Heat Capacities of Cyclohexane
+ Alkane Systems
The Journal of Physical Chemistry, Vol. 83, No. 23, 1979 2979
Excess Heat Capacities of Cyclohexane 4- Alkane Systems and Orientational Order of n-Alkanes Sallendra N. Bhattacharyya and Donald Patterson* Otto Maass Chemistry Building, McGiII University, Montreal H3A 2K6, Quebec, Canada (Received January 8, 1979; Revised Manuscript Received August 14, 1979) Publication costs assisted by the National Research Council of Canada
A Picker flow microcalorimeter has been used to obtain molar excess heat capacities (cPE) through the concentration range at 10 (except for n-CI6),25, and 55 "C for cyclohexane mixed with n-C, alkanes ( n = 8, 10, 12,16) and also With the corresponding branched-C, alkanes 2,2,4-trimethylpentane, 2,2,4,6,6-pentamethyIheptane, and 2,2,4,4,6,8,8-heptamethylnonane. The equimolar cpE values are small and negative (-0.5 J K-' mol-') for the branched-C, alkanes becoming more negative with increasing temperature. This is consistent with differences in free volume between the component liquids as treated by the Prigogine-Flory theory which gives cpE< 0 and dcpE/dTC 0. For the n-C, alkanes, however, cpE values are large and negative increasing in magnitude to an equimolar value of -7 J K-l mol-' for C-CS + n-C16 at 25 "C. Values of cpEare much larger than predicted by the Prigogine-Flory theory and decrease in magnitude with an increase of T. They are consistent with orientational order in the n-C, alkanes, which decreases with T and which is destroyed by the c-C6. Literature excess enthalpy values at 25 "C and an integration of cpEgives hE as a function of T and the derived Xlzparameter. Order contributions to cpE,hE, and X12are found to depend on T through the alkane volume; i.e., alkane order depends on the packing density of segments. Order in the n-C, alkanes gives anomalous values to duE/dT and to dhE/dPand dcF/dP which are respectively positive and negative for c-Cs + n-C16 while the Prigogine-Nory theory predicts the opposite signs.
Introduction During the mixing process like molecular pairs in the pure components are replaced by unlike following the quasi-chemical equation 1/2(1-1)+ '/z(2-2)
-
(1-2)
(1)
Equation 1 entails enthalpy and entropy changes corresponding to AwH and Aws in the regular solution terminology which appear at the macroscopic level in the excess thermodynamic quantities hE and sE. The interchange energy AWHor the X12parameter of the more recent Prigogine-Flory theory1 may be written2 in terms of contact energies qC,expressed per unit area of intermolecular contact with s1 the molecular surface/volume ratio for component 1. When the components are of different chemical nature and surrounded by spherically symmetrical dispersion forces, X12is positive. Recent work3 shows that even if the components are of identical chemical nature as in a mixture of n-alkanes, AwH and X12are still important, as indeed is Aws, implying the need for an entropic parameter Q12in the Prigogine-Flory theory. These quantities are now due to the presence of short-range correlations between the orientations of the molecules forming the 1-1, 2-2, and 1-2 contacts. There is a net change in the intensities of the correlations in the mixing process and this contributes to AwH and Aws. Generally, then, the viJin eq 2 contain contributions from both spherically symmetrical dispersion forces and correlations of molecular orientations, so that an apparent X12 is found, reflecting two thermodynamic effects, i.e. X12= X12(dispforces) + Xlz(correl) (3) Both thermodynamic and depolarized Rayleigh scattering measurements4 indicate that when components 1 and 2 are both n-alkanes, the contact energies qb1for the 1-1,
2-2, and 1-2 pairs are all subject to correlations of orientations, although there is a net decrease of order in eq 1; i.e;, X12is positive. However, if component 1 is a spherical-molecule liquid such as cyclohexane or 2,2-diniethylbutane while component 2 is again an n-alkane such as n-C16,both the 1-1 and 1-2 pairs are uncorrelated, so that the experimental X12reflects the order in component 2. The Prigogine-Flory theory introduces another contribution to the excess thermodynamic quantities due to changes in free volume during the mixing process. This may be calculated, and thus hE data may be translated into Xlz values, constituting a method of investigating alkane order. C a l ~ r i m e t r i cand ~ ~ depolarized Rayleigh scattering5 measurements have already shown that the n-alkane order decreases rapidly with increasing temperature. Since for cyclohexane mixed with the n-alkanes X12(correl)directly reflects the alkane order, it must also decrease with T, and cpE = d h E / d T should be negative. Measurements of cpE provide a means not only of revealing the presence of orientational order of the n-alkanes but also of studying its temperature dependence. The free-volumecontribution to hE is also temperature dependent. However, its effect in cpE is small and can be calculated. In the present work, cpEis obtained through the concentration range at 10,25, and 55 "C for c-Cs f n-C,, where n = 8, 10, 12, and 16 (at 20-55 "C) and also for c-C6 branched alkanes (br-C,) where little or no orientational order is found but where the free-volume contribution exists, viz., 2,2,4-trimethylpentane, 2,2,4,6,6-~entamethylheptane, and 2,2,4,6,8,8-heptmethylnonane. The decrease of orientational order with T is due to two causes: (a) thermal motion per se and (b) expansion of the liquid which brings about a loosening of the packing between chains. We will be concerned here with the relative importance of these two causes as well as with the testing of equations which describe the temperature dependence of the order and the associated thermodynamic functions. Heintz and LichtenthaleP have already measured hE for
0022-3654/79/2083-2979$01.00/00 1979 American Chemical Society
+
2980
The Journal of Physical Chemistry, Vol. 83, No. 23, 1979
Bhattacharyya and Patterson
negative and which becomes larger as the temperature is increased. Thus cpE dcpE/dT - (large) t - (small) -
hE
orientational order free volume
t
The Prigogine-Flory theory which takes account of the free-volume effect but not orientational order gives expression 4 for cpE,where the quantities with a tilde and C p E = (XlS1* X,S,*)c,(T) - Xls'*cp(Tl)X z ~ z * ~ p ( ~ (4) z;)
-0
-E
+
'Y
3U5
cp=
(4/J7-1/'3
- 1)-1
S* = P*V*/T*
25'
1
I
0
02
06
04
I
I
08
10
XI
Figure 1. Molar excess heat capacity as a function of cyclohexane mole fraction for cyclohexane n-C,, and cyclohexane 2,2,4,4,6,8,8-heptamethylnonane (br-C,,J at 25 and 55 "C. Prigogine-Flory ne,, at 25 " C (upper theoretical predictions: dashed curves for c-c, br-C,, at curve) and 55 "C (lower curve); dotted curves for c-c,, 25 OC (upper curve) and 55 " C (lower curve).
+
+
+
+
+ n-C, and + br-C, at 25 and 40 "C, concluding that the n-alkane order decreases rapidly with temperature in qualitative agreement with the prediction by Bendler' based on an analogy between these liquids and liquid crystals in the isotropic phase. Our cpE measurements support and extend this work. Experimental Section Materials. Cyclohexane was obtained from Aldrich and the normal and branched alkanes were from Chemical Samples Co. The stated purity of 99% was verified in a number of instances through gas-liquid chromatography. The impurities are mainly branched alkanes which we believe could cause a maximum error of 1% in the cpE values. Reduction parameters P*, V*, and T* of the pure components are needed for the application of the Prigogine-Flory theory. They have been obtained from equation-of-state data and are listed in ref 3a, together with values of the s parameter obtained from consideration of the geometry of the molecules. Methods. The excess heat capacities were measured by using a Picker flow microcalorimeter (Techneurop, Montreal and Setaram, Lyon). The instrument and procedure have been described previously.* We estimate the precision and accuracy to be within f0.02 for cpEvalues up to 1 J K-l mol-' and &2% for cpE values higher than this. Results and Discussion Temperature Dependence of cpEand hE. The cPEvalues for C-CG + higher n-alkanes are negative and extremely large, indicating the destruction of orientational order in the n-alkane by the C-Cg. Figure 1 shows cpE for C - c S + n-C16 and c-C + br-C16 at 25 and 55 "C. The two c16 isomers give c> values which are vastly different because of the order present in n-C16 and virtually absent from br-Cl6. Thus cpEfor C-CS + n-C16is strongly negative at 25 "C, indicating the rapid decrease of orientational order at this temperature. It is less negative at 55 "C where part of the order has already been destroyed. On the other hand, cpE for C - c S + br-C16is negative but small and becomes more negative with increasing temperature. This behavior is explained by the difference in free volumes between cyclohexane and the br-C16 which leads to hE
a star are, respectively, reduced quantitits and reduction parameters. The reduced temperature T and volume V of the solution are obtained from the corresponding quantities for the pure components, together with a value for the X l z parameter which has been fitted to hE. The P*, V*, and T" parameters obtained from the pure-component equation-of-state data at 25 "C are, in principle, independent of T and hence were used for the 55 "C as well as the 25 "C calculations. The Xlz parameter, fitted to the equimolar hE datum at -25 "C was also used at 55 "C. Figure 1 shows the theoretical cpE. Calculations at 55 "C were made for c-c6 n-C16 by using P*, V*, T* values obtained from 55 "C data but keeping the X12value fitted at 25 "C. The equimolar cpEvalue changes from -0.24 to -0.28 J K-' mol-l. The Xlz value was then also changed to fit hE at 55 "C, Le., lowered from 16.5 to 8.3 J ~ m - The ~. value of cpE was then -0.40 J K-l mol-'. It is thus clear that changes in the predicted cpE arising from a variation of the molecular parameters with T are negligible compared with the discrepancy between theory and experiment for C - c C + n-C16. On the other hand, the predictions for C-CS + br-CI6, also seen in Figure 1,are quite reasonable (all molecular parameters obtained at 25 "C). The success here can be ascribed to the lack of orientational order in br-Cl6. The cpEvalues for other c-c6 + n-C, systems follow the general pattern of Figure 1. As the length of the normal chain is decreased, cpEbecomes less negative, corresponding to decreasing orientational order in the liquid. The cpEvalues for all of the systems at the various temperatures were represented by the Redlich-Kister equation
+
CpE
=
+
CC,X1X2(X1
c=o
- X')L
C, = A, B,(t - 25) + C,(t - 25)' (5) where t is the temperature in degrees Celsius. The constants are listed in Table I while Figure 2 shows the equimolar cpEvalues (=c0/4) as functions of T for the systems. In Figure 2 it will be noticed that the cpE values at c-c6 n-C16 at 20, 25, and 35 "C do not vary smoothly with temperature. The runs a t these temperatures were repeated (several times at 25 "C), confirming the points as shown. We believe therefore that variation of the values does not represent experimental scatter. However, the deviations of the points from a smooth curve are close to the experimental error and hence that curve is taken here. Values of hE may be obtained as functions of T through an integration of cPEby using the Redlich-Kister equations (eq 5) and the constants of Table I. Redlich-Kister equations for hE a t 25 "C are available in ref 3a for all the systems except C-CS + n-Clo which is included in the recent work by Heintz and Lichtenthalera6 Analytical expressions for hE are not given here since they are easily obtained, but Figure 3 shows the equimolar
+
Heat Capacities of Cyclohexane
The Journal of Physical Chemistty, Vol. 83, No.
I- Alkane Systems
23, 1979 2981
TABLE I: Excess Heat Capacity Redlich-Kister Constants cj for Cyclohexane (Component 1) t Normal and Branched Alkanesa ~
C,
CO
alkane
A,
n-CEb n-C,,b n-C,,b n-C, b c 2,2,44rimethylpentane ( br-C,)c 2,2,4,6,6-~entamethylheptane (br-C, ? ) * 2,2,4,4,6,8,8-heptamethylnonane (br-C,,)'
-7.86 -11.17 -15.43 -27.43 -0.48 -0.91 - 2.21
a Units are as follows: ci and A;, J K - ' mol.
I;
1 0 2 ~ "
io^,
0.93 8.00 5.15 -14.81 17.24 --21.92 31.75 -28.11 -3.97 0 -4.79 0 - 5.52 0
A,
CZ
I O ~ B , 104c,
-2.54 -2.47 7.49 -2.83 -1.72 -2.54 -3.93 -5.62 13.07 -5.15 -21.31 47.67 1.40 - 6.51 5.30 0.36 -4.10 0 1 . 4 3 -6.18 0
Bi, J K" m o l - ' ; Ci,J K - 3 mol"'.
10-55 "C.
A, -1.28 -1.80 --2.24 -4.51 1.99 1.12 2.27
~o*B, 1 0 4 ~ ~ -3.09 -1.80 -1.03 -2.56 1.04 --1.62 -7.780
3.39 -1.09 -0.37 0.23 --10.37 0 0
20-55 "C.
br-Cn -1
--2 -
-6-
I
I -8
IO
30
20
40
50
60
tPC Figure 2. Equimolar excess heat capacity values at different temperatures for cyclohexane with the normal alkanes (as indicated) and branched alkanes (full lines reading from top: br-C,, br-Clp, and br-Cf6). The Wicgine-Flory theoretical curve is shown for cyclohexane nCI6 as a dashed line at the top of diagram.
+
hE as a function of T obtained with the 25 "C hE data of ref 6. These data are usually -1% and sometimes as much as 2% higher than those of ref 3a. Nevertheless this constitutes reasonable agreement and either set of data could be used. However, the 25 and 40 "C data of ref 6 give cpEvalues at 32.5 "C which are usually -15% less in magnitude than ours. This could be explained by an error of 2-3% in the 40 OC data relative to those at 25 "C. Our cpEdata for C-CG n-C16give an equimolar hE value at 50 "C which is within 1%of the datum of Lundberg? so that we believe that the present cpE data are correct. Direct hE(T) measurements have been undertaken for various systems between 25 and 65 "C, and they are in substantial agreement with the present cpE values. Figures 2 and 3 show the temperature dependence of the equimolar cpE and hE for these systems. The attempt can again be made to predict cpE values by using the Prigogine-Flory theory which considers free volume alone and not orientational order, and here hE values at 25 "C were used to obtain Xlz values for the various systems. The failure of the theory, made evident in Figures 2 and 3, reveals the importance of order in the n-C, liquids. The theory gives small negative c E values becoming more negative with increasing T wiich is approximately the correct behavior for C-CG + br-C,. However, the experimental cpE values for c-C, n-C, are an order of magnitude larger and become rapidly less negative with increasing T , contrary to the prediction of the free-volume-based theory. Furthermore, the experimental cpEvalues depend markedly on alkane chain length. These features indicate that n-alkane order depends on chain length and decreases with an increase of T , producing negative cpE,but the rate
+
+
I
t/"C Figure 3. Temperature dependence of the equimolar excess enthalpy for cyclohexane with n-alkanes and branched alkanes, obtained through integration of cPEand hE at 25 "C. The br-C,,, curve was obtained by interpolation between br-C, and br-Clp. The temperature ranges are as shown by the full curves in Figure 2. The Prigogine-Flory theory n-C, which decrease slightly with T , similar gives hE curves for c-C, to those for c-Ce -I- br-C,.
+
of decrease falls off with an increase of T. Effect of Packing Density on n-Alkane Order. Figures 4-6 show how certain direct measures of the orientational order in the n-C, alkanes vary with the fractional free or empty volume of the liquid. The contribution of orientational order to cpE and hE may be estimated in a number of ways. First, it may be assumed that the dispersion force and free-volume terms are the same for cyclohexane mixed with an n-alkane and with a branched isomer, leaving only the orientational order contribution in the former system. Thus, we may write that for the systems containing n-alkanes cpE(order) = cpE(n-C,) - cpE(br-C,) and correspondingly for hE(order). (Values of cpE have been obtained (unpublished) at 25 "C for c-C6mixed with other branched C8 isomers and with various branched C9 isomers. cpE(order),as defined, should not vary by more than -0.1 J K-l mol-l with the choice of the br-C, reference isomer.) The intercomparison of the various systems is facilitated by considering cPE(order)and hE(order)per unit molecular volume rather than per mole, i.e., cpE(order)/(xlVl* + xzV2*)and hE(order)/(xlVl* + zzVz*),where V* are the "hard-core" volumes or volume reduction parameters for the two components as listed in ref 3a. The above quantities are plotted in Figures 4 and 5. In Figure
Bhattacharyya and Patterson
The Journal of Physical Chemistry, Vol. 83, No. 23, 1979
2982
-
-0
--=----
___-------br-Clz
1.25
/
O
1.20
1.25
I
I
0.050
0.055
I a25
1.30
v
I 0.060
I
L
0.065
0.070
7 0.30
I 0.40
a35
I
0.45
dT Flgure 4. Temperature dependence of the order contribution to equimolar cPEdivided by the “hard-core volume” for cyclohexane n-C,, cyclohexane n-C,,, cyclohexane nC,,, and cyclohexane nC,,. Three scales of temperature are used, corresponding to the tegree of thermal expansion ?f each pure alkane: reduced volume (V) and reduced temperature ( T ) of the Prigogine-Flory theory and the experimental a T , where a is the thermal expansion coefficient. The range of temperature is exactly that of the curves in Figures 2 and 3 and 25 O C values have been indicated by points in the figure to provide a reference of real temperature.
+
+
+
+
“I
I
1.30
v
Figure 6. Temperature dependence of the X , , parameter for cyclohexane with normal and branched alkanes obtained from the equimolar hEin Figure 3 through the use of the Prigogine-Flory theory. The scale of temperature is V ; the reduced volume corresponds to the degree of thermal expansion for each alkane. The same range of temperature as in Figures 2-5 with 25 O C values shown here to indicate the real temperature for each alkane.
Figure 6js a plot of the interaction parameter Xlz of eq 3 against V . The X12parameter was obtained from the equimolar hE through the Prigogine-Flory theory (eq 6 of ref 3a) which subtracts the free-volume contribution leaving X12. As seen in Figure 6, X12for a c-c6 i-br-C, system is independent of temperature, corresponding to X12(correl) N 0 in eq 3. The br-C16and br-C12alkanes, with the same type of molecular surface, have the same X12value and the same interaction with the cyclohexane molecules. The difference between these Xlzvalues and that for br-C8 is surprising and was also noticeable in ref 3a. For c-C6 n-C,, Xlz decreases rapidly with an increase of T, tending to become asymptotic to the br-C, values. The temperature-dependent Xlz(correl) is clearly apparent and gives a measure of the decrease of alkane order with T. An empirical fitting of the Prigogine-Flory theory could be achieved by introducing contact energies in the n-C16 liquid which decreased with an increase of temperature, leading through eq 2 to decreasing X12,i.e.
+
dq22/dT 01
1.20
I
1.25
I
a
1.30
I
1.35
Flgure 5. Temperature dependence of the order contribution to the equimolar hEdivided by the “hard-core volume” for cyclohexane n-C,,cyclohexane n-Cl0, cyclohexane n-& and cyclohexane n-Cle. The same scales of temperature as in Figure 4 ccrrespond to the degree of thermal expansion of each pure nC,. Only Vis shown here. The range of temperature is as in Figures 2-4 with 25 OC values shown in order to indicate the real temperature for each alkane.
+
+
+
+
4, c E is plotted against three equivalent representations
of tke fraction of the nIC, volume which is free or empty: (a) V = V / V * ,where V is the reduced volume and V* is the hard-core volume, i.e., the molar volume at 0 K, (b) the experimental value of aT, where a is the thermal expansion coefficient, and (c) the reduced temperature T. According to the Prigogine-Flory theory, the three scales are related through
