Saturated liquid densities of polar and nonpolar pure substances

Scott W. Campbell, and George Thodos. Ind. Eng. Chem. Fundamen. , 1984, 23 (4), pp 500–510. DOI: 10.1021/i100016a021. Publication Date: November 198...
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Ind. Eng. Chem. Fundam. 1984, 2 3 , 500-510

5110

Saturated Liquid Densities of Polar and Nonpolar Pure Substances Scott W. Campbell and George Thodos" Northwestern University, Evanston, Illinois 6020 1

Experimental informationavailable in the literature for the density behavior of the saturated liquid state of 46 nonpolar p( 1 - T,)" , in which and polar substances has been modeled according to the expression, pR = 1 a(1 - T,)"' pR is the ratio of the density at reduced temperature T , to the critical density. This relationship has been found to apply over the complete range between the triple point and the critical point. Exponent rn is a universal constant (rn = I*). The critical constants and normal boiling points are needed to establish a,p, and n for nonpolar substances. For polar substances, the dipole moment is also required. Saturated liquid densities calculated with correlated values of a,p, and n have been compared with corresponding experimental measurements to yield an overall average deviation of 0.83% (4284 points) for the 46 substances used in this development. This generalized treatment has been applied to 16 additional substances to yield for them an average deviation of 1.14% (669 points).

+

The ability to predict saturated liquid densities for elements and compounds continues to play an important role in the treatment of thermodynamic properties. Although the calculation of such densities is possible through the use of an equation of state more often than not, the calculated saturated densities are not in good agreement with corresponding experimental measurements. Rather than relying on an equation of state to calculate saturated liquid densities, it is perhaps more expeditious to utilize a relationship that applies directly to the saturated liquid state. Early attempts express the dependence of density in the form of first, second, and third degree polynomials in temperature (82). Although the form of such relationships proves satisfadory for temperatures approaching the normal boiling point, it does not accommodate the density behavior at higher temperatures, particularly near the critical point. Riedel (161) presented a generalized relationship for the reduced saturated liquid density in terms of reduced temperature as follows PR

= 1 + 0.85(1 - TR) + (0.53

+ 0.2ac)(1

-

TR)"3

(1)

Equation 1is applicable to nonpolar and nonassociating polar liquids between their triple points and critical points. has been The Riedel factor, a, = (d In PR/d In TR)TR=l.oo, shown by Reid and Sherwood (160) to relate to the Pitzer acentric factor, w , as follows w = 0.2O3(ac - 7.00) + 0.242 (2) Thus, by substitution, eq 1may be expressed in terms of w through the relationship PR

= 1 + 0.85(1 - TR)

+ (1.6916 + 0.9846~)(1 TR)1'3 (3)

Equation 3 is somewhat more convenient to apply than eq 1, since values of w are more accessible than corresponding values of ac. Yen and Woods (222)investigated the density behavior of 62 pure compounds, whose critical compressibility factors range from z, = 0.21 to z, = 0.29, and proposed the relationship /JR

=1

+ A(1 - TR)'I3 + B(1 - TRI2l3+ D(1 - TR)4'3

(4) where A , B, and D are given as third-order polynomials in 2,. These investigators reported an average deviation of 2.1% (693 points) for the 62 substances.

+

Rackett (153) utilized the fact that pR depends on (1TR) and, in addition, takes advantage of the dependence of this reduced density on z, to define the relationship for the saturated liquid state as (5) where uR = l / p R This simple relationship applies to nonpolar substances and nonassociating polar compounds, but is claimed by Rackett to be inadequate for the estimation of densities for helium, hydrogen, alcohols, carboxylic acids, nitriles, and for water below T R = 0.82 (531 K). Spencer and Danner (184) utilized the simple form of the Rackett equation, eliminated the critical density, and replaced z, by the parameter, ,Z to define saturated liquid densities as

Values for the parameter ZRAwere presented by Spencer and Danner for 111 substances which included hydrocarbons, organic and inorganic compounds, and elements. A linear regression analysis was used by them to establish the parameter 2, for each substance. Spencer and Adler (183) updated the values reported by Spencer and Danner, included 54 additional compounds, and reported values of ZRA, for all 165 substances. Joffe and Zudkevitch (87) extended the development of Riedel (161) given by eq 3 to express the dependence of densities for both polar and nonpolar substances between their triple points and their critical points as follows PR = 1 + 0.85(1 - TR)+ (1.6916 0.9846$)(1 - TR)"~ (7) where ic, is a nongeneralized temperature dependent parameter for polar compounds and a constant for nonpolar substances, but varies from substance to substance. To establish ic, for polar compounds, two densities must be known while for nonpolar substances only one such value is needed. Despite the similarity of eq 3 and 7 , it was noted by Joffe and Zudkevitch (87) that parameter rarely has the same value as w , the acentric factor. Density Model for Saturated Liquids A comprehensive treatment for the densities of liquids in their saturated states between the triple point and critical point cannot be expressed by a simple relationship. This follows from the fact that although the dependence

0196-4313/84/1023-0500$01.50/00 1984 American Chemical Society

+

+

Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984 501

of density on temperature is essentially linear near the triple point, this dependence undergoes a continual change with increasing temperature and becomes exponential in nature as the critical point is approached. To gain an insight into the type of relationship applicable within this interval, use can be made of available expressions relating to the sum and differences of saturated liquid and vapor densities. The rectilinear diameter expression, first proposed by Cailletet and Mathias (14),can be expressed in reduced form as 1

z(PR1

+ PRv)

= 1 + c(1- TR)

(8)

Young (227)suggested that this mean density can be better expressed as a second- or third-order polynomial in temperature in order to account for the nonlinear behavior of (pR1 pRv)/2 in the vicinity of the critical point. For the difference of the coexistence densities, Goldhammer (65), using Young's experimental measurements for twelve polar and nonpolar compounds, proposed the expression

+

PR1 - PRv

= E(1 - TR)1/3 (9) The sum of eq 8 and 9 yields, for the reduced density of saturated liquids, the expression

Equation 10 resembles in form the expression suggested by Riedel (161),given by eq 1, where C = 0.85 and E = 2(0.53 0 . 2 4 . The general form of eq 10, although interesting, should be reassessed with regard to its constants and exponents in view of the fact that a large amount of experimental information has appeared in the literature since 1910 and particularly in the past 30 years. The availability of this recent information makes it mandatory to examine the ability of eq 10 to conform to the actual density behavior of the saturated liquid state. In order to make eq 10 more general, the following form has been assumed pR = 1 a(l - Tdrn p(1- TR)" (11)

+

+

+

Thus, this basic density model to be examined becomes more flexible without any restrictions imposed to constants a and /3 and exponents m and n. The form of eq 11 properly accommodates the extremes between T R = 0 and T R = 1. At the critical point, the reduced density becomes pR = 1while a t absolute zero, the hypothetical saturated liquid density takes the value, pRo = 1 a /3. Furthermore, the form of eq 11 properly accounts for the condition that dpR/dTR a as T R 1, if either of the exponents m or n is less than unity. Graphical Interpretation of Proposed Model From eq 11, the deviation of density from critical state conditions is a function of reduced temperature p~ - 1 = f(1 - TR) (12) In this regard, considerable information relating to this dependence can be obtained by plotting In (pR - 1)vs. In (1- TR). Figure 1 presents this functional dependence, obtained from experimental density measurements for n-butane, l-propanol, and water. The general pattern exhibited by these compounds is essentially the same near their critical points where this dependence is found to be linear. For reduced temperatures below T R = 0.85 this linear dependence no longer holds in general, but instead follows a course dependent upon the nature of the substance. For nonpolar and nonassociated compounds, the general tendency is for the density to be higher than

+ +

+

predicted from its linear extension, as depicted by n-butane. On the other hand, l-propanol exhibits a nearly linear dependence throughout its entire liquid-vapor coexistence region, while water follows a trend opposite to that exhibited by n-butane. The behavior for water is undoubtedly linked with its unexpected volumetric increase as its triple point is approached. Experimental saturated liquid densities available in the literature for 46 compounds and elements were treated in the manner depicted in Figure 1. These substances, along with their basic physical properties and sources of density information, are presented in Table I. For each of these substances, ar enlarged plot of log (pR - 1)vs. log (1- TR) was prepared covering the range available for the saturated liquid state. A careful examination of these relationships in the vicinity of the critical point showed that the slope, m, of the linear behavior in this region ranged from 0.347 for oxygen to 0.405 for ammonia. For water this slope was 0.378. Furthermore, the value of m appeared to be independent of the class of compound and its molecular complexity. Thus, for the liquid state of the monatomic elements, m = 0.391; for the diatomic compounds, m = 0.378; the alkanes, m = 0.375; the naphthenes, m = 0.379; the aromatics, m = 0.374; and the alcohols, m = 0.380. The average slope of the 46 substances examined in this study was found to be 0.379 and therefore the exponent m of eq 11was fixed at the value 3/8. Upon fixing this exponent, a becomes the intercept of the linear portion of the log (pR - 1)vs. log (1- TR) relationship when it is extended to T R = 0. Therefore, the constant /3 becomes the difference between the extension of the actual density curve to TR = 0 and this value of CY. To complete the definition of eq 11,the constants CY and /3 and the exponent n were established from available density measurements for each substance using a nonlinear regression analysis. The values of these parameters resulting from this treatment are listed in Table I. Positive values of /3 are indicative of the behavior shown by n-butane while negative values are associated with that typified by water. Although these calculated parameters best represent the density behavior for the substances included in Table I, these values have been used to develop generalized relationships capable of predicting a,p, and n for other substances. Correlation of Parameters a, 0, and n The parameter a exhibits the largest contribution to density over the complete saturated liquid region. For nonpolar substances with negligible quantum contributions, the characterization factor

+

s=

TRb

In

pc

- TRb

(13)

can be related to this density parameter. The factor s is the negative slope of a straight line between the critical point and the normal boiling point on In PRvs. 1/TR coordinates. Although this factor accommodates nonpolar substances, additional factors must be introduced to account for polarity, molecular association, and quantum effects. The parameter selected to account for quantum effects 1 A= (MTc)1/2uc1/3 is related to the quantum parameter advanced by Hirschfelder et al. (79)

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Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984

0

a m m w w d w 3mwmmmm w - o m ~ m m m m m m m I"""" """""c?" 0 0 0 0 0

0 0 0 0 0 0 0

0000

A T 0 0

Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984 503

which allows for deviations from the classical principle of correspondingstates. Upon substituting m = M / N , A and A, differ only by a constant. Contributions due to polarity and molecular association, because of the presence of hydroxyl groups, have been included to completely define the density parameter a as follows a

(e)'"'

= 1.278 + 1.092s - 198.1 -

ZC

+ A&lar + A 6 H

(16) To account for the special nature of polar compounds, A;ola the contribution due to polarity, has been introduced and was found to be where the modulus 6 represents a reduced dipole moment defined as

4 k

k2

6=(18) Tcu, In eq 18, the dipole moment, p, is expressed in Debye units, the critical temperature, T,, in Kelvins, and the critical volume, uc, in cm3/g-mol. For nonpolar compounds, the dipole moment, p = 0, and thus the modulus 0 = 0; consequently A;olar = 0. The final term A& is necessary to completely define parameter a. This term must be included to account for the associating nature of hydroxyl groups present in water, alcohols, and carboxylic acids. For these types of associated compounds, this contribution has been found to be A& = 3*277 - 13.509

(19)

ZC 0

3.9

a m tmA N

where z, is the critical compressibility factor. For all other classes of compounds, A& should be ignored. Values of parameter a have been calculated using eq 16 and are included in Table I1 along with values for s, p, A, and 6. Although the contribution of parameter /3 plays a minor role in the critical point region, its effect becomes progressively important at lower temperatures and is most pronounced at the triple point. For both polar and nonpolar compounds, this parameter is dependent upon the critical compressibility factor. The quantum contribution and that due to molecular association, because of the presence of hydroxyl groups, also influence 0. By including these contributions, this parameter becomes 1 - 4 9 . 5 9 ~ i l '-~ A$H l ~ (20) = 1.15 0.003319e20~472~ The contribution due to quantum effects is ordinarily small except for helium, hydrogen, and neon. The presence of hydroxyl groups in water, alcohols, and carboxylic acids has been found to relate to the modulus A$H = 0.01784zc(sTc/M)1~057 (21) and is zero for other classes of compounds. Values of /3 calculated with eq 20 are given in Table 11. The correlation of exponent n follows the general trend exhibited by the density parameter /3 for both polar and nonpolar substances. Thus, this exponent relates in the same manner to TRb and z, and has additional contributions due to quantum effects and the presence of hydroxyl groups in water, alcohols, and carboxylic acids. This exponent can be expressed as

'

% e

+

n = 0.5

TRb + -[1 + 258.2A2.088+ M H 1 2:J3

(22)

where the quantum contribution is again ordinarily small,

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Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984

-13

-2 n-BUTANE (M=58.123 zc=0.2736) TC=425.4 K

pc=0.228 g/cm3

P,=37.47 atm

-

1.0 -- 0.8

vc= 254.91 cmyg-mole Haynor and Hlza (1977) Kahn (1973) 0 Kay(1940) 0 Leqatshi, Nelson, Dean and Fruit (1942) t McClune (1976) 9 Rossini et a1 11953) X Save, Webster and Laccy (1937) 4

- 0.6

- 0.5 - 0.4 - 0.3 10.2

n-PROPANOL (M~60.096zc=0.2528) T= , 536.71 K

-2

= 51.02 atm

pc=0.2754g/cm3 ve= 218.21 cmyg-mole

-- 1.0 4 Ambmr and Townsend (1963) 0 Hales and Ellsnder (1976) 0 0

Ortega (1982) Ramsay and Young (1889)

+ Shaates and Kay (1964)

- & - 0.5 -0.8

- 0.6

- 0.4

9 Vogel (1948) x Young (19101

/**"*' -2

WATER (M.18.016 Tc=647.30 K

z,:O.2351)

e=218.3 atm

pe=0.315 g/cm3

+, kAr

v,= 5 7 9 cmj/g-mole

- '- g - 0.6 -1.0 -Q8

t Amagat (1693) X

Eastan, Mitchell and Wynrm-Jones (1952)

rn Eck(1939)

Hirn(1866) Kennedy, KnigM and Holster(1958) byes and Smith (1931) + Mmddejeff (1861) Osbarne, Stimson and Ginnings (19371 Ramsay and Yaung (1892) h Smith and Keyes (1934) 0 Wateraton (1861) 0

- 0.3

0

J

0.0010

1

0.002

I

I

I

1 1 I l l 0.010

0.004 006

I

0.02

1

I

I

I 1 1 1 1

0.04 .06

D8 (XI0

I

0.2

I

I

0.3 0.4

- 0.5 - 0.4

- 0.2

I l l I l 0 . 1 0

0.6

O.% 1.0

I -TR Figure 1. Types of density-temperature relationships resulting from experimental saturated liquid densities for n-butane, 1-propanol, and water.

except for helium, hydrogen, and neon. The contribution AZH due to the presence of hydroxyl groups was found to be AaH = 27.6&-342,6"O/PcTc

(23)

for water, alcohols, and carboxylic acids and is made A ~ H = 0 for all other classes of compounds. Values of n calculated with eq 22 are also given in Table 11. The calculated parameters a,p, and n shown in Table

I1 compare favorably with the original values obtained from the nonlinear regression analysis given in Table I. Therefore, eq 16,20 and 22, developed for the prediction of parameters a,0, and n are of a generalized nature and therefore become nonrestrictive in application to accommodate substances not used in this development. Applications to Other Substances In addition to the 46 substances involved in the development of eq 16,20, and 22, a number of randomly selected

Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984 505 Table 11. Auxiliary Constants and Density Deviations Resulting - from Predicted Parameters predicted density parameters S A X lo3 p,D 0 X lo4 a,eq 16 (3, eq 20 n, eq 22 monatomic helium 3.6567 56.837 0 0 1.4967 -0.5022 2.5089 neon 5.0956 9.628 2.0517 1.4250 0 0 0.3021 argon 5.3324 3.050 0 2.0827 0 1.3732 0.3872 krypton 5.3497 1.677 2.0455 0.4217 0 0 1.3683 xenon 5.3626 1.041 0 0 2.0660 1.3609 0.4143 didtomic n-hydrogen 4.0612 30.410 0 0 1.7416 -0.0069 1.5724 nitrogen 5.5659 3.761 2.1303 0 0 0.3989 1.4288 oxygen 5.4716 3.396 0 2.0881 0 0.4089 1.3851 carbon monoxide 3.617 5.6406 0.112 2.1903 0.3781 1.4253 0.010 fluorine 3.338 5.6513 0 2.1439 0.4097 0 1.3939 chlorine 5.8226 1.167 0 2.1000 0.4786 0 1.3819 2.124 6.0604 hydrogen chloride 1.08 1.9762 0.5810 1.4205 0.443 polyatomic carbon dioxide 6.6654 1.901 0 0 2.3488 0.4773 1.4368 ammonia 1.47 6.8282 2.892 0.740 2.3872 0.6048 1.4505 sulfur dioxide 1.63 6.8380 1.214 0.505 2.3772 0.5063 1.4471 water 1.85 7.3308 2.404 0.925 2.8468 -0.8859 3.7234 aliphatic hydrocarbons methane 5.3953 3.917 0 0 2.0469 0.4160 1.3918 ethylene 5.8582 2.213 0 0 2.1590 0.4455 1.4162 ethane 5.9202 1.982 0 0 2.1726 0.4499 1.4238 6.1773 propylene 1.424 0.366 2.2217 0.020 0.4691 1.4485 propane 6.2185 1.335 0.084 0.001 2.2487 0.4615 1.4566 2-methylpropane 6.3874 1.013 0.132 2.3331 0.002 0.4451 1.4762 n-butane 6.4678 1.003 0 2.2821 0 0.4854 1.4874 2-methylbutane 6.6259 0.815 0.13 0.001 2.2995 0.5014 1.5116 6.7482 n-pentane 0.802 0 2.3191 0.5097 1.5208 0 n-hexane 6.9854 0.666 0 2.3494 0 0.5290 1.5496 n-heptane 7.2457 0.571 0 2.3849 0 0.5473 1.5777 n-octane 7.5075 0.497 0 2.4534 0.5500 0 1.6001 naphthenes cyclopropane 6.0987 1.415 0 0 2.1773 0.4806 1.4289 cyclohexane 6.5500 0.686 0 2.3007 0 0.4897 1.4858 methylcyclohexane 6.6754 0.589 0 0 2.2995 0.5095 1.5137 aromatics benzene 6.5573 0.749 0 0 2.2869 0.4977 1.4711 toluene 6.8289 0.629 0.36 0.007 2.2991 0.5316 1.5120 o-xylene 7.0783 0.539 0.62 0.017 2.3636 0.5353 1.5349 rn-xylene 7.1514 0.541 0.005 0.34 2.3573 0.5478 1.5477 p -xy1ene 7.1280 0.540 0 0 2.3543 0.5461 1.5467 alcohols methanol 8.4597 1.592 1.70 0.479 2.6213 -0.0418 1.5925 ethanol 8.8334 1.177 1.69 0.331 2.6347 0.0189 1.5838 1-propanol 8.7501 0.925 1.68 0.241 2.6005 0.1214 1.5914 8.5732 1-butanol 0.753 1.66 0.178 2.5328 0.1647 1.5886 miscellaneous methyl chloride 6.2387 1.331 1.87 0.604 2.2791 0.5066 1.4278 6.4525 carbon tetrachloride 0.525 0 2.2625 0 0.4950 1.4710 6.9198 diethyl ether 0.822 1.15 2.3164 0.101 0.5351 1.5297 7.0699 0.981 acetone 2.88 2.4191 0.781 0.6478 1.5543 acetic acid 7.8435 0.955 1.74 2.5405 0.299 0.3810 1.6279 acetonitrile 7.0987 1.196 3.92 2.7399 1.619 0.7698 1.6390

compounds were used to test the capability of this method to predict saturated liquid densities. Altogether 16 additional compounds were considered, including nonpolar, slightly polar, and strongly polar liquids. These substances are listed in Table I11 and include para-hydrogen, l-octanol, pyridine, diethylamine, and dimethyl sulfide. For these 16 substances, the basic constants and parameters s, A and 0 are given in this table along with the calculated density parameters a,P, and n. Saturated liquid densities calculated with these parameters have been compared with corresponding experimental measurements obtained from the sources given in Table 111. Average and maximum percent deviations resulting from these comparisons are also presented in this table. These comparisons show that 2-propanol exhibits the highest average deviation of 2.81 % (45points) followed by pyridine with 2.66% (28 points). The overall average deviation for the 16 compounds was

% dev

points

av

max

164 58 128 81 67

0.28 1.07 0.30 0.34 2.28

0.99 2.74 1.28 1.24 5.78

54 158 45 103 106 53

2.72 0.93 0.55 0.49 1.24 1.48 1.22

3.14 1.59 1.36 1.43 1.57 2.38 2.20

89 127 56 148

1.94 0.42 0.70 0.37

3.23 1.77 1.27 0.90

115 64 89 55 152 116 84 95 99 86 126 69

0.22 0.20 0.35 1.09 0.35 0.16 0.68 2.12 0.91 0.39 0.89 0.34

1.16 0.71 0.96 2.24 2.04 0.95 1.17 2.45 1.31 1.87 1.81 0.97

25 95 59

0.87 1.40 1.32

1.31 2.20 1.90

202 125 103 105 83

0.25 0.20 0.56 0.70

1.00

1.08 0.65 1.51 2.83 1.66

102 78 74 95

0.82 1.67 1.32 0.88

3.28 2.25 11.53 1.74

50 137 77 78 51 47

0.66 0.54 0.82 2.68 3.24 0.88

5.75 0.81 1.12 3.60 3.59 1.57

4284

0.83

111

found to be 1.14% (669 points).

Comparison with Other Methods The ability of this method to predict saturated liquid densities has been compared with methods available in the literature, based on percent average deviations for the total of 62 substances listed in Tables I1 and 111. For these comparisons, the methods of Riedel(161),Yen and Woods (2221, Rackett (153),Spencer and Danner (184), and Joffe and Zudkevitch (87) were used. For the present study and the methods reported by Riedel, Yen and Woods, and Rackett, no actual saturated liquid density information is needed; however, for the application of the method of Spencer and Danner and also that of Joffe and Zudkevitch, some experimental densities are required. The average deviations resulting from this comparative evaluation are presented in Table IV, which also includes

506

Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984

00

0 0

000000

00

00

00

Lot-mt-NN

CDO

"?"1?"

-em

"ri

a m m a

O N Low

0 0

0 0

0 0

OmvP'rlN

0 0 - 0 0 0

000000

1? 19

the temperature range of the experimental data available and the number of experimental densities associated with each substance. The overall average deviations of all methods, except that of Spencer and Danner, have been based on a total of 4953 data points. The method of Riedel predicts reliable density values for both polar and nonpolar compounds, but it cannot accommodate as well the quantum liquids, alcohols, nitriles, and water. Riedel's method produces an overall average deviation of 2.13% (4953 points). The method of Yen and Woods is slightly less accurate than that of Riedel, but it accommodates the alcohols much better. Like Riedel, Yen and Woods could not predict reliable densities for the quantum liquids and organic nitrogen compounds. Their method gave an overall average deviation of 2.21% (4953 points). The method of Rackett performs well for nonpolar substances and, consistent with his claim, falls short with nitriles, organic acids, and liquid helium and hydrogen. Despite his claim of the inadequacy of his method to predict densities for alcohols, this comparative study does not show excessive deviations for this class of compounds, except for methanol. With no substances excluded, the method of Rackett predicts densities with an overall average deviation of 2.76% (4953 points). The method of Spencer and Danner given by eq 6 requires that the adjustable constant, ZRA,be obtained through a linear regression analysis spanning a specified temperature range, within which experimental densities must be available. If a value of 2, is not known, Spencer and Danner suggest the use of z, in its place with a corresponding loss in accuracy. Since values of ZRAare not given by them for helium, para-hydrogen, water, pyridine, dimethyl sulfide, isobutyl, sec-butyl, tert-butyl, and n-octyl alcohol, these substances could not be included in the evaluation of the Spencer and Danner method. With the exclusion of these substances, an overall averag2 deviation of 0.77% (4358 points) results from the use of their method. Slightly higher deviations are associated with the alcohols, nitriles, and acetic acid. The method of Joffe and Zudkevitch expressed by eq 7 involves the use of the adjustable parameter $. For nonpolar substances, this parameter is treated by them as a temperature-independent constant to be established from a single measured density. On the other hand, for polar compounds, parameter # is made to vary linearly with temperature and therefore two actual density values must be available to define eq 7 properly for such compounds. Applying these requirements to eq 7, densities were calculated for all 62 substances to yield an overall aierage deviation of 0.64% (4953 points). In general, the method of Joffe and Zudkevitch can accommodate all types of substances, including the quantum liquids for which two actual density values must be used. The method of the present study does not require the involvement of any experimental densities, but it requires the molecular weight, normal boiling point, and dipole moment, in addition to the critical constants. The contribution due to polarity is insignificant unless the reduced dipole moment 8 exceeds 0.5 X lo4. For values of 8 C 0.5 x lo4, this contribution can be neglected. This parameter has been found to vary from zero for nonpolar compounds for acetonitrile, with methanol and water to 1.619 X having values of 0.479 X lo4 and 0.925 X lo4, respectively. Since, for methanol, 8 represents a threshold value, the polar contribution of this substance can be neglected. The average deviations given in Table IV show that this method does not discriminate between types of substances

Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984

507

508

Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984

and is able to predict reliable density values for quantum liquids and polar substances, including the alcohols, water, and nitriles. With the exception of xenon, isopentane, normal-hydrogen, 2-propanol, pyridine, and acetic acid, the remaining substances exhibit deviations of less than 2.00%. The overall average deviation, with all substances included, is 0.87% (4953 points). Acknowledgment Due thanks are extended to the Exxon Education Foundation for the Exxon Teaching Fellowship awarded to Scott W. Campbell. Nomenclature A , B , D = density coefficients, eq 4 C = density coefficient, eq 8

E = density coefficient, eq 9 h = Planck constant m = molecular mass, M/N, g/molecule, eq 15 m, n = exponents, eq 11 M = molecular weight N = Avogadro number P = pressure, atm P, = critical pressure, atm PR = reduced pressure, P/Pc R = gas constant s = characterization factor, TRbIn Pc/(l- T R b ) T = temperature, K ?(b = normal boiling point, K T', = critical temperature, K TR = reduced temperature, T/Tc 71Rb = reduced temperature at normal boiling point, Tb/Tc u = molar volume, cm3/ mol u, = critical volume, cm /g-mol I I R = reduced volume, u / u , 2, = critical compressibility factor, P,u,/RT, Z R h = adjustable constant, eq 6 Greek Letters a , = density parameters, eq 11 CY, = Riedel factor, (d In PR/d In T R ) + ~ , eq ~,1 A& = contribution to density parameter a due to the presence of hydroxyl groups, eq 16 A&ar = dipole moment contribution to density parameter a, eq 16 A& = contribution to density parameter fl due to the presence of hydroxyl groups, eq 20 i!& = contribution to exponent n due to the presence of hydroxyl groups, eq 22 0 = reduced dipole moment, I.L~/T~U, K = Boltzmann constant, R/N A = quantum parameter, 1/(MT,)1/2~c1/3 A, = quantum parameter for deviations from classical principle of corresponding states, eq 15 I.( = dipole moment, D p = density, g/cm3 pc = critical density, g/cm3 pI = density of saturated liquid, g/cm3 pR = reduced density, p / p c p ~ = 1 reduced density for saturated liquid, p I / p c pRV = reduced density for saturated vapor, p , / p c pa = saturated liquid density, g/cm3, eq 6 p v = saturated vapor density, g/cm3 $ = parameter for eq 7 w = Pitzer acentric factor

f-

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Received for review August 12, 1983 Accepted April 3, 1984

COMMUNICATIONS Gas-Liquid Mass Transfer Data in a Stlrred Autoclave Reactor A novel experimental technique for measuring volumetric mass transfer coefficients in a stirred autoclave reactor is proposed. The method is based on pursuing the instationary pressure drop due to mass transfer under isochoric conditions. The usefulness of the technlque Is demonstrated for the Fischer-Tropsch slurry system.

In a recent communication to this journal, Huff and Satterfield (1982) gave a detailed description of a mechanically stirred autoclave which was used to study the Fischer-Tropsch synthesis in a slurry phase. As pointed out by these authors, in such an experimental unit all phases are completely mixed under appropriate operating conditions, which facilitates and simplifies data evaluation 0196-431318411023-051l$Ol.50/0

and interpretation. However, when determining the kinetics of gas-liquid-solid catalyst systems, rate data may be falsified by mass transfer limitations. In the case of Fischer-Tropsch synthesis it is particularly the mass transfer at the gas-liquid interface which may be important. In a study of this synthesis on a reduced fused magnetite catalyst, Satterfield and Huff (1982) showed 0 1984 American Chemical Society