A. BONDI
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occurring a t around 30% Au. The maximum in exchange rate disappears a t higher temperatures and rates calculated a t 130” from the kinetic parameters of the alloys show only a steady decrease with increasing Au content. The similarity between the variation in exchange rate and alloy surface area shown in Figure 4 is worthy of comment. Au-rich samples had appreciably lower surface areas than those containing high concentrations of Pd and the maximum areas were associated with the maximum specific exchange rate. In a previous study of the catalytic activity of Pt black,” it was shown that the dehydrogenation activity decreased more rapidly during sintering of the black than would be expected from the loss in surface area alone. This suggested that the sintering process eliminated active sites or defects responsible for the catalytic
activity. The decrease in density of active sites was also accompanied by an increase in activation energy for the dehydrogenation reaction. The exchange results indicate that alloys containing about 10 atom % Au have a greater resistance to sintering than Pd alone so that the density of catalytically active sites may be higher a t this composition. In a recent study of the electrochemical formation and reduction of oxygen layers on Pt-Au alloys,18 a similar maximum in activity was observed a t a composition of about 20 atom % Au. It is interesting that this effect was attributed to an increase in roughness factor of the Ptrich phase.
(17) D.W.McKee, J.‘Fhys. Chem., 67, 841 (1963). (18) M.W.Breiter, ibid., 69, 901 (1965).
A Method to Estimate the Bulk Modulus and the Thermal Expansion Coefficient of Liquids
by A. Bondi Shell Development Company, Emerwille, Cdifornia (Received August 31, 1066)
Simple corresponding states correlations are presented for the thermal expansion coefficient and for the zero pressure bulk modulus of liquids and polymer melts.
Purpose and Scope The improved theory of solutions by Flory and coworkers1 requires as input information equation of state data derivable from the thermal expansion coefficient and the zero pressure bulk modulus of the solvent and the solute. While the former is often available from the literature and can, if necessary, be determined in a comparatively simple experiment with the accuracy required by the Flory calculations, the bulk modulus data are less readily accessible. Isothermal zero pressure bulk moduli (KO)of fair accuracy can be obtained by extrapolation from high The Journal of Physical Chemistry
pressure p-v-t data.2 A more plentiful source is the sound velocity of liquids (and polymers).3-5 These data have the advantage of comparatively safe extrapolation to higher and lower temperature because (1) P. J. Flory, et al., J . Am. Chem. Soc., 86, 3515 (1964); 87, 1833, 1838 (1965). (2) S. D. Hamann, bibliography in “Physic0 Chemical Effects of High Pressure,” Academic Press, Inc., New York, N. Y.,1957. (3) L. Bergmann, “Der Ultraschall,” Hirsel, 1954, 1957. (4) V. F. Nosd’rev, “Use of Ultrasonics in Molecular Physics,” Pergamon Press Inc., New York, N. Y.,1965. (6)W. P. Mason, “Physical Acoustics,” Vol. 11, Academic Press Inc., New York, N. Y.,1964,1965.
53 1
ESTIMATION OF BULKMODULUS OF LIQUIDS
the sound velocity changes linearly with temperature over wide ranges. The major source of uncertainty can be the absence of expansion coefficient and specific heat data, required for conversion of the adiabatic to isothermal bulk modulus. Here a 10% uncertainty in the expansion coefficient or in the heat capacity causes 4 to 5% and 2 to 3% uncertainty, respectively, in the bulk modulus. If the ideal gas heat capacity of the compound in question is known, the conversion to liquid phase heat capacity by known methods6 could hardly introduce errors in excess of *5%. However, the crude guesses which may be made in the absence of gas-phase information could lead to errors in C , as large as 20% since there are at present no reliable simple methods available to estimate the heat capacity of liquids. Hence there is a need for additional estimation methods for the p-v-t properties of liquids and polymer melts. The present work constitutes an attempt to provide simple estimation methods for the thermal expansion coefficient and the zero pressure bulk modulus. The scope of the work is set by the desire to use molecular structure information as the sole data input, yet retain the option to produce a more accurate result by additional experimental data, especially of the density. The assumptions underlying the correlations used limit the validity to temperatures below the atmospheric boiling point for simple substances and to above the glass transition temperature for polymers.
General Principles The basis of the present work is a correspondingstates correlation6 developed from the theory of Prigogine, et aZ.,' for the equilibrium properties of liquid polymers. That theory is a forerunner of the recent equation of state for polymeric liquids by Flory, et aZ.,* which has provicled some of our correlating criteria. The main difference between the theories and the corresponding states correlation is the supply of the reference volume and the reference temperature (potential energy parameter) from independent experimental information in the correlation, whereas the theories derive these reference parameters from the p-v-t data. The dimensioiiless properties to be correlated are the packing density p* = V,p/M, reduced temperature T* = ZcRT/2Eo, and reduced zero pressure bulk modulus KO* = KoV,/Eo where V , is the van der Waals volume derived from X-ray diffraction datale E" is the energy of vaporization defined as E o = AH,,, - RT at that temperature at which p* = 0.588; the number of nearest neighbors Z is taken as 10, and 3c is the number of external degrees of freedom per molecule including those due to internal rota-
0M L f . 0 Q MEK EtCL
0 VF 0.3 0.3
I
0.4
I
0.5
I
0.6
I 0.7
I
1.8
I
0.9
I 1.0
I
1.1
I 1.2
3
Figure 1. Plot of reduced expansion coefficient us. temperature. References: (a) API-Res. Project 44, Tables; (b) API-Res. Project 42, Tables; (c) A. K. Doolittle, et al., A.Z.Ch.E. J., 6, 150, 157 (1960); (d) see ref 22; (e) D. I. Juravlev, Russ. J. Phys. Chem., 9, 875 (1937).
tion excited in the liquid state in the temperature range under consideration. One finds that the reduced density-temperature curve of most liquids with M 2 150 can be represented by the relation p* =
0.726
- 0.249T*
- 0.019T*'
(1)
The center of gravity of experimental points is at about p* = 0.555 corresponding to T* = 0.652. Hence this point is taken as reference point, and c is calculated from the observed density and energy of vaporization as c = 0.652E0/5RT(1.8), where T ( l . 8 ) is the temperature (OK) at which p* = 0.5556. For rigid nonlinear polyatomic molecules, of course, 3c = 6 . Empirical correlations for estimating 3c and for calculating E o from increments for flexible molecules have been presented so that the reference temperature 9~ = 5cR/Eo can be estimated from molecular structure information. When density data are available, one obtains directly OL = T(1.8)/0.652. The correlations are not applicable to alcohols with fewer than four carbon atoms per hydroxyl group. (6) A. Bondi and D. J. Simkin, A.1.Ch.E. J., 6 , 191 (1960). (7) I. Prigogine, et al., J. Chem. Phya., 26, 741 (1957). ( 8 ) P.J. Flory, et ol., J . Am. Chem. SOC.,8 6 , 3507 (1964). (9) A. Bondi, J. Phys. Chem., 6 8 , 441 (1964). (10) A. Bondi, A.I.Ch.E. J., 8, 610 (1962). (11) A. Bondi in "Rheology," Vol. 4, F. R. Eirich, Ed., Academic Press,Ino., New York, N. Y.,1965.
Volume 70, Number 2 February 1966
A. BONDI
532
The Thermal Expansion Coefficient The generalized expansion coefficient a* = a& = l/p*(dp*/bT*) can be obtained by differentiation of the generalized density-temperature curve over the range for which the latter has been plotted.12& The data in Figure 1 show that, as expected, only the low-temperature region is adequately represented in this fashion. Representation of the steep increase of a* as T , is approached requires inclusion of T, into the equation. Since for many substances T,* = 1.30, an equation in T* could be used as indicated in Figure 1. The meaning of the structure insensitive generalized expansion coefficient a* = aE0/5cR is basically that the expansion coefficient is proportional to the ratio of the configurational heat capacity to the “lattice” energy of liquids, a well-known result of the theory of lattice vibrations in crystals. For the expansion coefficient of van der Waals liquids composed of rigid molecules, Wall and Krigbaum obtained the very similar result that a R/AHvap.12b I n Figure 1, one can discern certain small but systematic deviations from the average behavior. However, in evaluating these differences one should keep in mind that few density measurements are sufficiently precise to assign significance to differences of 10% or less in expansion coefficients. The scatter of the data points is of that order of magnitude. Within this scatter there is a trend for highly branched paraffins to exhibit smaller than average values of a*, and several polar compounds show not only a slightly higher than average value of a* but also somewhat smaller than average variations of a* with temperature. The latter trend, of course, deals with a second derivative of density with respect to temperature and all comparisons can note “trends” at best. The rather good fit of the data of flexible molecules onto the general curve suggests that the parameter “c” in the reducing temperature E0/5cR accounts adequately for the effect of internal rotation on the thermal expansion coefficient. In the reduced temperature range 0.3 < T* < 0.7 high polymer melts are very near their glass transition (T,) temperature. The more rapid change in the number of excited external degrees of freedom over a giTren temperature near T g is probably reSPOnsible for the fact that Some Polymer melts exhibit a thermal expansion coefficient in excess of expectation from their low molecular weight analogs. The characteristic difference between the themal expansion of amorphous polymers a t T > T, and of simple liquids is apparent from the data Of Figure 2 . I n View of the notorious difficulty of dilatometric measurements on
-
The Journal of Physical Chemistry
1.0-
0 0
Poiyethylene (13)
v Polystyrene (h)
Polyethylene
V
x.161
(8)
Polyatyrenc (h) x = 8,200
0 0
0.3
0.4
0.5
0.6
0.7
T’
Figure 2. Reduced thermal expansion coefficient of polymer melts as a function of reduced temperature. References: (a) L. D. Moore, J . Polymer Sci., 36, 155 (1959); (b) D. Bradbury, Ph.D. Thesis, Harvard University, 1950; ( e ) G. S. Parks and J. D. Ferry, J . Chem. Phys., 4, 70 (1936); (d) T. G Fox and P. J. Flory, J . Phys. Colloid Chem., 55, 221 (1951); (e) F. Danusso, et al., Chim. Ind. (Milan), 41, 748 (1959); ( f ) Equals ref 13; (9) T. G Fox and P. J. Flory, J . A p p l . Phys., 21, 581 (1950); ( h ) K. Ueberreiter and G. Kanig, 2. Naturforsch., 6a, 551 (1951); (i) N. Bekkedahl, Rubber Chem. Technol., 14, 347 (1941); ( k ) A. Kovacs, J . Polymer Sci., 30, 131 (1958); (1) S. Furuya and M. Honda, ihid., 20, 587 (1956).
polymers, one should perhaps not try to rationalize the relations between the data of Figure 2 and molecular structure. It is noteworthy that the very accurate data on polyethylene melts by Pals13fall very near the curve for simple liquids. A question can arise regarding the appropriate expansion coefficient for the polymeric solute at T < Tg(polymer). The coefficient for polymeric glasses is nearly independent of molecular structure, a, = (2.2 f 0.1) X 10-4 However, the thermal expansion coefficient of the partial specific volume of polymer solutes a t T < Tg(polymer) is within a fairly wide error margin about 0.9 ut,l5,l6just as the partial specific volume is more nearly like the extrapolated liquid density than like that of the glass.” One may (12) (a) The Flory theory predicts for the similar reduced expansion coefficient(at = 0) a.o z= (3v4/9)/[(1 - 3v’/a) - 11, i.e,, a funtion of V , or T,only; (b) F. T.Wall and W. R. Krigbaum, J. Chem. Phys., 17, 1274 (1949). (13) D. T. pals, unpublished information* (14) A. J. Kovacs, I ~ % +Polymer. w . ForsCh.9 3, 394 C1gM). (15) G. V. Schulz and 31. Hoffmann, Makromol. Chem., 23, 220 (1957). (16) A. Schmitt and A. J. Kovacs, Compt. Rend., 255, 677 (1962). (17) A. Horth, et ai., J. Polymer Sci., 39, 189 (1959).
ESTIMATION OF BULKMODULUS OF LIQUIDS
conclude from these observations that the appropriate p-v-t propertks assigned to the polymer in the range Tg(polymer) > T > Tg(solution) are those extrapolated from its melt properties. The properties of polymer solutions a t T < Tp(solution) are only beginning to be examined.lB Hence generalizations cannot yet be made. Due to the compensating effect of opposing temperature trends one finds that the product a p = (bp/bT),, the temperature coeficient of density, is independent of temperature over the wide range 0.4 < T* < 0.8. This very useful fact had been known to petroleum chemist^,^^^^^ but is not widely appreciated. The incorrect notion that (dv/dT), is constant is rather widespread. Examples for the constancy of (bp/bT), and the lack thereof with (dv/bT), are given in Figure 3 for simple liquids as well as polyethylene melts. The data by Murphy, et al., show the constancy of (bp/bT), for many nonhydrocarbon liquids as well. It should be noted, however, that the above indicated rise in expansion coefficient of polymer melts near T , leads to (dv/bT), = constant as one approaches T , to within about 50’. Long extrapolation of density or specific volume, as practiced in crystallinity determination, should therefore be carried out with caution. Bulk Modulus of Liquids The bulk modulus of liquids KO (at atmospheric pressure) is far more sensitive to molecular structure than is the thermal expansion coefficient. Yet, as suggested by Flory’s theoryz1 and confirmed by the data in Figure 4, KO*= KoVw/EOis a simple function of p* or T* for a wide variety of substances, indicating that KO is primarily a function of Eo and V,, i.e., of the cohesive energy density, besides T*. The highly anisometric structure of long-chain compounds and the related anisotropy of “molecular” compressibility requires special consideration. The internal compressibility, of these molecules is only about l/loo as large as the compressibility (Ki-l) of the intermolecular space. Hence, in the limit of M + the bulk modulus of the polymer liquid is given by 1 --2 1 _1- -- 2 +-= K O 3Ki 3Ep 3 K i
ie., the compression takes place only in two dimensions, normal to the axis of the long molecule. For molecules of finite length, one has to consider also the compressibility of intermolecular space between ends of molecules, and the arithmetic becomes a bit more complicated
533
,
4 6
-
,
P :API
- R e e e i r e h Project 4 4 - T ~ b I e ~
b = D . T . F . Pais, Unpublmhcd Wormatlon c = L.D. Moore. jr., J . Polymer S a . lb.1 5 5 (1959) d = F. Danusso et P I . LBChlrn c L’hd.G. 748 (1959)
250
1
150
100
450
400
T. *K
Figure 3. Comparison of the temperature coefficients of density (open circles) with those of specific volume (solid symbols) for n-paraffins, polyethylenes, and polypropylene melts.
0 4-
- +0A 1-
A V
6 - v 0 2 -
-
4
0
I -
0
0.2
I
I
I
I
1.0
2.0
I
I
3.0
]IT‘
Figure 4. Plot of generalized bulk modulus of liquids us. temperature. References: (a) ref c of Figure 1 and ref 3 and 4; (b) H. Geelen, Ph.D. Thesis, Delft, 1956; (e) ref b of Figure 1; (d) J. W. M. Boelhouwer, private communication.
The general shape of the relation is indicated in Figure 5 . Typical E, data have been assembled in Table I. Polymers with cis configuration in the repeating unit of the backbone chain do not lose van der Waals compressibility parallel to the chain axis. Their bulk modulus in the melt state is therefore correlated as (18) K.Ueberreiter and W. Bruns, Ber. Bunsenges., 68, 541 (1964). (19) M. R. Lipkin and S. S. Kurtz, Ind. Eng. Chem. AnaE. Ed., 13, 291 (1941). (20) C. M.Murphy, et al., Trans. ASME, 71, 561 (1949). (21) According to which at p = 0
V o l u m e 70,N u m b e r 2
February 1966
A. BONDI
534
Table I : Spectroscopic Elastic (Young's) Modulus E p of Various Polymer Molecules (All in 10'0 dynes/cma) EP
EP (calcd)'
Polymer
Polyethylene Polytetrafluoroethylene Polyvinylchloride (syndiotactic) Polyvinylidene chloride Polyoxymethylene Polyisobut ylene Polypropylene (isotactic) Polystyrene Polyvinyl alcohol Polyethylene terephthalate Nylon 66 Cellulose I Cellulose I1 Polyethylene glycol ( M m) Poly-3,3-bis(fluoromethyl)oxacyc1~ butane Poly-3,3-bis( chloromethyl)oxacyclobutane Poly-3,3-bis(bromomethyloxacyclobutane Poly-3,3-bis(iodomethyl)oxacyclobutane
(obsd)b
182,d340 160 160 (or 2301) 220 70-84 49
122,d 146' 196,d 157'
-
7
260
3
V 41.5 54
0 0
2
3
Figure 5. Plot of generalized bulk modulus of liquids, including polymer melts, composed of flexible chain molecules. References: (a) ref c of Figure 1; (b) ref b of Figure 1; (c) K . H. Hellwege, et al., Kolbid-Z., 183, 110 (1962); (d) ref b of Figure 2; (e) L. A. Wood and G. M. Martin, J. Res. Nag. Bur. Std., 68A, 259 (1964); (f) E. Passaglia and G. M. Martin, ibid., 68A, 273 (1964).
4.8 110 100 92 77
KO*= 2'oo - 1.60 T*
and for long-chain compounds and polymer melts
(3)
Both equations are strictly empirical and are probably valid only in the range 1.0 > T* > 0.4. At the lowtemperature end there is an indication of a slower than 1/T* rise of KO*. At the high-temperature end the point KO*= 0 is attained a t T* = 1.25 which is somewhat less than the critical temperature, which as a rule is a t T* = 1.3. Hence exact expressions for both systems yield S-shaped curves rather than straight The Journal of Physical Chemistry
1
I IT*
KO,ie., like that of a monomeric liquid, however, on the polymer correlation curve. The equations for the two cases shown in Figures 4 and 5 are quite similar, for simple liquids
T*
3
I
137 90
= 2*30 - - 1.98
Poly8iloxm x=30
1
42 i2 255 76c
' From the spectroscopic force constants; M. Asahina, et al., J. Polymer Sci., 59, 93, 101, 113 (1962). By X-ray diffraction analysis of crystallite extension; I. Sakurada, et al., ibid., 57, 651 (1962); Makromol. Chem., 75,651 (1964). F. W. Dalmage and L. E. Contois, J. Polymer Sci., 28, 275 (1958), find 140 X 1010 dynes/cma by the same technique. d L . F. G . Treloar, Polymer, 1, 95, 279, 290 (1960). * W. J. Lyons, J . A p p l . Phys., 29,1429 (1958).
KO* (L)
6
lines. While this deficiency is not a serious problem with most monomeric liquids because they are used largely in the temperature range in which the straight line is a good approximation, it may well become a problem when one is concerned with solutions of polymers at or below their glass transition temperature. The example of poly(methy1 methacrylate) in Figure 5 illustrates this point. Since the thermal pressure coefficient y (=Koa) can presumably be measured more accurately than it can be calculated from the independently measured KO and a, an attempt has been made to correlate y directly. Two routes have been tried, as Pi* = rTV,/Eo and as P ~ * / Pbecause *~ the latter is often claimed to be temperature independent, a t least a t low temperatures.22 A few spot checks of room temperature measurements yield data in tolerable agreement with each other. However, data taken over an extended temperature range by several well-known investigators are in striking disagreement as shown by the curves of Figure 6. The other data in that figure offer little encouragement for the discovery of a uni* ~T*, since the properties versal correlation of P ~ * / Pvs. of n-paraffins are generally most amenable to correlation. Convergence of all curves into a single curve is indicated only for the higher molecular weight alkanes (No 2 16) at low temperatures (T* 5 0.7). Numerical * ~ been obtained for various comvalues of P ~ * / Phave (22) G . Allen, et al., Polymer, 1, 467 (1960).
535
ESTIMATION OF BULKMODULUS OF LIQUIDS
2.2
0
Table 11: Comparison of Flory’s Reducing Temperature TF* with That of the Present Work (eL) and of Flory’s Reducing Volume VF* with VO“
n-Heptnne (a) n-Heptane (b)
Q n-Heptme (e) Q n-OEtme lb)
0 n-Non.ne
2.0
(bl
t,
n-Dodscme (b)
V
Substance
0 n-Telradeenne (d)
Cyclohexane Methylcyclohexane Benzene
1.6
1.4 0.4
0.6
1 .o
0.8
TF*/@L
D*TF*/@L
VF*/VO
0 70 0 70 65
10.9 11.5 10.3 10.65 11.5
5.70 5.75 5.90 5.60 6.0
0.996 1.02 1.01 1.03
0 70 70 20 20 20 65
10.4 11.0 9.47 12.1 11.6 10.0 9.90
5.77 5.61 5.60 6.31 6.20 5.05 5.02
0.99 1.01
n-Hexsdecmc (b)
CCla 1.8
‘c
1 .t
4
T*
Figure 6. Plot of reduced internal pressure function us. reduced temperature for several alkanes. References : (a) J. S. Rowlinson, “Liquids and Liquid Mixtures,” Academic Press, New York, N. Y., 1959; (b) J. W. M. Boelhouwer, Physica, 26, 1021 (1960); (c) see ref c of Figurc 1; (d) see ref 22.
pounds and were found to cover too broad a range even within given chemical series to offer much hope for useful correlation. This lack of simple trends appears to hold even within data series obtained by single and very careful investigators. Comparison of the reducing parameters V* and T* of Flory’s generalized equation of state with the corresponding independently determined properties used in tthe present correlation yields the following results. The volume V* can be compared directly with V,, and T F = sv/2v*cR = (ijEEo/Nrc‘R)Fcan be compared with OL = E0/5cR, where the two energy terms E o and Eo are not identical by definition, but are likely to be very similar in numerical magnitude. Flory’s rc’ is only one-half of our c. Hence T * F / O L should be of the order 1 0 ~ . This turns out to be the case. Specifically, while the temperature drift of V* prevents a constant ratio, V,/V*F = 0.70 f. 0.02, which is of the order of the packing density at OOK. Typical comparisons with zero-point volume Vo are shown in Table 11. The same table also shows the ratio T*F/eL and the almost universal constant factor p * T * / B L , indicating how the two methods of correlation can be interchanged. . -
Limitations of the Methods Inspection of Figure 1 shows that prediction Of the expansion coefficient of monomeric liquids for a generalized curve might be good only to within *lo%. This uncertainty can often be reduced to f.3% by starting with a known expansion Coefficient Of a COm-
Diphenyl n-Hexane n-Heptane n-CBF14 CFa * CyClO-CsFi1
1.02 1.03
V,, = zero point volume, W. Biltz, “Raurnchemie der festen Stoffe,” 1934.
pound similar to that under consideration, e.g., a member of the same homologous series. Its experimental value of areL us. T* then locates the curve of areL (1.30 - T*)-”’ for the compound under consideration. The E o increments required for this calculation can be found in ref 10 and 11. The thermal expansion coefficients of polymer melts should be estimated from the correlation of Figure 2. However, care should be taken not to extrapolate into the immediate neighborhood of or below T,* where rather different relations p r e ~ a i l . ’ ~ ~ * ~ The average error of the bulk modulus correlations within the linear range discussed earlier appears to be of the order of =k5oja, and the maximum error range may be of the order of =kl5%. Here again the error range can be sharply reduced by sacrificing generality and specializing the correlations for families of compounds.
-
Conclusions The corresponding-states correlations presented here permit an estimate of the thermal expansion coefficient and of the bulk modulus of liquids and polymer melts to within =klO%. This error limit can be reduced to less than 3% by specializing the correlations for particular series of compounds. Further reduction of this error limit is probably unrealistic because few experimental data have a smaller error limit. The alternative route to the bulk modulus via the sound velocity is open only if heat capacity data with uncertainties of 5 10% are available. (23) A. Bondi, J . Polymer Sei., A2, 3159 (1964).
Volume 70, Number 2 February 1966
