2310
The Journal of Physical Chemistry, Vol. 82, No. 21, 1978
J. T. Edward, P. G. Farrell, and F. Shahidi
Partial Molal Volumes of Organic Compounds in Carbon Tetrachloride. 1. Alkanes. Conformational Effects John T. Edward," Patrick G. Farrell, and Fereidoon Shahldi Department of Chemistry, McGill University, Montreal, Quebec, Canada H3A 2K6 (Received June 20, 1978) Publication costs assisted by the Natural Sciences and Engineering Research Council of Canada
The average number of gauche arrangements (Zg)present in molecules of 30 straight- and branched-chain alkanes ranging in size from pentane to nonane has been calculated using Pitzer's steric partition function; 2, for seven longer straight-chain alkanes (n-decane to n-dotriacontane) has been calculated by an empirical equation. The partial molal volumes of all 37 alkanes in CC14 may be calculated with good accuracy by the addition of group increments (methyl, 26.85; methylene, 17.36;methine, 10.35;quaternary carbon, 3.40 mL mol-') to a covolume of 11.61 mL mol-', and subtracting 2.5 mL mol-' for each unit of 2,. The relation of the increments of the groups to their van der Waals volumes is discussed. The partial molal volume P of an organic solute exceeds its van der Waals volume V , (ENu,, N being the Avogadro number and v, the van der Waals of a single molecule) by the void or empty volume V, associated with the packing of solute molecules among the solvent mole c u l e ~ .Recently ~ two equations5s6have been advanced to take account of V,. The first5 assumes spherical solute molecules of radius r, (E (3~,/47r)'/~),with an empty volume proportional to the surface area of the molecule, so that Po= 47rN(r, + A)s/3 (1)
the parameter A which takes account of V , depending on solvent, being 0.57 A in water.7 The second equation, due to Terasawa et a1.6 P=aV,+b (2) has been found to hold with fair precision for aqueous solutions of alkyl halides, sulfides, ethers, alcohols, and glycols, the constants a and b depending on the type of compound. While experimental data for small molecules can be accommodated about equally well by either equation, (1) proved better for larger molecules of approximately spherical shape, and (2) for larger molecules of approximately cylindrical shape.s These two equations represent important first stages in attempting to relate P" to V,, but fail to account for the small differences in Poof the isomers. These differences have long been recognized in cyclohexane derivatives (the Auwers-Skita rule1@12),and recently Mann13J4has shown how the molecular volumes (or densities) of straight- and branched-chain alkanes are related to their conformations. We have applied and extended Mann's ideas to the partial molal volumes of organic molecules in carbon tetrachloride. This solvent, unlike water, has the advantage of dissolving a great variety of hydrocarbons of differing sizes and shapes. Such solute molecules, devoid of polar groups which can interact with solvent, are ideal substrates for investigating the effect of molecular geometry on molecular volumes. In this paper we report the effect of molecular conformation on the partial molal volumes of 37 straightand branched-chain alkanes listed in Table I.
Results and Discussion Effect of Average Number of Gauche Conformations (ZJ on 9". It is immediately apparent that isomeric alkanes can have significantly different partial molal volumes in carbon tetrachloride (compare the isomers C6H14 (compounds 2-6 in Table I), C7H16 (7-14), and CgHzo 0022-3654/78/2082-2310$01 .OO/O
TABLE I: Partid MOM Volumes (70, in mL mol-' of Alkanes in Carbon Tetrachloride at 25 C O
no. 1 2 3 4 5 6
alkane n-pentane n-hexane 2-methylpentane
2,:
0.66 0.96 1.31 2.29 2.00 2.64 1.25 1.64 2.47 3.58 2.00 2.00 3.55 4.00 1.54 1.84
-
-
Vexntl
vcalcd
116.2 i 0.3 132.3 t 0.6 134.1 i 0.4 131.6 i 0.5 134.5 t 0.6 132.6 t 0.6 149.3 i 0.3 151.7 i 0.4 149.0 f 0.4 146.6 i 0.4 153.7 i 0.5 153.6 t 0.5 148.9 I 0.5 148.2 t 0.4 166.0 f 0.4 181.8 i 0.4 183.1 i 0.2 180.7 i 0.2 181.0 f 0.4 179.3 %- 0.2 178.4 f 0.3 184.3 i 0.4 183.4 t 0.3 183.1 t 0.2 182.4 c 0.4 180.0 t 0.2 180.1 c 0.2 181.1 i 0.1 178.5 t 0.3
115.7 132.3 134.0 3-methy lpentane 131.5 2,2-dimethylbutane 134.8 2,3-dimethylbutane 133.1 7 n-heptane 149.0 8 2-methylhexane 150.5 9 3-methylhexane 148.4 10 3-ethylpentane 145.6 11 2,2-dimethylpentane 152.1 1 2 2,4-dimethylpentane 152.1 13 2,3-dimethylpentane 148.2 14 3,3-dimethylpentane 147.1 1 5 n-octane 165.6 16 n-nonane 182.2 1 7 2-methyloctane 2.22 183.8 1 8 4-methyloctane 2.92 182.0 19 3-methyloctane 3.08 181.6 20 4-ethylheptane 3.93 179.5 21 3-ethylheptane 179.1 4.08 22 2,6-dimethylheptane 2.59 185.3 23 2,2-dimethylheptane 2.66 185.2 24 2,4-dimethylheptane 3.38 183.3 25 2,5-dimethylheptane 3.50 183.0 181.0 26 2,3-dimethylheptane 4.31 27 3,3-dimethylheptane 4.38 180.9 180.6 28 3,5-dimethylheptane 4.48 179.1 29 3,4-dimethylheptane 5.07 30 3,3-diethylpentane 8.00 172.1 f 0.2 171.8 31 n-decane 2.13 198.9 f 0.3 198.9 232.1 2.72 231.6a 32 n-dodecane 33 n-hexadecane 298.6 3.89 29 8. 6a 365.1 5.06 364.5a 34 n-eicosane 398.4 5.65 398.3a 35 n-docosane 498.1 7.41 499.2a 36 n-octacosane 564.7 8.58 565.8a 37 n-dotriacontane a L. G. Longsworth, J. Colloid Interface Sci., 22, 3 (1966). (16-30)), just as Mann13*14has already found the neat liquids of the same compounds to have different molar volumes. Mann showed that these differences are related to Z, the number (zeitung) of gauche conformations found on an average in a molecule at a given temperature. The familiar picture of Pitzer,15 Flory,l6 and others was adopted: the conformations of a molecule were obtained by considering in turn the individual conformations about each C-C bond beyond the terminal C-C bond. These 0 1978 American Chemical Society
The Journal of Physical Chemistty, Vol. 82, No. 21, 1978 2311
Partial Molal Volumes of Organic Compounds in CCI4
Chart I
aa ag + a€!g'a g -a g+g+ g-g-
0.4176 0.1264 0.1264 0.1264 0.1264 0.0383 0.0383
0 1
2.0
0
0.1264 0.1264 0.1264 1 0.1264 2 0.0766 2 0.0766 2, = 0.6588
1 1
conformations are restricted by the familiar threefold bond rotational potential to a lowest energy minimum a t the anti (a) conformation and two gauche minima (g+ and g-) situated a t rotations 4 = f120" from the anti conformation. Each gauche conformation was associated by Mann with an enthalpy increase AH of 700 cal mol-' (at 20 "C), except for g+g- or g-g+ sequences, where steric interference becomes severe and AH large;" conformations containing such sequences were neglected. The shapes of the potential wells for a and g conformations are sufficiently similar16 that A S can be obtained solely from symmetry considerations, and the mole fraction y of each conformation calculated by Pitzer's steric partition f u n c t i ~ n . ' ~ JHence ~ J ~ 2, was calculated by the equation
where 2 : is the number of gauche arrangements counted in conformation k. Following Mann, we have adopted AH = 700 cal mol-' as the difference between a gauche and anti conformation, with the exception of the special case represented by vicinal dimethyl compounds (6, 13, 26, and 29 of Table I). Verma, Murphy, and BernsteinZ0have shown that the energy difference = 0 between the C2h and C2 forms of 2,3-dimethylbutane (61, in spite of the fact that the C2h form has only two gauche interactions and the C2 form has three. Consequently, the two forms are present in the ratio =l to 2, in agreement with their statistical weights. This result has been rationalized by molecular mechanics calculations,21 which show that in the C2h form of 6 methyl...methyl distances are short, since alleviation by small torsional or valency angle adjustments is not possible, while in the C2 form such alleviation is possible. We have accordingly set AH = 0 for the various conformations in 13, 26, and 29 involving the same sequence of atoms. Values of 2, calculated in this manner are given in the third column of Table I. As an example of the method, consider n-pentane, the first compound of the table. If we exclude g+g- and g-g+ conformations, this can exist in seven conformations, differing in enthalpy as shown: aa, AH = 0; ag', ag-, g'a, g-a, AH = 700; g+g+,g-g-, AH = 1400 cal mol-'. The mole fractions of these conformations, calculated in the usual way,13J9are shown in Chart I, hence 2, may be calculated as shown above. Even if we exclude conformations containing a g+g- or g g + sequence (which would be taken into account in more precise calculations16), the total number of conformations increases rapidly with chain length, going from 3 for nbutane to 3366 for n-dodecane. Obviously, calculation of 2 for n-dotriacontane (37 in Table I) would be impossibly lasorious by the procedure above. Fortunately, the values of 2, calculated for n-alkanes from n-pentane (seven conformations) to n-nonane (239 conformations) prove to be a linear function of n, the number of carbon atoms in the chain 2, = 0.38
:
+ 0.293(n - 4)
(4)
n
Figure 1. Dependence of the average number of gauche c o n f m t i o n s
(Z,)on the
number of carbon atoms (n) in an n-alkane.
Zg
Flgure 2. The dependence of Po on the average number of gauche of unbranched (+), singly branched (O),and doubly conformations (Z,) branched (W) alkanes (compounds identified by numbers in Table I).
as shown in Figure 1. The values of 2, for compounds 31-37 in Table I have been obtained from this equation. Most of the values of 2, for the branched alkanes are found in Mann's p a p e r ~ . ' ~ J ~ In Figure 2 we show 9" for the isomeric compounds C6H14, C7H16, and C9HZ0plotted against 2,. A linear relationship is apparent, but only for isomeric alkanes of similar structural type: one line for single branched C6H14, another for doubly branched CGH14, etc. However, all lines have a slope of -2.5 mL mol-' 2;l. This compares with a slope of -2.1 mL mol-' 2;l for the molar volumes of the neat hydrocarbons plotted against Zg,13J4and a reduction
2312
The Journal of Physical Chemistry, Vo/. 82, No. 21, 1978
of 3.5-3.9 mL in the molar volume of methyl-substituted cyclohexanes when an equatorial methyl becomes Le., for an increase of 2 in 2,. Allinger" has pointed out that both the decrease in volume and increase in enthalpy associated with change from an anti to a gauche conformation most probably come from the squeezing together of alkyl groups more closely than the sum of their van der Waals radii. An Additiue Scheme for 9". The results displayed in Figure 2 point to the dependence of 9" on both constitution and-2, and, as shown in Table I, the experimental values of V" are in excellent agreement with values calculated by the equation rn
9" = V , + C nili- 2.52, 1
(5)
where V, is a covolume of 11.61 mL mol-', Z is the volume increments due to the groups methyl (26.85 mL mol--'), methylene (17.36 mL mol-'), methine (10.35 mL mol-'), and quaternary carbon (3.4 mL mol-'), and n is the numbers of such groups in the hydrocarbon molecule. The values of V , and of the various increments, with the exception of quaternary carbon, were found by a leastsquares fitting of experimental data to eq 5; quaternary carbon was assigned the van der Waals volume'-3 of 3.4 mL mol-' as its increment. An examination of models indicates that the surface of alkane (but not of aromatic hydrocarbon) molecules is made up of hydrogen atoms, and that the carbon atoms of the backbone are essentially completely shielded from contact with the relatively large carbon tetrachloride molecules of the solvent. The empty volume u, (E V , / N associated with a single solute molecule is a function of the closeness of packing of solvent molecules about it, and consequently in setting up an additive scheme we have chosen to consider the empty volume of alkanes to be associated solely with hydrogen atoms, the contributions of carbon atoms being given completely by their van der Waals volumes. We have found, as have other^,^^^^^ that any additive scheme for the molecular volumes of alkanes requires a constant term, which following TraubeZ we have labeled "covolume" V,. This may be identified with Bondi's "fluctuation v01ume"~ for an infinitely small kinetic particle (CniZi = 0), and is required by scaled-particle theory.26
Variable Volume Increments f o r Hydrogen Atoms. Accepting a constant increment of 3.4 mL mol-' for carbon, an increment of 12.1 mL mol-' is required for each of the hydrogen atoms of methane in carbon tetrachloride,n and increments of 7.8, 7.0, and 7.0 mL mol-.' for the hydrogen atoms of methyl, methylene, and methine. These increments can be regarded as being made up partly of the van der Waals volume of the hydrogen atom and partly of empty volume. While a van der Waals radius rw of 1.2 8, for the hydrogen atom28 has found general acceptance, such an atom surrounded by a single-electron cloud can be expected to be comparatively soft, with a variable rw, and Bondi2 has adopted a value of 1.00 8, for rw of hydrogen attached to aromatic carbon and K i t a i g o r ~ d s k ya ~value ~ of 1.3 8, for hydrogen of aliphatic chains in a crystal. These radii are derived from contact distances a t which attractive and repulsive forces are in balance. While the repulsive forces rise steeply as the distance is lessened, and depend only upon the hydrogen and (in the present case) the chlorine atom of the solvent, the attractive forces rise gently, and extend to distances several Angstrom units beyond the minumum defining the contact d i ~ t a n c e Consequently, .~~~~~ the contact distance
J. T. Edward, P. G. Farrell, and F. Shahidi
between hydrogen of EC-H and chlorine of solvent will be affected by attractive forces between the carbon atom and the solvent molecule, and (to a much lesser extent) by attractive forces between the atoms joined to this carbon atom and the solvent molecule. Successive replacement of the hydrogen atoms of methane by more polarizable carbon atoms increases the attractive forces and decreases the contact distance; however, the steep region of the repulsive curve would seem to be reached with methylene hydrogen, so that no further shortening of distance occurs on going to methine hydrogen. Comparison with Other Additive Schemes. Previous additive scheme^^^*^^^' for the molar volumes V of alkanes32 have been based on a different assumption, viz., that the increment due to hydrogen is independent of the nature of the carbon atom to which it is attached. This accounts for the widely different values of the increments in these schemes32from ours. Two of them23,24 contain a term for a "residual volume" (-30 mL mol-'), not greatly different from Traube's covolumeZ of -25 mL mol-'. Our covolume is much smaller, because of the greater volume we attribute to methyl hydrogens as compared with methylene hydrogens. None of the other schemes takes account of 2,. Exner's scheme3' has no covolume. This necessitates a very large increment (14.90 mL mol-') for hydrogen, obtained by difference from V of CH3(CH2),CH3 and of (CH,),. This in turn requires a negative increment (-13.22 mL mol-') for carbon. Formally, a negative increment is unobjectionable, since the additive scheme works fairly well, but on intuitive grounds it is hard to comprehend. Values of 9" calculated from eq 1 with A = 0.74 8, are in reasonable agreement with experiment for n-alkanes up to n-hexane, whose shape does not deviate too severely from spherical,33but the agreement progressively worsens as the chain gets longer. On the other hand, values of V" calculated from eq 2 with a = 1.627 and b = 21.2 mL mol-' are in excellent agreement with experiment (correlation coefficient 1.0000) for the unbranched alkanes 1,2,7, 15, 16, 31-37. According to eq 2, the contribution aV, of a methylene or methyl group to 9" should be 16.64 and 22.4 mL mol-', respectively, in comparison with our increments of 17.36 and 26.85. The smaller value for methylene comes from the fact that it is based on experimental 9" values, and hence is lessened by the constant fraction of 2 associated with each methylene as a polymethylene ckain is lengthened. Similarly, the smaller value of the methyl group is compensated for by having b larger than our covolume. However, while eq 2 and 5 are equally satisfactory for the single series of unbranched n-alkanes, the application of eq 2 to isomeric branched alkanes (which, following Bondi,2 have identical values of V,) would be unsuccessful.
Experimental Section Materials. All the hydrocarbons were commercial products obtained from Chemical Samples Co. or Aldrich Co. and were used without further purification if their specified purity was 299.0%. If not, the compounds were doubly distilled and the middle fraction retained for experimental work. Fisher Scientific Co. spectrograde carbon tetrachloride was dried over calcium chloride overnight and then fractionally distilled twice, the middle fraction (60%) being stored over Linde A-type molecular sieves in the dark prior to use. The density for different batches of the solvent was 1.58430-1.58451 g mL-' (lit. 1.5843934and 1.58440-1.5845335). Measurements. Density measurements were carried out a t 25 "C using a Paar digital precision density meter, Model DMA Between 12 to 16 measurements were
Volume Change on Dilution and Protonation of Polycarboxylates
made within the concentration range 0.1-1.3% for each compound. Apparent molar volumes, &, of solutes were calculated from l / d - w,/do @v = M
w2
where the terms have their usual meaningsag The partial molar volumes, V", of solutes were obtained by linear extrapolation of the 4;s to infinite dilution, using a simple least-squares method.
Acknowledgment. We are grateful to Dr. S. C. Wong for computer analysis of our data, to the National Research Council of Canada for financial support, and to the Faculty of Graduate Studies and Research of McGill University for financial assistance toward the purchase of a precision density meter.
References and Notes (1) J. T. Edward, Chem. Ind. (London), 774 (1956). (2) A. Bondi, J. Pbys. Chem., 68,441 (1964). (3) J. T. Edward, J . Chem. fduc., 47,261 (1970). (4) A. Bondi, J . Pbys. Chem., 58,924 (1954). (5) J. T. Edward and P. G. Farrell, Can. J . Chem., 53,2965 (1975). (6) S.Terasawa, M. Itsuki, and S. Arakawa, J . Phys. Chem., 79,2345 (1975). (7) We have recently become aware that E. J. King,' using a different approach, obtained an equation essentiaHy identical with (l),and found that for a variety of solutes in water A = 0.55 A. (8) E. J. King, J. Phys. Chem., 73, 1220 (1969). (9) J. T. Edward, P. G. Farreil, and F. Shahidi, J . Chem. Soc., Faraday Trans. 7 , 73, 705 (1977). (10) K. von Auwers, Annalen, 84,420 (1920). (11) M. L. Allinger, Experientia, 10,328 (1954). (12) E. L. Eliel, N. L. Ailinger, S. J. Angyal, and G. A. Morrison, "Conformational Analysis", Interscience, New York, 1965, pp 172-177. (13) G. Mann, Tetrahedron, 23,3375 (1967).
The Journal of Physical Chemistry, Vol. 82, No. 21, 1978 2313
(14)0.Mann, M. Muhlstadt, J. Braband, and E. DZjring, Tetrahedron, 23, 3393 (1967). (15) K. S. Pitzer, J . Cbem. Pbys., 8,711 (1940);Cbem. Rev., 27,39 (1 940). (16) A. A b , R. J. Jernigan, and P. J. Fbry, J. Am. Chem. Sm.,88,631 (1966). (17) R. P. Smith, J . Cbem. Pbys., 92, 1162 (1965). (18) F. Becker, Z.Naturforsch. A, 14,547 (1959). (19) M. knack, "Conformation Theory", Academic Press, New York, N.Y., 1965,p 34. (20) A. L. Verma, W. F. Murphy, and H. J. Bernstein, J . Cbem. Pbys., 1540 (1974). (21) N. L. Allinger, J. A. Hirsch, M. A. Miller, I.J. Tyminski, and F. A. Vancatledge, J. Am. Chem. Sm.,90,1199 (1968);R. H. Boyd, bid., 97,5353 (1975). (22) G.Mann, M. MuMsWt, and J. Braband, TetraMon, 24,3607(1968). (23) S. S. Kurtz and M. R. Lipkin, Ind. fng. Chem., 33, 779 (1941). (24) M. L. Huggins, J . Am. Chem. Soc., 63, 116 (1941). (25)J. Traube, Samml. Chem. Vortr., 4,255 (1899). (26) H. Hoiland and E. Vikingstad, Acta Chem. Scand., Ser. A , 30,692 (1976). (27)R. A. Pierotti, J . Phys. Chem., 87, 1840 (1963). (28) L. Pauiing, "Natwe of the Chemical Bond", 2nd ed, C m l l University Press, Ithaca, N.Y., 1948,p 187 et seq. (29)A. I.Kitaigorcdsky, "Molecular Crystals and Molecules", Academic Press, New York, N.Y., 1973,pp 56, 164. (30) N. L. Allinger, M. A. Miller, F. A. Vancatledge, and J. A. Hirsch, J . Am. Chem. Soc., 89,4345 (1967); N. L. Ailinger and M. J. Hickey, ibid., 97, 5167 (1975). (31) 0.Exner, Collect. Czech. Chem. Commun., 32, 1 (1967). (32)These scheme^^^*^'^^' yield the molar volume V of the neat hydrocarbon; however, the excess molal volume (= V o - V ) is small, being 1.0 f 0.4 mL mol-' for compounds 2-6;3.0A 1.2 for 7-14; and 2.1 i 1.2 for 16-30. (33) +Pentane in its fuliy extended conformation, considered as a prolate ellipsoid of axial ratio 2.1,has a surface area 9% greater than that of a sphere of equal volume. However, with lengthening of the n-alkane chain the difference between the surface area and that of a sphere of equal volume increases rapidly. (34)J. A. Riddick and W. B. Bunger, "Techniques of Chemistry", Vol. 11, "Organic Solvents", Wiley-Interscience, New York, N.Y., 1970,pp
520-526. (35)L. G. Longsworth, J . Colloid Interface Sci., 22,3 (1966).
Volume Change on Dilution and on Protonation of Polycarboxylates in Aqueous Solutions J. Skerjanc Department of Chemistry, University of Ljubljana, 6 1000 Ljubljana, Yugoslavia (Received May 4, 1978)
The volume changes accompanying the dilution and the protonation of two polycarboxylates in water have been determined at 25 "C by dilatometry at various degrees of ionization, CY. The polyelectrolytes were the sodium salts of polyacrylic acid (PAA) and of the copolymer of maleic acid with ethylene (EMA). The volume change on dilution of solutions of both polycarboxylates increases continuously with increasing a. The two polyacids display a remarkable difference in the a dependence of the volume change on protonation. While the PAA curve first increases with increasing a and then decreases slowly, the EMA curve shows in addition to such behavior a well-defined discontinuity at a = 0.5. The experimental results have been compared with predictions based on the cell model. The agreement between theory and experiment is only fair.
Introduction Weak polyacids and polybases have been extensively investigated by polymer chemists. Much of this interest can be ascribed to the fact that the polyion charge of these materials can be varied by titration with base or acid; as a consequence, the properties of these polymeric species are gradually changing from those of an uncharged polymer to those of a typical polyelectrolyte. The majority of the weak polyacids are polycarboxylic acids, and practically all properties of interest in the field of polyelectrolytes have been accumulated for their simplest representatives, such as polyacrylic acid. Recently there 0022-3654/78/2082-2313$01 .OO/O
has been considerable research interest in the and v o l ~ m e changes ~ - ~ associated with the protonation of the carboxylate groups of various polycarboxylates. Most of the measurements of the volume change on protonation have been done in the presence of a simple salt which is the usual environment for naturally occurring polyelectrolytes. From a theoretical point of view, however, the study of salt-free polyelectrolyte solutions is more attractive, since the interpretation of results obtained with such systems is much simpler. In this paper we present the results of the dilatometric measurements obtained with aqueous solutions of the
0 1978 American
Chemical Society
