VOLUMETRIC AND THERMODYNAMIC PROPERTIES OF FLUIDS
July 5, 1955
virial coefficients conform within about 1% to the theory we are proposing for volumetric and thermodynamic properties generally. In later papers we
[CONTRIBUTION FROM
THE
3433
shall test this scheme against various experimental data and present general tables. BERKELEY, CALIF.
DEPARTMENT OF CHEMISTRY A N D CHEMICAL ENGINEERING, UNIVERSITYOF CALIFORNIA, BERKELEY]
The Volumetric and Thermodynamic Properties of Fluids. 11. Compressibility Factor, Vapor Pressure and Entropy of Vaporization1 BY KENNETH S. PITZER, DAVID2. LIPPMANN,~ R. F. CURL,JR., CHARLES M. HUGGINSAND DONALD E. PETERSEN RECEIVED JANUARY 21, 1955 The theoretical considerations of Part I suggested that the compressibility factor of a normal liquid in either gas or liquid state should be expressible as a function of just one parameter in addition to the reduced temperature and reduced pressure. The additional parameter is defined in terms of the vapor pressure a t T, = 0.7. This third parameter is required because the intermolecular force in complex molecules is a sum of interactions between various parts of the molecules-not just their centers-hence the name ucentricfuctor is suggested. The theory requires that any group of substances with equal values of the acentric factor should conform among themselves to the principle of corresponding states. This result is verified with relatively high accuracy. While a completely analytical expression for the compressibility factor was not obtained, power series expressions in the acentric factor proved satisfactory and the coefficients are tabulated for a wide range of reduced temperature and pressure. The reduced vapor pressure and the entropy of vaporization are also treated similarly. Agreement is obtained to 0.5% over most regions with maximum deviations of about 2%.
A general introduction together with discussion from the critical point. This in effect gives the of the theoretical basis for the correlation scheme slope of the vapor pressure curve, see Fig. 2, Part I. has been presented in Part I of this series3 Al- For a simple 3uid3 (e.g., A, Kr, Xe, CH4) the rethough exact theory was available only for the duced vapor pressure is almost precisely 0.1 a t a second virial coefficient, those results together with reduced temperature of 0.7. This point is well regeneral arguments indicated that a three parameter moved from the critical yet above the melting point correlation might well yield a t least a factor of ten for almost all substances. Consequently it is convengreater accuracy than is obtained from the simple ient to take 0.7 as our standard value of reduced hypothesis of ccrresponding states. As was indi- temperature for the determination of the acentric cated in Part I, we take two critical constants for factor which we define as two of our three independent parameters for each w = - l o g P , - 1.000 substance. The critical temperature is readily selected as the first parameter which characterizes the with Pr the reduced vapor pressure (P/Pc)a t Tr = 0.7. intermolecular interaction energy. The slope of the vapor pressure curve is, of course, While the critical volume would be the simple measure related to intermolecular distance, it is closely related t o the entropy of vaporization. unsatisfactory from the empirical viewpoint. The Thus we may regard our acentric factor as a measdifferential compressibility is infinite a t the critical ure of the increase in the entropy of vaporization point. Consequently the critical volume is not over that of a simple fluid. It was also shown in directly measureable with any accuracy. The val- Part I that the acentric factor would depend upon ues commonly given are extrapolated. The criti- the core radius of a globular molecule, the length cal pressure is a much more accurately determinable of an elongated molecule, or the dipole moment of a quantity and it suffices just as well for correlation slightly polar molecule. purposes. Hence we choose the critical pressure Table I lists the essential parameters for the as our second parameter. various substances which were given substantial The third parameter is to measure the deviation consideration in our correlations. On the basis of of the intermolecular potential from t h a t of a sim- the arguments in Part I it was not expected t h a t ple fluid. An important deviation arises from the the highly polar molecules, ammonia and water, fact that the sum of the inverse sixth power terms would conform to our scheme. Points for these subapplying to the various portions of a pair of com- stances are included on some graphs to illustrate plex molecules cannot be replaced by a single in- the magnitude of the deviations ; however, these verse sixth power term in the distance between mo- points were given no weight in preparing the final lecular centers. Since these forces between non- tables. The references to Table I include the central portions of the molecules must be consid- sources of data for the respective substances for the ered, the term acentricfactor is suggested. other tables of this paper.4 The most convenient empirical quantity is the Compressibility Factor.-The compressibility reduced vapor pressure at a point well removed factor was interpolated graphically to even values (1) This research was a part of the program of Research Project 50 of the American Petroleum Institute. (2) A portion of this paper is abstracted from the Ph.D. Dissertation of David 2. Lippmann, University of California, 1953. I T, , 3427 (1955). (3) K. S. Pitzer, THISJ O U R N A L
(4) In addition to the substances listed in Table I, supplementary use was made of d a t a for n-hexane in certain areas. T h e sources are S. Young, Scient. Proc. Roy. Dub. Soc., New Series 12, 374 (19091910); E. A. Kelso with W. A. Felsing, 2nd. Eng. Chem., 34, 161 (1942): E. A. Kelso with W. A. Felsing, THIS J O U R N A L62, , 3132 (1940)
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K . S.PITZER, D. 2.LIPPXWN,K . F. CURL,JR., C. 11. HUGGIKS AND D. E.PETERSEX 1-01, 77 TABLE I THE-*CENTRIC I‘ACTOR AND CRITICAL rjAT.4 ( / I r is reduced vapor prc>>urea t T,= I).?) - l u g PI
Substance
w
‘I‘, I OK )
I,,. ( a t m . )
Argon‘L 0.998 - ( I 002 150 72 48 00 Krypton” 0.998 - .002 209.39 34 182 Simple fluid (1.000) (.000) Xenon‘ 1.002 .002 289.75 57.636 Methane“ 1,013 ,013 190,iiii 45.795 Sitrogen‘ 1 040 .010 126.20 33,540 Hydrogen sulfide’ 1.100 ,100 373.55 88.868 EthaneQ 1,105 ,105 305 75 18 804 Propane’ 1 152 152 369 99 42.011 Seopentane’l 1,195 ,195 133 76 31.57 n-Butane‘ 1.201 201 425.17 37 470 Benzenet 1.215 ,215 562.66 18 (34 Carbon dioxide’ 1.225 , 2 2 5 304 16 72 800 n-Pentaneu 1.252 ,252 470.60 33.628 Ammoniak 1.250 ,250 406.00 112.31 ivater’ 1 548 ,348 617.66 219.513 n-Heptanem 1.352 ,352 539.94 26.882 a A . M . Clark, F. Din, J. Robb, A . Michels, T. LVassenaar and T h . Zwietering, Physica, 17, 876 (1951); A . Michels, Hub. IVijker and Hk. IYijker, ibid., 15, 627 (1949); L. Holborn and J . Otto, 2. Physik, 33, 1 (19251. t’ E. Llathias, C. A . Crommelin and J . J . Meihuizen, Physicu, 4, 1200 (1937); J . J. Meihuizen and C. -4. Cromnielin, i b i d . , 4 , 1 (1937). c A . Michels and T. IVassenaar, i b i d . , 16, 253 (1950); J . A . Beattie, R . J . Barriault and J. S. Brierly, J . Chem. Phys., 19, 1219 (1931); H . IV.Habgood and n‘. G. Schneider, Can. J . Chein. 32, 98 (1954). IV.H . Corcoran, R . R . Bowles, B . H . Sage and IY.S.Lace)-, I n d . Eng. Chenz., 37, 825 (19431; -4. Stock, F. Henning and E . Kuss, Ber., 54, 1119 (1921); A. Michels and G. LV. Sederbragt, Physicn, 2, 1000 (1935); ’4. Michels arid G . LV. Sederbragt, i b i d . , 3 , 569 (1936); R . H . Oltls, H. H . Reamer, B . H . Sage and IV. S. Lace!., I n d . Eng. Chenz., 35, 922 (1943). e A . S. Friedman and D . IYliite, THISJOURSAL, 72, 3931 (1950); 0. T. Bloomer and I;.S.Rag, “Thermodynamic Properties of Sitrogen, ” Institute of Gas Teclinology, Technology Center, Chicago (1952). 1 H . H. Reamer, B. H . Sage and LV. S. Lace)., I n d . Eng. Chetn., 42, 140 (1950). B. H . Sage and IV. N.Lace)-, “Thermodynamic Properties of the Higher Paraffin Hydrocarbons and Nitrogen,” American Petroleum Institute, Sew York, S. Y., 1950; B. H . Sage, D . C. IVebster antl IT.S. Lricey, Ind. Eng. Chem., 29, 658 (19:37). J . -4.Beattie, I). R . Douslin and S. IY.Levine, J . Chenz. P h y s . , 19, 918 (1931). E . J . Gornowski, E. H. Amick, Jr., and 4.S. Kison, I n d . Eng. Chem., 39, 1348 (1947). A . Michels, T. IVassenaar, T h . Zweitering antl P . Sinits, Physicu, 16, 501 (1950); A . Michels, B . Blaisse a n d C. Michels, f’roc. Roy. Soc. (London) A160, 358 (1937); A. Michels and C. llichels, i b i d . , A153, 201 (1936); A. Michels, C. Michels and H. n’outers, ibid.,A153, 214 (1936); B. J . Kendall and B. H . Sage, Petroleum ( L o i z d o n ) , 14, 181 (1951). i: F. G. Keyes and R . B . Brownlee, THISJOURSAL, 40, 25 (1918); F. G. KeJ-es, i b i d . , 53, 965 (1931); J . A . Beattie J . H . Keenajt and C. K. Lalnence, i b i d . , 52, 6 (1930). and F. G. Keyes, “Thermodynamic Properties of Steam, John %Xey and Sons, Inc., New York, N. Y . , 1936. *‘L. B. Smith, J. A . Beattie and IV.C. Kay, THISJOURK A L , 59, 1587 (1937); S. Young, Sei. Proc. Roy. Dub. S o c . , new series 12,374 (1909-1910).
+
Q
J
’
of reduced temperature and pressure for the substances listed in Table I . The region T , from 0.Sto 4.0 and P,from 0 to 9 was selected for detailed study. This region includes the interesting behavior near the critical point and covers the area of greatest practical interest. Outside of this region the available data become too sparse to give a good test of our theory. Indeed the data are relatively sparse above P,= 3 and T , = 2. For each point in a closely spaced matrix of values of P,and T,, the
compressibility factor was plotted as a function of the acentric factor. Most of the points fell on smooth curves (and usually on straight lines) within a few tenths of 1 5 , . The largest deviation of experimentally well established points is about The highly polar H?O and XH3 are excepted from this agreement, of course. Two typical sets of these curves are shown in Figs. 1 and 2 which are for P r 1.6 and 3.0, respectively, and the various values of T , indicated. Several sets of substances have nearly equal values of the confining factor. I n order to avoid undue confusion in Figs. 1 and 2, single points are shown for each set as indicated a t the bottom of the figure. The agreement of the individual substances within each set is excellent. For the data in Figs. 1 and 2, the average deviation is between 0.1 and 0.2%. Figure 1 shows very clearly one advantage of the present system over that of Meissner and Seferian.j Any attempt to include such highly polar substances as water and ammonia in a three parameter system will necessarily lead to deviations such as are shown by the open circles in Fig. 1. If such highly polar substances are to be included, a fourth parameter will be necessary for an accurate correlation. The compressibility factor is expressed generally as a function of three variables. ( P V l R T ) = z( r,, P,,w ) (2) The functional dependence on temperature and pressure is very complicated. Indeed this coxnplexity has retarded advances in this field very considerably. No simple analytical equation is adequate. The eight constant BenedicP equation is the simplest one which even approaches the desired accuracy. SI-hile we expect to examine completely analytical representations further, for the present we will expand the compressibility factor function as a power series in the acentric factor
-
3 =
z(0)
+ u ~ ( l l+
, , ,
(3)
where d o )d, l ’ , etc., are each functions of T,and P,. In almost all regions the first two terms in equation 3 are sufficient. This result corresponds to the straight lines in Figs. 1 and 2. iln attempt was made to evaluate the quadratic term d 2 )for the small region in which it appeared to be significant. While there was no difficulty a t any particular value of T , and P,(such as T, = 1.10 in Fig. l ) , the resulting values of z ( ? ) showed such irregular behavior as functions of Tr and P , that there arose considerable doubt as to their validity. Consequently, no values of d2’are reported a t this time, and the data in regions of apparent curvature are fitted with the best straight lines. The values of z(l) were plotted as functions of T, and P,and were adjusted within the limits of experimental error to yield a reasonably smooth function. The values of z(O) were initially much more accurate than those of z ( l ’ and were found t o be smooth functions of temperature and pressure without further adjustment. However, large graphs ( 3 ) H. P. hreissner and R. Seferian, C h e i n . Eng. Progiess, 47, 579 (1951). SI. Benedict. G . B . Webb a n d L. C. R u h i n . J C h r i n . Phys., 8, (1;) 33-1 ( I W O ) .
July 5, 1955 I
.o
VOLUMETRIC AND THERMODYNABLIC PROPERTIES O F
>
I
I
lrZ2.0
I .c
I
1
I
1
5
3435
FLUIDS I
I
T,= 3.0 2.5
I .4
1.30
1.6
1.25
1.5
I .20
0.8 I .4
-.PR VT 0.E 1.10
1.25
~w 120 I 15
0.4 0.4
0.2 N2
C3H8
"3
0.2
w.
::I
1.00
A
'ZH6
C4H10
c H4
H2S
co2
0
0.1
0.2
0.3
0.4
w.
I
I
0.I
=+=&
0.3
0.4
Fig. 2.-The compressibility factor as a function of the acentric factor at P , = 3.0 and the indicated values of Tr .
Fig. 1.-The compressibility factor as a function of the acentric factor, w, a t P, = 1.6 and the values of T, indicated.
straight lines even a t temperatures well removed from 0.7 where w is defined. The values are summarized in Table VI where log P, for w = 0 and were used to interpolate values of do)a t interme- ( b log P , / b w ) ~are tabulated. Second-order terms diate points. The final values of the functions in are not needed for the vapor pressure. The voluequation 3 are given in Tables 11-V. I n view of metric data are less accurate and the correlation the omission of the quadratic term one must re- plots show random deviations as high as 37, but gard the results in the region T , = 1.05-1.10; P, = averaging less than 1%. The greatest deviations occur in the range where the values are rounded to 1.4-2.0 as less precise. Outside of the critical region the spacing of points the hundredths place. The volumetric data for the two-phase region are in Tables I1 and I V are close enough to allow linear interpolation without significant error. Tables 111 repeated for convenience in Table VI1 as a function and V provide more closely spaced values in re- of reduced pressure. I t should be noted that the gions of large curvature. Even this close spacing functions Z T ( ~ ) and zp(l) are partial derivatives is not sufficient to allow linear interpolation a t all where the subscript indicates the variable held constant, thus points, but simple graphs should suffice. In certain regions either the data are poor or zTt1) (bz/bW)T zp(') = ( b z / b w ) p (4) more commonly the data deviate from the linear ZT(~) (b~/bT),(dT/bw)p correlation in w . In such regions values are given to only the second decimal place and correspond- The last equation gives a relationship between zp(l) ingly lower accuracy must be expected in the calcu- and z T ( I ) ; however, we found i t more convenient to lated results. evaluate each of the z functions directly from voluCoexisting Phases.-Separate studies were made metric data. of the vapor pressure and of the compressibility We expect to treat the volumetric data for liqfactor for both liquid and gas along the saturation uids a t low temperatures in greater detail later. curve. The vapor pressure data are generally The present approximate values in this region sufquite accurate and log P, vs. w plots give good fice, however, for calculations of Az of vaporization.
+
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K . S. PITZER, D. 2. LIPPMANN, R. F. CURL,JR., C. 11.HUGGINS AND D. E. PETERSENVol. 77
0.80 ,S5 .90
0.851 ,882 ,904
1.6 1.7 1.8 1.9 2.0 2.5 3.0 3.5
,985 ,988 ,991 ,993 ,995 1.000 1.001 1,002
1 ~
0.066
,067 ,778 1
,971 ,977 ,982 ,986 ,989 ,999 1.002 1,004
0.100 ,101 ,102
0.133 ,134 ,135
,957 ,966 ,974 ,980 ,984 ,999 1.003 1.006
,944 ,956 ,966 ,974 ,979 ,998 1.004 1.008
~
0.164 ,165 ,167
0.192 ,194 ,198
0.225 ,226 ,229
0.258 ,258 ,258
0.287 ,287 ,288
0.318 ,316 ,316
,930 ,946 ,958 ,968 ,975 ,998 1.005 1.011
,917 ,936 ,950 ,962 ,971 ,998 1.007 1.013
,904 ,926 ,944 ,958 ,968 ,998 1.008 1.015
,893 ,919 ,937 ,952 ,964 ,997 1 010 1.018
,882 ,911 ,931 ,948 ,961 ,999 1.012 1.020
,872 ,903 ,926 ,944 ,959 1.000 1.014 1 022
VOLUMETRICAND THERMODYNAMIC PROPERTIES OF FLUIDS
July 5 , 1955
3437
TABLE I1 (Continued) P,
1
Tr
0.8
3.6
3.8
4.0
4.5
5.0
6.0
7.0
8.0
0.547
0,576
0.605
0.675
0.746 ,730 ,718 ,709 ,702 ,700 ,699 ,705 ,714 ,726 ,740 ,781 ,826 ,860 ,895 ,925 ,950 ,972 1.035 1,058 1.067 1.068
0,883 ,861 ,842 ,828 ,819 ,814 ,810 ,809 ,810 ,816 ,824 ,844 ,877 ,904 ,930 ,955 ,976 ,996 1,055 1.077 1.086 1.086
1.017 0.990 ,966 ,947 ,932 ,923 ,916 ,911 ,907 ,907 ,910 ,921 934 ,953 ,972 ,993 1,010 1.027 1.079 1.10 1.105 1.104
1.15 1.115 1.089 1.066 1.048 1.032 1.019 1.008 1,000 0.994 ,992 ,994 1.000 1.010 1 ,023 1.039 1.051 1.064 1.105 I.124 1.126 1.124
,618 ,643 ,668 ,734
,601 629 ,659 ,727
,587 ,618 ,651 ,722
1.2 1.25 1.3 1.4
,664 ,682 ,701 ,754
~.
T* 0.90 .91 .92 .93 .94 .95 .96 .97 .98 .99 1.00 1.01 1.02 1.03 1.04 1.05
0.7 0.8 0 9 1 0 0.5 0.6 0.701 I 0.102 0 118 0.135 0.151 0,167 ,120 ,136 ,152 .168 ,104 ,715 ,- __ ,122 ,138 ,153 ,169 ,650 ,728 ,124 ,140 ,155 ,170 ,666 ,740 ,805 ,125 ,142 ,157 ,173 ,681 -,751 ,812 ,160 ,145 ,612 ,176 .697 ,782 ,819 ,632 -__ ,164 ,149 ,180 ,711 ,772 ,826 ,652 ,186 ,170 .56 ,724 ,832 ,782 177 ,193 669 ,591 -__ ,735 ,791 ,838 ,6113 ,514 1 ,205 . 685 ,746 ,800 844 ,291 ,638 ,554 ,699 ,757 ,807 ,849 ,476 ,654 ,583 ,713 ,767 ,813 ,854 . 608 ,525 ,672 ,726 ,776 ,820 ,860 ,558 ,630 ,687 ,737 ,784 ,826 ,865 ,648 ,586 ,701 ,748 ,793 ,833 ,870 ,605 ,609 .714 ,800 ,758 ,838 ,874
0.4 0.778 ,787 ,796
0.99 1.OO 1.01 I , 02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11
1
I
I
1
~
I
,
,205 ,291 ,476 ,525 ,558 ,586 ,609 ,628 ,645 ,663 ,677 ,691 ,703
,210 ,220 ,283 ,402 ,466 ,509 ,543 ,572 ,597 ,618 ,636 ,652 ,667
,223 ,231 ,243 ,273 .34 .41 ,470 ,505 ,534 ,562 ,587 ,607 ,625
,235 ,241 ,248 ,260 .29 .33 ,375 ,423 ,468 ,504 ,535 ,561 ,584
,247 ,250 ,259 ,270 .283 ,307 ,341 ,370 ,408 ,445 ,480 ,512 ,538
-
9.0
1.21 1.185 1.166 1.147 1.129 1.113 1.100 1.088 1.078 1.071 1.070 1.075 1.082 1.091 1.097 1.106 1.136 1.150 1.148 1.143
The substitution of our functional relationships for z and P yields a quadratic formula for the entropy. A S = As(o) f wAs(') f
(7)
Each of the coefficients is readily derived and, upon appropriate numerical calculations, yields the values given in the last three columns of Table VI. The unit is cal./degree mole. Since the available data for the volume of the saturated vapor a t very low reduced temperatures are scanty, use was made of calorimetric values of the entropy of vaporization in this region. However, once do)and z ( ' ) are fixed a t the lowest temperature to agree with entropy data, the remainder of their values follow from volumetric data and a requirement of reasonable smoothness.
,260 ,265 ,271 ,278 .288 ,302 ,324 ,349 ,379 ,412 ,443 ,473 ,502
,273 .278 ,283 ,291 ,297 ,307 ,320 ,336 ,358 ,388 ,412 ,442 ,469
,287 ,290 ,294 ,300 ,306 ,314 ,323 ,333 ,349 ,373 ,396 ,422 ,448
,301 ,304 ,307 ,311 ,316 ,324 ,332 ,343 .356 ,370 ,387 ,408 ,428
,315 ,317 ,319 ,323 ,328 ,334 ,341 ,348 ,358 ,369 ,383 ,400 ,418
,328 ,331 ,331 ,334 ,339 ,343 ,350 ,358 ,367 ,375 ,387 ,402 ,417
K. S. PITZER, D. 2. LIPPMANN, R. F. CURL,JR., C. M. HUGGINS AND D. E. PETERSENVol. 77
3438
TABLE IV VALUES OF dl)FOR COMPRESSIBILITY FACTOR CALCULATIOS (See Tables T'A and VB for additional data in the region enclosed by dotted lines.) Tr
0.80
1.10 1.15 1 20 1.25
1.7 1.8 1.9 2.0 2.5 3.0 3 . . '
1.6617 = 1.392 1.50~ w = 0.180
A similar calculation with the vapor pressure daturngba t IOOOF. yields w = 0.184, which is reasonable agreement. For the calculation of the compressibility factor of isobutane we take an example a t 340'F. and 800 lb./sq. in. since this is in the sensitive region a little above critical temperature and pressure. The reduced variables are Pr = 1.5533, Tr = 1.0919. Interpolation in Tables IIB and VB yields do) = 0.433 and z(') = 0.074, and substitution into equation 3 yields a calculated value z = 0.447 which may be with an experimental valueYb of 0.4517. SoTE ADDEDIs pRooF:-In additional papers Riedel ( C h e w . Ing. Tech., 26,259, 679 (1954)) has treated the volumetric behavior of liquid and vapor along the saturation curve and the entropy of vaporization. While the agreement with our results in these cases is not as precise as for vapor pressures,? it is satisfactory.
BERKELEY, CALIF. (9) (a) F. D. Rossini, et ai., "Selected Values of t h e Physical and Thermodynamic Properties of Hydrocarbons andRelated Compounds," Carnegie Press, Pittsburgh, p a . , lQ53; (b) R e f . g t o Table I.
