2714
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979 P. F. Fahey, D. W. Kupke, and J. W. Beams, Proc. Natl. Acad. Sci. U.S.A. 63, 548-555 (1969). D. S. Scharp, N. Fujita, K. Kinzie, and J. B. Ifft, Biopolymers, 17, 817-836 (1978). B. Jacobson, Ark. Kemi, 2, 177-210 (1950). Y. Miyahara, Bull. Chem. SOC.Jpn., 29, 741-742 (1956). W. Kauzmann, Adv. Protein Chem., 14, 1-63 (1959). K. Gekko and H. Noguchi, Biopolymers, 10, 1513-1524 (1971). K. Gekko and H. Noguchi, Macromolecules, 7, 224-229 (1974). T. Tsugo and K. Yamauchi, Bull. Agric. Chem. Soc. Jpn., 24, 96-100 (1960). M. G. &eenspan and C. E. Tschiegg, Rev. Sci. Instrum., 28,897-901 (1956). 0. Kratky, H. Leopold, and H. Stabinger, Methods Enzymol., 27, 98-1 10 (1973). W. M. Jackson and J. F. Brandts, Biochemistry, 9, 2294-2301 (1970). T. T. Herskovits, J. Biol. Chem., 240, 628-638 (1965). J. G. Davis, C. J. Mapes, and J. W. Donovan, Bhhemktry, 10,39-42 (1971). R. Townend, R. J. Winterbotton, and S. N.Timasheff, J . Am. Chem. SOC., 82, 3161-3168 (1960). W. Y. Wen and S. Saito, J . Phys. Chem., 68, 2639-2644 (1964). F. J. Millero, A. L. Surdo, and C. Shin, J. Phys. Chem., 82, 784-792 (1978). A. A. Yayanos, J. Phys. Chem., 76, 1783-1791 (1972). Y. Miyahara, Bull. Chem. SOC.Jpn., 26, 390-393 (1953). S. Goto and T. Isemura, Bull. Chem. SOC.Jpn., 37, 1697-1701 (1964). E. J. Cohn and J. T. Edsall, "Proteins, Amino Acids and Peptides", Reinhold, New York, 1943, pp 370-381. T. L. McMeekin and K. Marshall, Science, 116, 142-143 (1952). J. T. Edsall, "Protelns, Amino Acids and Peptides", Reinhold, New York, 1943, pp 155-165. J. T. Edsall, "The Proteins", H. Neurath and K. Bailey, Ed., Vol. 1, Academic Press, New York, 1953, Chapter 7. A. A. Zamiyatnin, "Progress in Biophysics and Molecular Biology", J. A. V. Butler and D. Noble, Ed., Vol. 24, Pergamon Press, New York, 1972, pp 109-123. E. J. King, J . Phys. Chem., 73, 1220-1232 (1969). A. I.Kitaigorodskii, "Organic Chemical Crystallography", Consultants Bureau, New York, 1961. J. D. Bernal and J. L. Finney, Discuss. Faraday Soc., 43, 62-69 (1967). G. D. Scott, Nature (London), 168, 908-909 (1960). J. D.Bernal and J. Mason, Nafure(London), 186, 910-911 (1960). J. D. Bernal, Nature (London), 165, 68-70 (1960). 1. D. Kuntz, J. Am. Chem. Soc., 94, 8568-8572 (1972). C. Tanford, J. Am. Chem. Soc., 84, 4240-4247 (1962). Y. Nozaki and C. Tanford, J . Biol. Chem., 248, 2211-2217 (1971). C. C. Bigelow and M. Channon, "Handbook of Biochemistry and Molecular Biology, Proteins", Vol. I, G. D. Fasman, Ed., CRC Press, Cleveland, 1976, p 209. C. C. Bigelow, J . Theor. Biol., 16, 187-211 (1967). H. 8. Bull and K. Breese, Arch. Biochem. Biophys., 128, 488-496 (1968). I. D. Kuntz and W. Kauzmann, Adv. Protein Chem., 28, 239-345 (1974).
D. W. Scott and A. G. Osborn (42) N. J. Hipp, M. L. Groves, and T. L. McMeekin, J. Am. Chem. SOC., 74, 4822-4826 (1952). (43) I.D. Kuntz, J . Am. Chem. Soc., 93, 514-518 (1971). (44) T. Yasunaga and T. Sasaki, NippOn Kagaku Zasshi, 72, 87-91 (1951). (45) B. E. Conway, J. E. Desnoyers, and A. C. Smith, Phil. Trans. R. Soc. London, Ser. A , 256, 389-437 (1964). (46) F. J. Millero, G. K. Ward, F. K. Lepple, and E. V. Mff, J. Phys. Chem., 78, 1636-1643 (1974). (47) H. Shiio, J. Am. Chem. SOC.,60, 70-73 (1958). (48) B. E. Conway and R. E. Verrall, J. Phys. Chem., 70, 3952-3961 (1966). (49) H. Noguchi, Prog. Polym. Sci. Jpn., 6, 191-232 (1975). (50) C. Tanford, S. A. Swanson, and W. Shore, J . Am. Chem. Soc., 77, 6414-6421 (1955). (51) Assuming 7 and 4 mol of hydrated water per mole of anionic and cationic amino acid residues, respectively, we calculated the electrostriction as (7 X 100
+ 4 X 96) X
18/65000 = 0.30 g/g
since there are 96 positive charges and 100 negative charges per 65 000 g of bovine serum albumin.50 Therefore, the volume change due to electrostriction is e ' - e = 0.30 X (-2.7)/18 = -0.045 mL/g e = 0.30/1 = 0.30 mL/g (52) K. Suzuki, Y. Taniguchi, and T. Watanabe, J . Phys. Chem., 77, 1918-1922 (1973). (53) M. Friedman and H. A. Scheraga, J. Phys. Chem., 69, 3795-3800 (1965). (54) H. B. Bull and K. Breese, Arch. Biochem. Biophys., 126, 497-502 (1968). (55) P. L. Privalov, N. N. Khechinashvili, and B. P. Atanasov, Biopolymers, 10, 1865-1890 (1971). (56) P. L. Privalov, FEBS Lett., 40, suppl. s140-153 (1974). (57) N. N. Khechinashvili, P. L. Privalov, and E. I.Tiktopulos, FEBS Left., 30, 57-60 (1973). (58) P. L. Privalov and D. R. Monaselidze, Bloflzlka, 6, 420-425 (1963). (59) "Handbook of Biochemistry and Molecular Biology, Proteins", Vol. 111, G. D. Fasman, Ed., CRC Press, Cleveland, 1976, p 595. (60) L. Btje and A. Hvidt, Biopolymers, 11, 2357-2364 (1972). (61) L. Bqe and A. Hvidt, J . Chem. Thermodyn., 3, 663-673 (1971). (62) W. Masterton and H. Seller, J. phys. Chem., 72,4257-4262 (1968). (63) F. M. Pohl, FEBS Lett., 3, 60-64 (1969). (64) K. Aune, A. Sakhuddin, M. Zarlengo, and C. Tanford, J. Bid. Chem., 242, 4486-4489 (1967). (65) C. Tanford and M. L. Wagner, J. Am. Chem. Soc., 76,3331-3336 (1954). (66) C. Tanford, Adv. Protein Chem., 17, 69-165 (1962). (67) R. K. Cannan, A. Kibrick, and A. H. Palmer, Ann. N . Y . Acad. Sci., 41, 243-266 (1941). (68) Y. Nozaki, L. Bunviele, and C. Tanford, J . Am. Chem. Soc., 81, 5523-5529 (1959). (69) W. G. Gordon, W. F. Semmett, R. S. Cable, and M. Morris, J . Am. Chem. Soc., 71, 3293-3297 (1949).
Representation of Vapor-Pressure Datat D. W. Scott and A. G. Osborn" BartlesviNe Energy Research Center, Department of Energy, Barflesville, Oklahoma 74003 (Received May 8, 1978) Publication costs assisted by the U.S. Department of Energy
Two equations are proposed for representing vapor-pressure data, one for the entire liquid range and the other for a limited temperature range. Their use is illustrated with selected sets of experimental vapor-pressure data. Each of these equations is shown to have definite advantages over some alternative equations used heretofore. Their use in detecting inconsistent data is emphasized. Original vapor-pressure data are reported for liquid benzene and solid hexafluorobenzene. Introduction Many equations have been proposed for representing vapor-pressure data. Partington,' in 1951, listed 56 such +ContributionNo. 231 from the thermodynamics research laboratory at the Bartlesville Energy Research Center.
equations sequentially and continued with about two dozen more forms of particular equations. Additional equations Proposed in the ensuing quarter of a century have swelled the ranks still further. However, no one equation, or small selection Of equations, has found Universal acceptance among investigators concerned with representing vapor-
This article not subject to US. Copyright. Published 1979 by the American Chemical Society
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979 2715
Representation of Vapor-Pressure Data
pressure data. In the past, the selection of an equation often was influenced by its mathematical simplicity and the ease of evaluating the adjustable constants by graphical or numerical methods. Now that digital computers are available for data reduction, mathematical simplicity no longer is a consideration, and a more mathematically complex equation should be used if it will provide a better representation of the vapor-pressure data than a simpler one. Thus, most of the vapor-pressure equations proposed in the past are obsolete and of historical interest only. The confused and unsatisfactory situation with respect to representation of vapor-pressure data led to the development in this laboratory of two equations different from any proposed previously. One is for use over the entire liquid range up to the critical point and admits as many adjustable constants as can be determined with significance from the vapor-pressure data. The other one is for a limited temperature range and admits either three or four adjustable constants, whichever the accuracy and extent of the vapor-pressure data justify. Experience with these two equations indicates that they have definite advantages over alternative equations used heretofore. The two equations are presented in what follows, and their use is illustrated by application to selected sets of vapor-pressure data.
Equation for the Entire Liquid Range The heading of this section requires a slight disclaimer. Scaling theory shows that the vapor pressure vs. temperature relationship at the critical point is nonanalytic. However, extremely accurate vapor-pressure data very close to the critical point are required to show the theoretical departure from an analytic relationship. Vaporpressure data that close to the critical point and of the required accuracy are extremely rare. Equations that provide for the nonanalytic behavior at the critical point seem to do so only with some degradation of the fit to the vapor-pressure data at temperatures well removed from the critical temperature. It seems better, then, to represent nearly all of the liquid range with an analytic equation. The equation proposed here, therefore, is an analytic one; as will be shown, it provides a satisfactory representation of currently available vapor-pressure data immediately below and even at the critical temperature. In the event of vapor-pressure data accurate enough to require a nonanalytic representation near the critical point, it would seem better to so treat only a small interval below the critical temperature, preferably with an equation made to merge smoothly with an analytic equation that represents the data satisfactorily over all of the remaining liquid range. The analytic equation proposed here is based on the Cox vapor-pressure equation,2 which may be written in various ways dependent on the choice of reference point. For the critical point as reference, for relative temperature and pressure, TR = T / T c and PR = P/Pc, with T and Tc temperature and critical temperature and P and Pc pressure and critical pressure, respectively, and for a choice of natural logarithms, the Cox equation is In PR = A ( 1 - l / T R ) in which In A = a0
+ alTR
i=2
azTR2 = CU,TR’ i=O
(1)
The basis is that the logarithm of A is approximately a quadratic function of temperature or relative temperature, and, in fact, as such is quite accurate for a short enough temperature interval far enough away from the critical
temperature. However, for accurate data over the entire liquid range, there is significant departure from the quadratic relationship. Attempts have been made to improve the fit by having the logarithm of A as a higher polynomial in temperature or relative temperature with more adjustable constants, i.e. In PR = A ( 1 - ~ / T R ) in which i=q
In A = CaiTRi i=O
with q greater than 2. Such attempts have forced a reasonably good fit over a moderate temperature range, but with abominably poor extrapolation properties outside of the range of the fit. (For examples, see subsequent discussion under headings of Hexafluorobenzene and Detection of Inconsistent Data and accompanying figures.) The fact is that the logarithm of A is not well represented by a polynomial in temperature or relative temperature but is well represented by a polynomial in a power of temperature or relative temperature greater than unity, i.e., TRnwith n > 1. The evidence for that statement is that when the exponent n is included as an adjustable constant in a least-squares fit to vapor-pressure data, the adjusted value typically is greater than 1.5 and often even greater than 2.0. The equation for representing the entire liquid range therefore has the logarithm of A as a polynomial in TRn rather than TR In PR = A ( l - l / T R ) in which 1=q
In A = Cu,TRnl
(3)
1=0
The coefficients, the ab’s,alternate in sign, with the first one positive. Experience shows that for a satisfactory fit to vapor-pressure data, the last coefficients must be positive too, so the number of terms in the summation, q 1, must be an odd number. Normally the critical temperature Tc is taken as known, and the critical pressure Pc and the exponent n are included with the a,’s as adjustable constants, so the total number of constants is odd. However, both Tc and Pc can be taken as known, in which case the number of constants is even. In the illustrations with actual data in the next section, the appropriate number of constants ranges from 11 down to 8. The ab’s and n are dimensionless numbers, so the vapor pressure in any desired units may be calculated with eq 3 by simply expressing the critical pressure in those units.
+
Illustrations of the Use of the Equation The use of eq 3 is illustrated with four sets of vaporpressure data over the entire liquid range, selected as typical of the kind of data to which the equation applies. To show the wide range of temperature for which data may be represented, all four sets include values for supercooled liquid obtained from data for the solid by use of calorimetric data. However, the equation applies equally well if the temperature range does not extend below the triple point. The values of the constants of eq 3 were obtained by an iterative, nonlinear, least-squares fit to the experimental data that minimized the sum of the squares of the weighted residuals, C1(AP,/u,)*, in which AP,is the difference between the observed and calculated vapor pressure, P,(obsd) - P,(calcd). The Q’S used for weighting initially were taken as the authors’ estimates of accuracy,
2716
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979
D. W . Scott and A.
G. Osborn
TABLE I: Constants of Eq 3 and Critical Temperatures and Pressures water
benzene
hexafluorobenzene
methane
2.266 088 4 f 0.063 2.171 383 * 0.005 1 -1.669 043 f 0.021 7.574 628 f 0.36 - 24.083 434 i 1.8 53.799 616 f 4.9 - 80.324 950 f 8.3 75.172471 r 8.5 -40.624 531 * 4.7 9.429 936 f 1.1 647.317 22122.42 f 0.63
1.574 414 1f 0.33 2.340 184 i 0.16 - 2.913 768 i. 0.68 8.389 532 * 1.3 - 14.508 140 i. 2.2 15.146 082 * 3.5 -8.568 283 i 2.9 2.023 096 * 0.95
2.055 188 3 f 0.36 2.317 378 i 0.053 - 1.722 665 f 0.074 4.276 182 * 0.96 -6.521 506 f 2.6 6.556093 f 3.1 - 3.995 053 f 1.8 1.156 317 i. 0.39
2.180 525 5 * 0.077 1.748 484 i. 0.002 2 -0.582 554 I 0.018 2.038 936 f 0.19 - 3.664 426 i- 0.61 4.109 262 f 0.93 - 2.644 974 i. 0.71 0.770 310 f 0.21
562.06 4882.8 (constrained)
516.729 3273.30
190.555 4599.10
if given and not unduly overoptimistic. Otherwise,the a’s were assigned from knowledge of the accuracy expected of the experimental method or from the scatter of the data points about a smooth curve through them. The weighting step in the least-squares adjustment is necessarily somewhat subjective. The initial values of the a’s eventually were multiplied by a common factor to obtain final values that gave a standard distribution of the weighted residuals; these final values of the u’s are the ones listed in later tables documenting the fit to the vapor-pressure equations. The variance-covariance matrix obtained from the least-squares procedure was used to calculate the standard deviations of the adjustable constants. If the standard deviation is less than half the value of the constant itself, that constant is significant at or above the 95% confidence level (null hypothesis with the Student t test). This consideration was used to ensure that no more adjustable constants were used than were justified by the experimental data. Of course, inclusion of more constants than are statistically significant usually results in failure of the iterative least-squares procedure to converge. If the vapor-pressure data were not reported initially with temperatures on IPTS-68, they were corrected to conform to that temperature scale. Values of critical temperature (and critical pressure, if needed), unless otherwise noted, were taken from the critical selection of Kudchadker, Alani, and Z~olinski.~ The statements in this and the foregoing two paragraphs apply not only to eq 3 for the entire liquid range but also to the equation for a limited temperature range to be discussed later and will not be repeated in that context. The four sets of vapor-pressure data now will be discussed in turn. Water. This is an almost obligatory compound with which to demonstrate a vapor-pressure equation. As a ubiquitous and easily purified substance and, moreover, the working fluid in steam power plants, it has had more careful investigations devoted to measuring its vapor pressure than any other compound. With a ratio of critical pressure to the lowest vapor pressure that has been measured of about 500 000, it provides an unusually wide range of relative pressure to be represented by an equation. Water is of specific interest in this laboratory as the primary reference substance for vapor-pressure determinations by comparative ebulliometry. Data for representing the utility of eq 3 were a critical selection from results of the most careful determinations, which presumably supersede results of earlier, less accurate measurements. Osborne, Stimson, Fiock, and Ginnings4 made particularly careful measurements between the critical point and only slightly above the normal boiling point by a dead-weight-gauge method. Their two data points at the high-temperature end of the range (0.15 and 1.15 K below the critical temperature) were eliminated
* 0.011
f
0.35
since they were slightly inconsistent with the remainder of the data. In view of the authors’ disclaimer about “the region near the critical, where the changing properties of water make the measurements somewhat less trustworthy”, the slight inconsistency is attributed to experimental difficulty rather than to any effect of scaling (nonanalytic) behavior. Stimson5made measurements by highly precise manometry between 298.15 K and the normal boiling point (373.15 K). Guildner, Johnson, and Jones6 made a definitive determination of the triple-point (273.16 K) pressure. Between the triple point and 298.15 K are two sets of data, those of Douslin by inclined-piston-gauge manometry7and those of Besley and Bottomley by precision mercury manometry.6 Besley and Bottomley provided only a tabulation of smoothed results and not their “raw” data; their smoothed values at 5 K intervals were used in lieu of raw data. Douslin and Osborne determined the vapor pressure of the solid (ice) by inclined-piston-gauge manometry. From calorimetric data,1° the relation between the vapor pressure of supercooled liquid water, PL,and that of the solid (ice), Ps, is In (PL/Ps)= 63.8754 - 1410.5/7’ - 11.040 In 7’ 0.01187‘ (4)
+
Unfortunately, the vapor-pressure determinations for ice were made before techniques for outgassing samples and ensuring good thermal contact for thermometric purposes had been quite perfected, as shown by deficiencies of the data reported for liquid water in the same publication relative to those obtained later with the same apparatus. Nevertheless, these data for ice are accepted as the best available for extending vapor-pressure data for liquid water into the supercooled region by use of eq 4. These data for water were found to justify an ll-constant version of eq 3, the constants of which are listed in Table I. In the least-squares adjustment, the data point at 373.15 K was weighted heavily enough to force agreement with the fixed point defining IPTS-68. Figure 1 contains a percentage deviation plot of the vaporpressure data relative to the equation; to accommodate the numerous data points with their different relative uncertainties, the plot is divided into two portions with different vertical scales. Wagnerll recently reviewed numerous previous representations of vapor-pressure data for water and presented a new equation. His equation fulfills the condition that d z P / d P be infinite at the critical point. Wagner’s equation is compared with the present one in the deviation plot of Figure 1; either the weighting used or the requirement of nonanalytic behavior at the critical point, or both, results in some degradation of the fit to the data well removed from the critical point. Wagner compared the values of a = (T/P)(dP/dT) and d a / d T at the critical point for various vapor-pressure equations for water. The value ac = 7.74 of the present equation is within the range
The Journal of Physical Chemistty, Vol. 83, No. 21, 1979 2717
Representation of Vapor-Pressure Data
377 Water (continued)
1
240
250
260
270
290
280
300
I
z
L
-
;I O 9 ' 0 I
8-10
IO 0 -IO
50 0
-50
50
75
100
125
r/u
I50
175
190555
Flgure 1. Percentage deviation plots of vapor-pressure data. Reference values, P,, are from eq 3 with constants of Table I. Water (separate plots for data above and below 300 K with 20-fold change of vertical scale): circles, ref 4; squares, ref 5; hexagon, triple-point value of ref 6; diamonds, ref 8; triangles, ref 7; inverted triangles, ref 9; dashed curve, equation of ref 11. Benzene (note 20-fold change of vertical scale at 290 K): circles, ref 12; squares, this research; triangles, ref 15. Hexafluorobenzene (note 20-fold change of vertical scale at 287.5 K and expanded horizontal scale around the critical temperature): circles, ref 17; squares, ebulliometric, ref 9; triangles, inclined piston, ref 9; inverted triangles, this research; fllled symbols, not used in least-squares adjustment of constants of equation; dashed curve, equation of ref 17. Methane (note 100-fold change of vertical scale at 79-89 K): circles, ref 19; squares, ref 20.
7.55-8.15 of the other equations; the value (da/dT)c = 0.096 K-l is, with two exceptions, greater than for the other analytic equations and thus is a better approximation to the scaling-law value of a obtained only with the nonanalytic equations. The two exceptions are formulations specific for water and not intended as general vaporpressure equations. Benzene. Besides providing an example of a compound whose vapor pressure has not been studied as thoroughly as has that of water, benzene is of specific interest in this laboratory as a secondary reference substance for comparative ebulliometry. Because of that interest, new determinations of the vapor pressure over the temperature range 308-389 K were made by comparative ebulliometry against the primary water standard. The results, which are reported in a later section, were taken as superseding the rather extensive body of data that has been obtained in investigations covering portions of that temperature range. For temperatures between the upper end of the ebulliometric range and the critical temperature, the results of Connolly and Kandalic12 were taken as being intermediate between those of Bender, Furukawa, and Hyndman13 and those of Ambrose, Broderick, and Townsend;14the data of the three investigations taken together have a scatter of about &0.1%. Connolly and Kandalic did not report their raw data but only an equation to represent them. Values of vapor pressure calculated from the equation at 10 K intervals between 400
and 560 K were used in lieu of raw data. For consistency, the critical pressure obtained from their equation was used instead of the selected value.3 Jackowski15reported vapor-pressure data for the solid. From calorimetric datal6 In (PL/Ps)= 84.3417 - 1050.O/T - 15.669 In T + 0.0274T ( 5 ) Jackowski did not describe any precautions to ensure that his sample was thoroughly dry. Near the triple point, traces of water impurity in nearly perfect solid solution would introduce very little error into the measured vapor pressure. However, with decreasing temperature, increasing solution imperfection and eventual phase separation would magnify the error. In the limit of no mutual solubility, the measured pressure would be the sum of the vapor pressure of solid benzene and that of solid water (ice). Whether for this suspected reason of water impurity or some other unknown reason, Jackowski's data below 260 K become increasingly more inconsistent with the data at higher temperatures, and only his results above 260 K were used for extending vapor-pressure data for liquid benzene into the supercooled region by use of eq 5. These data for liquid benzene were found to justify an eight-constant version of eq 3, the constants of which are listed in Table I. Figure 1contains a percentage deviation plot of vapor-pressure data relative to the equation. Hexafluorobenzene. This compound was selected because vapor-pressure determinations spanning the entire liquid range had been made in a single laboratory and were referred to the same temperature scale and a uniform set of standards. Douslin, Harrison, and Moorel' used a dead-weightgauge method for higher vapor pressures up to the critical point, although at widely spaced temperature intervals of 25 K except immediately below the critical temperature. Their two data points for the lowest temperatures and vapor pressures were eliminated from the least-squares adjustment of the constants of the equation because of a slight inconsistency with the remainder of the data. A dead-weightgauge calibrated at only a single pressure may not be entirely reliable when used to measure pressures about 1 order of magnitude less than the calibration pressure. Douslin and Osborn9 used both comparative ebulliometry and inclined-piston-gauge manometry to cover the intermediate and low-pressure ranges of the liquid. Their ebulliometric data were referred to the new representations for the reference substances water and benzene. One of the two inclined-piston data points very near the triple point was eliminated from the least-squares adjustment since it was marginally inconsistent. Douslin and Osborn9also made determinations for the crystals by inclined-piston-gauge manometry. As for solid water (ice) mentioned earlier, these determinations were made before techniques of inclined-piston-gauge manometry had been quite perfected. As a test for reliability, the data were examined by a "third-law" treatment. For the crystals, the published values of -(G"(T) - H o ( 0 ) J / P were corrected to IPTS-68; for the vapor, unpublished values calculated in this laboratory were used. The values obtained for the enthalpy of sublimation at the absolute zero, AtHo(0),from each data point are plotted vs. the temperature of the determination in Figure 2. It is seen that the values of A@" (0) at the higher temperatures are satisfactorily invariant, but at the lower temperatures below 245 K, they fall off precipitously with decreasing temperature. The discrepancies are in the right direction to be explained by water impurity in the sample as dis-
2718
The Journal of Physical Chemistty, Vol. 83, No. 21, 1979
D. W . Scott and A. G. Osborn
TABLE 11: Vapor Pressure of Crystalline Hexafluorobenzene TI K PlkPa
263.151 1.1534
268.151 1.7571
258.151 0.7447
a ~
7-
1
a
5441
225
250
r/K
275
Flgure 2. Enthalpy of sublimation of crystalline hexafluorobenzene at the absolute zero calculated from individual vapor-pressure data points: circles, ref 9; squares, Table 11. Filled circles denote data points eliminated from the least-squares adjustment of the constants of the vapor-pressure equation.
cussed in connection with Jackowski's data for solid benzene. Because of doubts about the quality of the data even above 245 K, repeat measurements by current techniques of inclined-piston-gauge manometry were made with a scrupulously dried sample, otherwise the same as used earlier. To avoid the necessity of changing the thermal-contact medium, we made no measurements below the freezing point of mercury used for that purpose. The new results are listed in Table 11,and values of A@'(O) obtained from them are plotted in Figure 2. For the new results, A@" (0) is satisfactorily invariant; the values are close to those from the previous results at higher temperatures. Therefore, only the previous results for temperatures below 245 K were rejected as unreliable. From calorimetric datal6 In (PL/Ps)= 76.6715 - 966.2/7' - 13.983 In T + 0.01987' (6) The vapor-pressure data for the crystals were used for extending data for liquid hexafluorobenzene into the supercooled region by application of eq 6. Because of the stated intent of using only data from one laboratory based on uniform standards, other data for liquid hexafluorobenzene (cited in ref 17) were not considered. Those data that were considered were found to justify a nine-constant version of eq 3, the constants of which are listed in Table I. Figure 1contains a percentage deviation plot of vapor-pressure data relative to the equation. Shown also is the curve resulting from the equation used in ref 17 to represent all of the data above the normal boiling point (eq 2 with q = 4). It is included to illustrate the hazard of using an unsuitable type of equation that is incapable of allowing inconsistent data points to be detected so that they can be eliminated before adjusting the constants of the equation. Despite the 61 K gap in reliable data between 387 and 448 K and the sparsity of data points above 448 K, the vapor-pressure vs. temperature relationship for hexafluorobenzene appears to be known at least as well as, and perhaps marginally better than, that of benzene itself. This situation is rather ironic, since samples of hexafluorobenzene suitable for studying its properties have been available only for the last 15 or 20 years, whereas the
253.149 0.4709
248.150 0.2912
243.151 0.1764
238.151 0.1060
properties of benzene have been studied since the very early days of organic chemistry. Methane. The previous three illustrations have been for compounds that are liquids at ordinary temperatures. For the fourth and final illustration, methane is selected as a representative of the simpler light gases. A thorough investigation of the vapor pressure of liquid methane was carried out by Prydz and Goodwin.lg By a combination of dead-weight-gauge and Bourdon-tube methods, they covered the range from the triple point to the critical point at 1 K intervals. Their results are taken to supersede various other earlier sets of data that Prydz and Goodwin cite in their publication. Vapor-pressure data for solid methane by a mass spectrometer method were reported by Tickner and Lossing.20 From calorimetric data21i22 In (PL/Ps)= 13.6992 - 77.09/T - 3.07 In T 0.0109T (7) Tickner and Lossing's data are not very accurate (measurements of very low vapor pressure seldom are); they report temperatures only to the nearest 0.1 K for even values of vapor pressure. Moreover, eq 7 may be of doubtful reliability for temperatures more than 40 K below the triple-point temperature. However, inclusion in the least-squares adjustment of the constants of the vaporpressure equation, with conservatively low weighting, of values for the liquid obtained from Tickner and Lossing's data by use of eq 7 at least provides a corridor through which the vapor-pressure equation must go in the supercooled-liquid region. The two sets of data taken together were found to justify a nine-constant version of eq 3, the constants of which are listed in Table I. Figure 1contains a percentage deviation plot of vapor-pressure data relative to the equation. Prydz and Goodwin used a nonanalytic equation to represent their data alone. Their equation is not plotted in Figure 1 because it would be largely obscured by the individual data points. Prydz and Goodwin comment on "slight, systematic deviations from the vapor-pressure equation". No systematic deviations from eq 3 are evident in Figure 1. Thus, again, a slight degradation of the fit results from requiring nonanalytic behavior at the critical point.
+
Application to Normal Fluids A final illustration of the application of eq 3, in addition to those for the four representative sets of vapor-pressure data, involves normal fluids. The results of this illustration will be applied later in connection with the equation for a limited temperature range. Pitzer and c o - ~ o r k e r correlated s~~ the vapor pressure of normal fluids with the equation log P R = (log PR)") + w(a log P R / a w ) T (8) in which w is the acentric factor. They tabulated values of -(log PR)C0) and -(a log P R / a w ) T as a function of TR but did not give analytical representations. For computer applications, analytical representations would be more convenient than the tabulated functions; eq 3 provides a suitable way of expressing them. For reasons to be made clear in a later section, an extension of the correlation to lower values of TR was desired for the n-alkanes as representative normal fluids. For TR between 0.40 and 0.68, values of the functions were obtained from vapor-pressure
The Journal of Physical Chemistty, Vol. 83, No. 21, 1979 2719
Representation of Vapor-Pressure Data
data for n-pentane (w = 0.252) and n-decane (w = 0.490). (They differed slightly from the tabulated values of ref 23 for TR between 0.56 and 0.68.) Values of PR(O) and antilog (a log &/aw)T, from the tabulation of ref 23 for TR between 0.70 and 1.00 and for the n-alkanes for TR between 0.40 and 0.70, were represented by six-constant and fourconstant, respectively, versions of eq 3. The result can be summarized as
\I
I
PR= exp([A(O)+ w ( a A / a w ) ~ ] [ l - ~ / T R ] ) (9) )~ as polynomials with In A(O) and In ( a A / a ~ represented in TR” with the following constants: n a, a, a2 a3
a4
In A ( ” )
In ( a A l a w ) T
1.784 176 9 1.736 104 1 -0.627 574 5 1.803 780 9 -1.965 181 4 0.808 762 5
1.511 195 5 2.570 200 2 - 2.288 929 6 1.311 687 9
In the least-squares weighting, the values at TR = 0.70 were weighted heavily enough to force agreement with the theoretical values. Consideration of the accurate vapor-pressure data that have been obtained for normal fluids since Pitzer and co-workers published their correlation in 1955 certainly would lead to an improved correlation and somewhat different numerical values for the analytical representation. Such an undertaking was beyond the scope of this research. The foregoing is intended merely to illustrate the approach and to provide provisional numerical values. Equation for a Limited Temperature Range Equation 3 has been shown to be suitable for representing vapor-pressure data over the entire liquid range. Often vapor-pressure data are available only over a limited temperature range well below the critical temperature, and a simple three-constant or four-constant equation is used to represent them. Here, an equation is proposed for representing data over such a limited range. It is developed first to apply to normal fluids whose volumetric and thermodynamic properties are functions of only three parameters: TR, the relative temperature, PR,the relative pressure; and w , the acentric factor as defined by Pitzer and c o - w o r k e r ~ .Since ~ ~ the necessary experimental data for developing the equation are available for the n-alkanes, it is based solely on them. Finally, the equation applicable to the n-alkanes as representative normal fluids is adapted for empirical application to any fluid, normal or abnormal. The temperature range over which the simple equation would be expected to apply may be taken as the range within which the state properties of the vapor depend only on the second virial coefficient, and third and higher virial coefficients make negligible contributions. This temperature range corresponds roughly with the practical range within which experiments can be conducted safely in ordinary glass apparatus. The starting point for developing the equation is the Frost-Kalkwarf equationz4 log P = A’+ B’/T + C’log T t D ’ P / P (10) Rewritten with natural logarithms and relative temperature and pressure, it becomes A ” + B”/TR + C”1n TR + D”PR/TR2 (11) In the derivation of the Frost-Kalkwarf equation, the temperature derivative of the enthalpy of vaporization, dAEHldT, is assumed constant; C’ in eq 10 is (dA,GH/dT)/R, where R is the gas constant. An evident In
PR =
040
045
I 050
1
1
1
I
055
060
065
070
075
TR
Figure 3. Temperature derivative of enthalpy of vaporization of nalkanes divided by the gas constant as a function of relative temperature. Curves for constant acentric factor: a, methane (0.013); b, n-pentane (0.252); c, n-hexane (0.294); d, n-heptane (0.349); e, n-octane (0.398); f, n-nonane (0.450);g, n-decane (0.490).
improvement is to remove the restriction that dA?H/dT is constant and allow it to be a function of relative temperature, TR, and acentric factor, w. For the n-alkanes, enthalpy-of-vaporization data are available for methane from the correlation of G ~ o d w i n ; ~ ~ for all members C5 through Clo at 298.15 K, from the work of Osborne and Ginnings;26for n-hexane and n-heptane at several temperatures above 298.15 K, from the work of Waddington and Douslin2’ and Waddington, Todd, and Huffman,28respectively; and for n-pentane and n-octane over a more extensive temperature range above 298.15 K, from unpublished work in this laboratory. Values of (dAEH/dT)/R calculated from those data are plotted in Figure 3. With decreasing relative temperature, dAEH/dT approaches ACpo,the difference between the heat capacity of the ideal gas and the liquid. Values of ACpofrom the ideal gasB and liquid-state%thermodynamic functions of n-octane, n-nonane, and n-decane, for temperatures taken to be adequately low, were used to add a few additional points to Figure 3. The curves in Figure 3 are calculated from the equation (dAfH/dT)/R = -3.013 t 11.023T~- 19.765TR2+ w(-62.645 + 154.785TR - 116.546TR2) (12) The numerical constants of eq 1 2 were obtained by least-squares fitting, with each point on Figure 3 weighted equally. The range of TR for which eq 1 2 was derived is essentially that for which a simple vapor-pressureequation would be expected to apply. Equation 12 is not intended for use much outside that range. It certainly is not valid up to T%= 1,where dAzH/dT = -m; in the other direction, dALH/dT is meaningless below the glassy transition temperature of the supercooled liquid because equilibrium times are too long on a laboratory time scale. Equation 12 provides the desired improvement in the third term on the right-hand side of the Frost-Kalkwarf equation. The last term of the Frost-Kalkwarf equation accounts for effects of gas imperfection and liquid volume and is derived approximately from the van der Waals equation of state. Thus it has no provision for the acentric factor, w. A second evident improvement is to express the last term, not as P[f(TR)],but as P[f(TR,w)],with the
2720
The Journal of Physical Chemisffy, Vol. 83, No. 21, 1979
function of relative temperature not necessarily proportional to 1/TR2. The search for a suitable function f(TR,w)required reliable vapor-pressure data, over an adequate temperature range, for at least two n-alkanes with sufficiently different values of w. For a member with a lower value of w , npentane (w = 0.252) was selected, and the data obtained by an ebulliometric method by Osborn and Douslin31were employed. For a member with a higher value of w , ndecane (w = 0.490) was selected, and the data obtained by Willingham et by a boiling-point method were employed (after elimination of the two data points at the lowest temperatures as slightly inconsistent). These data of Willingham et al. do not cover quite a wide enough temperature range. However, at 298.15 K, n-decane has a vapor-pressureof only about 0.2 kPa, and the vapor must behave very nearly as an ideal gas. So-called “third-law’’ values of vapor pressure, therefore, can be calculated from the tabulated thermodynamic functions of the ideal gas29 and the liquid30at 273.15 and 298.15 K and the enthalpy of vaporization determined at 298.15 K.26 The limitation on accuracy is that the thermodynamic functions (tabulated in calTHK-l mol-l) are uncertain in the second digit after the decimal point. Supplementing the results of Willingham et al. with the two third-law points provided an adequate temperature range for n-decane. The search for a suitable function f( TR,w)was largely a matter of trial and error with little guidance from theory. The function eventually selected as being in accord with the vapor-pressure data of both n-pentane and n-decane is f(TR,w) = 0.176 exp((1.65 0.8w)/T~- 0 . 9 ~ )(13) Incorporation of the two improvements of the FrostKalkwarf equation expressed by eq 12 and 13 gives the final equation. Since the acentric factor becomes one of the constants of the equation, it is well at this point to change symbolism and designate it C instead of 0. The result is In P R = A + B / T R - 3.013 In TR 5.512TR 3.294TR2+ C(-62.646 In TR + 77.392T~19.424TR2) 0.176PRexp((1.65+ 0 . 8 c / T ~- 0.9c) (14)
+
+
The applicability of eq 14 to other n-alkanes besides the two for which it was developed was demonstrated by adjusting constants to fit the vapor-pressure data of Willingham et and Forziati et al.33for the C6through C9 and C12members of the series. For the C5through Clo and C12members (w = 0.252, 0.294, 0.349, 0.398, 0.450, 0.490,0.576), the values obtained for the constant C were 0.252, 0.306, 0.351, 0.398, 0.438, 0.489, and 0.568. The trivial discrepancy for n-decane results from rounding the original numerical values of eq 13 to the convenient short values given. The values obtained for the constant C are seen to parallel closely the values of the acentric factor w. The comparison, of course, also is an indication of the internal consistency (or lack thereof) of the vapor-pressure and critical-point data for the n-alkanes. Estimation of Critical Temperature and Pressure Equation 14 is given in terms of the reduced variables T R and PR, and values of the critical temperature and pressure are needed for calculating them. For many substances, experimental values of Tc and Pc have been obtained. In the absence of experimental values, methods of estimation have been proposed, a summary of which through 1968 was given by Kudchadker, Alani, and Zwolin~ki.~
D. W. Scott and A. G. Osborn
A general and useful method of estimation for normal fluids is to require that, for two suitably spaced temperatures, eq 9 is satisfied when the constant C of eq 14 is substituted for w. In fact, data at lower values of T R for the n-alkanes as representative normal fluids were included in the adjustment of the constants of eq 9 with such an application in mind. The procedure is an iterative one. Provisional values of Tc and Pc are selected and used to generate provisional values of the constants of the vapor-pressure equation. The constant C of eq 14 so obtained is substituted for w in eq 9. Further substitution of pairs of values of temperature and calculated vapor pressure into eq 9 gives two simultaneousequations that are solved for improved values of Tc and Pc. These new values of Tc and Pc are used to generate new values of the constants of the vapor-pressure equation. The cycle is repeated until the changes in the values obtained for Tc and Pc are acceptably small. The iteration converges rapidly enough to consume little computer time. Version of the Equation with Four Constants Equation 14 would not be expected to apply to abnormal fluids, Le., compounds with large dipole moments, strong hydrogen bonding, or other sources of abnormality. In addition to the truly abnormal fluids, there are substances that behave as normal fluids at higher temperatures but develop specific structural effects in the liquid at lower temperatures. Benzene, for example, meets the criteria for a normal fluid at higher temperatures; however, in the liquid at lower temperatures there is structure, as shown by the X-ray diffraction pattern,34unlike any structure in the liquid n-alkanes. Substances such as benzene may be designated “quasi-normal”. The necessity, from the availability of data, of taking the n-alkanes as the prototypes of normal fluids down to low values of T R could categorize most other substances, at least to a small degree, as quasi-normal. Thus, eq 14 is of limited applicability, although for empirical use with limited data that justify only a three-constant equation it is preferable to the Rankine or Antoine equations. For vapor-pressure data that justify a four-constant equation, however, a modification of eq 14 can be used empirically for all fluids, normal, quasi-normal, or abnormal. The modification consists of separating the four terms involving the constant C into two pairs of terms, one with stronger and the other with weaker dependence on TR, and allowing the constants of those pairs of terms to adjust independently. Specifically, the modified equation is
+
In P R = A B/TR - 3.013 In TR + 5.512T~3 . 2 9 4 T ~+~D(-62.646 In T R + 77.392T~)1 9 . 4 2 4 E T ~+~0.176PR exp((1.65+ O.~E)/TR- O.9E) (15) If experimental values of critical temperature and pressure are available for calculating the reduced variables TRand PR,the use of eq 15 is straightforward. Otherwise, Tc and Pc can be estimated conveniently by the method of the previous section. However, for abnormal or quasi-normal fluids, the result will be empirical “effective” values, ecand IIc, that may have little relation to the true values. For such an empirical approach, it makes little difference which constant, D or E, of eq 15 is substituted for w in eq 9; in the illustrations to follow, E was selected. The application of eq 15 is illustrated with data for a normal fluid, a quasi-normal fluid, and an abnormal fluid. For the first illustration, the data for n-decane employed
Representation of Vapor-Pressure Data
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979 2721
TABLE 111: Constants o f Eq 15,Effective and Actual Critical Temperatures and Pressures, and Acentric factors
A
B
D E @JKa 11, / kPaa Tc/K
P,/ kPa W
n-decane
benzene
water
12.328096 t 0.21 - 18.97638 i 0.071 0.487066 3 i 0.004 5 0.487689 9 i 0.0034 618 2103.0
-7.161336 i 0.39 - 14.75739 i 0.14 0.325302 1 i 0.0085 0.327376 3 i 0.0067 520 2956.0 562.06 4898 0.209
- 2.994225
-
617.46 2099 0.490
a Rounded to nearest K and
1,2-pentadiene
10.038763 i 1.5 -15.36813 t 0.50 0.378555 9 t 0.031 0.358389 1 t 0.023
- 1.425096
-
0.041 - 10.89468 t 0.012 0.184178 4 i 0.00081 0.176201 8 i 0.00041 724 43996.6 647.377 22120 0.344 t
1,2-dimethylbenzene
* 1.0 -10.77272 i 0.29 0.161654 6 i 0.020 0.184869 2 t 0.011 524 6539.8
630.3 3733
0.1kPa.
to develop eq 13 and 14 are used. For the second, the previously unpublished data for benzene (ebulliometric method with water as standard) are used. For the third, values for water at 10 K intervals between 300 and 400 K, smoothed with eq 3, are used. Although values of Tc and Pc are known for all three compounds,the data are treated as though they are unknown, and effective values 0 c and IIc are estimated by the method of the previous section. In Table I11 are listed the values of the constants and effective critical temperature and pressure, and, for comparison, values of the actual critical temperature and pressure and of the acentric factor w. For the normal fluid, n-decane, the values of D and E are close to that of w, and the effective values OC and I I C , agree well with the actual values, Tc and Pc. The agreement is not exact, partly because of rounding and partly from the fact that eq 9 does not fit the vapor pressure of n-decane exactly. For the quasi-normal fluid, benzene, D and E are greater than w, and Oc and IIc are less than Tc and Pc. In contrast, for the abnormal fluid, water, D and E are less than w, and Oc and Ilc are greater than Tc and Pc. However, as shown in Table IV, with the empirically adjusted values of constants and effective critical temperature and pressure, the fit to the vapor-pressuredata is excellent and seemingly as good for the abnormal and quasi-normal fluids as for the normal fluid. Comparison of Vapor-Pressure Equations For higher molecular weight compounds that have low vapor pressure at room temperature, the vapor pressure often is determined over a convenient range of higher temperatures. An equation selected to represent the data then is used for calculating the enthalpy and entropy of vaporization at 298.15 K. The accuracy of the results obtained thus depends on how well the vapor-pressure equation extrapolates to lower temperature. n-Decane is a typical higher molecular weight compound, and the reliable representation of its vapor pressure down to room temperature and below with the equation of the previous section provides a means of testing how well other equations extrapolate to lower temperatures. Since an equation that extrapolates well outside of the temperature range over which it is determined also gives a good fit and reliable values of dP/dT or d In P / d T within that range, the test really is of the general merit of the equation. The procedure was to fit selected equations other than eq 15 to the same data for n-decane and with the same weighting of those data. Any bias from random scatter was avoided by using smoothed values [P(calcd) of Table IV] for the least-squares adjustment rather than the actual experimental values. Only those data for temperatures greater than 375 K were used and not the “third-law”
TABLE IV: Fit of Vapor-Pressure Equationsa
T
P(obsdl
P(calcdlb
P(0bsd) P(ca1cdl
0
n-Decane
448.303 103.921 103.921 447.719 102.401 102.395 447.063 100.701 100.703 446.476 99.207 99.207 445.842 97.609 97.611 439.950 83.722 83.724 431.594 66.757 66.759 423.888 53.654 53.653 416.662 43.318 43.315 409.663 34.892 34.892 403.852 28.951 28.952 397.531 23.451 23.451 392.797 19.913 19.917 387.695 16.609 16.609 382.679 13.812 13.811 378.270 11.686 11.684 375.010 10.292 10.292 371.501 8.954, 8.951 367.629 7.649‘ 7.644 298.15 0.1823d 0.1826 273.15 0.0264d 0.0264
0.000 0.006 - 0.002 0.000 -0.002 -0.002 - 0.002 0.001 0.003 0.000 - 0.001 0.000 - 0.004 0.000 0.001 0.002 0.000 0.003 0.005 -0.0003 0.0000
0.0041 0.0041 0.0041 0.0039 0.0039 0.0036 0.0031 0.0028 0.0025 0.0023 0.0021 0.0020 0.0020 0.0019 0.0019 0.0018 0.0018 0.00018 0.000025
Benzene (Original Data of This Resealrch)
388.847 382.798 376.795 370.839 364.927 359.061 353.242 347.469 341.741 336.061 330.426 324.834 319.289 313.787 308.332
270.03 232.02 198.49 169.03 143.24 120.791 101.325 84.532 70.120 57.817 47.375 38.565 31.177 25.023 19.933
400 390 380 370 360 350 340 330 320 310 300
245.5449 179.4787 128.7339 90.4528 62.1392 41.6477 27.1678 17.2021 10.5402 6.2282 3.5354
270.03 232.02 198.49 169.03 143.25 120.789 101.324 84.532 70.117 57.818 47.377 38.564 31.177 25.022 19.934
0.00 0.00 0.00 0.00 -0.01 0.002 0.001 0.000 0.003 -0.001 - 0.002 0.001 0.000 0.001 -0.001
0.0052 0.0045 0.0040 0.0034 0.0031 0.0025 0.0024 0.0020 0.0017 0.0015 0.0013 0.0012 0.0010 0.00091 0.00082
Water (Smoothed Data)
245.5443 179.4793 128.7342 90.4527 62.1389 41.6475 27.1678 17.2023 10.5404 6.2283 3.5351
0.0006 0.00098 - 0,0006 0.00077
-0.0003 0.0001 0.0003 0.0002 0.0000
-0.0002 -0.0002 - 0.0001 0.0003
0.00060 0.00048 0.00039 0.00032 0.00027 0.00024 0.00022 0.00020 0.00019
Temperatures are given in kelvins and pressures in kilopascals. P(ca1cd) from eq 15 with constants of Table 111. ‘ Excluded from least-squares adjustment. “Third-law” values,
D. W. Scott and A. G.Osborn
/-a /
* 40-
/
30/
/
/
/
-
-
\
-301
\\
0
- 501 I
I
I5
20
I
25
30
35
,
40
104 x I / ( T / K )
Flgure 4. Percentage deviation plot of vapor-pressure equations for ndecane: a, Rankine equation; b, Frost-Kalkwarf equation; c, Cragoe equation; d, Cox equation; e, eq 15; f, Antoine equation. Circles are data of Willingham et (smoothed), triangles are ”third-law’’ values at 298.15 and 273.15 K, and squares are data of Carruth and Kobaya~hi.~~
values a t 298.15 and 273.15 K. To have the equations in the forms most common in the literature, logarithms to base 10 and pressure units of torr were used. (1 torr = 0.133 322 kPa.) The equations obtained were the Rankine equation log (P/torr) = 26.24354 - 3364.837/T - 5.976108 log T (16) the Antoine equation log (P/torr) = 6.948036 - 1498.033/(T - 78.98645) (17) the Frost-Kalkwarf equation log (P/torr) = 32.61186 - 3707.198/T 8.096689 log T 4.716487P/P (18)
+
the Cragoe equation 15.07032 - 1.389201 X log (P/torr) = -3444.899/T 10-2T+ 8.626016 x 10-6P (19)
+
and the Cox equation log (P/torr) = A(l - 447.3052/T) in which log A = 0.9177971 - 7.964967
(20)
10-4T + 6.603375 X 10-7P Extrapolations to lower temperatures with these five equations are compared in Figure 4. There the percentage deviation from values given by the reference equation is plotted vs. 1/T. With this way of plotting, the slope of the curve for a given equation at a given temperature is approximately proportional to the error in d In P/d(l/T) (and thus to the error in the enthalpy of vaporization) calculated from the equation a t that temperature. The “third-law” values of the vapor pressure at 298.15 and 273.15 K are included on the plot to indicate how well the equations extrapolate to those two data points far outside of the temperature range on which the equations are based. None of these simple equations with only three or four constants would be expected to extrapolate well to higher temperatures up to the critical temperature, since equaX
tions with as many as 11 constants are required for the entire liquid range. However, extrapolations to higher temperatures are included in Figure 4 for completeness. Between 448 K and the critical temperature, the reference vapor pressure was obtained from the correlation of Pitzer and co-workers for normal fluids, and eq 15 was included among the equations compared. The sudden upsweep of eq 15 immediately below the critical temperature occurs well outside the range of intended validity for eq 1 2 and therefore the terms in eq 15 based on it. Figure 4 shows that the Cox equation, which has been used for many years to represent vapor-pressure data in this laboratory, gives by far the best extrapolation to lower temperatures, and certainly is the equation of choice among the five considered. The Cox equation also gives the best extrapolation to higher temperatures from 448 up to about 530 K. The two three-constant equations, the Rankine and the Antoine, give very poor extrapolations, both to lower and to higher temperatures, and obviously are not appropriate for representing accurate vapor-pressure data. The other two four-constant equations, the Frost-Kalkwarf and the Cragoe, give better extrapolations than the three-constant equations, but definitely are inferior to the Cox equation. Experimental values of the vapor pressure of n-decane a t lower temperatures reported by Carruth and K ~ b a y a s halso i ~ ~are plotted in Figure 4. The large deviations, one exceeding 50%, do not accord with the claimed accuracy of 2-370. The gas-saturation method employed by those investigators evidently is unsuitable for determining low vapor pressures. Detection of Inconsistent Data In fitting a suitable equation to vapor-pressure data, inconsistencies are revealed either by failure to obtain a satisfactory fit or by adjustment of the constants to physically unrealistic values. Often the inconsistent data points can be identified and rejected. With an unsuitable equation, inconsistent data may go undetected. Two examples will illustrate the detection and rejection of “sour” data points. The first example involves data for l,Zdimethylbenzene, for which Pitzer and reported results by a static method at lower temperatures and Forziati et al.33reported results by a boiling-point method at higher temperatures (in good accord with the earlier results of Willingham et al.32by the same method). As seen from the deviation plot of Figure 5, five data points at the low-temperature end of the boiling-point results are inconsistent with the remainder of the data. Those five points were eliminated from the adjustment of the constants of eq 15 listed in Table 111. The Antoine equation of the API RP-44 tables37 is plotted in Figure 5 for comparison; it is seen to be warped by the inclusion of the inconsistent data points in its derivation. The second, more extreme, example involves data for 1,2-pentadiene, for which Osborn and D o u ~ l i nreported ~~ results by inclined-piston-gauge manometry at lower temperatures and Forziati et al.39reported results by a boiling-point method a t higher temperatures. As mentioned in the discussion of hexafluorobenzene in a previous section, erroneous values of vapor pressure could be obtained at lower temperatures in the initial use of the inclined-piston gauge before the development of techniques that now yield consistently satisfactory results. As seen from the deviation plot of Figure 5, six data points at the low-temperature end of the inclined-piston-gauge results were eliminated as inconsistent. Also, as mentioned in the discussion of 1,2-dimethylbenzene in the first example, and
Representation of Vapor-Pressure Data 275
300
I
I
TIK 350
325
The Journal of Physical Chemistty, Vol. 83, No. 21, 1979 2723 375
400
a
2
425
A02
-
i
1.2- Pentodiene
,
1
225
250
275
300
325
T / K
Figure 5. Percentage deviation plots of vapor-pressure data and equations. Reference values, PrSf,are from eq 15 with constants of Table 111. Filled symbols designate data points eliminated as inconsistent. 1,2Dimethylbenzene (note 10-fold change of scale at 335 K): squares, ref 36; circles, ref 33; solid curve, Antoine equation at ref 37, Table 23-2433.1 IlOtk, issued April 30, 1954. 1,2-Pentadiene: squares, ref 38; circles, ref 39; dashed curve, six-constant equation of ref 38; solid curve, Antoine equations of ref 37, Table 23-2-(5.2202w, issued October 31, 1969.
earlier in connection with n-decane, the boiling-point method used by Forziati et a1.33,39and Willingham et could yield erroneously high values at the low-temperature end of the range. The difficulty seems to be exaggerated for temperatures near or below room temperature. In any event, as seen from the deviation plot of Figure 5, all data by the boiling-point method except in the immediate vicinity of the normal boiling point were eliminated as inconsistent. A situation in which more than half of the data points have to be rejected certainly is unsatisfactory. However, eq 15 with the constants listed in Table I11 can serve as an interim representation until really definitive vapor-pressure data are available for 1,2-pentadiene. Results of two attempts to fit equations to all of the data for 1,2-pentadiene also are shown in Figure 5. In the first,% a six-constant equation was employed (eq 2 with q = 4). However, with that equation, a t the estimated critical temperature of 524 K, the calculated pressure is kPa-an absurdly high value indicative of something amiss. In the second,37smoothly joining Antoine equations were used to represent the higher temperature and the lower temperature data separately. That attempt, like the first, shows the folly of trying to fit bad equations to bad data. In both of these examples, use of Antoine equations failed to detect and eliminate "sour" data points. Compilations of vapor-pressure data based on Antoine should be viewed with considerable suspicion and relied on only for the most crude kinds of calculations. For all careful work, users of vapor-pressure data should go to the original experimental results and represent them with a really suitable equation after elimination of the inconsistent data points revealed by the fitting process. Acknowledgment. The work upon which this research was based was sponsored by the Office of Basic Energy Sciences, Department of Energy, and was conducted at the
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