hydrate cell, Y23 for pentagonal dodecahedra and tetrakaidecahedra & = fugacity coefficient
3/23
for
Supercripts g = gas phase MT = empty hydrate lattice Subscripts c = methane i,j,k = component types m = cavity type m w = water Literature Cited
'
Bloch, M. G.,Lifland, P. P., Chem. Eng. Prog., 69 (9), 49 (1973). Claussen, W. F. J. Chem. Phys., 19, 259 (1951). Coan, C. R., King, A. D.. Jr., J. . Chem. Soc., 93, 1857 (1971). Culberson, 0. L., McKetta, J. J.. Petrol. Trans. AlME, 193, 223 (1951). Deaton, W. H., Frost, E. M., U S . Department of the Interior, Bureau of Mines, Monograph 8, 1946. Dodson, G. R.. Standing, M. B., Proc. A.P.I. Drill. Prod. Pract., 173 (1944). Dymond, J. H.. "Compilation of Second and Third Virial Coefficients," Physical Chemistry Laboratory, Oxford University, 1964.
Ertl, H.. Khoury, F., Slm, D., Kobayashi, R., Chem. hg. Techn. (1976) (in press). The article is available in English from this source. Galloway, T. J., Ruska, W.. Chappelear, P. S., Kobayashi, R., Ind. Eng. Chem., Fundam., 9, 237 (1.970). Hammerschmidt, E. G., Ind. Eng. Chem., 26, 851 (1934). Kobayashi, R., Katz, D. L., Trans. A I M , 204, 51 (Aug 1955). Laulhere. B. M., Briscoe, C. F., Gas, 15 (9), 21 (1939). Laughlin, A. R., "Water Content in Natural Gas," presented at the Gas Conditioning Conference, Norman, Okla., 1969. McCarthy, E. L., Boyd, W. L., and Reid, L. S., Trans. AIM€, 189, 241 (1950). McKetta, J. J.. Wehe. A. H.. Pet. Refiner, 37, 153 (Aug 1958). Olds, R. H., Sage, B. H., Lacey, W. N.,Ind. Eng. Chem., 34, 1223 (1942). Parrish, W. R., Prausnitz, J. M., Ind. Eng. Chem., Process Des. Dev., 11, 26 (1972). Records, L. R.. Seely, D. H., Trans. AIM€, 192, 61 (1951). Rigby. M., Prausnitz, J. M., J. Phys. Chem., 72, 330 (1968). Skinner, W., M. Ch.E. Thesis, University of Oklahoma, 1948. Stackelberg, M.. von Muller, H. R.. Z.Elektrochem., 58, 25 (1954). Walls, J. H. van der, Platteeuw, J. C., Adv. Chem. Phys., 2, 1 (1959).
Received for reuieu; February 6, 1976 Accepted July 22, 1976 Work reported here was supported in part by Northern Engineering
Services Co., Calgary, Canada, and by Columbia Gas System Service Corporation. Their support is gratefully acknowledged.
Empirical Description of the Liquid-Vapor Critical Region Based upon Coexistence Data Kenneth R. Hall' and Philip T. Eubank Chemical Engineering Department, Texas A&M University, College Station, Texas 77843
Empirical relationships from coexistence data form the basis for a description of the vapor-liquid critical region for pure fluids which is independent of but generally consistent with the scaling hypothesis. These relationships are: rectilinearity with temperature for mean density and for mean isochoric slope (dPld issuing from the coexistence curve; and power law behavior for the vapor-liquid differences of density, enthalpy, and (df / d 7&. The present description displays excellent agreement with data mapping divergences for various thermodynamic properties at the critical point. With one notable exception, this description also agrees with the theoretical predictions of the scaling hypothesis. The exception is: the present description produces 8 = 1 - 2 p as a lower bound which is somewhat larger than the scaling hypothesis assertion that 8 = a. Unfortunately, the data cannot distinguish between these results. Another interesting result is that the present description correctly predicts maxima with temperature for both mean enthalpy and mean entropy near the critical point.
nP.
Introduction A vast literature has developed during the past two decades concerning the vapor-liquid critical region of pure compounds; the recent work of Rowlinson (1972), Gielen et al. (1973), Levelt Sengers e t al. (1976), and Stanley (1971) exemplify these studies. The powerful scaling hypotheses proposed by Widom (1965) and Griffiths (1967) now form the basis for investigating critical phenomena. Essentially, the scaling hypothesis provides critical exponent equations t o describe the divergences of various thermodynamic properties along specific paths leading to the critical point (CP). In general, these relationships agree with the experimental data available, but in some instances agreement is uncertain causing controversy between experimentalists and theoreticians. Our purpose with this paper is to provide additional insight into the problem with a unique approach. Our description of the critical region derives solely from experimental evidence concerning the variation with temperature of coexisting densities (Figure l),isochoric slopes
issuing from the vapor pressure curve (Figure 2), and heat of vaporization. Although based in the two-phase region, this description correctly predicts critical exponents for the divergence of properties along paths in the single-phase region. We do not consider this description as a rival for the scaling hypothesis, but rather an inductive approach complementing the deductive theoretical model. In fact, only one serious disagreement appears to exist between the two approaches: the scaling hypothesis asserts that the exponent describing the curvature of the vapor pressure is 8 = LY (Vicentini-Missoni et al. (1969), Widom and Rowlinson (1970), and Green et al. (1971)), while the present description predicts 8 = 1 - 2 p as a lower bound. Direct experimental measurements of the slope of the vapor pressure curve are sparse and do not clearly indicate which value of 8 is correct; analysis of these older data as well as recent, yet unpublished data will be the subject of a subsequent paper. Ind. Eng. Chern., Fundarn., Vol. 15,No. 4 , 1976
323
15L
T 'i
i
Figure 1. Temperature-density graph with coexistence curve and normal rectilinear diameter, j7. Dashed line is that predicted by extended scaling hypothesis with (p - p c ) TI-^- near the CP.
-
,
5
I0
,
,
5
20
1
0
30
25
~
~
, I
35
~
Figure 3. Mean of the coexistence densities versus temperature for 3He (squares, Wallace and Meyer (1970)), 4He (circles, Kierstead (19'73)),and ethane (hexagons, Douslin and Harrison (1973)).3He and 4He show scatter about their respective critical densities indicating symmetric coexistence curves whereas the upper line illustrates the pronounced asymmetry of ethane. Each of the data sets is of high precision; the scatter is magnified by 1000 in the ordinate.
REDUCED TEMPERATURE, ?/?,
Figure 2. Pressure-temperature diagram with isochores the dashed curves. The dash/dot lines are the slope of these isochores at the coexistence curve.
Foundation of t h e Two-Phase Description Three empirical relationships form the foundation for our description of the critical region. These equations adequately represent data in the near critical region ( T < 0.01) but all lack higher order terms required to expand the region of agreement. The equations we have chosen are
(1)
and $a
=I ) ,
+ kgT F
kcTX
(3)
where p is density, h is enthalpy, $ is the isochoric slope, ( T,/P,).(~P/C~T),~, T is the reduced temperature difference ( T , - TIIT,, (P,,)K is ( P ~ / P , ) , ( A ~ ) Ris (hR- hi)/RT,, $o is ( T c / P,)[(aP/dT),]u, subscript c denotes a critical property, subscript u indicates a coexistence property, the upper signs in eq 1 and 3 denote vapor ( u = g) and the lower signs denote liquid ( u = 1). Equation 1 results from combining the "law" of rectilinear diameters (Cailletet and Mathias (l886,1887),,Cornfeld and Carr (1972), and Zollweg and Mulholland (1972)). (FIR E [(Pg)R
324
+ ( P 1 ) ~ ] / =2 1 + h 1 T
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
(la)
Figure 4. Difference of the coexistence densities vs. temperature on natural logarithmic coordinates for 3He (squares, Wallace and Meyer (1970)),4He(circles, Kierstead (1973)),and ethane (hexagons, Douslin and Harrison (1973)). The slope of each line is the critical exponent
a.
with the power law behavior of the coexisting density difference
Levelt Sengers et al. (1971) used a general equation which reduced to eq 1 to describe data for CO2, NOz, and CClF,; Douslin and Harrison (1973) used eq 1to describe their precise density measurements for ethane. All precise measurements of coexisting densities verify eq l a and it is common practice to establish p c using the equation and a known T,. A better procedure is to use eq l a and Ib t o establish p c and T , as demonstrated by Douslin and Harrison and previously by Davis and Rice (1967) with fl = %. Figures 3 and 4 demonstrate the validity of eq l a and l b . Gambill (1957) states that eq 2 has been proposed and tested a t least ten times in the literature after originating with Thiesen (1897). Thiesen used fl' = l/3. Narsimhan (1963) derives eq 2 with p' = % from the Eijtvos equation and that of the Parachor. Literature values of p' range from l/3 to '!/lo; Figure
T i
1m4
1 03
r
10-1
10-2
Figure 5. Reduced heat of vaporization vs. temperature on natural logarithmic coordinates for 3He (squares, Wallace and Meyer (1970)), carbon dioxide (diamonds, Michels et al. (193711,water (circles, Osborne et al. (1937)), and ethane (hexagons, Douslin and Harrison (1973)).The slope of each line is the critical exponent p’.
0.1
, ! l l \ l l
1
1
] , ] I l l !
1
1
IO-^
10-4
/ , 1 1 1 1 1
, ,
1 / 1 1 1
io-’
10-2
T
Figure 7. Difference of the reduced isochoric slopes vs. temperature on natural logarithmic coordinates for 4He (circles, Kierstead (1973)), argon (squares, Michels et al. (1958)),and methane (diamonds, Jansonne et al. (1970)).
i
142-
,
5
20
25
I 30
35
104xr
Figure 6. Mean of the reduced isochoric slopes vs. temperature for 4He (circles, Kierstead (1973)),argon (squares, Michels et al. (1958)), and methane (diamonds, Jansonne et al. (1970)).
l
o I
0
5 demonstrates t h e agreement between eq 2 and measurements. Equation 3 is analogous with eq 1 and results from combining the equations $E
$])/a
($g
= $c 4-
KST
(34
and
a$
($1
- $g)
= 2/26,’
(3b)
Figures 6 and 7 indicate the validity of eq 3a and 3b. Hall and Eubank (1976) have recently compiled a table listing values of A, $c, hg, and k6 for 3He, 4He, Ar, Xe, COz, C2H6, NP, 0 2 , CH4, and H20 for which precise data exist in the range 10-5 < T < 10-I. Interestingly, X has nearly its classical fluid value of (I/!!) for real fluids. T h e constants k l , k2, k4, k j , and k6 appear to be non-negative--k1 and k s are nearly zero for 3He and JHe but increase in value with molecular size, acentricity, and electrostatic complexity. T h e isochoric slopes usually result from differencing density data although Kierstead (1973) measured directly the slope for 4He.
The “Hook” of the Mean Density and Mean Isochoric Slope Current scaling models predict that (dif/dT) and (dF/dT) will diverge a t the C P with exponent cr (Po)R =
1
k1T
7 k2TB
+
k7T1-U
(4)
and $o
= $ c f kgT =F k6T’ 4-
k8Tl-O
(5)
2
I
I
4
6
II
10‘ x r
Figure 8. Test for rectilinearity of i j for ethane (Douslin and Harrison ~ deviations between data and least-squares fits; (1973)).The A i j are circles using eq la, squares using ZR = 1 -t k7T1-‘‘- with a- = lis. Clearly, eq la is superior. Equation 4 produces essentially identical results as eq la; however k7 = 0.0083 f 0.0539 which is statistically insignificant because the error is much greater than the coefficient. In addition, the least-squares truncation criterion of Hall and Canfield (1967) indicates that the best fit of the data is eq la not eq 4, which provides further statistical evidence that k7 is zero.
yet we have chosen to disregard this assertion in our basis. The reason is that no data exist which reliably demonstrate the hook for 7i and F. Because we seek an inductive description of the CP, we must use the data available and their consequences. Figure 8 illustrates for ethane the general observation that eq l a is superior t o eq 4 for fitting if^ us. T . Recently, Weiner e t al. (1974) claimed to have observed the “hook”, deriving this result from dielectric constant measurements of SF6. We feel this conclusion is premature for the following reasons: (a) SF6 would be completely out of correspondence with all other precisely observed substances under the same reduced conditions; (b) the gravitational correction, assumed equal for coexisting liquid and vapor (Feke e t al. (1973)) is questionable (Table I presents values of ( d p / & ) ~calculated from our basis for Ar which agree with the data presented by Mulholland e t al. (1975)); (c) Weber (1970) shows t h a t a 0.5 cm error in locating the meniscus would lead t o just such an apparent, but false, “hook.” Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
325
Table 1. ( d p ~ / d r =~ ()P ~~ K T ) for R Argon on the Coexistence Curve
curvature of the vapor pressure diverges to infinity as with 8=1-2/3
r
io-*
10-1
3.594 3.799
( d p ~ / d p ~ ) ~ l 0.2737 ~ ( d p ~ / d p ~ ) ~ ( 1 0.3191
10-3
54.48 51.18
10-5
This result is a stronger divergence than the scaling contention
775.2 736.8
11 110 10 610
$=a
The Vapor P r e s s u r e Equation Equations 1and 2, combined with the Clapeyron equation, provide the means to derive a vapor pressure equation for the critical region. The Clapeyron equation is an exact thermodynamic relationship among coexistence properties for a first-order phase transition Ah
(64
T(Au)
or in reduced notation d(Pu)R - -(Ah)R(PI)R(Pg)R (6b) d7 zc(l- T ) ( . ~ ) R where ( P u )is~the reduced vapor pressure, Pulp,,and 2, is P,/(Rp,T,). Substituting eq 1,2, and l b into eq 6b yields d(P,)R - -h4[(1 -k h . 1 ~ -) ~k 2 2 ~ 2 P ] 7 ( f l - 6 ) (7) dr hzZc(1 - 7) The slope of the vapor pressure a t the critical point is neither zero nor infinite but rather
P = P'
(9)
Table I1 contains experimental verification of eq 9. Taking the limit of eq 7 as r approaches zero provides an expression for the important quantity $c = hd(h2Zc)
(10)
Table I11 contains experimental verification of eq 10. Substituting eq 9 and 10 into eq 7 produces
+ klr)' - h 2 2 ~ 2 d ] (11) dr (1 - 7 ) Integrating eq 11 produces a vapor pressure equation valid near the C P while differentiating eq 11reveals the curvature of the vapor pressure -d(P,)R - -$,[(I
and
+ [(l+ k17)'-
h2'r"])
(13)
Taking the limit of eq 13 as r approaches zero reveals that the 326
(15)
with 1 - 2P x 0.29 and a = 0.13. The data cannot conclusively choose between eq 14 and 15; a logical theoretical approach would be to observe that eq 1,2, and l b are really only leading terms and that the higher order terms would resolve the conflict. When we insert any reasonable higher order terms into our derivation, even stronger divergences appear. We therefore conclude that 1 - 2/3 represents a probable lower bound for 8. Final resolution of the conflict between eq 14 and 15 must await conclusive experimental work. O t h e r Measurable P r o p e r t i e s in the Critical Region Table IV presents a summary of results from our analysis for various observable functions. Those properties above the dashed line derive from eq 1, 2, and 3 only. The properties below the dashed line require the additional assumption that the specific heat diverges in the same manner as the jump in specific heat a t the coexistence boundary. Derivations of these relationships have been deleted for brevity, but they are available from the authors upon request. The isothermal compressibility, KT p - l ( a p / d P ) ~ , is everywhere infinite within the two-phase region, but finite in the one-phase region (even along the coexistence curve) except a t the CP where it diverges to infinity. The divergence from below T , (along coexistence) is represented by the exponent y- while the exponent along the critical isochore from T > T , is y+.Our derivation indicates that y- = y f = y = 1 h - /3. The classical value of X = (l/2) together with the usual value of /3 = 0.35 then yields y = 1.15. Other properties diverging a t the CP with exponent y are: the thermal expansion coefficient, CY^ -p-'(ap/dT)p; the isothermal throttling coefficient, @T (ah/dP)T; and the specific heat a t constant pressure, Cp =
+
T (aS/aT)p.
Therefore, we conclude that
$c
(14)
10-4
We do not feel t h a t the existence of the "hook" is proven nor is it disproven. Although current data do not support this phenomenon, future data nearer the C P may reveal it. In fact, it is a moot point for our purposes-none of the results in this paper change if eq 4 and 5 replace eq 1 and 3, respectively.
dP, -=dT
T - ~
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
The shape of the critical isotherm, represented by IP - P,I p - pc I 6 , has an exponent 6 = (1 A)/@ Again with X = (%) and = 0.35, 6 is approximately 4.3 in agreement with observation. Roach (1968) defined the shape of the critical isobar as I T - T , I I p - pc I iT. Our analysis shows that P = 6. Equations 1 , 2 , and 3 provide a divergence for the jump in C, at the coexistence curve with an exponent of 1- h - (3 = a*. Using the normal values for X and /3 gives a* = 0.15 which closely agrees with values derived from data for the jump in C,, e.g., Amirkhanov and Kerimov (1963) for water and Moldover (1969) for 4He. Derivation of equations providing the exponents below the line in Table IV requires the assumption that C, itself diverges with the same exponent as the jump in C, a t coexistence, that is: a* = a-. This equality follows from the scaled equation of state of Vicentini-Missoni et al. (1969) and the works of Levelt Sengers and Greer (1972) and Moldover (1969). The last column in Table I1 shows approximate agreement between experimental values of a- (in parenthesis) and a* (calculated from 1 - h - p). The specific heat along the coexistence path, C,, has a divergence exponent of 1 - (3 (much stronger than a ) while the velocity of sound has an exponent of a/2. The latter exponent is supported by data from Thoen et al. (1971), Voronel (1974), and Barmatz (1970).
- I /3
+
-
Finite, Nonzero P r o p e r t i e s at t h e C P The Joule-Thomson coefficient, entropy, and enthalpy are all finite a t the CP but vary along either branch of the coex-
Table 11. Numerical Results for Critical Exponents and Constants of Various Compounds Data ref.o Wallace and Meyer (1970) (E) Wallace and Meyer (1979) (E) Brown and Meyer (1972) (E) Roach (1968) (E) El Hadi et al. (1969) (E) Kierstead (1973) (E) Kagoshima et al. (1973) (E) Moldover (1969) (E) Michels et al. (1958) (E) Gielen et al. (1973) (E) Voronel (1974) (E) Habgood and Schneider (1954) (E) Giglio and Benedek (1969) (E) Edwards et al. (1968) (E) Jacobsen and Stewart (1973) (C) Roder and Weber (1972) (C) Michels et al. (1937) (E) Lunacek and Cannel1 (1971) (E) Lipa et al. (1970) (E) Keenan et al. (1969) (Ci
Osbo'rne et al. (1937) (E) L. Sengers and Greer (1972) (C) Goodwin (1974) (C)
Range of
T
1.5 x 10-4 to 8.0 x 3 x 10-5 to 0.1 3.7 x 10-4 to 2.6 x 7.5 x 10-3 to 9.4 x 10-2 2.8 x 10-5 to 4.0 x 10-3 3.0 X 10-5 to 1.5 x 10-4 to 6.0 x 1.4 x 10-3 to 0.12 5.7 x 10-5 to 9.8 x 10-3 1.6 x 10-5 to 9.9 x 10-2 9.3 x 10-5 to 7.3 x 10-4 10-4 to 2 x 10-2 1.4 x 10-5 to 2.2 x 10-2 1.6 X to 0.13 3.7 x 10-3 to 9.4 x 10-2 8.9 X 10-5 to 2.0 x 10-2 10-4 to 3 x 10-2 1.1x 10-4 to 3.1 x 10-2 3.8 x 10-4 to 0.18 2.3 x 10-4 to 8.3 x
... ...
2.9 x 10-3 to
p
(E)
Eubank (1972) (C) Voronel (1974) (E)
rz-C4Hlo
Das and Eubank (1973) (C) Das et al. (1973) (C)
i-C4Hlo Das et al. (1973) (C)
4.1 x 10-4 to 8.1 x 10-3
3.46
0.278
a-=cY*=
Y=
l+A-P
(l+h)lP
1.10
4.04
(1.12)
(4.21)
...
1-A-p 0.18
...
...
...
(1.18)
0.354 0.357 0.48
3.94
0.292
1.13
4.18
0.17
0.360
...
...
0:280
...
...
...
...
...
...
...
...
...
...
...
...
.
...
(1.32)
...
...
...
.. .
...
...
...
(1.15)
...
(0.15)
0.362 0.352 0.52
5.83
0.276
1.16
4.20
0.12
0.363
.. .
...
...
0.274
(1.18)
(4.24)
...
...
...
...
...
...
...
...
(0.13)
0.35
...
0.57
5.91
0.30
1.22
4.49
0.08
...
(1.23)
...
...
...
...
...
(0.14 f 0.07)
...
...
o=
1,-2p
A
0.361 0.368 0.46
...
0.537 3.934
..
(0.105)
...
...
0.352 0.375 0.48
6.50
0.296
1.12
4.19
0.17
0.357 0.382 0.53
6.16
0.286
1.17
4.29
0.11
0.357 0.369 0.45
7.14
0.286
1.09
4.06
0.19
...
...
...
...
...
...
...
(1.22)
...
...
...
...
...
...
...
...
...
(0.125)
0.339 0.391 0.50
8.24
0.322
1.16
4.42
0.16
...
...
0.306
...
...
...
...
...
...
(1.20)
(4.45)
(0.1 f 0.05)
0.351 0.386 0.57
6.15
0.298
1.22
4.47
0.08
...
6.04
0.286
0.336 0.400 0.347
0.11
Jansoone et al. (1970) (E) Jansoone et al. (1970) E) Douslin and Harrison (1973)
p'
0.357
...
0.51
0.13
(0.06)
(1.29)
(4.45)
2.6 x 10-4 to 2.3 x
0.351 0.372 0.55
6.49
0.298
1.20
4.42
0.10
1.4 x 10-3 to 0.11 1.2 x 10-5 to 2.7 x 10-2 8.5 x 10-4 to 8.1 X lo-* 5.5 X 10W to 8.3 X 5.5 X 10-3 to 9.2 x 10-2
0.38
0.39
...
...
0.24
...
...
...
...
...
...
...
...
...
...
(0.13)
0.40
0.39
...
...
0.20
...
...
...
0.38
0.41
...
...
0.24
...
...
...
0.38
0.41
...
...
0.24
...
...
...
I' Under data ref.: (E) = experiment and (C) = correlational; the values in parentheses in the last four columns are from independent literature measurement or analysis.
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
327
Table 111. Constants and ComDarison of Their Combinations hi
kS ks -
kz k4 -
0
1.30 1.39 0.20 1.41 1.75 0.63 1.58 2.84 0.67 1.85 3.84
2 10 57 55
7 10 32.5 30.5
0.97 2.01 4.65 1.60 2.50 5.64
98 67
33 41
0.79 1.89 3.96
75
43
Data ref.
k3
Wallace and Meyer (1970) 2.07 Roach (1968) 2.5 Michels et al. (1958) 6.5 Jacobsen and Stewart 9.25 (1973) Michels et al. (1937) 10.5 14.4 Keenan et al. (1969) Douslin and Harrison (1973)
8.2
Data ref. ( k 3 ) Brown and Meyer (1972) Moldover (1969) Voronel (1974) Lipa et al. (1970)
1.36 1.70 2.66 3.47
1.0 1.5 5.4 5.7
0.63 0.70 0.50 0.54
Lipa et al. (1970) Amirkhanov and Kerimov (1963) Voronel(l974)
3.95 4.74
6.5 8.0
0.64 0.68
3.43
7.9
0.64
Table IV. Results for Various Observable Properties from Eq 1-3 Property
Path
Isothermal compressibility, (KT)R (P)R-’[(~PR/~PR)I]
- P,I Thermal expansion coefficient, Pressure vs. density, IP IP
- Pc16
Result derived from present basis
Coexistence, singlephase side Critical isochore, T T, Critical isotherm
Exponent y- = 1
>
+x -p
y + = Y- = Y
6 = (X
+ 1)/p
Coexistence, singlephase side Coexistence, singlephase side
y=1+x-p
y=1+x-p
Jump in isochoric specific heat, (AC\)R (AC,)/R
Critical isochore, T > Tc Coexistence, two-phase less single-phase
Gravitational correction
Coexistence, single-
1
Isochoric specific heat, C,
Two-phase along is to define k.3
( ~ P ) R= $(KT)R
Isothermal throttling coefficient, @Tz ( d h / d P ) ~ or (@T)R (P,/RT,)@T= ZcPK-’[l - (1 - 7)$(KT)R]
y=1+x-p
a* = 1 - A
-p
+ 2x - 3p
Coexistence, two-phase side Coexistence, singlephase side Specific heat along coexistence curve, C, Isobaric specific heat, Cp
Adiabatic compressibility, E
(K~)R
(P)R-’(~PRI~PR)s
Speed of sound, W ; WR ? W(P,/Pc)
Coexistence Coexistence, singlephase side Coexistence, singlephase side Coexistence, singlephase side
istence curve. At the CP the Joule-Thomson coefficient becomes iCc-l thus providing an alternate technique for determination of this important quantity. Integration of the equation for ( C u )of~ Table IV provides the change in entropy away from the C P
328
P-1
Ind. Eng. Chem., Fundam., Vol. 15,No. 4, 1976
T h e mean entropy (So)K
[(SI)R + (sg)R]/2
Z (Sc)R
+ hzk4T” - hs(X
+ p)-l~X+@(17)
has a “hook” a t the CP and further exhibits a maximum value
Figure 9 illustrates this behavior for water from Keenan et al. (1969) and nitrogen from Jacobsen and Stewart (1973). These authors find T , t o be -6 X 10-2 for water and -3 X for nitrogen. Comparison of these literature values with those predicted by eq 18 provides a sensitive check of both our empirical description of the two-phase region and the accuracy of the constants 0, A, k2, k3, and k4 of Tables 11and 111. The last column of Table I11 is ~ ~ ( k - for 8 ) 3He, 4He, Ar, N2, COz, H20, and ethane. With X = 0.48 and /3 = 0.352 (Table 11)for nitrogen, T, = 8.0 X However, this value of X is suspected to be low because (1)the careful C, measurements of Lipa et al. (1970) yield a- = (%) or X = 1- /3 - a- = 1 - 0.352 - 0.125 = 0.523, and (2) the value of X for oxygen is 0.52. With X = 0.523 and /3 = 0.352 for nitrogen, T , = 2.7 X lo-* in close agreement with Figure 9. For water with X = 0.50 and P = 0.347 (Levelt Sengers and Greer, 1972) T , = 8.0 X lo-* also agrees with Figure 9. Finally, values of T , can be estimated from the last column of Table I11 for any of the remaining five compounds. For example, T, = 7.6 X for CO2 from the measurements of Michels e t al. (1937) and using X = 1 - /3 a- = 1 - 0.357 - 0.125 = 0.518 based again upon the C, experiments of Lipa et al. The corresponding equation for the enthalpy along the coexistence curve is
( h , - h c )E~f
k
4
+ k & 4 ~ 2 J- k3(X + /3)-1~x+d - ZCICc7 knZclC/c(/3+ ~
~
1)-1~$+1
(19)
Close to the CP the last two terms may be neglected and ( h , - hc)R ( s c - S c ) R .
Exponent Inequalities Thermodynamics imposes certain inequalities among the critical exponents. Stanley (1971) lists these expressions in his fourth chapter. The important expressions are Rushbrooke: Griffiths:
oi
p(S
+ 2/3 + y I2
(20)
+ 1) I2 - a
(21)
- 1) y(6 + 1) I(2 - a)(6 - 1) 2 P(6
(22) (23)
As a direct result of our assumption that a- = a* = a , each of these expressions reduces to an equality thereby satisfying the thermodynamic requirements (other weak or seldom used inequalities are likewise satisfied).
Discussion The inductive approach t o the descripiton of the critical region we have presented generally agrees with the deductive scaling hypothesis with the exception of the value for 8. This discrepancy has its major consequence in evaluation of the curvature of the chemical potential with temperature. Using our basic equations and the assumption that a- = a* = a , we obtain
Equation 24 diverges to + a a t the CP because 1 - 26 > 1 h - 0,but it should go slowly because the negative divergence is not much weaker than the positive one. The lattice gas model (Yang and Yang, 1964) predicts that this curvature is zero and therefore 8 = oi. Moldover (1969) found the curvature to be less singular than the curvature of the vapor pressure “and possibly zero” when analyzing his C, data for 4He for T > Brown and Meyer (1972) found the curvature ap< T < lo-’ proximately constant for 3He in the range while Voronel(l974) found it to be finite but increasing in the range lop3 < T < 10-l. Barieau (1968) concluded that the
I
0
I
I
I
I
2
4
6
8
IO‘ x
10
I-
Figure 9. M e a n o f the coexistence entropies vs. temperature for water (circles, K e e n a n e t al. (1969)) a n d n i t r o g e n (squares, Jacobsen a n d Stewart (1973)). T h e m e a n e n t r o p y is here reduced b y t h e c r i t i c a l entropy f r o m t h e respective references rather t h a n b y t h e gas constant used in t h e text.
curvatures of both chemical potential and vapor pressure show discontinuities a t the CP. Indirect evidence comes from equations of state proposed by Goodwin (1970), Verbeke (1972), and Kreglewski (1975), all of which require 8 > a . Goodwin uses 0 > 0.5 while Gielen et al. (1973) find 8 = 0.1935 for Verbeke’s equation. Mulholland et al. (1975) show the breadth of the coexistence curve, k 2 , to increase with the slope of the mean density, k l , for a variety of compounds. The present values of k l and h z (Table 111) also show this trend. We feel that the present description possesses advantages for correlations in the two-phase region. Utilizing /3 and X rather than /3 and y has advantages because X is somewhat more convenient to measure than y and has very nearly its classical value. The insight provided for and 3 will be valuable for correlations in the critical region, especially the observation that they are not linear as often assumed [Eubank (1972), Das and Eubank (1973), and Das et al. (1973)]. Finally, we feel that the present description is complementary to the scaling hypothesis providing two-phase information in general agreement with the single-phase model. The sole discrepancy regarding values of 8 and a should provide valuable examination of this point.
Acknowledgment We gratefully acknowledge receipt of tabulated data from H. A. Kierstead, M. J. Buckingham, and H. Meyer. We also express appreciation to J. M. H. Levelt Sengers and coworkers for providing a preprint. We benefited from discussions with R. E. Allen, D. G. Naugle, G. D. Allen, and F. J. Narcowich. Financial support was provided by the National Science Foundation (ENG 74-23411 and GK-37097),American Gas Association (BR110-1, BR110-2), and Petroleum Research Fund (7594-AC7). Literature Cited Amirkhanov, Kh. I., Kerimov, A. M., Teploenergetika, I O , 61 (1963). Barieau, R. E., J. Chem. Phys., 49, 2279 (1968). Barmatz, M., Phys. Rev. Lett., 24, 651 (1970). Brown, G. R., Meyer, H., Phys. Rev., A6, 364 (1972). Cailletet, L., Mathias, E. C., C. R. Hebd. Seane. Acad. Sci. (Paris), 102, 1202 (1886); 104, 1563 (1887). Cornfeld, A. B., Carr, H. Y., Phys. Rev. Lett., 29, 28 (1972). Das, T. R., Eubank, P. T., Adv. Cryo. Eng., 18, 208 (1973). Das, T. R., Reed, C. O., Jr., Eubank, P. T., J. Chem. Eng. Data, 18, 244 253 (1973). Davis, B. W., Rice, 0. K., J. Chem. Phys., 47, 5043 (1967).
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Douslin, D. R., Harrison, R. H., J. Chem. Thermodyn., 5, 491 (1973). Edwards, C., Lipa, J. A., Buckingham, M. J., Phys. Rev. Lett., 20, 496 (1968); value of (r = 0.08 f 0.08 has been revised to 0.14 f 0.007: see J. A. Lipa, Thesis, University of Western Australia, 1970. El Hadi, Z. E. H. A,, Durieux. M., Van Dijk, H., Physica. 41, 289, (1969). Eubank, P.T., Adv. Cryo. fng., 17, 270 (1972). Feke, T. T., Lastovka, J. B.. Benedek, G. B., Langley, K. H., Elterman, P. 6.. Opt. Commun., 7, 13 (1973). Garnbill, W. R., Chem. Eng., 64, 261 (1957). Gielen. H.. Jansoone, V., Verbeke, O., J. Chem. Phys., 59, 5763 (1973). Giglio. M.. Benedek, G. B., Phys. Rev. Lett., 23, 1145 (1969). Goodwin. R. D., J. Res. Natl. Bur. Stand., 74A, 655 (1970). Goodwin, R. D., NBS Tech. Note 653, April 1974. Green, M. S.. Cooper, M. J., Levelt Sengers, J. M. H., Phys. Rev. Lett., 26,492 (1971). Griffiths. R. B., Phys. Rev., 158, 176 (1967). Habgood. H. W., Schneider, W. G., Can. J. Chem., 32, 98 (1954). Hall, K. R., Canfield, F. B., Physica, 33, 481 (1967). Hall, K. R., Eubank, P. T., lnd. fng. Chem., Fundam., 15, 80 (1976). Jacobsen, R. T., Stewart, R. B., J. Phys. Chem. Ref. Data, 2, 757 (1973). Jansoone, V., Gielen, H.. De Boelpaep, J., Verbeke, 0.B., Physica, 46, 213 (1970). Kagoshima, S., Ohbayashi, K., Ikushima, A., J. Low Temp. Phys., 11, 765 (1973). Keenan, J. H., Keyes, F. G., Hill, P. G., Moore, J, G., "Steam Tables", Wiley, New York, N.Y., 1969. Kierstead, H. A., Phys. Rev., A7, 242 (1973). Kreglewski, A., Therrno. Res. Center, Texas A&M University, private communication, 1975. Levelt Sengers, J. M. H., Straub, J.. Vicentini-Missoni, M., J. Chem. Phys., 54, 5034 (1971). Levelt Sengers, J. M. H., Greer, S. C., int. J. Heat Mass Transfer, 15, 1865 (1972). Levelt Sengers, J. M. H., Greer, W. L. Sengers, J. V., J. Phys. Chem. Ref. Data, in pressr(1976). Lipa, J. A,, Edwards, C., Buckingham, M. J., Phys. Rev. Lett., 25, 1086 (1970).
Lunacek, J. H., Cannell, D. S., Phys. Rev. Len., 27, 841 (1971). Michels, A . , Blaisse, B., Michels, C., Proc. Roy. SOC.,A160, 358 (1937); temperature scale corrected as recommended by J. M. H. Levelt Sengers, W. T. Chen, J. Chem. Phys.. 56, 595 (1972). Michels, A., Levelt, J. M. H., de Graaff, W., Physica, 24, 659 (1958); with correction of temperature scale as in J. M. H. Levelt Sengers, lnd, fng. Chem., Fundam.. 9, 470 (1970). Moldover, M. R., Phys. Rev., 182, 342 (1969). Mulholland, G. W., Zollweg. J. A., Levelt Sengers, J. M. H., J. Chem. Phys., 62, 2535 (1975). Narsimhan. G. J. Phys. Chem., 67, 2238 (1963). Osborne, N. S., Stimson, H. F., Ginnings, D. C., J. Res. Mtl.Bur. Stand., 18,389 (1937). Roach, P.. Phys. Rev., 170, 213 (1968). Roder, H. M., Weber, L. A.. NASA, SP-3071, Vol. I(1972). Rowlinson, J. S., Ber. Bunsenges. Phys. Chem., 76 (3/4). 281 (1972). Stanley, H. E., "Introduction to Phase Transitions and Critical Phenomena," Chapter 2, Oxford University Press, Oxford, 1971. Thiesen, M.. Verhl. Deut. Phys. Ges., 16, 80 (1897). Thoen, J.. Vangeel, E., Van Dael, W., Physica, 52,205 (1971). Verbeke, 0. B., J. Res. Natl. Bur. Stand., 76A, 207 (1972). Vicentini-Missoni, M., Levelt Sengers, J. M. H., Green, M. S., J. Res. Mtl. Stad., 73A, 563 (1969). Voronel, A. V., Physica, 73, 195 (1974). Wallace, B., Meyer, H., Phys. Rev., A2, 1563 (1970); "Tabulation of the Original P-V-T Data for 3He-4He Mixtures and for 3He", Physics Dept., Duke University, 1971. Weber, L. A,, Phys. Rev., A2, 2379 (1970). Weiner, J., Langley, K. H., Ford, N. D.. Phys. Rev. Len., 32, 879 (1974). Widom. B., J. Chem. Phys., 43, 3898 (1965). Widom, B., Rowlinson, J. S., J. Chem. Phys., 52, 1670 (1970). Yang. C. N., Yang, C. P., Phys. Rev. Lett., 13,303 (1964). Zollweg, J. A,, Mulholland, G. W., J, Chem. Phys., 57, 1021 (1972).
Received for review M a r c h 11, 1976 Accepted June 8,1976
Solubilities of Gases and Volatile Liquids in Polyethylene and in Ethylene-Vinyl Acetate Copolymers in the Region 125-225 O C David D. Liu and John M. Prausnitz' Chemical Engineering Department, University of California, Berkeley, Berkeley, California 94 720
Gas-liquid chromatography was used to measure low pressure solubilities of nine volatile solutes in polyethylene and in copolymers of ethylene and vinyl acetate containing 3.95, 9.2, and 30.3 wt % vinyl acetate. Solubilities in ethylene-free poly(viny1 acetate) were reported earlier. The solutes studied are methyl ethyl ketone, acetone, isopropyl alcohol, vinyl acetate, sulfur dioxide, methyl chloride, ethane, ethylene, and carbon dioxide. While gas-liquid chromatography provides a rapid and simple method for measuring solubilities, experimental precautions must be observed to assure reliable results. Such precautions are particularly important for sparingly soluble solutes (e.g., ethane, ethylene, carbon dioxide) where the measured solubilities tend to be less accurate than those attained with higher-boiling solvents.
Introduction Ethylene-vinyl acetate copolymers are widely used in competition with plasticised vinyl resins or vulcanized rubbers. They are also used as modifiers of wax and similar materials and as strength-contributing polymers in hot-melt adhesives. Since ethylene is a gas as well as a sluggish monomer, high pressure must be used in the synthesis of ethylene-containing copolymers rich in ethylene and to achieve reasonably rapid reaction rates. Typical reaction conditions are in the range 100-300 "C and 1000-2500 atm. Since the reaction is rarely complete, low pressure separators are used to separate unreacted monomers from the copolymer for recycling. These separators also separate other volatile ad330
Ind. Eng. Chern., Fundarn., Vol. 15, No. 4, 1976
ditives in the reaction mixture. For engineering design, therefore, it is necessary to have available accurate solubilities of volatile compounds in the polymer.
Thermodynamic Analysis The equation for vapor-liquid equilibrium for volatile component i is
where y , is the vapor phase mole fraction, P is the total pressure, w ,stands for the weight fraction in polymer phase, R is the gas constant, and T is the absolute temperature. The
