A Simple "Back of the Envelope" Method for Estimating the Densities and Molecular Volumes of Liquids and Solids Gregory S. Girolami The University of Illinois at Urbana-Champaign, 505 South Mathews Avenue. Urbana, IL 61801 Liquids are commonly measured out by volume rather than bv mass. so to establish the number of moles of liauid used wc must'know the llqurd's density. Usually, this is n;,t a problem because the needed value cun uften be found in reference texts, such as the CRC handbook, in supplier's catalogues, or in the original literature. Occasional1y, however, the density of a liquid is not available, and one is left with the task of measuring the value experimentally. Sometimes, the desired value wn.k estimated bv mterpolaiion. For example, the density of din~ethylethylphosphine1'M~k:t wits recently estimated bv interiolation of the known densities of trimethyl- an; triethilphosphine (1).Usually, no such interpolative method is suitable because appropriate reference compounds cannot be identified. Over the years, several sophisticated methods have been developed that can be used to calculate the densities of organic and inorganic species, and recent developments in this area have dealt with the develo~mentof computerbased algorithms ( 2 4 ) .Here I describe a simple and remarkably accurate method for estimating the density of liquids "on the back of a n envelope". The method described has proven to be very useful in our own research. Despite its simplicity, it is accurate to within 0.1 g ern3 for a wide variety of liquids, and oftengives estimates that are within 2 or 3% of the actual density. I also describe a modification of the method that allows one to calculate the molecular volumes of solids for use in crvstalloma~hic ., . investieations. The molecular weight I W )of a liquid is easily calculated from its molecular formula; the molecular weight is related to the mass of a n individual molecule by ~ v o ~ a d r ocon's stant. Because density is equal to mass divided by volume, we need only find a simple method for estimating the volume of a n individual molecule. This can be done a s follows. Assume that a hydrogen atom has a volume of 1. On this scale, carbon and the other elements in its row have relative rolumes of about 2, whereas silicon and the other elements in its row have relative volumes of about 4. A summary of the relative volumes of all of the elements is shown in the table a t the top of the next column. The scaled volume of a molecule is then simply the sum of the volumes of its constituent atoms. If we denote this sum by the symbol Vs then the density of a given liquid can be calculated from the following simple formula.
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element
rel. volume
H
1st short period, Li - F 2nd short period, Na - CI I st long period, K - Br 2nd long period, Rb - l 3rd long period. Cs - Bi where the factor of 5 allows the density to be expressed in units of g cm". Example Density Calculations For example, the density of PMezEt can be calculated from eq 1a s follows. Dimethylethylphosphine (C4H11 P) W=90 Vs=(4x2)+(11xl)+(lx4)=23 d = ---- -0.78~cd 5x23
interpolated value (1):0.76 g emA
To a first approximation, only the stoichiometry of the liauid is i m ~ o r t a nand t not its actual molecular structure. ~ o n s e ~ u e n iisomers l~, of a compound should all have approximately the same density. This often turns out to be the case. For example, acetone has a density of 0.79 g c d , whereas propionaldehyde has a density of 0.80 g cm3. The formula predicts 0.83 gem" for both. Increased Densities Certain classes of liquids have densities that are consistently higher than those predicted by the formula. Among these (not surprisindy) are molecules that have hydrogenbonding groups or rings. More accurate densities-for &ch liquids can be obtained if the density calculated from eq 1 is increased by 10% for each of the following structural features present in the molecule. a hydmxyl group
a carboxvlie acid -m 0 u .D a pnrnary or secondary amino group an arnide group (includmgN-subsiituted amides1
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For fused rings, the density should be increased by 7.5%per fused ring, so the density of naphthalene derivatives should be increased by 15%. If a molecule contains several of the structural features in the list above, the density should be increased by no more than 30% of the density calculated from eq 1. The following examples illustrate these rules. Cyclohexanol (C6H120) A ring and one hydroxyl group; correction factor = 1.2
1.2 x 100 d=-- 0.92 gem3 5x26
actual value: 0.962 g em3 Ethylenediamine (C2HeN2)
Correlation between observed densaies and those calculated from eq 1 after correcting for the presence of functional groups. The line drawn is the least squares fit (y = 1 . 0 1 ~ 0.006); R' = 0.982.
Two primary amine groups; correction factor = 1.2 W=60 Vs=(2x2)+(8x1)+(2x2)=16
actual value: 0.899 g em3 Sulfolane (C4&02S) Aring and two S=0 bonds; correction factor = 1.3 W= 120 Vs=(4x2)+(8x1)+(2x2)+(1x4)=24
actual value: 1.262 g cm3
Accuracy of the Method The agreements between the observed and calculated densities for the compounds above are excellent, but from these exam~lesalone it is not nossible to evaluate whether the agreemknts are fonuitous'or generally applicable. The reference book The Chemist's Comoanion eives a list of 180 organic solvents and their densities.' ~flthese,the solids (mp > 30 "C) and gases (bp < 40 "C)were excluded from c~nsideration.~ The experimental densities of the remaining 166 liquids were compared with the densities calculated from eq 1 after correction for the presence of groups in the list above. The correlation is excellent (see the figure); in only two cases (acetonitrile and dibromochloromethane) does the error exceed 0 . 1em3. ~ The rms error for all 166 liquids is only 0.049 gem . The table in The Chemist's Companion also lists 23 inorganic liquids, and eq 1yields densities that are accurate to within 7%for nearly all of these.4 Remarkably accurate densities are calculated for such diverse inorganic liquids as those given in the table below.
1-Bromonaphthalene(CloHIBr) Two fused rings; correction factor = 1.15 W = 207
vs=(lox2)+(7xl)+(lx5)=32
actual value: 1.483 g em3 'Molecules that have two oxygen atoms doubly bonded to sulfur (sulfones)count as having two S=0 groups and should have their densities increased by 10%for each oxygen atom, or 20%in all. 2Gordon,A. J.; Ford, R. A. The Chemist's Companion;,Wiiey:New York; 1972, p 4. Three errors in the table (the densities of Paminopropane and iron pentacarbonyl, and the molecular weight of CC13F) were wrrected. 3The room temperature densities of liquids with low boiling points generally are less than those predicted by the formula. This is no surprise: As liquids approach their boiling points, the intermolecular forcesweaken considerably. 4For inorganic acids, the density should be increased by 10%for each oxygen atom that bears an acidic proton. Interestingly, the predicted density of water is 0.99 g cm3.
calculated Iron pentacarbonyl tin tetrachloride bromine fluorosulfonic acid
1.56 2.22 3.20 1.69
experimental 1.49 2.23 3.12 1.73
Certain inorganic liquids, such as SO3 and SbF5, are extensively associated via oxygen or halogen bridges. (SO3is a cyclic trimer, whereas SbF5 is polymeric in the liquid state.) These liquids should be considered inorganic ring compounds and the densities calculated from eq 1 should be increased by 10%. Derivation of Equation 1 The relative volumes of the elements and the factor of 5 in the denominator of eq 1 can be derived as follows from tables of van der Waals radii. Specifically, the van der Waals radii of nitrogen, oxygen, and fluorine are 1.5, 1.4, and 1.35 A (5).However, if we make the simplification of adopting a common van der Waals radius for the elements C, N, 0 , and F, then a reasonable value is the average of Volume 71 Number 11
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963
the van der Waals radii of nitrogen and oxygen: 1.45 A. Taking 1.15 A for the van der Waals radius of a hydrogen atom, we can compute the relative volumes of these elements a s in the table below where the scale factor relating the last two columns is 6.40 A3. element H
1st short period
rvdW 1.15 1.45
4
$r"d 6.40
12.8
dm
re!, vol. 1.OO 2.00
Similar simplifications can be made to deduce average van der Waals volumes for the heavier congeners, and it is found that their scaled volumes are near those given in the full table presented previously. Note that the average van der Waals radius of the first short-period elements is slightly smaller than the van der Waals radius of carbon; the van der Waals radius of 1 . 1 5 A listed above is also slightly smaller than Pauling's 1.2 A. These features make it unnecessary to correct for overlap of the van der Waals spheres involving atoms that are covalently bonded to each other. The volume of a molecule in cubic h g s t r o m s will then he given by 6.40Vs, where Vs is the sum of the relative volumes of the constituent atoms. However, Pauling's van der Waals radii were derived from solid state data, and a correction must he made for liquids. Jf we assume that liquids have effective van der Waals radii that are about 10% larger than the solid state values, then the volumes of molecules in a liquid are about 30% larger than the solid state values. The volume in cubic Angstroms for a molecule in a liquid is therefore (1.3 x 6.40)Vs, or 8.30Vs. Once the molecular volume has been calculated, it is trivial to compute the density. The mass of a n individual molecule is given by WIN.,,, where N,. is Avogadro's constant, 6.022 x loz3.Dividing this molecular mass by the molecular volume gives.
where the result is expressed in g cm3. Application to X-Ray Crystallography These results are also of use in crystallographic studies. Specifically, the initial solution of a crystallographic data set is often facilitated if the number of molecules residing in the unit cell (symbolized by Z) can be established. The
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molecular volumes of both organic and inorganic solid.; can be calculated from the 6.41'; f o ~ m u l nFrom . ~ this number and the total volume of the unit cell (which can be directly determined from the diffraction pattern), it is simple to determine Z. For example, some years ago we synthesized a coordination complex of titanium that formed crystals clearly containing several molecules per unit cell (6).The number of molecules per unit cell was estimated a s follows.
molecular volume = 6.4Vs = 614 hi3 unit cell volume (determined experimentally) = 3822 A3 molecules per unit cell = 3822 1614 = 6.2 actual value: 6 Conclusion The methods described above for the estimation of densities and molecular volumes are surprisingly accurate and very simple. The information necessary to compute these physical properties is easily memorized: equation 1; the series 1, 2, 4, ... for the relative volumes of hydrogen and the succeediue rows in the oeriodic table: and a short list of functional group corrections. Apart from their utility, the methods mav also orovide interesting insights into the molecular origins of the physical pope&es oTliquids and solids.
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Acknowledgment I thank Ken Suslick, Tom ~ a u c h f u s sand , Victor Day for helpful discussions. Literature Cited 1. Poii, R.: Gordon, J. C . J . A m . Chem S o c 1992, 111,6723. 2. Gauerrofti. A. J.A m Chem Sac. 1983,105,5220-5225. 3. Mingos. D. M. P.; Roh1.A. L. J. Cham. Soe.Dallon Pans. 1991,34193425. 4. McGowan, J. C.; Mellors, A . Moiecular Voiume in Chemistry and Biology: Halsted: New York,1986. 5. Pauling. L. TheNalum ofthe Chemicol Bond; Cornell University: Ithsea, NY;1960. p 260. Gimlami, G. S. Orgonomofallics 1987.6.2551. 6. Oardnar, T. C.;
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.
jFor cooro natfon complexes, cryslallograpners have ong Jseo Tne mo ec-.ar vo m e of a sol a n c ~ co evens mpler rL es of th~mo: Angstroms is approximately equal to 20 times the number of non-hydro en atoms. Alternatively, we have noted that molecular volumes in A are often close to 10 times the total number of atoms. The 6.4Vformula gives more accurate molecular volumes than either of these rules.
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