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Statistical thermodynamics of group interaction in n-alkane-n-alkanol and n-alkanol-n-alkanol solutions
ChemicolEngineering Science, 1973, Vol. 28, pp. 1005-1011.
Pergamon Press.
Printed in Great Britain
Statistical thermodynamics of group interaction in n-alkane-n-alkanol and n-alkanol-n-alkanol solutions TSUNG-WEN LEE, ROBERT A. GREENKORN and KWANGEHU CHAO School of Chemical Engineering, Purdue University, Lafayette, lndiana 47907, U.S.A. (First received 13 December 197 1; in revisedform 26 June 1972) Abstract- The statistical thermodynamics previously developed for pure liquids of non-polar and polar chain molecules is extended to their mixtures. An orientation factor is introduced into the quasi-chemical relation of the hydrogen Bond to reduce the calculated preferential Bonding. Excess properties are calculated for n-alkane-n-alkanol-1 and n-alkanol-I-n-alkanol-1 solutions. The heat of mixing and excess free energy calculated with the group properties reported previously are in good agreement with data for solutions in which the alkanol is higher than ethanol. INTRODUCTION
paper [6] we developed a partition function for the description of the configurational properties of non-polar and polar chain molecules by considering group interactions. A group is an identifiable structural unit of a molecule, such as a methyl group or a hydroxyl group. The interactions of the groups were expressed in terms of the cell theory and the quasi-lattice theory of liquids. The partition function developed gives a quantitative representation of the volume and heat of vaporization of pure n-alkane and nalkanol-1 liquids over a wide temperature range. The interaction properties of methyl, methylene, and hydroxyl groups were determined. Our present objective is to extend the statistical thermodynamics of group interactions to liquid solutions. Formulas for the excess properties are derived from the partition function and applied to n-alkaneln-alkanol and n-alkanollnalkanol solutions and are compared to experimental data on heat of mixing and excess free energy. Large deviations from idea1 solution behavior is observed when the hydrogen bonds of the alkanols are broken by dilution with alkanes. Comparison of the calculated excess properties of n-alkaneln-alkanol solutions with data therefore provides sensitive tests of the theory and of the values of the group interaction parameters. The early work on group contribution to soluIN A PREVIOUS
tion properties is due to Pierotti[8], Deal[ l] and Wilson[lS]. The Analytical Solution of Groups developed by Derr and Deal[2] provides a flexible means for representing excess free energy data which is the scope of their work. Parameters are required to be developed from experimental activity coefficients at various temperatures of interest. Some group parameters are found to vary with the solution system. Renon and Prausnitz[ 1 l] and Wiehe and Bagley [ 141 adopted a molecular association model in their studies of alkane-alkanol solutions. The formation of linear polymers from alkanol monomers was expressed in terms of an association constant. The excess Gibbs free energy is expressed as the sum of a physical contribution and a chemical contribution. One adjustable parameter needs to be determined for each binary system for which data on the solution behavior must be available. The association model is capable of quantitative representation of the excess properties except for dilute alkanol solutions. The present work is an extension of the treatment by Kuo er a1.[5] who applied the quasilattice theory to alkane-alkanol solutions. Even though quantitative representation of the excess heat and free energy was obtained, their energy parameters had to be determined at each temperature of interest. In this work we use tempera-
1005
TSUNG-WEN
LEE, ROBERT
A. GREENKORN
ture-independent properties of methyl, methylene and hydroxyl groups previously reported]61 for the representation of properties of pure liquid alkanes and alkanols. THE
EXCESS
=
gA
,rr(er3,$)NAwf
i
CHAO
AH”=U--I:_,.
(5) A
The mixing enthalpy is equal to excess enthalpy. The internal energies are simply related to the interaction energies. Rewriting Eq. (5),
PROPERTIES
To obtain the excess properties of solutions we generalize the partition function previously described [6] for a heterogeneous chain ZA
and KWANG-CHU
(z?q1/3-
AHE= z NUeU-x i5j
exp
l)wA+ACA
I
-If; xx4 X
kT
[
1
(1)
’
Equation (1) describes pure liquid A as indicated by the subscript and superscript. Subscripts i and j denote group species. For a mixture of various heterogeneous chains Eq. (1) is generalized into
I: N&j.
A
iH
The pairwise interaction energy lU between groups i and j were given previously [6] for the various modes of interaction among the methyl, methylene and hydroxyl groups. Gibbs free energy does not differ much from the Helmholtz free energy at low pressures. We accordingly express the Gibbs free energy of mixing by AG”=
F-
x FA.
X [
kT
1 ’
(2)
The combinatorial factor &!Aand g are, as suggested by Guggenheim [4],
(7)
A
The Helmholtz free energy is related to the partition function of Eq. (2) by F=-kTlnZ
-2 Nil%
(6)
(8)
with similar equations for the pure liquids. Substituting Eqs. (2) and (8) into Eq. (7) and rearranging, AG”=RTx
MAln+A A
-RTx -
izj
[(N~lnN~-~ZVJAlnN,*A) A
(N, In NU - T Ni In Ni)]
- RT(ln 2) T zt KNU-- 5 N$-CN$ Equations (3) and (4) express the difference between the values of g for the real system and the hypothetical system in which preferential orientation is absent. (The hypothetical values are indicated with stars). The difference in g and g* is related to the corresponding pair contact numbers NU and N:, which are calculated according to equations presented previously [6]. The heat of mixing of liquids at low pressures is closely approximated by the change in internal energy,
-T
N:A)l
in which +A stands for the group fraction of A in solution
1006
4A = XANA/~ xi&~.
(10)
Statistical thermodynamics of group interaction
The relative size of a molecule is measured in terms of the number of equivalent methylene groups, x, defined by XA =
x
niAyi
of Guggenheim’s quasi-chemical approximation, according to which the pair contact number NU are related to the corresponding like-pair nurnbers by
(11)
1:
where yf is the core volume of i relative to that of a methylene. The average degrees of freedom per group in molecule A is denoted by CA and is defined by XACA =
2 i
X*ACf.
(12)
The first three terms on the right hand side of Eq. (9) are contributions due to Guggenheim’s combinatorial factor g. The first term corresponds to g* for which we adopt Flory’s expression[3]. The fourth term is the contribution due to cell motion, and the last term is due to heat of mixing. The detailed derivation of Eq. (9) is given elsewhere [71. The excess Gibbs free energy is directly related to Gibbs free energy of mixing by hGE=AGM-RTE
MAlllitCf, A
(14)
In summary, the excess properties of interest are determined according to Eqs. (6), (13) and (14). The pairwise contact numbers NU required in these equations are calculated by Guggenheim’s quasi-chemical relation. THE QUASI-CHEMICAL ORIENTATION
RELATION FACTOR
AND
The formation of contacts between groups depends on the relative abundance of the group species involved and the energy associated with the formation of contacts. Rigorous expressions of the preferential orientation effect due to energy differences have not been derived for systems of 3-dimensions. We represent this effect in terms
(15)
Equation (15) expresses that the formation of ij contacts are favored by a negative value of the exchange energy wQ, and are disfavored by a positive value of w,. The quasi-chemical relation of Eq. (15) is shown by Prigogine and coworkers[9] to be an excellent approximation for small values of )w/kT (, but becomes more in error at high values of Iw/kTI. (The hydrogen-bonds of the alkanols have a large negative value of (w/kT) at usual temperatures.) To use the quasi-chemical relation in representing the formation of hydrogen bonds we must introduce an empirical orientation factor t in Eq. (15). NO.eH = 4 exp (- 2fo...Hw0...H/kT). (16) NO,..ONH...H
(13)
where MA denotes the mole fraction of A. The excess entropy is obtained by combining Eqs. (6) and (9) according to ASE = (AHE - AGE)/T.
N’ = 4 exp (-2 wu/kT). NtiNU
The orientation factor t for hydrogen bonds of alkanols is approximately equal to O-7. This value is used in the quasi-chemical relation of the hydrogen-bond of alkanols. The exchange energies of the other group pairs encountered in this work are much lower than that of the hydrogenbond; no orientation factor is needed in their quasi-chemical relations. The value of t G O-7 for the alkanol hydrogen bonds was established from fitting the heat of mixing of alkane-alkanol systems. The calculated heat of mixing would be much too small without t. The model for hydrogen bonds would fail to dissociate upon dilution by an alkane. In addition the sensitive dependence of heat of mixing on concentration in dilute alkanol solutions would be absent. These same effects were observed in all the alkane-alkanol systems studied[7]. The calculation of an excess property in this work requires the property value to be known for the solution of interest as well as for the pure
1007
TSUNG-WEN
LEE, ROBERT A. GREENKORN
components. The same orientation factor was used for the hydrogen bond both in the solution and in the pure alkanol. While the calculated properties of the solution responded sensitively to the value of t, the pure alkanols are relatively insensitive. Figure 1 shows the calculated molar volumes of n-decanol for t = 1, and t = 0.7. There is a negligible difference between the two at temperatures up to about 100°C. Above 100°C the difference increases appreciably with increasing temperature. The excess properties studied in this work are all at temperatures below 100°C. In spite of the use of the orientation factor for hydrogen bonds t = 0.7, the relevant pure alkanol properties which were calculated in this work remain the same as described previously[6] with t = 1.0. 280 I
I
I
I
I
I
_c 260 m\" GB
--
to H
-0.7
240
and KWANG-CHU
CHAO
temperature range. In a related study we made of the alkane-alkanone systems, we found t = 1. RESULTS
AND
DISCUSSION
The properties of methyl, methylene and hydroxyl groups are given in Table 1[6]. The property values of methyl and methylene groups were developed based on analysis of data on pure n-alkane liquids. The properties of hydroxyl were developed by referring to data on pure n-alkanol1 liquids, and their solutions. The orientation factor t = 0.7 is used in all calculations described here. The calculated heat of mixing is illustrated in Fig. 2 with the n-pentanol-n-tetradecane system for comparison with data. Quantitative representation is obtained, including the low alcohol concentration range where heat of mixing is highly sensitive to the concentration. Comparison of the calculated heats of mixing and data has been made for numerous other n-alkane-nalkanol-1 systems [7]. The comparison shows that the heat of mixing of n-alkane-n-alkanol-1 systems are generally well represented over a wide range of molecular weight of the alkanes as well as of the alkanols. The molecular weight of the alkane appears to have little effect on the Table 1*. Interaction parameters of hydroxyl, methylene and methyl groups
Fig. 1. Comparison of calculated molar volume of n-decanol.
That a correction factor needs to be introduced into Guggenheim’s quasichemical relation was suggested by Prigogine et a1.[9]. But their correction factor was complex and inconvenient. The orientation factor we employ here is an empirical simplification of the correction factor of Prigogine et al. It has so far been found to be needed only for the alkanol hydrogen-bond, for which [w/H] is very large. It seems reasonable to expect t to be described by a general function of w/kT. Clearly, as Iw/kTI + 0, t + 1; and, as (w/kT 1 increases, t decreases. There is insufficient information to establish this function quantitatively. Excess heat data for the alkanealkanol systems are available in a very limited
Group
Degrees of freedom
Core volume (cmVg-mole)
Hydroxyl Methylene Methyl
0.410 0*1203 0.3212
10.58 14.101 22.10
Interaction energy constants Energy constant, r) (cal cm3’15g-mole-*.ls) Type of interaction (CH, or CHs) - * . (CH, or CHz) (CH, or CH,) * . .O (CHsorCH$...H ...0 O-.-H H...H
10,140 10,240 9,950 21,500 57,400t 21,500
*Taken from Lee et aI.[6]. tThe complete hydrogen bond energy is given by lHs = 11”SlVI”‘~ - oHBwhere nHBis 57.408 cal cm1.15g-mole-*‘15and cHBis 3,100 Cal/g-mole.
1008
Statistical thermodynamics of group interaction
600 400
-
Thewetical o Experimental
[i2]
200
0
0.2
0.4
0.6
Mole fraction,
n-Pantanal
0.6
I.0
Fig. 2. Heat of mixing for b-pentanol-n-tetradecane system.
quality of fitting, while that of the alkanol has a substantial effect. The highest alkanol for which heat of mixing data are available is n-octanol. Our calculations deviate from Savini and coworkers’ data[ 121 on n-octanolln-heptane by about 10 per cent on the average. The deviation tends to become larger for the lower alkanol solutions starting from propanol, for which deviations up to 15 per cent are observed. The increase in deviation for solutions of the lower alkanols in alkane is due to an apparent “anomaly” in their behavior shown in Fig. 3. The heats of mixing of various alkanols
‘5
E
m__
200-
(I)
Cz H. OH
(2)
C, H, OH
(3)
C, H,, OH
(4)
C, H,,OH
(5)
C,, H,,OH
in n-heptane show a common pattern of behavior for all the alkanols except ethanol and methanol for which the group parameter values of Table 1 are not recommended. Excess free energy and entropy data are available for a few n-alkanol-n-alkane systems. Figure 4 illustrates the calculated values with n-propanol-n-heptane for comparison with data. The deviation in AGE amounts to about 50 cal/ g-mole in the middle composition range. Assuming approximately equal contribution from both components to this deviation, the error in the calculated activity coefficients would be a maximum of about 9 per cent. Since propanol is marginal to the anomalous behavior of methanol and ethanol we expect predicted free energies for higher alkanol solutions to be better, and the corresponding calculated activity coefficient can be within acceptable limits for many practical applications. Heat of mixing data have recently become available for alkanol-alkanol solutions [ lo]. Figure 5 illustrates calculated values with npropanol systems for comparison with data. The agreement is reasonable. We have extended group contribution theory to liquid solutions. The same property values for methyl, methylene and hydroxyl groups (Table 1) are applied to the calculation of pure liquids as well as their solutions. An empirical orientation factor t = O-7 is necessary in the quasichemical relation of the alkanol hydrogen-bonds
\ Fl
-
\
Theoretical o v Experimental
\
[IO]
100 -
0
I
I
0.2
0.4 Mole fraction
I
I
I 0.6
0.6
0
I.0
I
I
I
0.4
06
0.6
Mole fraction
alcohol
Fig. 3. Heat of mixing data for n-alkanol-n-heptane system at 30°C.
I
I.0
alcohol
Fig. 4. Excess free energy and entropy of n-propanol-nheptane system at 20°C.
1009 CFS Vol. 28. No. 4-B
I 0.2
TSUNG-WEN
LEE, ROBERT A. GREENKORN
and KWANG-CHU
CHAO
number of repeating groups (with subscript and superscript) P pressure R universal gas constant coordination number : entropy T temperature t orientation factor u configurational energy V molar volume v cell volume W exchange energy total number of group per molecule configurational partition function ; coordination number with subscript z n
0
0.2
0.4
Mole froctlon,
0.6
0.6
I.0
n-Prcqmol
Fig. 5. Heat of mixing for n-propanol-n-alkanol systems at 30°C.
in order to achieve quantitative representation of the excess properties of alkanol-alkane solutions. Pure alkanol liquids are insensitive to the orientation factor. The solution behavior of the two lowest alcohols appears not to be well described by the group interaction model.
Greek symbols E interaction
energy (interaction energy between groups with subscript) 5 geometric constant r) characteristics of pairwise interaction constants, characteristics of orderliness 4 group fraction c chemical reaction energy y ratio of core volumes (with subscript) relative to the base group
Acknowledgement-This research was supported by NSF Grant GK-16573. R. S. Ramalho made available heat of mixing data on alkanol-alkanol systems before publication.
NOTATION
external degrees of freedom total potential energy Helmholtz free energy combinatorial factor of Flory-Huggin (combinatorial factor of Guggenheim without superscript) Gibbs free energy enthalpy Boltzmann constant mole fraction number of molecules (group contact pairs with subscript)
E”, g* F
G H k M N
Subscripts ij interaction property i group species
A B HB
between groups i andj
type of molecule base group hydrogen bond
Superscripts * characteristic
property reduced property type of molecule E excess property A4 mixing property A
REFERENCES DEAL C. H., DERR E. L. and PAPADOPOULOS M. N., Ind. ‘Engng Chem. Fundls 1962 117. DERR E. L. and DEAL C. H., Distillation 1969. pp. 37-47, Brighton, Sept. 1969. FLORY P. J., Principles ofPolymer Chemistry, p. 497. Cornell University Press, Ithaca, N.Y. 1953. GUGGENHEIM E. A., Proc. R. Sot. 1944 A183 226. KU0 C. M., ROBINSON Jr. R. L. and CI-IAO K. C., Ind. Engng Chem. Fundls 1970 9 564. LEE T. W., GREENKORN R. A. and CI-IAO K. C., Ind. Engng Chem. Fundls 1972 11293. LEE T. W., Ph.D. dissertation, Purdue University 1971. 1010
Statistical thermodynamics of group interaction [8] PIEROTT G. J., DEAL C. A. and DERR E. L., Ind. Engng Chem. 1959 5195. [9] PRIGOGINE L., MATHOT-SAROLEA and VAN HOVE L., Trans. Faraday Sot. 1952 48 485. [lo] RAMALHO R. S. and RUEL M., Can. J. Chem. Engng 1968 46 456. [l 11 RENON H. and PRAUSNITZ J. M., Chem. Engng Sci. 1967 22 299. [121 SAVINI C. G., WINTERHALTER D. R. and VAN NESS H. C.,J. Chem. Engnglhta 1965 10 168; 171. [131 VAN NESS H. C., SOCZEK C. A. PELGUIN G. L. and MACHDO R. L.,J. Chem. Engng Data 1967 12 217. 1141 WIEHE 1. A. and BAGLEY E. B., Ind. Engng Chem. Fundis 1967 6 209. 1151 WILSON G. M. and DEAL C. A., Znd. Engng Gem. Fundls 1962 120.
1011
Pergamon Press.
Printed in Great Britain
Statistical thermodynamics of group interaction in n-alkane-n-alkanol and n-alkanol-n-alkanol solutions TSUNG-WEN LEE, ROBERT A. GREENKORN and KWANGEHU CHAO School of Chemical Engineering, Purdue University, Lafayette, lndiana 47907, U.S.A. (First received 13 December 197 1; in revisedform 26 June 1972) Abstract- The statistical thermodynamics previously developed for pure liquids of non-polar and polar chain molecules is extended to their mixtures. An orientation factor is introduced into the quasi-chemical relation of the hydrogen Bond to reduce the calculated preferential Bonding. Excess properties are calculated for n-alkane-n-alkanol-1 and n-alkanol-I-n-alkanol-1 solutions. The heat of mixing and excess free energy calculated with the group properties reported previously are in good agreement with data for solutions in which the alkanol is higher than ethanol. INTRODUCTION
paper [6] we developed a partition function for the description of the configurational properties of non-polar and polar chain molecules by considering group interactions. A group is an identifiable structural unit of a molecule, such as a methyl group or a hydroxyl group. The interactions of the groups were expressed in terms of the cell theory and the quasi-lattice theory of liquids. The partition function developed gives a quantitative representation of the volume and heat of vaporization of pure n-alkane and nalkanol-1 liquids over a wide temperature range. The interaction properties of methyl, methylene, and hydroxyl groups were determined. Our present objective is to extend the statistical thermodynamics of group interactions to liquid solutions. Formulas for the excess properties are derived from the partition function and applied to n-alkaneln-alkanol and n-alkanollnalkanol solutions and are compared to experimental data on heat of mixing and excess free energy. Large deviations from idea1 solution behavior is observed when the hydrogen bonds of the alkanols are broken by dilution with alkanes. Comparison of the calculated excess properties of n-alkaneln-alkanol solutions with data therefore provides sensitive tests of the theory and of the values of the group interaction parameters. The early work on group contribution to soluIN A PREVIOUS
tion properties is due to Pierotti[8], Deal[ l] and Wilson[lS]. The Analytical Solution of Groups developed by Derr and Deal[2] provides a flexible means for representing excess free energy data which is the scope of their work. Parameters are required to be developed from experimental activity coefficients at various temperatures of interest. Some group parameters are found to vary with the solution system. Renon and Prausnitz[ 1 l] and Wiehe and Bagley [ 141 adopted a molecular association model in their studies of alkane-alkanol solutions. The formation of linear polymers from alkanol monomers was expressed in terms of an association constant. The excess Gibbs free energy is expressed as the sum of a physical contribution and a chemical contribution. One adjustable parameter needs to be determined for each binary system for which data on the solution behavior must be available. The association model is capable of quantitative representation of the excess properties except for dilute alkanol solutions. The present work is an extension of the treatment by Kuo er a1.[5] who applied the quasilattice theory to alkane-alkanol solutions. Even though quantitative representation of the excess heat and free energy was obtained, their energy parameters had to be determined at each temperature of interest. In this work we use tempera-
1005
TSUNG-WEN
LEE, ROBERT
A. GREENKORN
ture-independent properties of methyl, methylene and hydroxyl groups previously reported]61 for the representation of properties of pure liquid alkanes and alkanols. THE
EXCESS
=
gA
,rr(er3,$)NAwf
i
CHAO
AH”=U--I:_,.
(5) A
The mixing enthalpy is equal to excess enthalpy. The internal energies are simply related to the interaction energies. Rewriting Eq. (5),
PROPERTIES
To obtain the excess properties of solutions we generalize the partition function previously described [6] for a heterogeneous chain ZA
and KWANG-CHU
(z?q1/3-
AHE= z NUeU-x i5j
exp
l)wA+ACA
I
-If; xx4 X
kT
[
1
(1)
’
Equation (1) describes pure liquid A as indicated by the subscript and superscript. Subscripts i and j denote group species. For a mixture of various heterogeneous chains Eq. (1) is generalized into
I: N&j.
A
iH
The pairwise interaction energy lU between groups i and j were given previously [6] for the various modes of interaction among the methyl, methylene and hydroxyl groups. Gibbs free energy does not differ much from the Helmholtz free energy at low pressures. We accordingly express the Gibbs free energy of mixing by AG”=
F-
x FA.
X [
kT
1 ’
(2)
The combinatorial factor &!Aand g are, as suggested by Guggenheim [4],
(7)
A
The Helmholtz free energy is related to the partition function of Eq. (2) by F=-kTlnZ
-2 Nil%
(6)
(8)
with similar equations for the pure liquids. Substituting Eqs. (2) and (8) into Eq. (7) and rearranging, AG”=RTx
MAln+A A
-RTx -
izj
[(N~lnN~-~ZVJAlnN,*A) A
(N, In NU - T Ni In Ni)]
- RT(ln 2) T zt KNU-- 5 N$-CN$ Equations (3) and (4) express the difference between the values of g for the real system and the hypothetical system in which preferential orientation is absent. (The hypothetical values are indicated with stars). The difference in g and g* is related to the corresponding pair contact numbers NU and N:, which are calculated according to equations presented previously [6]. The heat of mixing of liquids at low pressures is closely approximated by the change in internal energy,
-T
N:A)l
in which +A stands for the group fraction of A in solution
1006
4A = XANA/~ xi&~.
(10)
Statistical thermodynamics of group interaction
The relative size of a molecule is measured in terms of the number of equivalent methylene groups, x, defined by XA =
x
niAyi
of Guggenheim’s quasi-chemical approximation, according to which the pair contact number NU are related to the corresponding like-pair nurnbers by
(11)
1:
where yf is the core volume of i relative to that of a methylene. The average degrees of freedom per group in molecule A is denoted by CA and is defined by XACA =
2 i
X*ACf.
(12)
The first three terms on the right hand side of Eq. (9) are contributions due to Guggenheim’s combinatorial factor g. The first term corresponds to g* for which we adopt Flory’s expression[3]. The fourth term is the contribution due to cell motion, and the last term is due to heat of mixing. The detailed derivation of Eq. (9) is given elsewhere [71. The excess Gibbs free energy is directly related to Gibbs free energy of mixing by hGE=AGM-RTE
MAlllitCf, A
(14)
In summary, the excess properties of interest are determined according to Eqs. (6), (13) and (14). The pairwise contact numbers NU required in these equations are calculated by Guggenheim’s quasi-chemical relation. THE QUASI-CHEMICAL ORIENTATION
RELATION FACTOR
AND
The formation of contacts between groups depends on the relative abundance of the group species involved and the energy associated with the formation of contacts. Rigorous expressions of the preferential orientation effect due to energy differences have not been derived for systems of 3-dimensions. We represent this effect in terms
(15)
Equation (15) expresses that the formation of ij contacts are favored by a negative value of the exchange energy wQ, and are disfavored by a positive value of w,. The quasi-chemical relation of Eq. (15) is shown by Prigogine and coworkers[9] to be an excellent approximation for small values of )w/kT (, but becomes more in error at high values of Iw/kTI. (The hydrogen-bonds of the alkanols have a large negative value of (w/kT) at usual temperatures.) To use the quasi-chemical relation in representing the formation of hydrogen bonds we must introduce an empirical orientation factor t in Eq. (15). NO.eH = 4 exp (- 2fo...Hw0...H/kT). (16) NO,..ONH...H
(13)
where MA denotes the mole fraction of A. The excess entropy is obtained by combining Eqs. (6) and (9) according to ASE = (AHE - AGE)/T.
N’ = 4 exp (-2 wu/kT). NtiNU
The orientation factor t for hydrogen bonds of alkanols is approximately equal to O-7. This value is used in the quasi-chemical relation of the hydrogen-bond of alkanols. The exchange energies of the other group pairs encountered in this work are much lower than that of the hydrogenbond; no orientation factor is needed in their quasi-chemical relations. The value of t G O-7 for the alkanol hydrogen bonds was established from fitting the heat of mixing of alkane-alkanol systems. The calculated heat of mixing would be much too small without t. The model for hydrogen bonds would fail to dissociate upon dilution by an alkane. In addition the sensitive dependence of heat of mixing on concentration in dilute alkanol solutions would be absent. These same effects were observed in all the alkane-alkanol systems studied[7]. The calculation of an excess property in this work requires the property value to be known for the solution of interest as well as for the pure
1007
TSUNG-WEN
LEE, ROBERT A. GREENKORN
components. The same orientation factor was used for the hydrogen bond both in the solution and in the pure alkanol. While the calculated properties of the solution responded sensitively to the value of t, the pure alkanols are relatively insensitive. Figure 1 shows the calculated molar volumes of n-decanol for t = 1, and t = 0.7. There is a negligible difference between the two at temperatures up to about 100°C. Above 100°C the difference increases appreciably with increasing temperature. The excess properties studied in this work are all at temperatures below 100°C. In spite of the use of the orientation factor for hydrogen bonds t = 0.7, the relevant pure alkanol properties which were calculated in this work remain the same as described previously[6] with t = 1.0. 280 I
I
I
I
I
I
_c 260 m\" GB
--
to H
-0.7
240
and KWANG-CHU
CHAO
temperature range. In a related study we made of the alkane-alkanone systems, we found t = 1. RESULTS
AND
DISCUSSION
The properties of methyl, methylene and hydroxyl groups are given in Table 1[6]. The property values of methyl and methylene groups were developed based on analysis of data on pure n-alkane liquids. The properties of hydroxyl were developed by referring to data on pure n-alkanol1 liquids, and their solutions. The orientation factor t = 0.7 is used in all calculations described here. The calculated heat of mixing is illustrated in Fig. 2 with the n-pentanol-n-tetradecane system for comparison with data. Quantitative representation is obtained, including the low alcohol concentration range where heat of mixing is highly sensitive to the concentration. Comparison of the calculated heats of mixing and data has been made for numerous other n-alkane-nalkanol-1 systems [7]. The comparison shows that the heat of mixing of n-alkane-n-alkanol-1 systems are generally well represented over a wide range of molecular weight of the alkanes as well as of the alkanols. The molecular weight of the alkane appears to have little effect on the Table 1*. Interaction parameters of hydroxyl, methylene and methyl groups
Fig. 1. Comparison of calculated molar volume of n-decanol.
That a correction factor needs to be introduced into Guggenheim’s quasichemical relation was suggested by Prigogine et a1.[9]. But their correction factor was complex and inconvenient. The orientation factor we employ here is an empirical simplification of the correction factor of Prigogine et al. It has so far been found to be needed only for the alkanol hydrogen-bond, for which [w/H] is very large. It seems reasonable to expect t to be described by a general function of w/kT. Clearly, as Iw/kTI + 0, t + 1; and, as (w/kT 1 increases, t decreases. There is insufficient information to establish this function quantitatively. Excess heat data for the alkanealkanol systems are available in a very limited
Group
Degrees of freedom
Core volume (cmVg-mole)
Hydroxyl Methylene Methyl
0.410 0*1203 0.3212
10.58 14.101 22.10
Interaction energy constants Energy constant, r) (cal cm3’15g-mole-*.ls) Type of interaction (CH, or CHs) - * . (CH, or CHz) (CH, or CH,) * . .O (CHsorCH$...H ...0 O-.-H H...H
10,140 10,240 9,950 21,500 57,400t 21,500
*Taken from Lee et aI.[6]. tThe complete hydrogen bond energy is given by lHs = 11”SlVI”‘~ - oHBwhere nHBis 57.408 cal cm1.15g-mole-*‘15and cHBis 3,100 Cal/g-mole.
1008
Statistical thermodynamics of group interaction
600 400
-
Thewetical o Experimental
[i2]
200
0
0.2
0.4
0.6
Mole fraction,
n-Pantanal
0.6
I.0
Fig. 2. Heat of mixing for b-pentanol-n-tetradecane system.
quality of fitting, while that of the alkanol has a substantial effect. The highest alkanol for which heat of mixing data are available is n-octanol. Our calculations deviate from Savini and coworkers’ data[ 121 on n-octanolln-heptane by about 10 per cent on the average. The deviation tends to become larger for the lower alkanol solutions starting from propanol, for which deviations up to 15 per cent are observed. The increase in deviation for solutions of the lower alkanols in alkane is due to an apparent “anomaly” in their behavior shown in Fig. 3. The heats of mixing of various alkanols
‘5
E
m__
200-
(I)
Cz H. OH
(2)
C, H, OH
(3)
C, H,, OH
(4)
C, H,,OH
(5)
C,, H,,OH
in n-heptane show a common pattern of behavior for all the alkanols except ethanol and methanol for which the group parameter values of Table 1 are not recommended. Excess free energy and entropy data are available for a few n-alkanol-n-alkane systems. Figure 4 illustrates the calculated values with n-propanol-n-heptane for comparison with data. The deviation in AGE amounts to about 50 cal/ g-mole in the middle composition range. Assuming approximately equal contribution from both components to this deviation, the error in the calculated activity coefficients would be a maximum of about 9 per cent. Since propanol is marginal to the anomalous behavior of methanol and ethanol we expect predicted free energies for higher alkanol solutions to be better, and the corresponding calculated activity coefficient can be within acceptable limits for many practical applications. Heat of mixing data have recently become available for alkanol-alkanol solutions [ lo]. Figure 5 illustrates calculated values with npropanol systems for comparison with data. The agreement is reasonable. We have extended group contribution theory to liquid solutions. The same property values for methyl, methylene and hydroxyl groups (Table 1) are applied to the calculation of pure liquids as well as their solutions. An empirical orientation factor t = O-7 is necessary in the quasichemical relation of the alkanol hydrogen-bonds
\ Fl
-
\
Theoretical o v Experimental
\
[IO]
100 -
0
I
I
0.2
0.4 Mole fraction
I
I
I 0.6
0.6
0
I.0
I
I
I
0.4
06
0.6
Mole fraction
alcohol
Fig. 3. Heat of mixing data for n-alkanol-n-heptane system at 30°C.
I
I.0
alcohol
Fig. 4. Excess free energy and entropy of n-propanol-nheptane system at 20°C.
1009 CFS Vol. 28. No. 4-B
I 0.2
TSUNG-WEN
LEE, ROBERT A. GREENKORN
and KWANG-CHU
CHAO
number of repeating groups (with subscript and superscript) P pressure R universal gas constant coordination number : entropy T temperature t orientation factor u configurational energy V molar volume v cell volume W exchange energy total number of group per molecule configurational partition function ; coordination number with subscript z n
0
0.2
0.4
Mole froctlon,
0.6
0.6
I.0
n-Prcqmol
Fig. 5. Heat of mixing for n-propanol-n-alkanol systems at 30°C.
in order to achieve quantitative representation of the excess properties of alkanol-alkane solutions. Pure alkanol liquids are insensitive to the orientation factor. The solution behavior of the two lowest alcohols appears not to be well described by the group interaction model.
Greek symbols E interaction
energy (interaction energy between groups with subscript) 5 geometric constant r) characteristics of pairwise interaction constants, characteristics of orderliness 4 group fraction c chemical reaction energy y ratio of core volumes (with subscript) relative to the base group
Acknowledgement-This research was supported by NSF Grant GK-16573. R. S. Ramalho made available heat of mixing data on alkanol-alkanol systems before publication.
NOTATION
external degrees of freedom total potential energy Helmholtz free energy combinatorial factor of Flory-Huggin (combinatorial factor of Guggenheim without superscript) Gibbs free energy enthalpy Boltzmann constant mole fraction number of molecules (group contact pairs with subscript)
E”, g* F
G H k M N
Subscripts ij interaction property i group species
A B HB
between groups i andj
type of molecule base group hydrogen bond
Superscripts * characteristic
property reduced property type of molecule E excess property A4 mixing property A
REFERENCES DEAL C. H., DERR E. L. and PAPADOPOULOS M. N., Ind. ‘Engng Chem. Fundls 1962 117. DERR E. L. and DEAL C. H., Distillation 1969. pp. 37-47, Brighton, Sept. 1969. FLORY P. J., Principles ofPolymer Chemistry, p. 497. Cornell University Press, Ithaca, N.Y. 1953. GUGGENHEIM E. A., Proc. R. Sot. 1944 A183 226. KU0 C. M., ROBINSON Jr. R. L. and CI-IAO K. C., Ind. Engng Chem. Fundls 1970 9 564. LEE T. W., GREENKORN R. A. and CI-IAO K. C., Ind. Engng Chem. Fundls 1972 11293. LEE T. W., Ph.D. dissertation, Purdue University 1971. 1010
Statistical thermodynamics of group interaction [8] PIEROTT G. J., DEAL C. A. and DERR E. L., Ind. Engng Chem. 1959 5195. [9] PRIGOGINE L., MATHOT-SAROLEA and VAN HOVE L., Trans. Faraday Sot. 1952 48 485. [lo] RAMALHO R. S. and RUEL M., Can. J. Chem. Engng 1968 46 456. [l 11 RENON H. and PRAUSNITZ J. M., Chem. Engng Sci. 1967 22 299. [121 SAVINI C. G., WINTERHALTER D. R. and VAN NESS H. C.,J. Chem. Engnglhta 1965 10 168; 171. [131 VAN NESS H. C., SOCZEK C. A. PELGUIN G. L. and MACHDO R. L.,J. Chem. Engng Data 1967 12 217. 1141 WIEHE 1. A. and BAGLEY E. B., Ind. Engng Chem. Fundis 1967 6 209. 1151 WILSON G. M. and DEAL C. A., Znd. Engng Gem. Fundls 1962 120.
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