7736
J. Phys. Chem. 1993,97, 1136-1142
Molecular Hard-Sphere Volume Increments Dor Ben-Amotz’ and Kelly G. Willis Department of Chemistry, Purdue University, West Lifayette, Indiana 47907-1393 Received: February 26, 1993; In Final Form: April 29, 1993
Molecular hard-spherevolumes derived from the analysis of the high-pressure equation of state data are extended to include a total of nearly 100 compounds. A subset of these are converted to molecular subgroup increments, which allow the prediction of effective hard sphere volumes and diameters for a wide variety of polyatomic molecules. Shorter chain compounds, which can be viewed as substituted methanes (or silanes) with halide, CN, OH, and CHs substituents are represented by one set of 11 volume increments. Long-chain compounds (with chain lengths of 4-1 8) including linear and branched alkanes, alkenes, ethers, alcohols, haloalkanes, ketones, and benzene derivatives are described by an independent set of 20 volume increments. Molecular volumes determined using these increments reproduce the input equation of state derived volumes with a standard deviation of f 1%. Less accurate correlations are derived for intermolecular attractive interaction coefficients, as well as the temperature derivatives of these and molecular diameters. Distinctions between the present hard-sphere volume increments and previously proposed liquid molar volume and “van der Waals” volume increments are discussed.
1. Introduction
Accurate estimates of molecular size are required in modeling a wide variety of dynamic and structural properties of vapor and fluid systems,14 including equations of state,”’ gas solubilities,I2-16 heat of ~aporization,12J~-1~ and transport propertie~,2&2~ as well as solvent perturbations of optical spectra,28vibrational frequencies,z9-36 isomerization equilibrium ~onstants,3~-38 and chemical r e a c t i v i t i e ~ .The ~ ~ ~effective hard-sphere diameters of atoms and molecules are often key parameters in predictive models for such chemical properties. Recent work has shown that molecular hard sphere diameters can be determined, with a precision of the order of 1%, from high pressure equation of state (or compressibility) data.6 The resulting diameters are generally found to decrease mildly with increasing temperature (reflecting the greater interpenetration of molecules at higher kinetic energy) and to be essentially pressure insensitive (at least over a few thousand atmosphere pressure range). Here we extend previous work by deriving hard sphere diameters for numerous other compounds from published equationof state data, and use these to define molecular subgroup volume increments, from which the effective hard-sphere volumes of a wide variety of compounds can be predicted. There are severalimportant distinctions between the subgroup volume increments defined in this work and previously proposed liquid molar volume*J and “van der Waals” volume increm e n t ~ . ~Liquid . ~ ~molar ~ volume increments,2v3for example, have a marked pressure dependence and typically increase rather than decrease with increasing temperature. This more complex behavior reflects the combined influence of repulsive and attractive intermolecular interactions on the equilibrium volume of fluids, while the molecular hard-sphere volumes described in this work represent only the size of molecular repulsive hard cores. Furthermore, unlike previously derived *van der Waals” volume i n c r e m e n t ~ , l . ~which 2 ~ represent the space-filling volumes of molecular subgroups, hard-sphere volume increments represent the contributions which molecular subgroups make to the overall effective hard-sphere volume of a parent compound. This distinction may seem subtle, but the corresponding subgroup volume increments are in fact found to be significantly different, in spite of the fact that the molecular volumes predicted using the two types of increments are quite similar. A more detailed discussion of these interesting disparities and correlations will be returned to in section 4, after briefly describing the procedure
used to extract molecular hard sphere diameters from equation of state data, in section 2, and the translation of these into subgroup volume increments, in section 3, which also includes attractive force and temperature derivative correlations.
2. Equation of State Data Analysis The effective hard-sphere diameters of atoms and molecules are determined by fitting high pressure equation of state data to a perturbed hard-sphere expression of the CarnahanStarlingvan der Waals (CS-vdW) form (eq 1). In this equation theusual
van der Waals excluded volume contribution is replaced by the CarnahanStarling hard-sphere equation of ~ t a t e . ~The 5 result is an equation which very accurately represents both excluded volume and attractive mean field interactions in real fluids, up to their freezing den~ities.~.6This equation relates the compressibility factor, 2,or pressure, P, of a fluid to its packing fraction, 7,temperature, T,and attractive mean field coefficient, T (ke is Boltzmann’s constant). The packing fraction, 7, is in turn related in a simpleway to the hard-sphere diameter, u, hardsphere volume, VHS,and number density, p, of the molecules in the system: ” 79 = ( = PV,, (1b) The attractive mean field coefficient, T, which has units of temperature and is roughly equal to the Boyle temperature of the l i q ~ i d , ~represents 6 the influence of long-range dispersive and multipolar intermolecular interactions.4 Previous comparisonsof eq 1with high-pressure compressibility data for a variety of compounds have shown that both a and T can be treated as density independent (and only mildly temperature dependent) constants.6 Thus, u and T may readily be determined by fitting isothermal high-pressure equation of state (compressibility)data. In practice, two fitting procedures may be conveniently used to determine these parameters. One involves directly fitting pressuredensity data to eq 1, and the other simultaneouslyfitting the density and isothermalcompressibility, & = (l/p)(dp/dP)T, of a liquid at a single pressure to eqs 1 and 2. Details of the procedures used to extract u and 7 values using
QQ22-3654/93/2Q97-7136%04.00/00 1993 American Chemical Society
Molecular Hard-Sphere Volume Increments
The Journal of Physical Chemistry, Vol. 97, No. 29, I993 7737 in this work, can yield accurate estimates of the effective hardsphere volumes of a large variety of other compounds.
these two methods have been described previously.6 The results of performing such fits to experimental equation of state data are collected in Table I. The absolute precision of the hard sphere diameters reported in Table I is estimated to be of the order of *I%. This estimate is based on a comparison of the diameters in Table I with those derived for some of the same compounds using independent high pressure equation of state data.6 The agreement between the two sets of hard sphere diameters are in all cases within f l % (and typically within less than &OS%). This uncertainty thus reflects the typical experimental imprecision of high pressure equation of state measurements. The temperature derivativesof u and T have also been included in Table I for those compounds for which temperature dependent data (over a greater than 5 OC range) was available. These derivatives can be used to reproduce u and T values over a temperature range of the order of f 1 0 OC about the 20 OC reference temperat~re.~'Alternatively [ T du/dTjzo and [ T dT/ dTj20 can be converted to the following expressions, which more accurately represent u and T over a wider temperature range+
7(T)
= To[ 1
+;I
The parameters uo, TO,TO,and T, in the above expressions are related to those in Table I, which pertain to 20 OC, by6,48
+ (4a) l + [ ~ ~ ~ / 1 2 ] / [ T d u / c W J ~(4b) ~)
uo = (a20)7/6{u20 1 2 [ ~ d a / d ~ ] , , ) - ' / ~
fig= - m
{
T , = -(293.15 K){1
3. Determination of Volume Increments Hard-sphere volume increments represent the contribution made by molecularsubgroupsto the effective hard-spherevolume of a parent compound. These allow the predictionof hard-sphere volumes for compounds directly from their chemical structure, without the need for isothermal compressibilityor high-pressure equation of state data. To create a table of such volume increments, a convenient set of molecular subgroups must first be selected. For example, only two subgroups, a methyl group CH3 and a methylene group CH2, may be used to describe all the possible linear alkanes. Two additional groups, a CH and a C group, must be added in order to describeall the possible branched alkanes. There is a certain amount of trial and error, as well as chemical intuition, which must be involved in the selection of a minimum set of volume increments to represent a large variety of molecular structures. For instance, comparisonof hard-sphere volumes for cyclic (and polycyclic) alkanescontainingfive- and six-membered carbon rings, suggests that the methylene (CH2) groups in these compounds are sufficiently different from methylene groups in linear (and branched) alkanes to warrant inclusion of two types of methylene groups in the increment tables. Similarly,thevolume increments for short-chain compounds, which can be described as substituted methanes, are found to be systematicallydifferent from the corresponding group increments in longer chain compounds. On the other hand, distinctions made in previous volume increment tables$* between primary, secondary, and aromatic halogen substituents, do not appear to be warranted by the current analysis. Thus, for example, the hard-sphere volume increment contributed by, each of the chlorine atoms in CH3C1, CH2C12 and CHCl3 and CC14are found to be essentially identical. Once an appropriate set of subgroups has been selected then any compound of interest may be represented by a list of its subgroups. The number of occurrences of each subgroup, nt, in the compounds may be treated as variables, and the volume increments, ui,as coefficients in the following linear expression for the effective hard-sphere volume, VHS,of each molecule:
+ T ~ ~ / [ T ~ T / ~ T ] (4d) ~~]-~
Finally, the parameters in Table I may also readily be used to estimate Lennard-Jones parameters, ULJ and ~ L J ,by way of the followingexpressions derived from the analysisof Lennard-Jones fluid computer simulation measuremenW6
Naturally, the physical significance of these Lennard-Jones parameters cannot be taken too seriously for real atoms and molecules whose interactions may at best only roughly be approximated by a Lennard-Jones potentials. Nevertheless, the above relations do offer a convenient estimate of effectiveLennardJones constants for molecules and, at least for atoms and small molecules, these estimates have been found to agree very well with those obtained by other meam6 Several alternative procedures for determining hard-sphere diameters, u, and attractive mean-field coefficients, T , in the absence of compressibilitydata have previously been described.6 These include using linear correlations between hard sphere volumes and space filling volumes obtained from "van der Waals" volume increment tables.42 As shown in the following section, such correlations, along with the volume increment tables derived
The task of creating a table of subgroup increment volume is thus reduced to that of deriving values for the coefficient, ut, in eq 6 . This is done using a least-squares fitting procedure,49 with the total volumes, VHS,and the subgroup Occurrence numbers, nt, as input parameters and the volume increments, u1, as output parameters. In this work a commercial software package (Igor by WaveMetrics, P.O.Box 2088, Lake Oswago, OR 97035) has been used to generate the appropriateuser defined multiparameter function and volume increments. Short-ChainCompounds. Short-chain compounds are defined here as any compounds which have exactly four atoms (or pseudoatomic groups such as OH, CN, and CH3) attached to a central carbon (or silicon) atom. These include any compounds traditionally defined as a substituted methanes (or silanes) as well as short-chain linear and branched alkanes such as ethane, propane, and neopentane (which can be viewed as methane with one or more methyl substituents). The increment volumes for these compounds at 20 OC are given in Table 11. These incrementsare obtained from correlations involving 69 different compounds (from Table I and ref 6 ) . Most of the increments listed in Table I1 have been derived from correlations involving two or more compounds containing the corresponding group. The only exceptions are the CN and Si group volumes which have each been derived from single compounds (CH3CN and (CH3)4Si, respectively).
Ben-Amotz and Willis
7738 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 TABLE I: Molecular Parameters from Fit to CW-vdW Equation of State (Evaluated at 20 "C) compound ~ 2 (A) 0 [Tdu/dTlio (A) 720 (K) [Tdr/dT]u, (K) T r a n g e ("C) ethane n-heptane n-octane n-nonane n-decane n-tridecane 2,3-dimethylbutane 2,2,4- trimethylpentane cyclohexane cis-decahydronaphthalene
trans-decahydronaphthalene spiro(4,S)decane spiro( 5,S)undccane cis-octahydroindene trans-octahydroindene 1-hexene 1,Chexadiene 1,s-hexadiene 2,4-hexadiene acetone 2-butanone 2-pentanone 2-hexanone 2-heptanone 2-octanone methyl acetate ethyl acetate 1-hexanol 1-heptanol 1-octanol 1-nonanol diethyl ether benzene toluene-ds chloromethane bromomethane iodomethane dichloromethane trichloromethane chlorodifluoromethane dichlorodifluoromethane bromoethane bromopropane bromobutane bromopentane bromohexane bromoheptane ammonia 0
4.23 1 6.264 6.561 6.836 7.081 7.743 5.926 6.539 5.646 6.618 6.666 6.645 6.891 6.393 6.388 5.839 5.719 5.730 5.710 4.809 5.235 5.623 5.963 6.280 6.568 5.012 5.421 6.113 6.424 6.696 6.960 5.388 5.277 5.680 4.156 4.324 4.573 4.635 5.052 4.436 4.906 4.850 5.300 5.668 5.998 6.298 6.575 3.255
-0.1315
-0,2066 -0.1818 -0.1325 -0.1324 -0.1417 -0.1688 -0.1318 -0.1709 -0.2662 -0.1359
-0.1442 -0.1684
-0.3564 -0,1307 4.1556 -0.2016 -0,2379 -0.1898 -0.2472 -0,2007 4.3342 -0.3298 -0,2249 -0,1951 -0.2427 -0.2527 -0.2600 -0.2830 -0.2358
1009.0 2200.7 2435.8 2632.0 2836.6 3349.9 1978.0 2278.2 2156.9 3176.9 3078.4 3083.2 3346.6 3071.5 2774.8 1923.4 1927.7 1896.2 2041.7 1717.3 1951.9 2154.4 2365.3 2571.9 2768.4 1898.2 2029.9 2630.7 2882.4 3058.5 3280.3 1696.2 2121.9 2350.2 1339.2 1530.3 1772.8 1821.0 1951.8 1202.7 1304.2 1751.1 2056.8 2262.2 2439.6 2606.3 2764.2 1223.7
-124.8
100-400 20-25 20-25 20-25 20-25 35-75 3683 3683 36-80 15-115 15-115 15-115 15-115 15-115 15-115 20-25 20-25 20-25 20-25 20-25 20-25 20-25 20-25 20-25 20-25 -2040 -20-40 20-25 20-25 20-25 20-25 0-35 37-87.5 -35-50 -2040 -2040 -2040
-1614.8 -695.6 -661.9 -380.8 -578.3 -697.4 -624.6 -615.3 -1368.1 -468.9
-308.9 -459.4
-637.8 410.3 423.9 -199.5 -272.1 -144.6 -536.3 -434.0 -515.4 -550.6 -38 1.9 -345.0 -591.5 -602.4 -606.2 -603.8 -223.2
0-55 0-5 5 -20-40 -20-40 -2040 0-50 0-50 0-50
0-50 0-25 -2040
P r a n g d (atm) 2000-9000 1 1 1 1 200-5000 l-6o00 1-6000 500-1200 400-3600 400-3600 400-3600 400-3600 4W3600 400-3600 1 1 1 1 1 1 1 1 1 1 1-1600 1-1600 1 1 1 1 1 20-1 600 1-2000 1-1600 1-1600 1-1600 1-1600 1-1600 1-1600 1-1600 1-1600 10-5000 10-5000 10-5000 10-5000 10-so00 2-1600
ref 51 52 52 52 52 53 54 54
55 56 56 56 56 56 56 57 57
57 57 52 52 52 52 52 52 58 59 52 52 52 52 60
55 61 58 58 58 62 62 58 58 58 63 63 63 63 63 64
Temperature and pressure range of experimental equation of state data used in fits.
e
TABLE 11: Hard-Shere Volume Increments for Short-Cham Compounds (at 20 " ) CH3 CHz CH C
F CI
19.64 16.45 12.37 9.23 7.83 18.20
Br
I OH CN Si
23.17 30.46 9.74 20.38 29.70
Figure 1 illustrates the predictive accuracy of the molecular volumes obtained using the volume increments in Table 11. The lower portion of the figure represents the absolute deviation of the predicted molecularhard sphere volumes from those obtained using compressibility data. The upper portion of the figure represents the percent deviation of the corresponding effective hard spherediameter, u = ( ( 6 / ~VHS)'/~. ) The resulting standard deviation of the calculated hard-sphere volumes is about i l % , and that for the diameters is less than f0.4%. Figure 2 shows a comparison of the predicted total molecular hard sphere volumes for the short chain compounds with the volumes obtained using Bondi's "van der Waals"volume increment tables.42 Clearly there is an excellent correlation between the
TABLE III: Hard Sphere Volume Increments for
Long-Cham Compounds (at 20 "C) CH3 CHI CH C CHz (Cycle-) CH (cycle) c (cycle) CHWH CH4H2
17.71 18.81 19.79 17.09 15.73 13.54 13.06
CsH5 (phenyl) c1 Br
I -0OH
0c-o
3 1.03
C=O
30.37
Si
78.38 15.02 20.26 28.36 8.83 7.49 29.85 19.60 30.81
two measures of molecular volume. The solid line in this figure represents a linear fit to the data points, with a slope of 1.001, an intercept of -4.48 A3, and a correlation coefficient of 0.992. This correlation is sufficiently good to allow the translation of Bondi's "van der Waals" volumes into molecular hard-sphere volumes with an accuracy nearly as good as that obtained using the hard-sphere volume increments in Table I1 (The standard deviation of the hard-sphere diameters predicted in this way, relative to the equation of state derived molecular diameters, is fl%).
Molecular Hard-Sphere Volume Increments
I
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7739
,ILong Chaln
Hab-Alknnes
Molecules
0
4
Alkanes
x Cycbalbnes
- 7
5
4
3
6
Hard Sphere Diameter, u / A
--7 I
.
0
.
.
I
.
20
.
.
I
.
40
.
.
I
,
.
60
,
I
.
,
EO
.
700
Hard Sphere Volume, V, / A3
Figure 1. Deviation of the molecular hard sphere diameters(upper panel) and volumes (lower panel) predicted using the short chain volume increments, ut, listed in Table I1 from those derived directly from compressibility data (Table I and ref 6 ) . %
.
Short Chaln Molecules
700 200 300 Hard Sphere Volume, V, / A3
0
Figure 3. Deviation of the molecular hard sphere diameters(upperpanel) and volumes (lower panel) predicted using the long chain volume increments, ut, listed in Table 111 from those derived directly from compressibility data (Table I and ref 6). Long Chaln Molecules
v,
0
x 0
20
40
60
Alcohols Halo-Alkanes
--
80
100
"van der Waals" Volume, VmdI/h'
Figure 2. Correlation between the molecular hard spherevolumes, Vm, for short chain compounds predicted using the incrementsin Table I1 and the corresponding molecular "van der Waals" volumes predicted using Bondi's volumeincrcments.42 Note that Bondi's volumeincrements,which are presented in units of cm3/moI, must be converted to A3 (multiplied by 1.66) before applying the linear correlation expression given in this figure.
Long-chain Compounds. Long-chain compounds are defined here as any compounds containing at le@ one chain of four or more atoms, other than hydrogen. Thus, for example, n-butane and 1-propyl chloride as well as star alkanes such as 2,2diethylpentane are classified as long-chain compounds. The increment volumes obtained from a least-squares fit to the hard-sphere volumes of these compounds at 20 OC (from Table I and ref 6) are given in Table 111. Most of the increments have again been obtained from correlations involving two or more compounds containing the corresponding group. The only exemptions in this case are the 0, I, and Si increment volumes which have each been derived from a single compound (diethyl ether, 1-iodobutane,and tetraethylsilane, respectively). Figure 3 illustrates the predictive accuracy of the molecular volumes obtainedusing the volume increments in Table 111. The standard deviations of the calculated hard-sphere volumes and diameters are about & l % and less than &0.4%, respectively. Figure 4 shows a comparison of the molecular hard-sphere volumes of these long-chain compounds with the corresponding "van der Waals" volumes derived from Bondi's increment tables." Once again the excellent correlation between the two measures of molecular volume is apparent. The solid line in this figure represents a linear fit with a slope of 1.095, and intercept of -13.10 A', and a correlation coefficient of 0.9997. The dashed line in Figure 4 represents the correlation obtained for short chain compounds (see Figure 2). The long-chain and short-chain correlations coincide for molecules with hard sphere volumes of about 90 A'. The correlation is again sufficiently good to allow the translation of Bondi's "van der Waals" volumes into molecular
-
7.093
vmndl.72.79
/
4
Alkanes
oL.:Y,, , , , , , I , , , , , , , , , I , , , , I , , , , 0
100
200
"van der Waals" Volume, V,,
I,,
,
.A
300
/
A3
Flgure 4. Correlation between the molecular hard sphere volumes, Vm, for long chain compounds predicted using the increments in Table 111 and the corresponding molecular "van der Waals" volumes predicted using Bondi's volume increments.42 The dashed line representsthe short chain molecule correlation (see Figure 2). hard sphere volumes with an accuracy nearly as good as that
obtained using the hard-sphere volume increments in Table I11 (with a diameter standard deviation of about f l % ) . Other Compounds. Most molecules representedin Table I (and ref 6) have been used in generating the volume increments listed in Tables I1 and 111. Exceptions include the rare-gas atoms, as well as N2, CO, C02, CS2, NH3, water, methane, ethylene, propylene, acetone, pyridine, benzene, and methylcyclohexane. These compounds were excluded either because they represent unique compounds that are not members of a longer chain series or because they cannot readily be described as a sum of the subgroupsin Tables I1 and 111. For example, methylcyclohexane is not well described as the sum of cyclic CHz, CH, and a methyl group, unless a unique methyl group volume is assigned to methyl substituents on cyclic alkanes. On the other hand, the hardsphere volumes of such compounds can be accurately estimated using Bondi's volume increments (includingcrowding corrections for ring compounds) along with the linear correlation expression given in Figure 4. This procedure yields a hard sphere diameter of 6.034 A for methylcyclohexane, which differs by only 0.2% from the 6.021-A diameter derived directly from compressibility dataa6 AttractiveMean-FieldCoefficient Increments. The same fitting procedure used to determinevolume increments can also be applied to the determination of incrementalcontributionsto the attractive mean field coefficient, T (appearing in eq l a and Table I). The results of such an analysis, for the same set of short-chain and long-chain compounds used to derive the volume increments, are collected in Table IV.
Ben-Amotz and Willis
1140 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 TABLE IV: Attractive Mean-Field Coefficient Increments (at 20 "C) short long long short chain chain chain chain 555
212 -223 -821
627 187 -302
-670 341 283 -198 387 75 1 1696
F
340 730
801
Br
980
I
1171
1044 1124
910
1102
c1 -0-
68
OH
oc=o
C=O CN Si
/L
Molar Volume Correlatlon
616
547 1025 -506
-535
The standard deviation of the parent molecule 7 values derived from the increments in Table IV is less than f5% (with a maximum deviation of less than &lo%),when compared with molecular 7 values obtained directly from the analysis of equation of state data. Although these deviations are greater than those for the volume increments, previous equation of state studies suggest that the resulting errors in thermodynamic property calculations are comparable to those introduced by the uncertainty in the volume increments.6 One of the striking features of the 7 increment results is that some of the increments are negative, suggesting that incorporation of the corresponding groups into a compound has the effect of decreasing the overall attractive intermolecular interaction strength of the compound. Notice that the only groups with such negative T increments are those with three or more bonding sites. This suggests the followingplausible explanation. Stericcrowding of the groups attached to groups with multiple bonding sites may be expected to reduce the surface contact area of the attached groups (relative to the contact area they would have in less sterically confined environments). Thus the net effect of introducing multiply bonded groups into a parent molecule is to reduce the exposed surface area and therefore the intermolecular attractive interaction strength of the parent molecule, and thus to contribute a negative 7 increment. Temperature Derivatives of u and 7. The softness of intermolecular interaction potentials gives rise to a slight temperature dependence of the effective hard sphere diameters and the attractive mean field coefficients of polyatomic molecules. The derivatives of g and 7 obtained from equation of state data, however, are in general quite noisy and only roughly correlated with other molecular parameters. The results in Table I (and ref 6) suggest that the following expressions may be used for rough estimates of the corresponding derivatives:
[Td~/dT],~=-0.15*0.5~ [ T ~ ] 2 0 = - ( 7 2 0 / 1 0 5 ) 2300K +
(74 (7b)
The derivative of the hard sphere diameter is invariably negative, reflecting the greater interpenetration of molecules at high temperature. There does not, however, appear to be any systematic dependence of this derivative on the size of the compound. The derivative of the attractive mean field coefficient is usually, but not always, negative, and its magnitude increases roughly quadratically with the attractive mean field coefficient.
4. Discussion
Correlation of Hard-Sphere and Molar Volumes. The significant temperature and pressuredependenceof fluid molar volumes clearly pose a problem in using molar volume based increments as measures of molecular size, particularly for supercritical fluids whose molar volumes can vary continuously between that of a gas and a liquid over a narrow pressure range. The physical
50
100
I50
Hard sphere volume,
200
A3
Figure 5. Correlation between liquid molar volumes (at 20 "C) and
molecular hard-sphere volumes for n-alkanes and primary substituted linear alkanes (see discussion in section 4).
Bondi Volume Increment /
A3
Figure 6. Comparison of the hard sphere volume increments in Tables
I1 and I11 with Bondi's "van der Waals" volume increments. The dashed line is that obtained from the molecular volume correlation for short chain compounds shown in Figure 2 (see discussion in section 4).
basis for the sensitivity of molar volumes to pressure and temperature is clearly the competition between repulsive and attractive intermolecular forces. Thus, for example, molecules of the same size but different attractive interaction strength are expected to have different molar volumes. This expectation is born out by results shown in Figure 5 , which illustrates the correlation between molar and hard-sphere volumes for four classes of compounds. The molar volumes of the less strongly interacting alkanes are larger than those of comparably sized haloalkanes, which are in turn larger than the strongly associated alcohols. Thus although there is a good correlation between molecular size and molar volume for each class of compounds, the correlation is different for different classes, in a way that reflects differences in their attractive interaction strengths. Correlation of Hard-Sphere and van der Waals Volume Increments. As illustrated in Figures 2 and 4, Bondi's molecular "van der Waals" volumes correlate very closely with effective hard sphere volumes derived from compressibility data. In spite of this good correlation, however, the corresponding volume increments themselves are not nearly as well correlated with each other. This somewhat surprising circumstance is illustrated in Figure 6,which shows a plot of the poor correlation between hard sphere volume increments and Bondi's "van der Waals" volume increments (from Tables I1 and III).42 The dashed line in this figure represents the best-fit correlation between the hard-sphere and "van der Waals" volumes of short-chain compounds (see Figure 2). Given this poor correlation, it is indeed difficult to understand how the resulting total molecular volumes could be as well correlated as they clearly appear to be in Figures 2 and 4. The explanation may rest in the different physical significance of the two types of volume increments. Whereas Bondi's "van der
Molecular Hard-Sphere Volume Increments Waals” volume increments represent the space-filling volumes of the corresponding molecular groups, hard-sphere volume increments represent the contributions which these groups make to the overall hard-spherevolumeof a parent molecule. Thus groups of the same “van der Waals” volume may contribute more or less significantly to the total hard-sphere volume of a molecule, depending on the tertiary structure which they impose on the molecule. For example, although the “van der Waals” volume increment for a methylene (CHz) group is smaller than that for a methyl (CH3) group, as expected from the difference in the number of atoms in the two groups, the corresponding hard sphere volume increment for CH2 is larger than that for CH3. This difference may arise from the fact that a methylene group is attached to two other groups while an methyl group is attached at a terminal end of the molecule. Thus the local structural rigidity imposed on a molecule by the addition of a methylene group producesa larger increasein the overall effective hard spherevolume of the molecule than does the addition of a terminal methyl group. Alternatively, differences between hard sphere and “van der Waals” volume increments may be viewed as resulting from differences in symmetry of the parent compounds. Lower symmetry implies a larger surface area to volume ratio and therefore a relatively larger volume excluded to the centers of neighboring molecules. This suggesting is consistent with previous observations that the effective hard-sphere diameters of prolate and oblate hard ellipsoids are larger than those of hard spheres of the same volume.6 Thus, groups which impose a more asymmetric structure on a parent compound may be expected to contribute more significantly to the effective hard-sphere volume of the compound. On the other hand, caution should be exercised in invoking such simplistic explanations for observed differences between hard sphere and “van der Waals” volume increments. If it is indeed the case than there is an intimate relationship between molecular structure, or symmetry, and the incremental contribution of subgroups to the overall hard-sphere volume of a molecule, then the question that might better be asked is; Why should a universal volume increment analysis work at all? In other words, shouldn’t every molecule be expected to have a unique set of volume incrementsassociatedwith its particular tertiary structure? From this perspective, and from the standpoint of practical utility, the most significant result of the present work is the finding that it is in fact possible to accuratelyrepresent the effective hard sphere volumes of a wide class of compounds using a relatively small set of universal subgroupvolume increments,although the underlying reason for this success, and thus the answer to the above question, is not altogether clear. Practical Applications. Several practical approaches to the determination of molecular hard-sphere volumes have emerged from this work. The simplest involves summing the appropriate volume increments from Table I1 or 111. In cases where the appropriate volume increments do not appear in these tables, several other procedures may be employed. One involves direct determination of a molecular hard-sphere diameter by fitting equation of state data (as described in section 2). Alternatively, correlations to volumes derived from Bondi’s “van der Waals” volumes increments may be used to estimate molecular hard sphere volumes (as described in section 3). For compounds containing groups not included in either the hard sphere or the “van der Waals” volume increment tables, it is probably safer to use the “van der Waals” volume increment tables as a basis for extrapolation of missing increments, since these are more nearly additive in the sense that differencesbetween group volumes are proportional to the sizes and number of atoms in the groups. Finally, extending somewhat beyond the central focus of this paper, the attractive mean-field coefficient, T , and the corresponding molecular increments appearing in Table IV, as well as
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7741 the derivative of u and T with respect to temperature given in Table I (or estimated using eq 7) may find a wide variety of applications. The temperature derivative of u allows the estimation of small changesin molecular sizewith temperature, which are not readily obtainable from other measures of molecular volume. Furthermore, the combination of parameters appearing in Table I can be used, in conjunction with eqs 1 4 , to predict the density, compressibility6and other thermodynamicproperties% of fluids as a function of pressure and temperature. More generally,any theory of condensed phase chemistry must require, at a minimum, knowledge of the strength of repulsive and attractive intermolecular interactions, such as, for example, the effective Lennard-Jones parameters, uu and q,~,which can be estimated using eq 5.6 Thus it is hoped that the results of this work will complement the increasingly sophisticated abilities of computers and fundamental statistical mechanics, in allowing the convenient estimation of molecular interaction parameters for use in modeling the properties of chemical systems.
Acknowledgment. We would like to thank Joseph Bizzelli for suggesting the conversion of molecular hard-sphere volumes derived from equation of state data to subgroup volume increments. The support of the Exxon Education Foundation and the Office of Naval Research (Contract No.N00014-92-1559) is gratefully acknowledged. References and Notes (1) Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses; John Wiley and Sons: New York, 1968. (2) Reid, R. C.; Prausnitz, J. M.;Poling, B. E. The Properties of Gases and Liquids, 4th 4.;McGraw-Hill Company; New York, 1987. (3) Exner, 0.In Organic High Pressure Chemistry;le Noble, W. J., Ed.; Elsevier: New York, 1988; p 19. (4) Longuet-Higgins, H. C.; Widom, B. Mol. Phys. 1964,8, 549. (5) Chandler, D.; Weeks, J. D.; Anderson, H. C. Science 1983,220,781. Barker, J. A.; Henderson, D. Rev. Mod. Phys. 1976, 48, 587. (6) Ben-Amotz, D.; Herschbach, D. R. J . Phys. Chem. 1990,94, 1038. (7) Bryan, P. F.; Prausnitz, J. M.Fluid Phase Equilibria 1987,38,201. ( 8 ) J o s h , C. G.; Gray, C. G.; Chapman, W. G.; Gubbins, K. E. Mol. Phys. 1987,62, 843. (9) Oellrich, L. R.; Knapp, H.; Prausnitz, J. M. Fluid Phase Equilibria 1978, 2, 163. (10) Jackson, G.; Chapman, W. G.; Gubbins, K.E. Mol. Phys. 1988,65, 1. (1 1) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. Eng. Chem. Res. 1990,29, 1709. (12) Reiss, H.; Frisch, H. L.; Helfand, E.; Lebowitz, J. L. J . Chem. Phys. 1960, 32, 119. (13) Snider, N. S.; Herrington, T. M. J. Chem. Phys. 1967, 47, 2248. Staveley, L. S. K. J. Chem. Phys. 1970, 53, 3136. (14) Wilhelm, E.; Battino, R. J . Chem. Phys. 1971, 55, 4012. (15) Johnston, K. P.; Eckert, C. A. AIChE J. 1981, 27, 773. (16) Shucla, K. P.; Lucas, K. Fluid Phase Equil. 1986, 28, 211. (17) Yosim, S. J.; Owens, B. B. J . Chem. Phys. 1963,39, 2222. (18) Boublik, T. J. Chem. Phys. 1970, 53, 471. (19) Wilhelm, E.; Battino, R. J . Chem. Phys. 1973, 55, 4012. (20) Cohen, M.H.; Turnbull, D. J . Chem. Phys. 1959, 31, 1164. (21) Hanky, H. J. M.;McCarty, R. D.; Cohen, E. G. D. Physica 1972, 60, 322. Hanky, H. J. M.;Cohen, E. G. D. Physica 1976,83A, 215. (22) van Loef, J. J. Physica 1974, 75, 115. (23) Chandler, D. J. Chem. Phys. 1975,62, 1358. (24) de Santis, A.; Ricci, F. P. Physica 1976,83A, 425. (25) Jonas, J. Acc. Chem. Res. 1984, 17, 74 and references therein. (26) Dymond, H. J. Chem. Soc. Rev. 1985, 14, 317. (27) Li, S.F. Y.; Trengove, R. D.; Wakeham, W. A.; Zalaf, M.Int. J . Thermoohvs. 1986. 7.273. (28) ‘Dobrosavljkvb, V.; Henebry, C. W.; Stratt, R. M. J . Chem. Phys. 1988. 5781. -. - -, 88. .- ... (29) Schweizer, K. S.; Chandler, D. J. Chem. Phys. 1982, 76, 2296. (30) Zakin, M.R.; Herschbach, D. R. J . Chem. Phys. 1986.85, 2376. Zakin, M. R.; Herschbach, D. R. J . Chem. Phys. 1988, 89, 2380. (31) Ben-Amotz, D.; Zakin, M. R.; King, H. E., Jr.; Herschbach, D. R. J . Phys. Chem. 1988, 92, 1392. (32) LeSar, R. J . Chem. Phys. 1987.86.4138. (33) Ben-Amotz, D.; LaPlant, F.; Shea, D.; Gardecki, J.; List, D.
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