Nomenclature
B
cell factor total molar concentration = molar concentration of species i = initial concentration difference between two cell chambers = final concentration difference between two cell chambers = main diffusion coefficient in a multicomponent mixture = binary mutual diffusion coefficient = friction coefficient = activation energy for diffusion process = activation energy for viscous process = Planck constant = Boltzmann constant = Avogadro's number = gas constant = absolute temperature = time = mole fraction of species i = =
C C,
ACI ACF
Dll D,,
Fi, AGl,
AG,
h
k N R T
t
Xi
GREEKLETTERS =
AD
characteristic distance for diffusion process
= characteristic distance for viscous process
A,
x x,
characteristic distance applicable t o diffusion or viscous process = characteristic distance for pure species i = solution viscosity = viscosity of species i =
17 7%
SUBSCRIPTS i,j = component i , j 1, 2, 3 = component 1, 2, 3 SUPERSCRIPTS 0 = infinite dilution of indicated species literature Cited
Barnes, C., Physics 5 , 4 (1934). Bearman, R. J., J . Phys. Chem. 65, 1961 (1961). Bidlack, D. L., Anderson, D. K., J . Phys. Chem. 68, 3790 (1964). Burchard, J. K., Toor, H. L., J . Phys. Chern. 66, 2015 (1962). Cullinan, H. T., IND. ENG.CHEM.FUNDAMENTALS 5, 281 (1966). Cullinan, H. T., Cusick, AI. R., A.I.Ch.E. J . 13, 1171 (1967). Cullinan, H. T., Toor, H. L., J . Phys. Chem. 69, 3941 (1965). Fischler, J., 2.Elektrochem. 19, 126 (1913). Grunberg, L., Trans. Faraday SOC.50, 1293 (1954). Holmes, J. T., Rev. Sei. Instr. 36, 831 (1965). Holmes, J. T., Orlander, D. R., Wilke, C. R., -4.I.Ch.E. J . 8 , 646 (1962). Leffler. J.. Cullinan. H. T.. IXD.ENG.CHEM.FUSDAVENTALS 9, 84 (1970). Stokes, R. H., J . Am. Chem. SOC.72, 763 (1950). Trevoy, D. J., Drickamer, H. G., J . Chern. Phys. 17, 1117 (1949). 5 , 189 (1966). Vignes, A,, ISD.ENG.CHEM.FUSD.UIZNT.LLS Yajnik, N. A., Bhalla, 31. O., Talwar, R. C., Soofi, A., Z. Phys. Chem. A118, 305 (1925). RTXEIVED for review April 1, 1969 ACCEPTEDOctober 2, 1969 Work supported by the Kational Science Foundation under NSF GK-1747. The first author (J.L.) was the recipient of a NASA Traineeship. ~
Internal Pressure Measurements and Liquid-State Energies E. B. Bagley, T. P. Nelson,' J. W. Barlow, and S-A. Chen Department of Chemical Engineering, Washington University, St. Louis, MO.63130
Internal pressure, P I = (bE/bV),was measured on cyclohexane over a 70°C temperature range. These data are compared with energies of vaporization, AEV. The ratio, (PtV/AEV),is often interpreted as an w exponent, n, in the energy-volume relation E = -a/Vn. Present data, however, show that ( P z v - A€') is equal to 3RT/2, within experimental error with n equal to unity and a a function of T. This difference can b e attributed to the potential energy associated with the external vibrational modes of the cyclohexane molecules in the liquid state. It appears that the total liquid-state energy can b e given as the sum of the various energetic contributions, allowing separability of dispersion, polar, and even hydrogen-bonding energies in agreement with treatments given by Blanks and Prausnitz and by Hansen.
WHATEVER may be the theoretical limitations of regular solution theory, the solubility parameter has proved in practice to be most useful in considering compatibility problems. I n fact, the concept has been very effectively extended not only to systems involving London dispersion (nonpolar) forces but to systems in which specific interactions can occur, as when polar and hydrogen-bonding forces are present. This in effect was done by Burrell (1955) and more quantitatively in recent years by Blanks and Prausnitz (1964) and Hansen 1
On academic leave from Llonsanto Co., St. Louis, Mo.
(1967), among others. Blanks and Prausnitz calculated a nonpolar (or dispersion force) solubility parameter for polar molecules by assuming that the experimental energy of vaporization, A E V , was the sum of a nonpolar energy, A E n p V , and a polar contribution, AEpV,so that the uwally reported r 2 , where A 2 = solubility parameter, P , was given by X 2 ( A E n p V / P ) and is calculated from the energy of vaporization of a nonpolar homomorph of the polar molecule being considered. Hansen also employed this homomorph concept in extending this approach to systems showing both polar and hydrogen-bonding effects. The success of this approach in
+
VOL. 9 NO. 1 FEBRUARY 1970
IlLEC FUNDAMENTALS
93
Table 1. Experimental Pressure and Temperature Data and least Squares Fit to a linear Relation for Cyclohexane Temp. Observed,
Pressure, PSI
OC
Obsd.
Calcd.
Diff.
24.5500 24.7744 25.0682 25,3795 25.6645 26.0023 26.3088 26.5681 26.8664
78.0380 113.7242 160.2546 210.1696 255.8130 308.7887 357.5777 399.3061 445.2127
78.2331 113.8659 160.5189 209.9507 255,2063 308.8460 357.5157 398.6903 446.0578
-0.1951 -0.1417 -0.2643 0.2189 0,6067 -0.0573 0.0620 0.6158 -0.8451
%
Diff.
-0.249 -0,124 -0.165 0.104 0.238 -0.019 0.017 0.154 -0.189
= 10.8051 atm/"C.
dealing with phase separation problems is well illustrated in Hansen's monograph (1967). The use of homomorph data is unsatisfying, however, and Wiehe and Bagley (1967) proposed that the product of internal pressure ( P J and molar volume of a solvent (8)was a measure of the London dispersion energy per mole of liquid, independent of the presence of strong specific interactions. This proposal, if correct, implies an experimental method by which the nonpolar solubility parameter can be determined without resorting to the questionable and awkward use of homomorphs. This method has been applied by Chen (1969) in conjunction with a theory of hydrogen-bonding effects, to obtain what appear to be reasonable estimates of nonpolar, polar, and hydrogen-bonding energies in alcohols. I n practice, however, even in systems in which no polar interactions exist, and for which P , 8 should be equal to AEv, P , p IS greater than AEJ'. This means that a nonpolar solubility parameter calculated from P , 8 differs appreciably from that tabulated by Blanks and Prausnitz or by Hansen from homomorph considerations.
-PRESSURE
SAMPLE
CELL
TUNGSTEN WIRE
MERCURY
I n carrying out studies on the internal pressure of alcohols and binary alcohol solutions we have had occasion to determine internal pressure data very precisely over a wide temperature range on solvents in which only nonpolar liquid energy effects are believed to be significant. The results indicate the reason for the discrepancy b e h e e n P , 8 and AEv. Experimental Procedure
The basic experimental procedure is that due to Westaater et al. (1928). The method has been used in recent years by
several other workers in studying small molecules (Bianchi et al., 1965), intermediate size molecules such as plasticizers
(Bagley and Wood, 1966), and polymer molecules (Allen et al., 1960a).
Equipment. The equipment used to obtain internal pressure measurements consists of a constant volume bomb, a thermometer, a pressure gage, and a constant temperature bath. The bomb and thermometer are shown in Figure 1. The constant volume bomb is a modified long-weldingneck flange made of carbon steel with a 6Oo-psi rating and a 1-inch nominal i.d. The liquid sample is placed in a borosilicate glass cell which is then put into the bomb. Mercury is used to fill the bomb around the glass cell and the experiment is arranged so that the mercury rises up the capillary neck of the glass cell, forming a mercury-solvent interface. An electrical circuit inaiiitains constant volume in the glass cell. As the mercury rises up the capillary neck of the glass cell, it makes contact with a sharpened tungsten wire, completing an electrical circuit and denoting a n ostensibly constant volume. Corrections to true constant volume must be made for thermal expansion and compressibility effects as noted below. The temperature of the bomb is measured by a HewlettPackard quartz thermometer, Model 2801A, with a Model 2850B probe. The probe screws into the bottom of the bomb, directly measuring the mercury temperature. This thermometer gives a visual display of the temperature with a maximum resolution of O.OO0loC. The large inass of the bomb and contents minimizes temperature fluctuations. The pressure is measured by a Texas Instrument Xodel145 Precision pressure gage, with a Type 1 capsule. The capsule is a quartz Bourdon tube with a range of 0 to 500 psi. The pressure gage gives a visual &play of the pressure measured with a resolution of 0.0001 psi. Calculation Procedure. A typical set of pressure us. temperature measurements is shown in Table I. The temperature span of t h e d a t a is small (2.31°C) and the d a t a can be fitted adequately by a straight line. A least-squares fitting procedure is employed in double precision on an IBM 7072 computer to obtain the slope of the line, (dP/bT)v. After the straight line is fitted to the data, the original data points are compared to the line, and any points which deviate by lY0 or more are discarded. The fitting procedure and comparison are repeated until no data deviate from the straight line by lY0 or more. All the data points in Table I were within 1% of the straight line and did not have to be discarded. I n fact, the largest deviation was only 0.249%, while the average deviation was 0.147,, and in the initial data only one point in several hundred will deviate from the first straight-line fit by more than 1%. Before the internal pressure can be calculated, the slope of the line ( B P / ~ T ) has v to be corrected for nonconstant volume of the glass cell. The correction is
TEMPERATURE PROBE
Figure 1. Schematic diagram of bomb, glass cell, and thermometer assembly used in determination of internal pressure of liquids 94
l&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
where ( b P / d T ) v o and (dP/bT)vC are the observed and corrected slopes, respectively, olg is the thermal expansion and pi are the conipressibilkies of coefficient of glass, and glass and the liquid sample, respectively (Restwater et al., 1928) I
Table II. Comparison of PiV and At?
r,
OK
298.87 309.71 319.84 329.30 338.87 349.23 358.84 369.05
for Cyclohexane n = - Pi?
A€",
V, Cc/Mole
Pi Col/Cc
PiG, Cal/Mole
Cal/Mole
108.78 110.24 111.66 113.05 114.51 116.15 117.73 119.48
76.292 73.974 71.940 69.867 67.658 65,684 63.079 60.521
8299.0 8154.9 8032,8 7898.5 7747.5 7629.3 7426.3 7231.1
7290 7130 6980 6850 6710 6550 6410 6270
A€"
AEV
1.14 1.14 1.15 1.15 1.15 1.16 1.16 1.15
+
Difference
3 / 2 RT
(4-7)
8180 8050 7930 7830 7720 7590 7480 7370
120 100 100 70 30 40 - 60 - 140
m,
piv2 = Cal X CclMole2 10-6
x
9.0277 8.9900 8.9694 8.9292 8.8717 8.8614 8.7429 8.6397
Table 111. Internal Pressure and Energies of Vaporization for +Heptane (Westwater ef ab, 1928) AEv
f, Cc/Mole
r, oc 15 20 25 35
145 146 147 149
P,, CaI/Cc
62 61 60 59
68 59 50 37
29 45 69 15
+ 312 RT,
Col/Mole
P t f , Cal/Mole
Diff.
9120 9080 9030 8920
9074 9008 8952 8835
- 50 -70 - 80 -80
Once the corrected slope is obtained, the internal pressure of t h e sample can be calculated from the equation
pi =
(g$)T
=
T
(E)
-p
where 5" is the average temperature and P is the average pressure of the data.
Pi
= n (c.e.d.)
pi
= n ( 7 )
(5)
E T L = E,, i- Etnt -I- E P E-I- E K E
Table I1 gives the values of internal pressure, P,, in calories per cubic centimeter obtained over the temperature range 25" to 95°C for the reagent grade cyclohexane used in these experiments. Values of molar volume, P,obtained from the International Critical Tables and energies of vaporization obtained from enthalpies of vaporization (API Project 44) by AEV = AHV
- RT
(2)
are also tabulated here. The values we used were interpolated and rounded off to three significant figures. No corrections were made for deviations from the ideal gas behavior of the cyclohexane vapor. Values of n, the ratio of internal pressure to cohesive energy density, are also given in Table 11. The data clearly are compatible with the 20" values reported by Allen, Gee, and Wilson (1960b) for cyclohexane, their value for n being 1.14 a t 20°C. Various workers have obtained P, values for cyclohexane and these values are summarized by Bianchi et al. (1965). Our extrapolated result a t 2OoC falls in the middle of the reported range. I n the past n has been interpreted as a n empirical exponent in an equation of state for the liquid for which the internal energy is given as (Hildebrand and Scott, 1950) =
-a/pn
=
;(k)
(7)
Here E,, represents the equilibrium energy of a molecule in a nonpolar liquid due to the presence of all the other molecules in the liquid. This energy is the result of the London forces between molecules. E,,,t represents the energy due to internal molecular modes of motion. E P Eand E K Erepresent the potential and kinetic energies of molecules vibrating in the liquid about their equilibrium position associated with E f l p . E P E and E K Emay each be equal to 3(RT/2) (Hildebrand and Scott, 1962, page 167). Assuming that the gas obeys the ideal gas law and the internal modes are the same in the liquid and gas, the total energy of the gas in equilibrium with the liquid will be
ETG= 3(RT/2)
+
EInt
(8)
The over-all rotational modes of the molecule are assumed the same in the liquid and the gas state and are included in E,,t. In some systems there may be restricted molecular rotation in the liquid different from that in the gas, but this does not appear to be the case in cyclohexane. This leads to AE" AEV =
(3)
- ET& -Enp - 3(RT/2) = ETc
(9) (10)
Now from Table I1 we see empirically that for cyclohexane AEv (3RT/2) agrees with P,Pwithin 100 cal per mole. The same holds true for n-heptane (Table 111). Thus, within experimental error we can set P , P = -Eflp and the proposal of Wiehe and Bagley (1967) Seems to be confirmd Further, since P,by definition is (bETI,/bV)r aiid siiice a t relatively low pressures ELntis probably a function of T alone,
+
The internal pressure i, then
p, =
1 3219 X 10: 1 3205 1 3204 1 3199
Another way to look a t the problem is to assume that the total energy of the liquid, E T &is ,
Results and Discussion
E
P 2 G 2 , Col Cc/Mole2
(4)
If the cohesive energy density (c.e.d.) is taken as ( A E v / P ) , then since only London forces are acting,
VOL. 9 NO. 1 FEBRUARY 1970
ILEC FUNDAMENTALS
95
Table IV. Internal Pressure and Energies
of Vaporization for Nonpolar Systems (Allen et al., 1960b) P,?
- 3/2 RT,
AEV,
Material
v, Cc/Mole
Pi, Cal/ Cc
Cal/Mole
Cal/Mole
Pentane Heptane Decane Tetradecane 2-Methylhexane Zllrlethylheptane &&Dimethylhexane
115.22 100.21 142.29 260.09 100.21 163.68 160.39
54.8 61.2 67.0 73.0 59.4 60.4 65.2
5,441 8,095 12,186 18,112 7,893 9,012 9 ,583
5,784 8,222 11,794 16,515 7,808 8 , 969 8,757
Pi also equals ( b E n p / b V )and ~ since P i p is equal to E,, we have
which leads directly to
Difference, Cal/Mole
-343
- 127
+392 +1597 +85 $43 +826
then
AEV
-Enp
- Eint (T,V)- 3
(y) -
(17)
We still have the experimental observation, for cyclohexane specifically, that 3 P i p= P = AEV + -. RT T 2 As before,
?$)
where a ( T ) is an integration constant depending on T only. Hence
and combining Equation 19 with Equations 17 and 18 we have
Eint is expected to be a function of temperature alone for nonpolar systems over small pressure ranges, but it may well vary with volume (or pressure) in specific cases even a t normal low pressure conditions and will certainly vary with pressure and volume under extreme conditions. The integration constant a ( T ) is readily obtained as being identically equal to ( P c P z ) Values . are tabulated for cyclohexane in Table I1 and for n-heptane in Table 111. The nheptane data are smoothed points taken from Westwater et al. (1928) and the temperature range is very limited. In this range a ( T ) is constant, independent of T . It has often been reported, for materials other than heptane, that P , p 2 is constant, independent of temperature, and hence a ( T ) is temperature-independent. This is probably a consequence of the fact that for many materials studied the temperature range covered was very narrow. The behavior of a ( T ) for cyclohexane is probably generally typical of liquidstate behavior, decreasing with increasing rapidity as the temperature rises. In any event, if we put a ( T ) as the sum of a constant term, a,, plus a temperature-dependent term, b ( T ) , the equation relating E to V and T can be written as
ETL
-(a,/P)
+ b(T)/V + Eint + 6RT/2
(14)
A very uncertain extrapolation of our data gives a value of
a,
of about 40 atm liter2/mole2, comparable, a t least, to the van der Waals a for cyclohexane of 23 atm liter2/mole2. In Equation 14 a, is much larger than b(T) for cyclohexane. It is interesting to speculate on the origins of the temperature coefficient, b(T). Suppose the internal energy, Emt, can be separated into a term depending on T and V , E,nt(T,V) and a term dependent on T only, E,,t’(T). Then one might write
a(Enp
+ Eint(T1v)= - [E,, b In V
+ Eint(T,V)]
(20)
Thus
The temperature-independent part could be associated with E,,, while the temperature-dependent part could be associated with Ei,t(T,V). At this stage more experimental data are required (heat capacity data in the liquid and gas phases might be particularly enlightening) to decide this question. However, the matter will not be pursued further in this communication. Allen and coworkers (1960a,b) have provided an extensive list of materials for which both P , and AEV are known, and then have tabulated values of n which, for nonpolar systems, range from 1.05 to 1.22. In Table IV some representative values of AEV and (P,P - 3/2 R T ) for nonpolar liquids are tabulated from their work (Allen et al., 1960b). For some of these materials the correction term (3/2)RT effectively brings P , and AEV into line. I n some cases the difference is too large to be neglected. Here a careful reanalysis of the reliability of the P , data appears to be required, though the explanation is probably related to the choice of 6(RT/2) as the effective classical vibrational energy of the molecules in the liquid state and 3(RT/2) as the energy in the gas. These may be reasonable for effectively spherical molecules but would be expected t o fail for more complex molecules. The approach given above can easily be extended to polar systems. Bottcher (1952) gives an expression for the dipole contribution to the cohesive energy of a liquid as
E,
=
-RT
(E
-
1)(E 4nD2
- nD2)
+ 2)-
where E is the dielectric constant of the liquid and nD is the index of refraction. If E and nD are not dependent on volume (or pressure) in a narrow range (say pressures less than 20 atm), then E , = E , (T only) and will not contribute to P,. 96
I&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
-
Taking P , p for acetone as (80.5)(74) = 5957 cal per mole, AEn,V = 5957 (3RT/2) 5000 cal per mole. The total AEV for acetone is 7000 cal per mole, so that the dipole contribution is 2 kcal per mole. Bottcher’s theoretical calculations, based on spherical molecules, give 2.4 to 2.8 kcal per mole for the polar contribution to liquid energy, good enough agreement to suggest that this basic approach for separation of polar and nonpolar effects is valid. This method of separating various contributions to the total energy of vaporization can also be applied to hydrogen-bonding systems. Thus for ethanol a t 30°C, the theoretical approach described in Chen’s thesis (1969) for hydrogen-bonding effects coupled with the inclusion of the 3(RT/2) term in AEV leads to the following nonpolar, polar, and hydrogenbonding contributions in ethanol a t 30’ :
-
E,,
=
3112 cal per mole; E , = 3415 cal per mole; EHa= 2858 cal per mole
Based on the honiomorph concept and a different approach to evaluation of E H B ,Hansen’s values for these quantities would be : En,
=
3136 cal per mole; E , = 1063 cal per mole; E H B= 5190 cal per mole
Hansen’s method appears to us to underestimate the polar effects, but a detailed discussion of the discrepancies warrants a separate paper. -4t the moment our modification of Hansen’s approach appears to describe realistically the magnitudes of the three contributions, nonpolar, polar, and hydrogen bonding, to the system energy. I n terms of future work it seems clear that careful and extensive determinations of ( b E / d V )T over as wide a temperature range as possible will be most informative, particularly in systems showing polar and hydrogen-bonding effects, even over relatively low pressure ranges, say 0 to 20 atm. These measurements should be coupled with very accurate AEV data along with C V data on both the liquid and gas phases. C, data are readily converted to C V data when internal pressure information is available, since C P
- cv
=
T
g)p
The values of P , and C V will permit the separation of the various energetic contributions to the liquid-state properties. We feel this approach should be rather general for low pressure studies of liquids (below 20 atm) and may well be useful a t higher pressures. Term interaction effects may occur at higher P (50 to 5000 atm), which may complicate the results but at the same time give considerable insight into liquidstate behavior. Acknowledgment
The authors express appreciation of partial financial support from the National Science Foundation (NSF Grant No. GI< 1971) and the Paint Research Institute. They are happy to acknowledge the support of the Washington University Computing Facilities through NSF Grant G-22296.
Nomenclature a, a,
=
empirical constants
a ( T ) ,b ( T ) = temperature-dependent empirical constants at constant pressure heat capacity a t constant volume E = molar energy EHB = liquid energy associated with presence of hydrogen bonds, per mole Eint = energy due to internal molecular modes of motion per mole Ei,t’(T) = energy due to internal molecular motions dependent on T only, per mole Ei.t(T,V) = energy due to internal molecular motions dependent on T and V,per mole E“, = internal liquid energy due to nonpolar forces E, = internal liquid energy due to polar interaction EPE = potential energy of vibration of molecules, per mole . EKE = kinetic energy per mole AEv = molar energy of vaporization AEnPV = molar energy of vaporization due to nonpoIar forces AE,V = molar energy of vaporization due to polar forces AHV = molar enthalpy of vaporization n = constant in potential law E = - a / V n nD = index of refraction P = pressure R = gas constant T = temperature V,B = volume, molar volume CP
= heat capacity
CV
=
GREEKLETTERS UP
= coefficient of thermal expansion of glass
Po
=
isothermal compressibility of glass
e
= isothermal compressibility of liquid = solubility parameter = dielectric constant
T
= =
PI
6
x
nonpolar contribution to solubility parameter polar contribution to solubility parameter
literature Cited
Allen, Geoffrey, Gee, Geoffrey, Managaraj, Duryodhar, Sims, David, Wilson, G. J., Polymer 14, 467 (1960a). Allen, Geoffrey, Gee, Geoffrey, Wilson, G. J., Polymer 14, 456 (1960b). Bagley, E. B., Wood, H. H., Polymer Eng. Sci. 6 (No. 2), 1 i1966). \ - - - - I -
Bianchi, Umberto, Agabio, Giuseppe, Turturro, Antonio, J . Phys. Chem. 69,4392 (1965). Blanks, R. F.. Prausnitz, J. RI., IND. ENG.CHEM.FUNDAMENTALS 3, 111964): Bottcher, C. J. F., “Theory of Electric Polarisation,” Elsevier, New York, 1952. Burrell, H., O$lc. Digest 27, 726 (1955). Chen, Show-An, “Thermodynamics of Associated Solutions,” doctoral thesis, Washington University, St. Louis, 310. (1969). Hansen, C. N., “Three-Dimensional Solubility Parameter and Solvent Diffusion Coefficient,” Danish Technical Press, Copenhagen, 1967. Hildebrand, J. H., Scott, R. L., “Solubility of Non-Electrolytes,” Reinhold. Kew York. 1950. Hildebrandj J. H., Scdtt, R. L., “Regular Solutions,” PrenticeHall, Englewood Cliffs, N. J., 1962. Westwater, W., Frantz, H. W., Hildebrand, J. H., Phys. Rev. 31, 135 (1928). Wiehe, I., Bagley, E. B., A.I.Ch.E. J . 13, 836 (1967). RECEIVED for review May 12, 1969 ACCEPTED October 29, 1969
VOL. 9 NO. 1 FEBRUARY 1970
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97
