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Speed of sound, molar volume, and molar isobaric heat capacity for binary liquid mixtures: analysis in terms of van der Waals's one-fluid theory
M-1693 J. Chem. Thermodynamics 1%4, 16, 1181-1189
Speed of sound, molar volume, and molar isobaric heat capacity for binary liquid mixtures: analysis in terms of van der Waals’s one-fluid theory a M. K. KUMARAN,b CARL J. HALPIN
GEORGE
Division of Chemistry, National Ottawa, Canada KlA OR6
C. BENSON,
PATRICK
J. D’ARCY,
and
Research Council of Canada,
(Received 21 February 1984; in revised form 18 June 1984) The speed of sound and excess molar isobaric heat capacity at 298.15 K, and the excess molar volume at 303.15 K are reported for (n-dodecane + n-hexane). The results, in combination with the excess.molar volume at 298.15 K published previously, are used to obtain the interaction parameters a, 2 and b, 2 for the mixture treated as a pure fluid obeying a generalized van der Waals’s equation of state. Similar analyses, using results from our previous investigations, are presented for (di-n-propylether + n-heptane) and (benzene + cyclohexane).
1. Introduction Van der Waals, in extending his equation of state for a pure fluid to describe the properties of a binary liquid mixture, assumed that the mixture could be represented by a hypothetical pure fluid which had thermodynamic properties identical to those of the mixture. Thus for a mixture, van der Waals’s equation of state for a pure fluid may be rewritten as P = wKm-~,)-42>
(1)
where p is the pressure of the mixture at temperature T, and V, is its molar volume. The parameters a, and b, of the mixture at mole fraction x of component 1 are defined by: a, = x%,+2x(1-x)a,,+(l-x)%2,,
(2)
and b, = x2b, +2x(1 -x)b,,+(l
-x)2b2,
(3) where ui and bi (i = 1,2) are the van der Waals constants for pure component i. The parameters a,, and b,, characterize interactions between unlike molecules in the mixture. as NRCC No. 23634. h National Research Council of Canada Research Associate.
’ Issued
0021-9614/84/121181+09
%02.00/O
0 1984 Academic Press Inc. (London) Limited
M.K.KUMARANET
1182 Subsequent workers equations of the type:
AL.
have further generalized
this treatment
P = (R T/ &$#4Y) - 4zl Kk
by considering (4)
where 4(y) is a function only of the reduced density variable: Y = &/4Kn,
(5)
with b, interpreted as a molar co-volume for a hard-particle investigation two particular functional forms were used: vdW:
system. In the present
4(y) = l/(1 -4y),
(6)
and cs:
4(y) = (1+y+yz-y3)/(1-y)3.
(71
These correspond respectively to van der Waals’s original equation of state and to the equation suggested by Carnahan and Starling.“’ Conventionally the interaction parameters ur2 and blZ are calculated from the parameters for the pure components on the basis of some combining ruIes.‘2*3’ Those adopted most commonly are the geometric mean: a 12
and the arithmetic
-
(al
(8)
a2)“*,
mean: b12 = (b, +b,P.
Some modifications
(9)
of (8) and (9) which have been used(z*4’ are a 12 = 5(a,
a 12 = 5h2ha21blb2)“*,
(10) (11)
b;i3 = (b:‘3 + b;‘3)/2.
(12)
a,)“*,
and The quantity 5: in (10) and (11) is close to unity, and Hudson and McCoubrey@) have given an expression to calculate it from properties of the pure fluids. It is generally believed that expressions similar to (8) to (12) are unavoidable in the theoretical treatment of mixtures. However, it appears that reliable estimates of a, and b, at a selected temperature can be obtained from a knowledge of the molar volume, molar heat capacity, and speed of sound for the mixture, thus providing an opportunity to check the utility of the various combining rules. For a fluid which obeys the generalized van der Waals’s equation of state, it can be shown that 2 = (YRT/M){Y~‘(Y)--(Y)+2p~~RT},
(13)
where u is the speed of sound in the fluid, y is the ratio of isobaric and isochoric heat capacities of the fluid, and A4 is its molar mass. The ratio y can be calculated from the relation: Y = 1-t Twq)2/c,,
m,
(14)
VAN
DER
WAALS’S
ONE-FLUID
THEORY
OF
1183
MIXTURES
where tlP is the isobaric expansivity and C,,, is the isobaric molar heat capacity. Equations (13) and (14) are directly applicable to a van der Waals one-fluid mixture. Thus experimental measurements of speed of sound, heat capacities, and excess molar volumes (the latter at two different temperatures) can be used to obtain estimates of a, and b, through the simultaneous solution of (4) and (13). To illustrate this approach, we have chosen three binary mixtures: (i), (n-dodecane + n-hexane), where the components belong to the same homologous series and therefore are structurally and chemically related; (ii), (di-n-propylether + n-heptane), where the components are chemically dissimilar but have some structural resemblances, and (iii), (benzene + cyclohexane), where the components are chemically and structurally dissimilar. The requisite thermodynamic properties for (ii) and (iii) are already available from our earlier investigations.(6-1 ‘) For (i) we have reported (12) the excess molar volume V,” at 298.15 K. We have now measured the excess molar heat capacity CF,, and speed of sound at 298.15 K, and V,E at 303.15 K for this mixture.
2. Experimental The materials used for the measurements on (n-hexane + n-dodecane) were the same as those used in our earlier study. Cl‘) Densities determined at 303.15 K with an TABLE
1. Results
of measurements
x
of property
x
0 for {xC,,H,,+(l
@
@= 0.07399 0.09558 0.12078 0.14643 0.20076 0.22408 0.24477
-0.316 - 0.420 - 0.554 -0.625 -0.783 -0.905 -0.981 - 1.076
0.28919 0.34634 0.39631 0.44614 0.44852 0.45121 0.49164
0.0 1549 0.02867 0.04695 0.07322 0.10522 0.12888 0.15874
2.89 4.59 6.80 9.77 12.98 15.09 17.46
0.19199 0.22864 0.25498 0.26225 0.28233 0.29240
-0.0956 -0.2092 - 0.2779 -0.3287
0.2843 0.3383 0.3802 0.4178
C;,,(T
= 298.15 -
x K)/(J.K-‘.
1.214 1.257 1.325 1.295 1.280 1.299 1.285
@= Au(T = 298.15
0= 0.0434 0.1095 0.1640 0.2188 “Au
= u-(1278.96x-1076.49(1-x)}
19.73 21.82 23.00 23.40 24.17 24.52 VJ(T
0.49485 0.52414 0.54845 0.57198 0.60079 0.65032 0.69777
at temperature x
T CD
mol-‘) -
1.372 1.329 1.280 1.214 1.181 1.128 1.083
0.75298 0.77108 0.79551 0.85393 0.89704 0.91357 0.94362
-0.941 -0.839 -0.833 -0.600 -0.426 - 0.362 - 0.226
25.13 26.07 26.76 26.65 25.49 23.54
0.68239 0.74610 0.77707 0.83186 0.88789 0.94723
20.82 17.61 15.88 12.50 8.69 4.23
-0.3861 -0.3742 -0.3441
0.6678 0.7836 0.8986
-0.2938 -0.2081 -0.1029
K)/(m.s-I)” 0.31275 0.35159 0.40844 0.47438 0.54845 0.61587
= 303.15 K)/(cm3.mol-I)
-0.3681 -0.3853 -0.3926 -0.3919 rn.s-’
-x)&H,,}
0.4622 0.5090 0.5826
M. K. KUMARAN
1184
ET AL.
TABLE 2. Coefficients cj, skewing factor k, and standard deviation s for least-squares representations of CF.,,, Au, and Vz for {xC,,H,,+(l-x)C,H,,} at temperature T by equation (12) CD
C:.,/(J,K-‘.mol-‘) Au/(m.s-‘) Vl/(cm” mol- ‘)
T/K
Cl
c2
298.15 298.15 303.15
- 5.282 105.414 - 1.5067
-0.864 - 75.307 -0.5944
C3
-20.451 -0.2167
C4
k
s
- 4.407 - 0.0807
0.9807
0.03 0.04 0.0005
Anton-Paar densimeter were 650.52 kg.m-3 and 741.75 kg. m- 3 for n-hexane and n-dodecane, respectively. A stepwise procedure was used to measure CpE,m at 298.15 K in a Picker flow microcalorimeter. (13, 14) The initial reference material was n-hexane, and 195.80 J. K- ’ *mol-’ was adopted for its isobaric molar heat capacity.‘15) The speed of sound was determined at 298.15 K by the pulse-echo-overlap method.‘r6-‘*I A tilting dilution dilatometer”’ was used to measure the excess molar volume at 303.15 K.
3. Results and discussion The measurements on {xC,,Hz6 +(l -x)&H,,} are summarized in table 1, where at 298.15 K, the deviation Au of the speed of sound from linearity in x at CL 298.15 K, and V,” at 303.15 K are given. Each set of results was fitted with an equation of the form: qx)=
x(l-x){l-k(l-2x))-’
jiI
cj(l-2xy’-‘.
(15)
Values of the coefficients Cj and the skewing factor k (where appropriate), determined by the method of least-squares with all points weighted equally, are listed in table 2 along with the standard deviations s of the representations. In analysing the results according to van der Waals one-fluid theories C,, m, U, and LYEfor a mixture with mole fraction x of component 1 were obtained by calculating CF. m, Au, and V,” (the latter at two temperatures) from the smoothing equations, and TABLE 3. Density p, isobaric expansivity tlpr isobaric molar heat capacity C,.,, and speed of sound u for component liquids at 298.15 K Component
d(kg.m-? .-~--
adkK-’
n-Dodecane n-Hexane Di-a-propylether n-Heptane Benzene Cyclohexane
745.30”2’ 654.93(15’ 741.94’6’ 679.56 16’ 873.66”‘,‘9’ 773.83”‘~r9’
0.953 a 1.347” 1.261 ‘6’ 1.210@’ l.21Soo’ 1.216”“’
a Present work.
C,,,/(J.K-‘.mol-‘) 375.30” 195.80°S 221.45’@ 224.764 ‘6’ 135.76’13’ 156.07”3’
u/(m.s-I) 1278.96 ’ 1076.49 ’ 1086.54’61 1129.84’@ 1298.90°0 1253.28””
VAN TABLE
4. Summary
of calculations
vdW: x
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
a Pa.m6.molm2
2.1858 2.3430 2.5064 2.6744 2.8466 3.0227 3.2024 3.3856 3.5721 3.7618 3.9547 4.1509 4.3504 4.5533 4.7595 4.9693 5.1825 5.3991 5.6190 5.8417 6.0708
DER
b cm3.mo1-’
111.96 116.68 121.42 126.17 130.93 135.68 140.44 145.20 149.96 154.73 159.49 164.25 169.01 173.78 178.55 183.33 188.10 192.88 197.66 202.45 207.24
WAALS’S
ONE-FLUID
2.7413 2.7195 2.7063 2.6932 2.6802 2.6672 2.6544 2.6417 2.6291 2.6166 2.6042 2.5919 2.5798 2.5677 2.5558 2.5440 2.5323 2.5207 2.5092 2.4978 2.4895
127.81 127.33 126.91 126.49 126.06 125.63 125.19 124.76 124.32 123.88 123.43 122.98 122.53 122.08 121.62 121.16 120.70 120.23 119.77 119.30 118.84
1185
OF MIXTURES
for mixtures (i), (ii), and (iii) at 298.15 K using van der Waals 4(y) given by the vdW and CS functional forms
equation
(6)
CS: equation
aI2 Pa.m6.mol-*
3.7381 3.7512 3.7592 3.7652 3.7701 3.7739 3.7768 3.7789 3.7803 3.7812 3.7816 3.7816 3.7815 3.7812 3.7810 3.7806 3.7796 3.7763 3.7610
2.5192 2.5612 2.5750 2.5818 2.5859 2.5885 2.5903 2.5915 2.5924 2.5931 2.5935 2.5937 2.5937 2.5935 2.5929 2.5918 2.5897 2.5852 2.5712 -
a
b 12 cm3.mol-’
Pa.m6.mol-2
(ib W12H2,+(1-
W,H,,J
159.12 159.22 159.28 159.32 159.34 159.36 159.37 159.37 159.38 159.37 159.37 159.36 159.36 159.35 159.35 159.35 159.34 159.33 159.28
(ii), {xC,H,OC,H,+(I0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
THEORY
123.01 123.31 123.41 123.46 123.49 123.50 123.52 123.52 123.53 123.53 123.54 123.54 123.54 123.53 123.53 123.52 123.50 123.47 123.36
4.2824 4.6510 5.0395 5.4433 5.8615 6.2935 6.7386 7.1962 7.6660 8.1478 8.6416 9.1475 9.6658 10.197 10.740 11.297 11.867 12.450 13.045 13.652 14.281
b cm3.mol-’
one-fluid
theories
with
(7)
aI2 Pa.m6.mol-2
263.96 277.25 290.69 304.21 317.79 331.41 345.07 358.75 372.47 386.21 399.98 413.78 427.62 441.49 455.40 469.35 483.35 497.38 511.44 525.52 539.72
7.8995 7.9329 7.9524 7.9673 7.9789 7.9878 7.9942 7.9984 8.0006 8.0012 8.0003 7.9984 7.9958 7.9929 7.9898 7.9863 7.9810 7.968 1 7.9154
309.12 307.62 306.48 305.33 304.18 303.03 301.88 300.72 299.56 298.40 297.24 296.07 294.91 293.74 292.57 291.40 29022 289.05 287.87 286.70 285.64
5.0440 5.1629 5.2023 5.2217 5.2332 5.2406 5.2457 5.2494 5.2520 5.2539 5.2551 5.2557 5.2558 5.2551 5.2536 5.2505 5.2447 5.2321 5.1927
b 12 cm3.mol-’
396.57 397.16 397.48 397.70 397.86 397.98 398.05 398.10 398.12 398.12 398.11 398.09 398.07 398.05 398.03 398.02 397.99 397.91 397.51
-xGH,.J 5.6096 5.5544 5.5235 5.4929 5.4627 5.4328 5.4033 5.3742 5.3454 5.3171 5.2891 5.2616 5.2344 5.2076 5.1812 5.1552 5.1297 5.1045 5.0796 5.0552 5.0392
293.99 295.76 296.35 296.63 296.80 296.91 296.98 297.03 297.07 297.09 297.11 297.11 297.11 297.10 297.07 297.02 296.93 296.73 296.11
M. K. KUMABAN
1186
ET AL.
TABLE 4-continued vdW: equation (6) x
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1
a
Pa.m6.mol-2
1.9697 1.9438 1.9226 1.9016 1.8810 1.8608 1.8407 1.8210 1.8015 1.7823 1.7632 1.7444 1.7259 1.7075 1.6894 1.6714 1.6537 1.6362 1.6189 1.6019 1.5905
b
cm3~mol-’
93.88 93.08 92.30 91.51 90.71 89.90 89.09 88.27 87.44 86.60 85.76 84.91 84.05 83.18 82.31 81.43 80.54 79.64 78.74 77.83 76.96
CS: equation (7) b 12
al2
Pa.m6.mol-*
a
cm3,mol-’
b
Pa.m6~mol-’
110,{xC,H, +(l -x)(c-&Hi,} 3.9894 1.7073 85.86 3.9286 1.7289 86.02 3.8810 1.7363 86.06 3.8346 1.7401 86.08 3.7894 1.7424 86.10 3.7451 1.7439 86.10 3.7019 1.7450 3.6596 86.10 1.7457 86.10 3.6182 1.7462 86.10 3.5777 1.7464 86.10 3.5380 1.7464 86.09 3.4991 1.7461 86.09 3.4609 1.7456 86.08 3.4236 1.7447 86.07 3.3870 1.7432 86.06 3.3511 1.7407 86.03 3.3161 1.7364 86.00 3.2818 1.7275 85.93 3.2482 1.7004 85.72 3.2155 3.1987
- (“’
b 12
al2
cm3.mol-’
Pa.m6.mol-’
225.73 223.52 221.49 219.46 217.42 215.38 213.33 211.28 209.22 207.15 205.08 203.00 200.92 198.83 196.73 194.63 192.53 190.41 188.30 186.17 184.29
cm3~mol-’
3.3705 3.4313 3.4523 3.463 1 3.4699 3.4744 3.4775 3.4797 3.4811 3.4819 3.4820 3.4814 3.4800 3.4775 3.4734 3.4666 3.4546 3.4299 3.3543
203.56 204.49 204.80 204.95 205.04 205.09 205.12 205.14 205.15 205.15 205.14 205.12 205.09 205.04 204.97 204.86 204.66 204.26 203.04
TABLE 5. Comparison of selected values of the van der Waals interaction parameters aI2 and b,, with values estimated from the following combining rules for ai2: I, (al a,)‘/*; II, {(b, + b,)/2)](a, a,/b, b,)“f III, {(bij3 + bl/3)/2}3(a, a,/b, b,)“‘, and for b,,: IV, (b, + bJ2; V, {(bij3 + bi13)/2}3 a,,/(Pa.m6.mol-2) I II
Eqn. of state
Selected ’
vdW cs
3.78 8.00
vdW cs
2.59 5.25
3.643 3.817 3.701 7.820 8.326 7.988 (ii), {xC,H,0C3H,+(l-x)C,H,,} 2.612 2.614 2.613 5.317 5.321 5.318
vdW cs
1.75 3.48
1.770 3.572
(9. W,A6+(1
III
b,,/(cm’.mol-*) IV
V
-x)WW
(iii), {xC,H,+(l-x)c-&Hi,) I.779 1.773 3.591 3.578
a Selected as representative of the results in table 4.
Selected ’
159.4 398.1
159.60 401.84
159.75 385.56
123.5 297.0
123.33 297.38
123.27 297.23
86.1 205.1
85.42 205.01
85.14 204.31
VAN DER WAALS’S ONE-FLUID
THEORY OF MIXTURES
1187
FIGURE 1. Comparison of excess molar volumes of mixtures at 298.15 K: (i), (xC,,H,,+(~-x)C,H,,}; (ii), {xC,H,OC,H,+(l-x)C,H,,j; (iii), {xC,H,+(l-x)c-C,H,,}. -, Experimental results;(6~‘2~1Y’ - - -, calculated from van der Waals’s one-fluid theory and the vdW-form using the selected a,, and b,, values; .., calculated from van der Waals’s one-fluid theory and the vdW-form with a,2 and b,, obtained from equations (8) and (9).
adding the values for the corresponding ideal mixture (or in the case of Au, a hypothetical mixture for which u varies linearly with x). The properties of the pure components at 298.15 K are listed in table 3. Equation (14) was used to calculate y and its value was inserted in equation (13). An iterative steepest-descent technique was used to solve the latter for y and consequently to obtain b,. Substitution of the result in equation (4) then yielded a,. The van der Waals constants aj and bi (i = 1,2) for the pure components were calculated in the same way. The interaction parameters uI2 and b,, were obtained from equations (2) and (3). Results of the calculations at 0.05 intervals of x are given in table 4 for the vdW- and CS-forms defined by equations (6) and (7). The results of similar calculations for mixtures (ii) and (iii) are also given in table 4.
M. K. KUMARAN
1188
ET AL
TABLE 6. Excess molar enthalpies Hi and excess molar Gibbs functions Gk calculated for mixtures at 298.15 K using van der Waals one-fluid theories in the vdW- and CS-forms (equations 6 and 7) with the selected values of a,2 and b,, from table 5 ”
Expt
HE/(J mol- ‘) vdW
Expt
GE/(J mol- ‘) vdW
cs
-17.2b -23.5 b -17.5b
- 334.0 - 387.0 - 256.8
- 245.6 - 279.7 - 184.2
cs
(9, {G&L6+(1 -x)C,H,,} -364.1 -412.9 - 269.7
-496.5 - 565.7 - 373.4
0.25 0.50 0.75
27.4 a 39.1 a 30.1 a
0.25 0.50 0.75
154.8 ’ 204.2 c 153.0’
(ii), jxC,H,OCsH,+(l 81.9 219.3 111.2 298.2 84.9 228.1
57.4 d 84.8’ 74.0 d
67.0 90.7 69.2
160.0 217.1 165.8
0.25 0.50 0.75
590.0 e 800.0 e 621.5 e
(iii), {xC,H,+(l -x)c-C,H,,} 178.3 544.2 231.8/ 249.1 761.6 317.31 196.5 601.4 343.7 f
133.9 186.9 147.5
383.6 536.0 423.1
-x)C,H,,)
’ Reference 12; b Reference 20; ’ Reference 7; ’ Extrapolated from resultso” at 343.15 K using excess enthalpies and heat capacities;“’ ’ Reference 22; ’ Reference 23.
It can be seen for the three mixtures, that the interaction parameters of both the vdW- and CS-forms remain nearly constant throughout the composition range. Table 5 provides a comparison between nominal selected interaction parameters, assigned on the basis of the results in table 4 for the mid-range 0.3 < x < 0.7, and values obtained from the various combining rules (equations 8 to 12 with t = 1). The simple arithmetic mean gives a slightly better estimate of b,, than does the more elaborate relation (12) and subsequent estimates of ui2 obtained from (11) deviate more from the selected values than do those obtained from the geometric mean. Although the geometric and arithmetic means are close to the selected values of a12 and b,, for all three mixtures, the small deviations of these estimates are quite significant since the values of V,” obtained from van der Waals one-fluid theories are very sensitive to the particular ai2 and b12 used in their calculation. This is illustrated in figure 1, where there is reasonably good agreement between the experimental V,” curves and those calculated from the vdW-form of the equation of state using the selected values of aI2 and b,,. In contrast, the curves calculated with the ai2 and b,, estimated from the geometric-mean and arithmetic-mean rules, show relatively large deviations from the experimental curves. Qualitatively similar results were obtained when the CS-form of equation of state was used. The selected values of ai2 and b,, obtained in the present work were also used in van der Waals one-fluid theories to estimate the excess molar enthalpy Hi and the excess molar Gibbs function GE from the equation of state for each of the three mixtures. Not unexpectedly, as shown in table 6, whether we use the simple vdW equation of state or the more complex CS equation of state, there is generally poor agreement between the calculated and experimental(7~‘2*20-23) values. It is clear that
VAN DER WAALS’S ONE-FLUID
THEORY OF MIXTURES
further adjustable parameters would have to be introduced these properties as well as V,“.
1189
in order to represent
REFERENCES 1. Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. 2. McGlashan, M. L. Trans. Faraday Sot. 1970, 66, 18. 3. Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures. 3rd Edition. Butterworths: London. 1982, Chapter 8. 4. Ewing, M. B.; Marsh, K. N. J. Chem. Thermodynamics 1977,9, 351. 5. Hudson, G. H.; McCoubrey, J. C. Trans. Faraday Sot. 1960, 56, 761. 6. Kimura, F.; Treszczanowicz, A. J.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1983, 15, 503. 7. Kimura. F.; D’Arcy, P. J.; Benson, G. C. J. Chem. Thermo&amics 1983, 15, 51 I. 8. D’Arcy, P. J.; Hazlett, J. D.; Kiyohara, 0.; Benson, G. C. Thermochim. Acta 1977, 21, 297. 9. Kumaran. M. K.; McGlashan, M. L. J. Chem. Thermodynamics 1977,9, 259. IO. Kiyohara, 0.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1978, 10, 721. 1984. 16. 183. Il. Kumaran. M. K.: Benson. G. C. J. Chem. Thermodvnamics 1984, 16, 537. 12. Hamam, S. E. M:; Kumaran, M. K.; Benson, G. C:J. Chem. Thhrmodynamics 13. Fortier, J.-L.; Benson, G. C.; Picker, P. J. Chem. Thermodynamics 1976, 8, 289. 14. Fortier, J.-L.; Benson, G. C. J. Chem. Thermodynamics 1976, 8, 411. 15. Aicart, E.; Kumaran, M. K.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1983, 15, 919. 16. Kiyohara. 0.; Grolier, J.-P. E.; Benson, G. C. Can. J. Chem. 1974, 52, 2287. 17. Kiyohara, 0.; Halpin, C. J.; Benson, G. C. Can. J. Chem. 1977, 55, 3544. 18. Kumaran. M. K.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1983, 15, 1071. 19. Smith, B. D.; Muthu, 0.; Dewan, A.; Gierlach, M. J. Phys. Chem. Ret Data 1982, 11, 1151. 20. Ott, J. B.; Marsh, K. N.; Stokes, R. H. J. Chem. Thermodynamics 1981, 13, 371. 21. Treszczanowicz, T. Bull. Acad. Pal. Sri., Ser. Sci. Chim. 1973, 21, 107. 22. Smith, B. D.; Muthu, 0.; Dewan, A.; Gierlach, M. J. Phys. Chem. Rej Data 1982, 11, 1127. 23. Smith. B. D.: Muthu, 0.; Dewan, A.; Gierlach, M. J. Phys. Chem. Ref Data 1982, 11, 1099.
Speed of sound, molar volume, and molar isobaric heat capacity for binary liquid mixtures: analysis in terms of van der Waals’s one-fluid theory a M. K. KUMARAN,b CARL J. HALPIN
GEORGE
Division of Chemistry, National Ottawa, Canada KlA OR6
C. BENSON,
PATRICK
J. D’ARCY,
and
Research Council of Canada,
(Received 21 February 1984; in revised form 18 June 1984) The speed of sound and excess molar isobaric heat capacity at 298.15 K, and the excess molar volume at 303.15 K are reported for (n-dodecane + n-hexane). The results, in combination with the excess.molar volume at 298.15 K published previously, are used to obtain the interaction parameters a, 2 and b, 2 for the mixture treated as a pure fluid obeying a generalized van der Waals’s equation of state. Similar analyses, using results from our previous investigations, are presented for (di-n-propylether + n-heptane) and (benzene + cyclohexane).
1. Introduction Van der Waals, in extending his equation of state for a pure fluid to describe the properties of a binary liquid mixture, assumed that the mixture could be represented by a hypothetical pure fluid which had thermodynamic properties identical to those of the mixture. Thus for a mixture, van der Waals’s equation of state for a pure fluid may be rewritten as P = wKm-~,)-42>
(1)
where p is the pressure of the mixture at temperature T, and V, is its molar volume. The parameters a, and b, of the mixture at mole fraction x of component 1 are defined by: a, = x%,+2x(1-x)a,,+(l-x)%2,,
(2)
and b, = x2b, +2x(1 -x)b,,+(l
-x)2b2,
(3) where ui and bi (i = 1,2) are the van der Waals constants for pure component i. The parameters a,, and b,, characterize interactions between unlike molecules in the mixture. as NRCC No. 23634. h National Research Council of Canada Research Associate.
’ Issued
0021-9614/84/121181+09
%02.00/O
0 1984 Academic Press Inc. (London) Limited
M.K.KUMARANET
1182 Subsequent workers equations of the type:
AL.
have further generalized
this treatment
P = (R T/ &$#4Y) - 4zl Kk
by considering (4)
where 4(y) is a function only of the reduced density variable: Y = &/4Kn,
(5)
with b, interpreted as a molar co-volume for a hard-particle investigation two particular functional forms were used: vdW:
system. In the present
4(y) = l/(1 -4y),
(6)
and cs:
4(y) = (1+y+yz-y3)/(1-y)3.
(71
These correspond respectively to van der Waals’s original equation of state and to the equation suggested by Carnahan and Starling.“’ Conventionally the interaction parameters ur2 and blZ are calculated from the parameters for the pure components on the basis of some combining ruIes.‘2*3’ Those adopted most commonly are the geometric mean: a 12
and the arithmetic
-
(al
(8)
a2)“*,
mean: b12 = (b, +b,P.
Some modifications
(9)
of (8) and (9) which have been used(z*4’ are a 12 = 5(a,
a 12 = 5h2ha21blb2)“*,
(10) (11)
b;i3 = (b:‘3 + b;‘3)/2.
(12)
a,)“*,
and The quantity 5: in (10) and (11) is close to unity, and Hudson and McCoubrey@) have given an expression to calculate it from properties of the pure fluids. It is generally believed that expressions similar to (8) to (12) are unavoidable in the theoretical treatment of mixtures. However, it appears that reliable estimates of a, and b, at a selected temperature can be obtained from a knowledge of the molar volume, molar heat capacity, and speed of sound for the mixture, thus providing an opportunity to check the utility of the various combining rules. For a fluid which obeys the generalized van der Waals’s equation of state, it can be shown that 2 = (YRT/M){Y~‘(Y)--(Y)+2p~~RT},
(13)
where u is the speed of sound in the fluid, y is the ratio of isobaric and isochoric heat capacities of the fluid, and A4 is its molar mass. The ratio y can be calculated from the relation: Y = 1-t Twq)2/c,,
m,
(14)
VAN
DER
WAALS’S
ONE-FLUID
THEORY
OF
1183
MIXTURES
where tlP is the isobaric expansivity and C,,, is the isobaric molar heat capacity. Equations (13) and (14) are directly applicable to a van der Waals one-fluid mixture. Thus experimental measurements of speed of sound, heat capacities, and excess molar volumes (the latter at two different temperatures) can be used to obtain estimates of a, and b, through the simultaneous solution of (4) and (13). To illustrate this approach, we have chosen three binary mixtures: (i), (n-dodecane + n-hexane), where the components belong to the same homologous series and therefore are structurally and chemically related; (ii), (di-n-propylether + n-heptane), where the components are chemically dissimilar but have some structural resemblances, and (iii), (benzene + cyclohexane), where the components are chemically and structurally dissimilar. The requisite thermodynamic properties for (ii) and (iii) are already available from our earlier investigations.(6-1 ‘) For (i) we have reported (12) the excess molar volume V,” at 298.15 K. We have now measured the excess molar heat capacity CF,, and speed of sound at 298.15 K, and V,E at 303.15 K for this mixture.
2. Experimental The materials used for the measurements on (n-hexane + n-dodecane) were the same as those used in our earlier study. Cl‘) Densities determined at 303.15 K with an TABLE
1. Results
of measurements
x
of property
x
0 for {xC,,H,,+(l
@
@= 0.07399 0.09558 0.12078 0.14643 0.20076 0.22408 0.24477
-0.316 - 0.420 - 0.554 -0.625 -0.783 -0.905 -0.981 - 1.076
0.28919 0.34634 0.39631 0.44614 0.44852 0.45121 0.49164
0.0 1549 0.02867 0.04695 0.07322 0.10522 0.12888 0.15874
2.89 4.59 6.80 9.77 12.98 15.09 17.46
0.19199 0.22864 0.25498 0.26225 0.28233 0.29240
-0.0956 -0.2092 - 0.2779 -0.3287
0.2843 0.3383 0.3802 0.4178
C;,,(T
= 298.15 -
x K)/(J.K-‘.
1.214 1.257 1.325 1.295 1.280 1.299 1.285
@= Au(T = 298.15
0= 0.0434 0.1095 0.1640 0.2188 “Au
= u-(1278.96x-1076.49(1-x)}
19.73 21.82 23.00 23.40 24.17 24.52 VJ(T
0.49485 0.52414 0.54845 0.57198 0.60079 0.65032 0.69777
at temperature x
T CD
mol-‘) -
1.372 1.329 1.280 1.214 1.181 1.128 1.083
0.75298 0.77108 0.79551 0.85393 0.89704 0.91357 0.94362
-0.941 -0.839 -0.833 -0.600 -0.426 - 0.362 - 0.226
25.13 26.07 26.76 26.65 25.49 23.54
0.68239 0.74610 0.77707 0.83186 0.88789 0.94723
20.82 17.61 15.88 12.50 8.69 4.23
-0.3861 -0.3742 -0.3441
0.6678 0.7836 0.8986
-0.2938 -0.2081 -0.1029
K)/(m.s-I)” 0.31275 0.35159 0.40844 0.47438 0.54845 0.61587
= 303.15 K)/(cm3.mol-I)
-0.3681 -0.3853 -0.3926 -0.3919 rn.s-’
-x)&H,,}
0.4622 0.5090 0.5826
M. K. KUMARAN
1184
ET AL.
TABLE 2. Coefficients cj, skewing factor k, and standard deviation s for least-squares representations of CF.,,, Au, and Vz for {xC,,H,,+(l-x)C,H,,} at temperature T by equation (12) CD
C:.,/(J,K-‘.mol-‘) Au/(m.s-‘) Vl/(cm” mol- ‘)
T/K
Cl
c2
298.15 298.15 303.15
- 5.282 105.414 - 1.5067
-0.864 - 75.307 -0.5944
C3
-20.451 -0.2167
C4
k
s
- 4.407 - 0.0807
0.9807
0.03 0.04 0.0005
Anton-Paar densimeter were 650.52 kg.m-3 and 741.75 kg. m- 3 for n-hexane and n-dodecane, respectively. A stepwise procedure was used to measure CpE,m at 298.15 K in a Picker flow microcalorimeter. (13, 14) The initial reference material was n-hexane, and 195.80 J. K- ’ *mol-’ was adopted for its isobaric molar heat capacity.‘15) The speed of sound was determined at 298.15 K by the pulse-echo-overlap method.‘r6-‘*I A tilting dilution dilatometer”’ was used to measure the excess molar volume at 303.15 K.
3. Results and discussion The measurements on {xC,,Hz6 +(l -x)&H,,} are summarized in table 1, where at 298.15 K, the deviation Au of the speed of sound from linearity in x at CL 298.15 K, and V,” at 303.15 K are given. Each set of results was fitted with an equation of the form: qx)=
x(l-x){l-k(l-2x))-’
jiI
cj(l-2xy’-‘.
(15)
Values of the coefficients Cj and the skewing factor k (where appropriate), determined by the method of least-squares with all points weighted equally, are listed in table 2 along with the standard deviations s of the representations. In analysing the results according to van der Waals one-fluid theories C,, m, U, and LYEfor a mixture with mole fraction x of component 1 were obtained by calculating CF. m, Au, and V,” (the latter at two temperatures) from the smoothing equations, and TABLE 3. Density p, isobaric expansivity tlpr isobaric molar heat capacity C,.,, and speed of sound u for component liquids at 298.15 K Component
d(kg.m-? .-~--
adkK-’
n-Dodecane n-Hexane Di-a-propylether n-Heptane Benzene Cyclohexane
745.30”2’ 654.93(15’ 741.94’6’ 679.56 16’ 873.66”‘,‘9’ 773.83”‘~r9’
0.953 a 1.347” 1.261 ‘6’ 1.210@’ l.21Soo’ 1.216”“’
a Present work.
C,,,/(J.K-‘.mol-‘) 375.30” 195.80°S 221.45’@ 224.764 ‘6’ 135.76’13’ 156.07”3’
u/(m.s-I) 1278.96 ’ 1076.49 ’ 1086.54’61 1129.84’@ 1298.90°0 1253.28””
VAN TABLE
4. Summary
of calculations
vdW: x
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
a Pa.m6.molm2
2.1858 2.3430 2.5064 2.6744 2.8466 3.0227 3.2024 3.3856 3.5721 3.7618 3.9547 4.1509 4.3504 4.5533 4.7595 4.9693 5.1825 5.3991 5.6190 5.8417 6.0708
DER
b cm3.mo1-’
111.96 116.68 121.42 126.17 130.93 135.68 140.44 145.20 149.96 154.73 159.49 164.25 169.01 173.78 178.55 183.33 188.10 192.88 197.66 202.45 207.24
WAALS’S
ONE-FLUID
2.7413 2.7195 2.7063 2.6932 2.6802 2.6672 2.6544 2.6417 2.6291 2.6166 2.6042 2.5919 2.5798 2.5677 2.5558 2.5440 2.5323 2.5207 2.5092 2.4978 2.4895
127.81 127.33 126.91 126.49 126.06 125.63 125.19 124.76 124.32 123.88 123.43 122.98 122.53 122.08 121.62 121.16 120.70 120.23 119.77 119.30 118.84
1185
OF MIXTURES
for mixtures (i), (ii), and (iii) at 298.15 K using van der Waals 4(y) given by the vdW and CS functional forms
equation
(6)
CS: equation
aI2 Pa.m6.mol-*
3.7381 3.7512 3.7592 3.7652 3.7701 3.7739 3.7768 3.7789 3.7803 3.7812 3.7816 3.7816 3.7815 3.7812 3.7810 3.7806 3.7796 3.7763 3.7610
2.5192 2.5612 2.5750 2.5818 2.5859 2.5885 2.5903 2.5915 2.5924 2.5931 2.5935 2.5937 2.5937 2.5935 2.5929 2.5918 2.5897 2.5852 2.5712 -
a
b 12 cm3.mol-’
Pa.m6.mol-2
(ib W12H2,+(1-
W,H,,J
159.12 159.22 159.28 159.32 159.34 159.36 159.37 159.37 159.38 159.37 159.37 159.36 159.36 159.35 159.35 159.35 159.34 159.33 159.28
(ii), {xC,H,OC,H,+(I0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
THEORY
123.01 123.31 123.41 123.46 123.49 123.50 123.52 123.52 123.53 123.53 123.54 123.54 123.54 123.53 123.53 123.52 123.50 123.47 123.36
4.2824 4.6510 5.0395 5.4433 5.8615 6.2935 6.7386 7.1962 7.6660 8.1478 8.6416 9.1475 9.6658 10.197 10.740 11.297 11.867 12.450 13.045 13.652 14.281
b cm3.mol-’
one-fluid
theories
with
(7)
aI2 Pa.m6.mol-2
263.96 277.25 290.69 304.21 317.79 331.41 345.07 358.75 372.47 386.21 399.98 413.78 427.62 441.49 455.40 469.35 483.35 497.38 511.44 525.52 539.72
7.8995 7.9329 7.9524 7.9673 7.9789 7.9878 7.9942 7.9984 8.0006 8.0012 8.0003 7.9984 7.9958 7.9929 7.9898 7.9863 7.9810 7.968 1 7.9154
309.12 307.62 306.48 305.33 304.18 303.03 301.88 300.72 299.56 298.40 297.24 296.07 294.91 293.74 292.57 291.40 29022 289.05 287.87 286.70 285.64
5.0440 5.1629 5.2023 5.2217 5.2332 5.2406 5.2457 5.2494 5.2520 5.2539 5.2551 5.2557 5.2558 5.2551 5.2536 5.2505 5.2447 5.2321 5.1927
b 12 cm3.mol-’
396.57 397.16 397.48 397.70 397.86 397.98 398.05 398.10 398.12 398.12 398.11 398.09 398.07 398.05 398.03 398.02 397.99 397.91 397.51
-xGH,.J 5.6096 5.5544 5.5235 5.4929 5.4627 5.4328 5.4033 5.3742 5.3454 5.3171 5.2891 5.2616 5.2344 5.2076 5.1812 5.1552 5.1297 5.1045 5.0796 5.0552 5.0392
293.99 295.76 296.35 296.63 296.80 296.91 296.98 297.03 297.07 297.09 297.11 297.11 297.11 297.10 297.07 297.02 296.93 296.73 296.11
M. K. KUMABAN
1186
ET AL.
TABLE 4-continued vdW: equation (6) x
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1
a
Pa.m6.mol-2
1.9697 1.9438 1.9226 1.9016 1.8810 1.8608 1.8407 1.8210 1.8015 1.7823 1.7632 1.7444 1.7259 1.7075 1.6894 1.6714 1.6537 1.6362 1.6189 1.6019 1.5905
b
cm3~mol-’
93.88 93.08 92.30 91.51 90.71 89.90 89.09 88.27 87.44 86.60 85.76 84.91 84.05 83.18 82.31 81.43 80.54 79.64 78.74 77.83 76.96
CS: equation (7) b 12
al2
Pa.m6.mol-*
a
cm3,mol-’
b
Pa.m6~mol-’
110,{xC,H, +(l -x)(c-&Hi,} 3.9894 1.7073 85.86 3.9286 1.7289 86.02 3.8810 1.7363 86.06 3.8346 1.7401 86.08 3.7894 1.7424 86.10 3.7451 1.7439 86.10 3.7019 1.7450 3.6596 86.10 1.7457 86.10 3.6182 1.7462 86.10 3.5777 1.7464 86.10 3.5380 1.7464 86.09 3.4991 1.7461 86.09 3.4609 1.7456 86.08 3.4236 1.7447 86.07 3.3870 1.7432 86.06 3.3511 1.7407 86.03 3.3161 1.7364 86.00 3.2818 1.7275 85.93 3.2482 1.7004 85.72 3.2155 3.1987
- (“’
b 12
al2
cm3.mol-’
Pa.m6.mol-’
225.73 223.52 221.49 219.46 217.42 215.38 213.33 211.28 209.22 207.15 205.08 203.00 200.92 198.83 196.73 194.63 192.53 190.41 188.30 186.17 184.29
cm3~mol-’
3.3705 3.4313 3.4523 3.463 1 3.4699 3.4744 3.4775 3.4797 3.4811 3.4819 3.4820 3.4814 3.4800 3.4775 3.4734 3.4666 3.4546 3.4299 3.3543
203.56 204.49 204.80 204.95 205.04 205.09 205.12 205.14 205.15 205.15 205.14 205.12 205.09 205.04 204.97 204.86 204.66 204.26 203.04
TABLE 5. Comparison of selected values of the van der Waals interaction parameters aI2 and b,, with values estimated from the following combining rules for ai2: I, (al a,)‘/*; II, {(b, + b,)/2)](a, a,/b, b,)“f III, {(bij3 + bl/3)/2}3(a, a,/b, b,)“‘, and for b,,: IV, (b, + bJ2; V, {(bij3 + bi13)/2}3 a,,/(Pa.m6.mol-2) I II
Eqn. of state
Selected ’
vdW cs
3.78 8.00
vdW cs
2.59 5.25
3.643 3.817 3.701 7.820 8.326 7.988 (ii), {xC,H,0C3H,+(l-x)C,H,,} 2.612 2.614 2.613 5.317 5.321 5.318
vdW cs
1.75 3.48
1.770 3.572
(9. W,A6+(1
III
b,,/(cm’.mol-*) IV
V
-x)WW
(iii), {xC,H,+(l-x)c-&Hi,) I.779 1.773 3.591 3.578
a Selected as representative of the results in table 4.
Selected ’
159.4 398.1
159.60 401.84
159.75 385.56
123.5 297.0
123.33 297.38
123.27 297.23
86.1 205.1
85.42 205.01
85.14 204.31
VAN DER WAALS’S ONE-FLUID
THEORY OF MIXTURES
1187
FIGURE 1. Comparison of excess molar volumes of mixtures at 298.15 K: (i), (xC,,H,,+(~-x)C,H,,}; (ii), {xC,H,OC,H,+(l-x)C,H,,j; (iii), {xC,H,+(l-x)c-C,H,,}. -, Experimental results;(6~‘2~1Y’ - - -, calculated from van der Waals’s one-fluid theory and the vdW-form using the selected a,, and b,, values; .., calculated from van der Waals’s one-fluid theory and the vdW-form with a,2 and b,, obtained from equations (8) and (9).
adding the values for the corresponding ideal mixture (or in the case of Au, a hypothetical mixture for which u varies linearly with x). The properties of the pure components at 298.15 K are listed in table 3. Equation (14) was used to calculate y and its value was inserted in equation (13). An iterative steepest-descent technique was used to solve the latter for y and consequently to obtain b,. Substitution of the result in equation (4) then yielded a,. The van der Waals constants aj and bi (i = 1,2) for the pure components were calculated in the same way. The interaction parameters uI2 and b,, were obtained from equations (2) and (3). Results of the calculations at 0.05 intervals of x are given in table 4 for the vdW- and CS-forms defined by equations (6) and (7). The results of similar calculations for mixtures (ii) and (iii) are also given in table 4.
M. K. KUMARAN
1188
ET AL
TABLE 6. Excess molar enthalpies Hi and excess molar Gibbs functions Gk calculated for mixtures at 298.15 K using van der Waals one-fluid theories in the vdW- and CS-forms (equations 6 and 7) with the selected values of a,2 and b,, from table 5 ”
Expt
HE/(J mol- ‘) vdW
Expt
GE/(J mol- ‘) vdW
cs
-17.2b -23.5 b -17.5b
- 334.0 - 387.0 - 256.8
- 245.6 - 279.7 - 184.2
cs
(9, {G&L6+(1 -x)C,H,,} -364.1 -412.9 - 269.7
-496.5 - 565.7 - 373.4
0.25 0.50 0.75
27.4 a 39.1 a 30.1 a
0.25 0.50 0.75
154.8 ’ 204.2 c 153.0’
(ii), jxC,H,OCsH,+(l 81.9 219.3 111.2 298.2 84.9 228.1
57.4 d 84.8’ 74.0 d
67.0 90.7 69.2
160.0 217.1 165.8
0.25 0.50 0.75
590.0 e 800.0 e 621.5 e
(iii), {xC,H,+(l -x)c-C,H,,} 178.3 544.2 231.8/ 249.1 761.6 317.31 196.5 601.4 343.7 f
133.9 186.9 147.5
383.6 536.0 423.1
-x)C,H,,)
’ Reference 12; b Reference 20; ’ Reference 7; ’ Extrapolated from resultso” at 343.15 K using excess enthalpies and heat capacities;“’ ’ Reference 22; ’ Reference 23.
It can be seen for the three mixtures, that the interaction parameters of both the vdW- and CS-forms remain nearly constant throughout the composition range. Table 5 provides a comparison between nominal selected interaction parameters, assigned on the basis of the results in table 4 for the mid-range 0.3 < x < 0.7, and values obtained from the various combining rules (equations 8 to 12 with t = 1). The simple arithmetic mean gives a slightly better estimate of b,, than does the more elaborate relation (12) and subsequent estimates of ui2 obtained from (11) deviate more from the selected values than do those obtained from the geometric mean. Although the geometric and arithmetic means are close to the selected values of a12 and b,, for all three mixtures, the small deviations of these estimates are quite significant since the values of V,” obtained from van der Waals one-fluid theories are very sensitive to the particular ai2 and b12 used in their calculation. This is illustrated in figure 1, where there is reasonably good agreement between the experimental V,” curves and those calculated from the vdW-form of the equation of state using the selected values of aI2 and b,,. In contrast, the curves calculated with the ai2 and b,, estimated from the geometric-mean and arithmetic-mean rules, show relatively large deviations from the experimental curves. Qualitatively similar results were obtained when the CS-form of equation of state was used. The selected values of ai2 and b,, obtained in the present work were also used in van der Waals one-fluid theories to estimate the excess molar enthalpy Hi and the excess molar Gibbs function GE from the equation of state for each of the three mixtures. Not unexpectedly, as shown in table 6, whether we use the simple vdW equation of state or the more complex CS equation of state, there is generally poor agreement between the calculated and experimental(7~‘2*20-23) values. It is clear that
VAN DER WAALS’S ONE-FLUID
THEORY OF MIXTURES
further adjustable parameters would have to be introduced these properties as well as V,“.
1189
in order to represent
REFERENCES 1. Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. 2. McGlashan, M. L. Trans. Faraday Sot. 1970, 66, 18. 3. Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures. 3rd Edition. Butterworths: London. 1982, Chapter 8. 4. Ewing, M. B.; Marsh, K. N. J. Chem. Thermodynamics 1977,9, 351. 5. Hudson, G. H.; McCoubrey, J. C. Trans. Faraday Sot. 1960, 56, 761. 6. Kimura, F.; Treszczanowicz, A. J.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1983, 15, 503. 7. Kimura. F.; D’Arcy, P. J.; Benson, G. C. J. Chem. Thermo&amics 1983, 15, 51 I. 8. D’Arcy, P. J.; Hazlett, J. D.; Kiyohara, 0.; Benson, G. C. Thermochim. Acta 1977, 21, 297. 9. Kumaran. M. K.; McGlashan, M. L. J. Chem. Thermodynamics 1977,9, 259. IO. Kiyohara, 0.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1978, 10, 721. 1984. 16. 183. Il. Kumaran. M. K.: Benson. G. C. J. Chem. Thermodvnamics 1984, 16, 537. 12. Hamam, S. E. M:; Kumaran, M. K.; Benson, G. C:J. Chem. Thhrmodynamics 13. Fortier, J.-L.; Benson, G. C.; Picker, P. J. Chem. Thermodynamics 1976, 8, 289. 14. Fortier, J.-L.; Benson, G. C. J. Chem. Thermodynamics 1976, 8, 411. 15. Aicart, E.; Kumaran, M. K.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1983, 15, 919. 16. Kiyohara. 0.; Grolier, J.-P. E.; Benson, G. C. Can. J. Chem. 1974, 52, 2287. 17. Kiyohara, 0.; Halpin, C. J.; Benson, G. C. Can. J. Chem. 1977, 55, 3544. 18. Kumaran. M. K.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1983, 15, 1071. 19. Smith, B. D.; Muthu, 0.; Dewan, A.; Gierlach, M. J. Phys. Chem. Ret Data 1982, 11, 1151. 20. Ott, J. B.; Marsh, K. N.; Stokes, R. H. J. Chem. Thermodynamics 1981, 13, 371. 21. Treszczanowicz, T. Bull. Acad. Pal. Sri., Ser. Sci. Chim. 1973, 21, 107. 22. Smith, B. D.; Muthu, 0.; Dewan, A.; Gierlach, M. J. Phys. Chem. Rej Data 1982, 11, 1127. 23. Smith. B. D.: Muthu, 0.; Dewan, A.; Gierlach, M. J. Phys. Chem. Ref Data 1982, 11, 1099.