliable second derivatives from experimental data, considerable care was taken in evaluating the isothermal inflection densities. At first we found least squares polynomials o f P versus d, Z versus d, and K versus d. (K = (Z- 1)/d = K VB, where VB = kB/do). Where available, we took densities up to or a little beyond critical. In this range, orders of five or more in the pressure polynomial provided satisfactory fits of the data. Orders o f four or more were satisfactory for Z, and three or more for K, which is consistent with the progressive division by d in changing from P to Z to K. At first, inflection densities were calculated from polynomials of orders 5, 6, and 7 for P; 4, 5, and 6 for Z; and 3, 4, and 5 for K. The most self-consistent results were found with the K isotherms, and the densities for the 4th and 5th orders agreed to within a few hundredths of a mole 1-1 in most cases, Larger differences (0.1 or 0.2 mole 1"1 occasionally resulted from adding or subtracting data points. We decided to calculate all inflection densities from the fourth power polynomials o f K. As the lowest-order reliable polynomial this had the advantage of depending the least on curve-fitting, o f avoiding the possible introduction o f extraneous wiggles, and of requiring fewer data points per isotherm. We decided to use only those isotherms for which at least eight data points were available. To avoid the possible distortion o f curvature at the ends of the data range, we required at least two experimental points beyond (above and below) the calculated inflection density. Isotherms above TB have no inflection points (see below), and usually we had to stay below about 0.85 TB to obtain inflection densities in the data range. Finally, no reliable results could be found for isotherms near the critical temperature, where inflection densities are near the critical density. Only isotherms with temperatures at least twenty degrees above the critical temperature could be included in most cases. Table 2 lists the inflection densities calculated in this way for the twelve gases for which the isotherms satisfied all the above conditions. We estimate most o f them to be correct to within about -+ 0.05 mole 1-1. Using the TB and do values in Table 1, we plotted the reduced temperatures, 0, and the reduced inflection densities, 8. The points for all the gases fell very close to the c o m m o n curve shown in Fig.3. The next step was to find an empirical representation of this c o m m o n 0 dependence of the reduced inflection densities. We made use o f the fact that the 0 versus ~ curve begins at zero 6 with 0 = 1 and 30/0/5 = - 3 . This can be seen as follows. The function K (0, 8) can be expressed in the virial series K =/3 +~,6 + A 6 2 + ....
(8)
in which 13= B/VB, 7 = Cdo/VB, A = D doZ /Va ...... and B, C, D, .., are the ordinary virial coefficients, which for a given fluid are functions of temperature only. Equation 7 shows that at zero density the inflection curve begins with K = O, so that/3 = O and 0 = 1. That is, the curve begins at the Boyle temperature, with T-- TB, where the second virial coefficient is zero. At this point d/3/dO = 1, because VB =
212
Table 1. Characteristic constants for the HH equation of state Gas
kB
TB
do
6
1.20
7 8
1.81 1.81
109.5 575.6 795.2
50.75 37.45 29.04
Reference
H2 Kr Xe Ar CH3F CH4
9
1.87
10 11
1.89 1.94
407.6 841 509.7
46.8 25.1 35.58
02
12
1.98
408.7
Ne
13
2.09
115.3
N2
14
2.14
324.3
47.9 90.1 40.67
F2 CO
15 16 17
2.21 2.22 2.24
366 335 715
55.03 41.0 27.8
18 20
2.36 2.36 2.37
926 898 759.2
37.7 19.3 25.48
C3H 8 CO2 neo- C5H12
21 22 23
2.54 2.75 2.75
C4H10 CF 4 cyclo-C4F8
24 25 26
2.80 3.02 4.31
C2H4 H2S C3 H6 C2H 6
iso-
°
19
894 714 1011 955 518.7 792
18.7 40.4 11.95 15.1 28.07 14.08
The gases listed are those for which good (few parts per thousand) to excellent (few parts per ten thousand) fits of super critical P V T data have been found thus far. It should be emphasized that these values are not unique, and rather different sets can still provide satisfactory data fits
T(dB/dT) at TB. 5 Evaluating K 'and t~" from (8), and substituting in (7) gives / 3 + 3 7 8 +6A62 + .... = 0
(9)
as the equation for the 0 versus 6 inflection point curve in terms of the reduced virial coefficients. Differentiating (9) we find that on this curve,
(10)
so that the initial slope, at 6 = O, is given by
('0)
(11)
(It is interesting to note that the inflection point curve follows roughly, but not exactly, the equation 0 = ( 1 - 6 ) . 3) Similarly, it can be shown that the 0 versus/5 curve connecting the isothermal Z versus 6 minima begins with a slope of - 2 , and roughly obeys the equation 0 = ( 1 - 8 ) 2 ; and, of course, the Z = I (UC) line obeys the relation, 0 = ( 1 - 6 ) . Various empirical forms were investigated for the inflection point curve. The following equation, explicit in 6, was finally adopted: 6 = ( 1 - 0 ) / 3 + C2 ( 1 - 0 ) 2 + C3 ( 1 - 0 ) 3
(12)
CRYOGENICS. APRIL 1975
Table 2. Observed and calculated inflection densities
1.0
UCL
Gas
7", K
Kr
298.15 323.15 348.15 373.15 398.15 423.15 448.15 473.15
~
7.20 6.40 5.62 4.86 4.17 3.52 2.90 2.29
0.5180 0.5614 0.6048 0.6483 0.6917 0.7351 0.7786 0.8220
0.1921 0.1710 0.1500 0.1300 0.1113 0.0941 0.0775 0.0610
7.21 6.40 5.63 4.90 4.20 3.54 2.91 2.31
0.01 0,00 0.01 0.04 0.03 0.02 0.01 0.02
Xe
323.15 348.15 373.15 398.15 423,15
7.45 6.86 6.31 5.81 5.34
0.4064 0.4378 0.4693 0.5007 0.5321
0.2567 0.2361 0.2174 0.2001 0.1839
7.40 6.86 6.35 5.86 5.38
-0.05 0.00 0.04 0.05 0.04
Ar
173.15 188.15 203.15 223.15 248.15 273.15 298.15 323.15 348.15
11.37 10.35 9.41 8.22 6.87 5.64 4.49 3.39 2.35
0.4248 0,2428 0.4616 0,2214 0.4984 0,2011 0.5475 0,1756 0.6088 0,1469 0,6701 0,1206 0.7315 0,0960 0.7928 0.0724 0.8541 0.0503
11.42 10.43 9.49 8.32 6.95 5.68 4.50 3.39 2.34
0.05 0,07 0,08 0,10 0,08 0,04 0,01 0,00 -0.01
CH 4 273.15 298.15 323.15 348.15 373.15 398.15 423.15
6.45 5.63 4.84 4.10 3.42 2.72 2.10
0.5359 0.1813 0.5849 0.1581 0.6340 0.1358 0.6830 0.1151 0.7320 0.0960 0,7811 0.0765 0.8301 0,0589
6.53 5.68 4.88 4.12 3.41 2.73 2,09
0.08 0.05 0.04 0.02 -0.01 0,01 -0.01
10.95 10.60 10.26 9.93 9.30 8.68 8.11 7.54 6.98 6.48 6.03 5.50 5.04 4.63
0.4527 0.2286 0.4649 0.2214 0.4771 0.2143 0.4894 0.2074 0.5138 0.1942 0.5383 0.1813 0.5628 0.1692 0.5872 0.1573 0.6117 0.1456 0,6362 0.1353 0.6606 0.1258 0.6851 0.1148 0.7096 0.1053 0.7340 0.0966
10.92 10.59 10.27 9.95 9.33 8.73 8.15 7.59 7.05 6.52 6.01 5.51 5.02 4.55
-0.03 -0.01 0.01 0.02 0.03 0.05 0.04 0.05 0.07 0.04 -0.02 0.01 -0,02 -0.08
8.15 5.71 2.21
0,04 0.01 -0.04
(D
0.6
0.4
O
O.I
6
0,2
0.3
Fig.3 The reduced inflection point curve. The common locus of 0 versus 6 for the isothermal inflection points of ¢r versus ~ for all corresponding gases is shown, along with the unit compressibility line (0 = 1--6) and the two-phase boundaries for gases A and B (methane and ethane). The three marks show the predicted location of the critical point on the inflection point curve for values of k B of 2, 3, and 4. The dotted line shows the approximate location of some reported critical points; others are more scattered
A least squares fit of (12) to the experimental points gave C2 = 0.0324, and 6"3 = 0.2180 for the range of 6 from O to about 0,26. Equation 12 is related in Fig.3. Table 2 includes values of inflection densities calculated from (12) using TB and do values from Table 1. The agreement with the experimental values is very good. The rms deviaition is 0.04 m 1q and the average absolute deviation is 0.03 m 1-1. An even better agreement can be obtained by changes in TB and do which are within their experimental uncertainty. For example, the calculated and experimental inflection densities for CF4 can be made to agree within 0.05 m 1-1 (0.03 average) by using TB = 519 K and do = 28.25 m 1"1 , and these constants still permit a satisfactory P V T data fit by the HolleranHammes (HH) equation. We conclude that the inflection densities of all the gases listed in Table 2 lie, within experimental error, on the same reduced temperature versus reduced density curve, as required by (1). If the more common corresponding-states reduction by the critical constants were employed, then the T/Tc versus did e curves of all the gases would include the common point (1, 1). But at lower densities they would all diverge and intercept the zero-density axis at the various values of TB/Tc = 1/0e, which range from 2.02 to 2.75 for the gases in Table 2. This is another example of the inadequacy of the critical constants in corresponding-states correlations. 4 Reduced inflection densities can also be calculated directly from the HH equation of state with given n (0, 5). When this is done it is found that the HH values of 6 agree to well within 0.001 with those calculated from (12) for 0 from 1.0 down to 0.5. At lower 0 the agreement worsens, with the difference reaching 0.004 at 0 = 0.4. Evidently the HH equation of
CRYOGENICS. APRIL 1975
d (calc), Ad, ml 1 m1-1
0
0.8
dm1-1
02
185 190 195 200 210 220 230 240 250 260 270 280 290 300
N2
163.15 203.15 273.15
8.11 5.70 2.25
F2
175.02 180.03 185.03 190.03 195.03 200.04 210.03 220.03 230.03 240.02 250.01 260.01
11.83 11.40 10.99 10.58 10.19 9.81 9.06 8.32 7.68 7.07 6.38 5.71
0.5031 0.6264 0.8423
0.1993 0.1401 0.0553
0.4782 0.2149 0.4919 0.2072 0.5055 0.1997 0.5192 0.1923 0.5329 0.1852 0.5465 0.1783 0.5739 0.1646 0,6012 0.1512 0.6285 0.1396 0.6558 0.1284 0.6831 0.1160 0.7104 0.1038
"11.76 -0.07 "11.35 -0.05 "10.96 -0.03 "10.57 -0.01 '10.18 -0.01 9.80 -0.01 9.07 0.01 8.36 0.04 7.68 0.00 7.01 -0.06 6.37 -0.01 5.75 0.04
213
Gas
T,K
d m 1-] 0
(5
d (calc), Ad, m 1-1 m 1l
270.00 280.00 289.99 299.99
5.18 4.57 4.03 3.41
0.7377 0.7650 0.7923 0.8197
0.0941 0.0832 0.0733 0.0619
5.15 4.56 3.99 3.44
-0.03 -0.01 -0.04 0.03
C2H4 348.15 373.15 398.15 423.15
5.82 5.29 4.80 4.34
0.4869 0.5219 0.5569 0.5918
0.2092 0.1904 0.1727 0,1560
5.81 5.30 4.81 4.35
-0.01 0.01 0.01 0.01
C2H6 348.15 373.15 398.15 423.15 448.15 473.15 498.15 523.15
5.75 5.26 4.83 4.38 3.98 3.60 3.23 2.89
0.4586 0.4915 0.5244 0.5574 0.5903 0.6232 0.6562 0.6891
0.2255 0.2065 0.1897 0.1719 0.1562 0.1412 0.1269 0.1132
5.72 5.26 4.82 4.40 4.00 3.61 3.24 2.89
-0.03 0.00 -0.01 0.02 0.02 0.01 0.01
iso- 448.15 C4H10473.15 498.,15 523.15 548.15 573.15
3.38 3.14 2.91 2.71 2.50 2.31
0.4693 0.4954 0.5216 0.5478 0.5740 0.6002
0.2238 0.2076 0.1924 0.1793 0.1656 0.1528
3.30 3.09 2.88 2.68 2.49 2.30
-0.08 -0.05 -0.03 -0.03 -0.01 -0.01
CF4 273.15 298.15 303.15 323.15 348.15 373.15 398.15 423.15 448.15
5.37 4.68 4.54 4.03 3.44 2.88 2.35 1.85 1.35
0.5266 0.5748 0.5844 0.6230 0.6712 0.7194 0.7676 0.8158 0.8640
0.1912 0.1666 0.1618 0.1436 0.1224 0.1027 0.0838 0.0657 0.0480
5.28 .4.61 4.49 3.98 3.39 2.83 2.30 1.79 1.30
-0.09 -0.07 -0.05 -0.05 -0.05 -0.05 -0.05 -0.06 -0.05
cyclo-423.15 C4F8 448.15 473.15 498.15 523.15 548.15
2.67 2.44 2.22 2.00 1.80 1.61
0.5343 0.5658 0.5974 0.6290 0.6605 0.6921
0.1898 0.1730 0.1574 0.1422 0.1281 0.1146
2.59 2.38 2.16 1.96 1.77 1.58
-0.08 -0.06 -0.06 -0.04 -0.03 -0.03
0.00
The observed values are f r o m 4th order polynomials of (Z--1)/d versus d. The data references are the same as in Table 1. For Ar, the points at T = 248.15 K and lower are from reference 9b; the others are f r o m 9a. The calculated densities are from (12)
state, though accurate for PVT points down to 0 = 0.4 or less, nevertheless begins to deviate in its second derivative, tf 7r , for 0 < 0.5 and 8 > 0.2.
Critical densities We have already noted that the critical points of all fluids obeying (1) with the same K (0, 8) must lie on this same inflection point curve (or its extension to somewhat higher densities). Thus the curve in Fig.3 is also a plot of 0 c versus 6c. Any attempt to verify this is confronted with the wide variation in reported values o f critical densities. Differences of several percent are common. An accurate correlation is
214
not possible under these circumstances. However, if the des given in reference 27 are used, the 0c, 8c points for Kr, Ar, Xe, 02, N2, CO, and CII 4 fall within 1% on a common curve. This line is nearly straight and lies above the extension of (12) by about 2% at 0 = 0.40 and about 4% at 0 = 0.36, as shown by the dotted curve in Fig.3. It is possible that the extended reduced inflection point curve could bend up to match this line. But points for the other fluids are even higher (except for H2 which falls below our line), and scattered. Perhaps our simple representation is not adequate for the critical region, and r (0, 5), which corresponds so well in the supercritical region and the liquid region, goes its separate way for each fluid around its critical point. But the disagreement among reported critical densities is too great to permit any defmite conclusion.
Critical temperatures Fortunately, critical temperatures are known much more accurately than critical densities. Reported values o f Tc usually agree to within a few tenths of a degree. We can therefore hope to correlate these. In the section on pressure inflection points we saw that the location o f the critical point on the inflection point curve depends on the value of the characteristic constant kB. In this section we investigate the dependence of Oc on k a. The HH equation for r (0, 8) will not be useful here, since 8e < 0.5 for the fluids under consideration. Table 3 lists Tc for sixteen fluids. Fig.4 shows a plot o f 0c versus k a for these fluids, found by using T~ and k B from Table 1. Once again, these points fall close to a common curve. By a least squares fit, omitting CO, the following dependence o f 0 c on ka was derived 0c = 0.1839 + 0.1230 kB -- 0.0128 kB2
(13)
for ka from 1.8 to 3.0. The solid curve in Fig.4 represents this equation. To show the closeness o f the correlation, Table 3 includes values of Tc calculated from (13) and the constants of Table 1. The average difference between calculated and observed critical temperatures for these fluids is less than 0.4 degree. Hydrogen with k B = 1.20 and 0 c = 0.301, and C4F8 with kB --- 4.31 and Oc = 0.490, lie far beyond the limits o f ( 1 3 ) but on reasonable extensions of it. They could be used to extend the curve if necessary, though they are isolated points at the present time. Points for the other fluids o f Table 1, namely methyl fluoride, propene, and neopentane, lie above the curve. Still, they are within 1 or 2% of(13), and so could be brought into agreement by corresponding changes in Ta. Such changes may be possible within the experimental uncertainty of this parameter, as discussed in the next section. Thus the correlation o f reduced critical temperatures predicted by (1) is excellent for many substances, and at least fair for some others.
Critical pressures Table 3 also lists Pc and lre for the sixteen gases that follow the smooth 0 c versus ka curve. It can be seen that 7rc is nearly constant. Iflr c is taken as 0.0308, then
CRYOGENICS. APRIL 1975
Table 3. Observed and calculated critical temperatures and critical pressures Fluid
CP ref
Tc, K
0c(calc)
Tc,K (calc)
ATc, K
Pc, atm
¢rc
Pc, atm (calc)
APc, atm
Kr
27
209.4
0.3646
209.9
+ 0.5
54.3
0.00307
54.5
+ 0.2
Xe
27
289.8
0.3546
289.9
+ 0.1
58.0
0.00306
58.4
+ 0.4
Ar
150.9
0.3691
150.4
-0.5
48.3
0.00309
48.3
0.0
CH 4
27
9b
190.7
0.3743
190.8
+ 0.1
45.8
0.00308
45.9
+ 0.1
02
12
154.6
0.3773
154.2
-0.4
49.8
0.00310
49.5
-0.3
Ne
28
44.4
0.3851
44.4
0.0
26.2
0.00307
26.3
+ 0.1
N2
27
126.2
0.3885
126,0
-0.2
33.5
0.00310
33.4
-0.1
F2
15
144.3
0.3932
143,9
-0.4
51.4
0.00311
51.0
-0.4
CO
27
133.0
0.3939
132,0
-1.0
34.5
0.00306
34.7
+ 0.2
C2H 4
27
283.1
0.3952
282.6
-0.5
50.5
0.00310
50.3
-0.2
H2S
27
373.6
0.4029
373,1
--0.5
88.9
0.00310
88.3
-0.6
C2H 6
20
305.3
0.4035
306,3
+ 1.0
48.1
0.00303
48.9
+ 0.8
C3H 8
27
369.9
0.4138
369,9
0.0
42.0
0.00306
42.3
+ 0.3
CO2
27
304.2
0.4253
303.7
-0.5
72.9
0.00308
73.0
+ 0.1
iso-C4H10
27
408.1
0.4280
408.7
+ 0.6
36.0
0.00304
36.5
+ 0.5
CF 4
29
227.5
0.4386
227.5
0.0
37.0
0.00310
36.8
-0.2
Pc = 0.00253 doT B
(14)
for kB from 1.8 to 3.0. Pc values calculated from (14) with the constants from Table 1 agree with observed values to within a few tenths o f an atmosphere. The average deviation is less than 0.3 arm. Since reported critical pressures often disagree by this amount, the agreement may be considered to be within experimental error. Actually, 7rc versus k B appears to curve slightly, and to have a maximum o f about 0.0309. This impression is enhanced if the 7rc, kB points for Ha (0.285, 1.20)and C4F 8 (0.299, 4.31) are taken into account. Once again, these are isolated points, but they might be used to extend the curve if necessary.
0.44
~o.4o
The three fluids (propene, neopentane, methyl fluoride) which fell one or two percent above the 0 e versus k B curve, now fall from 4 -9% above the 7rc versus k B curve, well beyond the usual experimental error in Pc- Evidently these fluids do not fit with the others, although conceivably the deviations could be due to inaccuracies in the constants TB, do, and kB. Ambiguities
in t h e constants
It is significant t h a t the above correlations o f critical temperatures and critical pressures were achieved with the use of values o f TB, do, and kB found by fitting supercritical PVT data to the Htt equation of state. Data near the critical point are not needed, and in fact must be disregarded in the fit even if available, as already noted. The results are even more remarkable when the uncertainties in the three constants are considered. If we assume a group of fluids that exactly obey (1) with the same ¢ (0, 6), then each has a true set o f values for the three constants. However, it turn~ out to be very difficult to find these true
CRYOGENICS. APRIL 1975
0.36
118
~
21.2
~
216
~
310
kB
Fig.4 Reducedcritical temperaturesversus the characteristic constant, k B. The points are for the sixteen fluids in Table 3. The solid curve is for (13) found by a least squaresfit omitting CO, the point abovethe curve
215
values from P V T data. The reason for this is the general similarity in behaviour o f r (0, 8) in neighbouring regions o f temperature and density. From data o f limited range or accuracy it is therefore difficult to pinpoint the exact O, 8 region, and the exact values o f Ta and do and hence kB. For this reason the values o f the constants listed in Table 1 are not unique. Other sets o f values will still produce satisfactory P V T fits. Changes in one constant can largely be compensated for by related adjustments in the others. Generally speaking if TB i~ increased, do and kB decrease. Fortunately the magnitudes and directions o f these compensating changes are such that, if one set o f constants places an inflection density on the c o m m o n 0 versus 6 curve, another set will usually displace the point to a new location which is still approximately on the curve. If one set locates a point off the curve, other sets tend to move the point nearly parallel to the curve. If the points are not too far off, then acceptable constants can be found that will bring them into line, as was seen with CF4. An example o f two rather different compensating sets of constants is provided by the values o f TB and do found for two different sets o f P V T data for ethane 2o,3o to be 759.2K, 25.48 m 1-1 , and 753.OK, 25.65 m 1-1 . Both sets locate the inflection densities approximately on the common 0 versus 8 curve, though at different positions. A similar compensation occurs with rrc and with the 0 c versus k B curve. For example, two different sets o f P V T data from neon 28,31 gave k~, TB, do values o f 2.09, 115.3K, 90.1 m 1"1 . The critical temperature calculated from (13) is 44.4 K for the first set and 44.3 K for the second. The critical pressure calculated from (14) is 26.3 atm for the first set and 26.1 atm, for the second. Thus it appears that it is not necessary for the present purpose to know the true values o f the three constants o f fluids obeying (1). Even though these constants cannot easily be determined unambiguously, nevertheless, a set o f values that provides a good fit of supercritical P V T data will also correlate the critical points.
References 1
216
Holleran, E. M. Cryogenics 11 (1971) 19
2 3 4 5 6 7 8 98 9b 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3t
Holleran, E. M., Hammes, J. Cryogenics 15 (1975) 95 Engelstaff, P. A., Ring, J.W. [Temperley, H.N.V., Rowlinson, J. S., Rushbrooke, G. S. (eds)] Physics of Simple Liquids (North-Holland, Amsterdam, 1968) Ch 7, p 256 Holleran, E. M. Cryogenics 15 (1975) 137 ltolleran, E. M. JPhys Chem 73 (1969) 167 Goodwin, R. D. JResNBS: A Phys and Chem 71A (1967) 203 Beattie, J. A., Brierly, J. S., Baxriault, R. J. J Chem Phys 20 (1952) 1613 Michels, A., Wassenaar, T., Louwerse, P. Physica 20 (1954) 99 Michels, A., Wijker, H., Wijker, H. K. Physica 15 (1949) 627 Michels, A., Levelt, J. M., DeGraaff, W. Physic¢ 24 (1958) 659 Michels, A., Visser, A., Lunbeck, R. J., Wolkers, G. J. Physica 18 (1952) 114 Douslin, D. R., Harrison, R. H., Moore, R. T. J Chem Eng Data 9 (1964) 358 Weber, L. A. JResNBS: A Phys and Chem 74A (1970) 93 Sullivan, J. A., Sonntag, R. E. Advances in Cryogenic Engineering 12 (1967) 706 Cmin, Jr, R. W. Sonntag, R. E. Advances in Cryogenic Engineering 11 (1966) 379 Prydz, R., Straty, G. C. JResNBS:APhysandChem 74A (1970) 747 Bartlett, E., Hetherington, H., Kvalnes, H., Tremearne, T. JACS 52 (1930) 1374 Miehels, A., Geldermans, M. Physica 9 (1942) 967 Reamer, H. H., Sage, B. H., Lacey, W. N. IndandEngChem 42 (1950) 140 Miehels, A., Wassenaar, T., Louwerse, P., Wolkers, G. J. Physica 19 (1953) 287 Douslin, D. R., Harrison, R. H. J Chem Thermo 5 (1973) 491 Beattie, J. A., Kay, W. C., Kaminsky, J. JACS 59 (1973) 1589 Michels, A., Michels, C. Proc Roy Soc A 153 (1935) 201 Beattie, J. A., Douslin, D. R., Levine, S. W. J Chem Phys 20 (1952) 1619 Beattie, J. A., Marple, Jr, S., Edwards, D. G. J Chem Phys 18 (1950) 127 Douslin, D. R., Harrison, R. H., Moore, R. T. J Phys Chem 71 (1967) 3477 Douslin, D. R., Moore, R. T., Waddington, G. J Phys Chem 63 (1959) 1959 Reid, R., Sherwood, T. K. The Properties of Gases and Liquids, 2nd edn (McGraw-Hill, 1966) MeCarty, R. D., Stewart, R. B. Advances in Thermophysical Properties 84 (ASME, 1965) Martin, J. B., Bhada, R. K. AIChE J 17 (1971) 683 Reamer, H. H., Sage, B. H., Lacey, W. N. Ind & Eng Chem 36 (1944) 956 Gibbons, R. M. Cryogenics 9 (1969) 251
CRYOGENICS. APRIL 1975