THERMODYNAMIC PROPERTIES OF SOLUTIONS
OF ALCOHOLS IN INERT SOLVENTS 1. A. W I E H E ’ A N D E. B. B A G L E Y Department of Chemical Engineering, School of Engineering and Applied Science, Washington University, St. Louis, Mo.
Hydrogen bonding in alcohol-inert solvent solutions can b e taken into account to give equations for the activity coeffiicients in closed form. The excess functions of Gibbs free energy, enthalpy, and entropy can b e readily calculated. This theory is used to calculate the isothermal vapor-liquid equilibrium composition curves and the heats of mixing of binary alcohol-aliphatic hydrocarbon solutions, based on a single experimental point for each system.
solutions in which specific interactions occur are of considerable engineering interest, most theories, such as the regular solution theory, fail when applied to these systems. Alcohol-inert solvent solutions, where the term “inert solvent” refers to liquids which do not form any type of complex with alcohol complexes or with molecules of their own kind, show the most serious deviations from ideal behavior. Dolezalek (1908), utilizing a quasichemical theory, was the first to account for deviations from ideal solution behavior by considering specific interactions. Since alcohols are capable of forming two hydrogrn bonds per molecule, many investigators have treated the hydrogen bonding of alcohols as the formation of large linear polymeric complexes (Coggeshall and Saier, 1951; Ginell, 1955; Kretschmer and Wiebe, 1954 ; Lassettre, 1937 ; Mecke 1948 ; Prigogine and Defay, 1962; Sarolea, 1953; Tobolsky and Thach, 1962). Once the complexes are correctly identified, the assumption is often made, following Dolezalek (1908), that the alcohol complexes form an ideal solution with an. inert solvent and with other alcohol complexes. This model is known as the “ideal associated solution” (Prigogine and Defay, 1962). A partial list of those using the ideal associated solution model includes Mecke (l948), Prigogiiie and Defay (1962), Lassettre (1937), Coggeshall and Saier (L951), Ginell (1955), and Tobolsky and Thach (1962). I t is a characteristic of most ideal associated solution models that several equilibrium constants for hydrogen bonding must be used to fit the experimental data. Tobolsky and Thach have one equilibrium constant for the monomerto-dimer reaction and ,another to represent all other hydrogen bonding reactions. Lassettre has even used eight different equilibrium constants. Sarolea (1953) has taken into account the contribution to the entropy of mixing of sizes of the complexes by using the equation of Guggenheim (11944). However, Sarolea requires the use of spectroscopic data. Kretschmer and Wiebe (1954) considered the effect of size of the complex by using the FloryHuggins equation. However, they did not develop their equations to the point where a reasonably direct calculation of thermodynamic functions was possible. An important immediate engineering application of a theory of alcohol-inert solvent solutions would be the estimation or correlation of vapor-liquid equilibrium data of alcoholaliphatic hydrocarbon solutions for distillation column design.
A
LTHOUGH
1 Present address, Department of Chemical Engineering, University of Rochester, Rochester, N. Y.
Since experimental vapor-liquid equilibrium data are not often readily available, it is desirable to be able to estimate such data from a minimum of experimental measurements. Further, even if extensive data are available, one often would like to put the equilibrium data in a convenient mathematical form for design calculations which use a computer. I n the past alcohol-aliphatic hydrocarbon vapor-liquid equilibrium data have been extremely difficult to correlate with any of the equations that fit the general Wohl (1946) form, such as the van Laar (1910, 1913), the Margules (1895), and the Scatchard-Hamer (1935) equations. Recently, however, Orye and Prausnitz (1965) have shown that the Wilson (1964) equation can be used to correlate the vaporliquid equilibrium data of a variety of solutions, including alcohol-aliphatic hydrocarbon solutions. However, the constants of the Wilson equation have rather obscure physical significance and depend both on the temperature and on the components. Alcohol-Inert Solvent Solution Model
Our physical picture is similar to the quasichemical theory theory of Dolezalek. Alcohols in the pure liquid state are considered to be a mixture of linear polymers with a wide distribution of molecular weights. A dynamic equilibrium exists as hydrogen bonds are broken and formed at a rapid rate. The addition of an inert solvent to the pure alcohol acts as a diluent, decreases the rate of formation of hydrogen bonds, and thus shifts the equilibrium to a lower alcohol number-average molecular weight. As more inert solvent is added, this process continues until, in the limit of an infinitely dilute solution, the alcohol is completely dissociated. Absorption of heat is required to break the hydrogen bonds, producing the large positive heat of mixing observed in alcohol-inert solvent solutions. For the purpose of mathematical development the following assumptions are made : The volume change upon mixing is zero. The alcohol complexes form a Flory-Huggins athermal solution with each other and with the inert solvent. This second assumption implies that there are no changes in the intermolecular forces between pairs of molecules when hydrogen bonds are broken or formed. A basis of one stoichiometric mole of solution will be used, so that nB
+ nA = 1 VOL. 6
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(1) 209
where
The chemical potentials may be calculated in the usual way, obtaining
= number of moles of inert solvent
nB
nA =
number of moles of alcohol monomeric formula units or the number of moles of alcohol neglecting the presence of any complexes, equal to the weight of the alcohol divided by monomer formula weight of the alcohol
and
Therefore XB
=
nB;
XA
(2)
= n.4
where xB = apparent mole fraction of inert solvent disregarding the existence of complexes x A = apparent mole fraction of alcohol disregarding the existence of complexes
for solvent and alcohol, respectively. N represents the total number of moles, taking into account the complexes and is equal to ( n B f nk)/(nB f nA]. Furthermore X A = knk. The activity coefficient of the solvent becomes
and, following Prigogine (1957), the activity coefficient of the alcohol is found to be
Chemical Potentials and Activity Coefficients
According to the Flory-Huggins athermal model, the Gibbs free energy of mixing is given by m
GM
= - T s M = RT [ne In $'e f 1
(nk
In
$'k)]
(3)
where $'B = volume fraction of inert solvent nk = number of moles of the alcohol complex with k-mer units, g. moles $k = volume fraction of the alcohol complex with k-mer units
Since any polymer length is permitted, summations will be from one to infinity. The volume fractions are given by VBnB
=
VBnB
f
Vknk
(4)
where 2 1 * = nl*/N*, the mole fraction of monomer molecules in pure liquid alcohol, and N* = the total moles of complexes in pure liquid alcohol per unit formula weight. Furthermore, Z is the mole fraction of alcohol monomer in the solution, equal to n l / N . Chemical Equilibria
The chemical equilibria between alcohol complexes can be represented by
At f A1
Af+l
At constant temperature and pressure, the condition for equilibrium is given by Pf
(5)
*
+
Pl
=
(14)
Pt+l
Substitution of Equation 11 into Equation 14 for the appropriate i values gives:
where
VB = molar volume of the pure inert solvent, cc./g. mole v k
= molar volume of the pure alcohol complex with k-mer
units, cc./g. mole If all alcohol complexes have the same density, equal to the density of the alcohol, dA, then
(15)
The left side of Equation 15 is the standard free energy of reaction and may be represented by: p(+i*
where = molecular weight of the alcohol complex with k-mer units, M A = molecular weight of the alcohol monomer unit and VA = molar volume of the pure alcohol, cc./g. mole. Flory (1944) has made a similar assumption with the more conventional polymers. If we define the symbol p as VB
P = -
VA
the volume fractions can now be expressed as
(7)
- pf* - Mi*
ILEC F U N D A M E N T A L S
hA
- RTsA
(16 )
where
hA = standard enthalpy of formation of a hydrogen bond, cal./g. mole sA = standard entropy of formation of a hydrogen bond divided by R We define an equilibrium constant, KA, dependent only on temperature, as
K A = exp ( P l *
+ PiT-
or from Equations 15 and 16
210
=
Pf+l*
f 1)
HM = b ( G ” / T ) / b ( l / T ) =
Rearranging Equation 18 and letting i = 1, we obtain
If i
= 2, we obtain:
z3
2 KANZlZp (PXB
+
-
XA)
3
(PXB
f
(20)
XA)’
where
I n general the mole fraction of any complex, Zi, can be represented by:
z4
=
7
(-3-J-l
XA
Mass Balance Equations
The balance on alcohol monomeric units is given as
iv
izt =
(22)
XA
The over-all mass balance can be written as
ZB
+c
zt =
1
(23)
where ZB = mole fraction of inert solvent recognizing the existence of alcohol complexes = XB/N. If Equation 21 is substituted for Z i in Equations 22 and 23, the series are convergent and can be put in closed form, provided
KANZ~ PXB
+
In (1
+ KA)]
(30)
The quantity H E M represents the contribution owing to the changing number of hydrogen bonds. I t may be alternatively evaluated by multiplying the difference in the number of hydrogen bonds between the pure and solution states by the standard enthalpy of formation of a hydrogen bond. Comparison with Wilson Equations
Algebraic manipulation of Equations 24 and 25 for the activity coefficients puts them in a form similar to that of the Wilson equations (1964) : PRESENT THEORY
< 1
XA
The resulting equations can be solved for Z4,N , Z1*, and N*.
1
Final Equations
By substitution of these quantities into the equations for the activity coefficients, we obtain In
YB
= In
WILSONEQUATION
[- si. + e XA
PXB
XA:]
PXB
XA
PRESENT THEORY
1 - XB
1
P
The intercepts of the activity coefficient-mole fraction curves are given by c
1 4-K A [ln(l; lim y A = -- exp x*+o
KA) -
P
‘1 P
XB
+
f - In KA
1 - 2.4
xA)
(33)
P
+ - + CXA CXB
XA)
XB
The excess Gibbs free energy of mixing is
+
WILSON EQUATION In Y A = -In (DxB
(27)
(PXB
’
DXB DxB
+
XA
(34)
where C and D are constants depending on the temperature and the components of the solution. The similarity of the first two terms of each of the equations is even more pronounced, because for alcohols with extensive hydrogen bonding we would expect 1 / K A and PIKA to be small as compared with unity. Thus the term p / ( l K A ) is similar to D and 1 / p is similar to C. However, there is no direct relation between the constants of the two models because of the difference in the third terms.
+
Heat of Mixing
Comparison of Theory with Experimental Data
The heat of mixing is mainly due to the heat of formation of hydrogen bonds and is, represented by the equations:
According to the theory, no adjustable parameters should be necessary to calculate the vapor-liquid equilibria and the VOL. 6
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MAY 1967
211
heat of mixing of alcohol-inert solvent solutions. One constant, p, should be the ratio of the molar volumes of the components, and the other constant, K A , should be calculable using Equation 18 from the enthalpy and entropy of hydrogen bonding. Unfortunately, there are no precise methods for determining the entropy or the enthalpy of hydrogen bonding (Pimentel and McClellan, 1960). Therefore, we originally attempted to determine K A by the value that would best fit the experimental data with p remaining as the molar volume ratio, However, even an infinite value of KA would not account for the large deviations from ideality of alcoholaliphatic hydrocarbon solutions as long as p remained the molar volume ratio. Thus it became necessary to consider p as an adjustable parameter.
the change of vapor pressure of the pure components with temperature, one can determine the constants of the present theory in a method similar to the isothermal case. Calculation of Heat of Mixing
The heat of mixing can be calculated as a function of mole fraction of alcohol by means of Equation 30 if constants K A and p have been previously evaluated from vapor-liquid equilibrium data of the same system. As is mentioned above, there is an additional group of terms that should be added to Equation 30 for the contribution to the heat of mixing by the change of p with temperature. However, in all cases that we have investigated, the magnitude of these terms is small compared to the terms in Equation 30.
Determination of Constants Results
Isothermal Data. With the approximation of ideal gas vapors, the values of K A and p were selected for each set of isothermal vapor-liquid equilibrium data by a trial and error procedure to give the least square deviation from the experimental data. The 60' C. range of the experimental data was not large enough to determine accurately both h A and S A from the temperature variation of KA. Therefore the value of hA = -5.9 kcal./g. mole was selected as the mean value given by Coulson (1952) as the energy of the O H . . . .O hydrogen bond. With this value of hA and the value of K A , sA values were calculated by means of Equation 18. The values of sA for ethanol solutions varied in a random fashion by less than 10% of their aver age. The arithmetic average of the S A values for ethanol was used to calculate new values of KA. New values of p were selected to give the least square deviation from the experimental data. This adjustment to a common S A value actually slightly decreased the average deviation from the experimental data. This is probably due to the inherent inaccuracies in a trial and error fitting procedure. Isobaric Data. Since the temperature variation of p is not known, one would normally not expect the present theory to be able to correlate isobaric vapor-liquor equilibrium data. However, for solutions of the lower carbon alcohols and aliphatic hydrocarbons, the equilibrium temperature does not vary greatly at constant pressure except for the very dilute and the very concentrated regions. Thus by using an equation for
Table 1.
Systems Investigated. The systems investigated are: methanol-n-hexane a t 45 ' C. ; n-butyl alcohol-n-heptane a t 50' C.; ethanol-cyclohexane a t 5', 20°, 35', 50°, and 65' C.; ethanol-methylcyclohexane at 35' and 55 ' C.; ethanolisooctane a t 25' and 50' C.; and ethanol-n-hexane at 760 mm. of Hg. Values of Constants. I n Table I are the values of sA hA, KA, and p which were determined or assumed for each of the solutions, based on isothermal vapor-liquid equilibrium data. The averages of the absolute values of the differences between the experimental and calculated values of the mole fractions of alcohol in the vapor are also given. I n Table I1 similar data are presented for the ethanol-n-hexane solution at a constant pressure of 760 mm. of Hg. Vapor-Liquid Equilibrium Data a n d Theory. In Figures 1, 2, and 3 we compare the isothermal experimental data with the theory in three different ways for three different systems: In Figure 1 experimental and calculated alcohol mole fractions in the vapor are compared for ethanol-cyclohexane solutions at 5', 35', and 65' C.; in Figure 2 experimental and calculated activity coefficients are compared for ethanol-isooctane solutions at 50' C.; and in Figure 3 experimental and calculated excess Gibbs free energies of mixing are presented for ethanolmethylcyclohexane solutions at 35' C. I n Figure 4 are shown the isobaric experimental and calculated alcohol mole fractions in the vapor for ethanol and n-hexane at 760 mm. of mercury.
Values of Constants for Isothermal Vapor-Liquid Equilibrium Data Gal./ G.-Mole
Ay. Dev.
hA,
System
Ethanol-methylcyclohexane
Ethanol-isooctane Ethanol-cyclohexane
Methanol-n-hexane n-Butyl alcohol-n-heptane
System
Ethanol-n-hexane
212
T,
a
C.
35.0 55 025.0 50.0 5.0 20.0 35.0 50.0 65.0 45.0 50.0
- 5900
-5900
-5900 -5900 -5900 -5900 -5900 -5900 -5900 -5900 -5900
SA
KA
P
in Y A
Source of Data
-4.94 -4.94 -4.94 -4.94 -4.94 -4.94 -4.94 -4.94 -4.94 -2.93 -6.18
297.4 165.4 411.1 190.2 487.2 487.2 297.5 190.2 126.6 1I640 55.1
3.99 3.88 5.67 4.61 4.59 4.19 3.87 3.66 3.49 25.59 2.91
0.0036 0.0030 0.0060 0.0033 0.0018 0.0021 0.0019 0.0018 0.0029 0.0088 0.0063
Kretschrner and Webe, 1949 Kretschmer and U'iebe, 1949 Kretschmer et al., 1948 Kretschmer et al., 1948 Scatchard and Satkiewicz, 1964 Scatchard and Satkiewicz, 1964 Scatchard and Satkiewicz, 1964 Scatchard and Satkiewicz, 1964 Scatchard and Satkiewicz, 1964 Ferguson, 1932 Smyth and Engel, 1929
Table II. Values of Constants for Isobaric Vapor-liquid Equilibrium Data hA, Au,. Dev. P1 M m . of Hg Cal./G.-Mole SA P yA 760.0
l&EC FUNDAMENTALS
-5900
-4.94
3.74
0.009
Source of Data
Sinor and Weber, 1960
Heat of Mixing Data and Theory. In Figures 5, 6, and 7, the experimental and calculated heats of mixing are compared for the systems: isooctane-ethanol a t 25' C. ; methylcyclohexane-ethanol a t 35' C.; and n-hexane-methanol a t 45" C. Discussion
Van Thiel, Becker, and Pimentel (1757) and Liddel and Becker (1757, 1758), :have presented indirect evidence that I."
00-
0.7-
0.8
A 3.C. e 35.c. 8 65'C*
-
0.6
-
A' 0.5
o'lL00
0.1
0.2 0.3 0.4 0.5 0.6
0.7 0.8
0.9
1.0
*A
.
Figure 1 Vapor-liiquid compositions of ethanol-cyclohexane system at three temperatures Points taken from experimental data of Scatchard and Satkiewicz (1 964). Curves calculoted using S A = - 4 . 9 4 and h A = - 5 9 0 0 cal./g. mole. Empirical values of p required to fit data: 4.59, 3.87, and 3.49 at 5 O , 35', and 6 5 ' C., respectively
alcohols form cyclic structures. Originally we included an equilibrium constant in our theory to take into account the presence of such structures. However, the best fit of the experimental data was obtained with this equilibrium constant equal to or close to zero. Thus we have concluded that there are few or no cyclic alcohol structures present a t the temperatures which we have investigated. Tables I and I1 show that by using the same value of hA for all alcohols and the same value of SA for any particular alcohol, it is possible to calculate the vapor-liquid equilibria in the range of accuracy of the experimental data for four different ethanol solutions at temperatures from 5' to 65' C. and for one methanol and one butanol solution a t one temperature each. Thus sA can be established for a given alcohol from vaporliquid equilibrium data of that alcohol and an aliphatic hydrocarbon at one temperature. Then K A can be calculated at any temperature and any aliphatic hydrocarbon solution of the alcohol, leaving only p to be determined. Thereafter in order to calculate the entire isothermal vapor-liquid equilibrium curve of an aliphatic hydrocarbon solution of that alcohol, only one experimental point is required. The value of sA is a measure of the steric hindrance of the hydrocarbon portion of the alcohol molecule toward the formation of a hydrogen bond. As the hydrocarbon portion of the alcohol lengthens, sA becomes more negative and K A becomes smaller. The physical significance of the observed numerical value of p is not understood. By allowing p to become an adjustable parameter rather than the theoretically constant molar volume ratio, we have modified both of the theoretical assumptionsnamely, no volume of mixing and the Flory-Huggins athermal solution. Instead of the entropy of mixing being expressed in terms of volume fractions, they have been replaced by empirical fractions. Somehow, using the empirical fractions seems to adjust for all assumptions including the Flory-Huggins lattice model. However, Table I indicates that for all solutions D
I
01
0
Figure 2. at 50" C.
0.1
0.2
0.3
0.4
0.5 XA
0.6
0.7
0.8
0.9
1.0
Activity coefficients vs. composition for ethanol-isooctane system
Points determined from experimental data of Kretschmer ef 01. (1 948). = -4.94, ha = - 5 9 0 0 Cal./g. mole, and p 4.61
Curves calculoted using
SA
VOL. 6
NO. 2
MAY 1967
213
OY 0
I
I
I
0.1
0.2
0.3
I
0.4
I
f5 A
t
0.6
I
0.7
I
I
0.8
0.9
3
Figure 3. Gibbs free energy of mixing vs. composition for ethanol-methylcyclohexane system at 35 O C. Points determined from experimental data of Kretschmer ef al. (1 949). using S A = -4.94, h~ = 5 9 0 0 cal./g. mole, and p = 3.99
-
0 0
,
0.1
I
0.2
I
I
I
I
0.3 0.4 0.5
0.6
0.7 0.8 0.9
XA
Figure 4. Vapor-liquid compositions of hexane system at 760 mm. of Hg
ethanol-n-
Points taken from experimental data of Sinor and Weber (1960). Curve colculoted using sa = -4.94, ha = - 5 9 0 0 cal./g. mole, and p = 3.74
decreases with increasing temperature, decreases with increasing carbon number of the alcohol, and varies with the solvent. Since the temperature variation of p is not well defined, one would not expect to be able to correlate isobaric vapor-liquid equilibrium data as easily as isothermal data. However, as is shown in Table I1 and Figure 4 for ethanol-n-hexane at 760 214
I&EC FUNDAMENTALS
Curve calculated
mm. of Hg, an average value of p still can fit the isobaric data fairly well. If we compare our equations with only that portion of the data between liquid mole fractions of 0.045 and 0.980, we obtain an average deviation in mole fraction of alcohol in the vapor of 0.005 instead of 0.009 obtained with the entire data. Orye and Prausnitz (1965) used only the data between liquid mole fractions of 0.045 to 0.980 to obtain an average deviation of 0.006 with the Wilson equation (1964). The Wilson equation is the only two-constant equation of which we are aware that correlates binary vapor-liquid equilibrium data of alcohol-aliphatic hydrocarbon solutions as well as the present model. The similarity of our equations to the Wilson equations has been pointed out. The advantage of the present model is that one of the constants, K A , is independent of the aliphatic hydrocarbon solvent, and its variation with temperature is known once S A is determined for the particular alcohol. Actually, since KA is a property of the alcohol, the present model contains only one solution constant, p . O n the other hand, the Wilson equation has the distinct advantage of being able to correlate vapor-liquid equilibrium data of a great variety of systems while the present model is limited to alcohol-inert solvent solutions. Aliphatic hydrocarbons are the only examples of inert solvents that we have studied to date. Although many investigators, such as Pimentel and McClellan (1960, p. 39), Prigogine and Defay (1962, p. 415), Mecke (1948), and Sarolea (1953), have indicated that carbon tetrachloride is an inert solvent, the negative heats of mixing measured by Otterstedt and Missen (1962) for alcohol-carbon tetrachloride solutions indicates that alcohols and carbon tetrachloride must form complexes. Pimental and McClellan (1960, p. 202) have discussed the evidence that aromatic hydrocarbons form hydrogen bonds. The fact that methanol is immiscible with most aliphatic hydrocarbons but is miscible with benzene and carbon
30 e
*ool
e
e
e
1
0
0-1
0.2
0.3
0.4
0.5
0.6
0.r
ac
0.u
1.0
XA
Figure 5.
Heat of mixing vs. composition for ethanol-isooctane system at
25” C. Points taken from experimental d a t a of Brown and Fock (1955). Curve calculated using SA = -4.94, h A = - 5 9 0 0 cal./g. mole, and p = 5.67. p determined from vapor-liquid equilibrium data a t same temperature
200
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.e
0.9
10
*A
Figure 6. Heat of mixing vs. composition for ethanol-methylcyclohexane system at 35” C. Paints taken from experimental data of Brown et al. ( 1 956). Curve calculated using SA = -4.94, and h A = - 5 9 0 0 cal./g. mole. p = 3.99 determined from vapor-liquid equilibrium data a t some temperature
tetrachloride a t room temperature offers more evidence that aromatic hydrocarbons and carbon tetrachloride are not inert solvents. Hildebrand and Scott (1964) have written: “ I t has been a common observation that the free energy of mixing calculated from various statistical mechanical models (such as the “quasichemical” theory) is almost invariably in better agreement with the experimental results than is the heat or entropy compared separately.”’ By investigating the heat of mixing, we have found it particularly informative in determining how closely our model approximates the real system, because the present theory permits the calculation of the heat of mixing from vapor-liquid equilibrium data a t one temperature. Ordi-
narily vapor-liquid equilibrium data a t several temperatures are required to do this. In Figures 5, 6, and 7 the experimental and calculated heats of mixing are compared for the systems: isooctane-ethanol a t 25’ C., methylcyclohexane-ethanol at 35’ C., and n-hexanemethanol at 45’ C. The deviation between experiment and theory is greater in the isooctane-ethanol solution than in either the methylcyclohexane-ethanol or the n-hexane-methanol solutions (31% average deviation as compared with 20 and 22%). Although these estimates are far from being exact, the agreement between experimental and calculated values is encouraging, considering that the van der Waals forces were neglected and that the value used for hA is not well established. VOL. 6
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MAY 1967
215
200
x I
too
0
0
Figure
45"
c.
I
I
I
,
I
01
0.2
0.3
0.4
0.5
I
0.8
0.7
I
I
0.a
0.0
1.0
7. Heat of mixing vs. composition for methanol-n-hexane system a t
Points smoothed experimental data of Savini e l al. (1965). Curve calculated with sA = -2.93 = -5900 cal./g. mole, and p = 25.59. p determined from vapor-liquid equilibrium data at same temperature
hA
molecular weight of alcohol complex with k-mer units, g./g. mole total number of moles taking into account identity of complexes, g. moles number of moles of alcohol mer units or number of moles of alcohol disregarding any complexes being present, g. moles number of moles of inert solvent, g. moles number of moles of alcohol complex with k-mer units, g. moles gas constant, cal./g. mole O K . entropy of mixing, cal./g. mole " K. standard entropy of hydrogen bonding divided by R, dimensionless absolute temperature, " K. molar volume of pure alcohol, cc./g. mole molar volume of inert solvent, cc./g. mole molar volume of alcohol complex with k-mer units, cc./g. mole mole fraction of alcohol disregarding existence of any complexes mole fraction of inert solvent disregarding existence of any complexes mole fraction of inert solvent recognizing existence of alcohol complexes = nB/N mole fraction of alcohol complex with i-mer units = ni/N
In all cases the calculated maximum heat of mixing is at a lower alcohol concentration than the experimental maximum. Since the van der Waals heat of mixing is usually symmetric, we would expect the addition of this portion of the heat of mixing to shift the calculated maximum closer to the experimental maximum. Both the present model and the Wilson model fail to predict liquid-liquid phase separation. The addition of a heat of mixing term to the Flory-Huggins model to take into account the potential type forces between molecules and complexes will be required before the limitation to miscible solutions can be removed. However, the effect of the van der Waals forces either is small compared to the effect of the hydrogen bonding forces or an adjustable p compensates for it, because the present theory seems to represent the thermodynamic properties of alcohol-inert solvent solutions throughout the miscible region. This is shown by our ability to represent the thermodynamic properties of the methanol-n-hexane solution at 45" C., only 12" C. above its upper critical solution temperature (Kiser et al., 1961). Ac knowledgrnenl
The authors are grateful to the Washington University computing facilities, supported through National Science Foundation Grant G-22296, for providing the required computer time.
yA yB
= activity coefficient of alcohol = activity coefficient of solvent
pA pB pi
= chemical potential of inert solvent, cal./g. mole = Chemical potential of alcohol complex with i-mer
= alcohol complex with i-mer units
P
= VB/VA
constant in Wilson equation constant in Wilson equation density of pure liquid alcohol, g./cc. excess Gibbs free energy of mixing, cal./g. mole Gibbs free energy of mixing, cal./g. mole heat of mixing, cal./g. mole heat of mixing due to heat of reaction, cal./g. mole standard enthalpy of hydrogen bonding, cal./g. mole number of mer units in an alcohol complex hydrogen bonding equilibrium constant = number of mer units in an alcohol complex = molecular weight of alcohol mer unit or of alcohol disregarding existence of complexes, g./g. mole
+B
=
Gk
= volume fraction of alcohol complex with k-mer units
Nomenclature = = = = = = = = = =
216
GREEKLETTERS
l&EC FUNDAMENTALS
=
chemical potential of alcohol, cal./g. mole units, cal./g. mole volume fraction of inert solvent
SUBSCRIPTS A = alcohol B = inert solvent = number of mer units in a particular alcohol complex i = number of mer units in any alcohol complex k SUPERSCRIPTS E = excess function M = mixing function * = pure liquid state
References
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MEASUFREMENT OF DIFFUSION COEFFICIENTS FOR CONCENTRATED BINARY POLYMER
SO LUT IO NS D . R. P A U L Chemstrand Research Center, Znc., Durham, N.C.
A simple interferometer technique has been used to determine the concentration profiles in an infinite field of polymer-solvent diffusion for various diffusion times. These profiles were combined by the Boltzmann transformation into a master profile which was analyzed by a method of Duda and Vrentas to give the binary diffusiion coefficient as a function of polymer concentration. Techniques which improve accuracy are discussed. The systems used were an acrylonitrile-vinyl acetate copolymer in the solvents dimethylacetamide, dimethylformamide, and dimethyl sulfoxide. Concentrations ranged from pure solvent to 25% polymer by weight. The concentration dependence of the diffusion coefficient is small for each solvent.
VERY few experimen.ta1 data exist in the literature on binary diffusion of polymer-solvent pairs in concentrated solutions. Such information would be very useful when considering many important polymer processing steps such as mixing, dissolution, and fiber formation by :solution spinning. Over a period of several years a simple optical interference method has been reported in the literature by Ambrose (1948), Berg (1938), Crank and Robinson (1950), Robinson (1950), and Searle (1946). This technique is ideally suited for measuring the concentration profile in a polymer-solvent system in a free diffusion situation and was used by Nishijima and Oster (1956 and 1961) and Secor (1965) for this purpose. If one obtains a concentration profile by a method such as this, then in principle the binary diffusion coefficient can be calculated for all concentrations represented, as Jost (1960) has discussed. However, this analysis does not include the convective transport that occurs as a result of the volume change on mixing. Recently Duda and Vrentas (1965 and 1966) have presented a mathematical development that properly considers this effect.
I n this work the simple interferometer mentioned earlier has been used to generate experimental concentration profiles in an infinite diffusion field of polymer and solvent. Profiles observed at various times are combined into a master plot using the Boltzmann transformation. This profile is then analyzed to give the binary diffusion coefficient by use of a simplification of the general equations given by Duda and Vrentas (1965) which is valid for the frequently encountered case where the solution density is linear in the weight fraction of the solute. Techniques for data analysis are used which improve the accuracy of the results. The single polymer, employed in these studies, was a vinyl acetate-acrylonitrile copolymer containing 7.770 vinyl acetate by weight with a weight average molecular weight of about 200,000. The diffusion coefficients were measured in three solvents-dimethylacetamide (DMAc), dimethylformamide (DMF), and dimethyl sulfoxide (DMSO). I n each case the composition ranged from pure solvent to a polymer weight fraction of either 0.20 or 0.25. VOL. 6
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