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The Journal of Physical Chemistry, Vol. 83,
No. 2, 1979
R. Thomas Myers
(30) F. J. Adrian, V. A. Bowers, and E. L. Cochran, J . Chem. Phys., 63, 919 (1975). (31) We consider the "complex" rather as a TME- * -CCI,' pair in which TME interacts very weakly with CCI,'. The pair would equilibrate with a TME+*..CCI,. pair. Thermochemical data shows the reaction TME CCI,' TME' -t CCI,., is endothermic by 0.38 eV. (32) A. Carrington and A. D. MacLachlan, "Introduction to Magnetic Resonance", Harper and Row, New York, N.Y., 1967. (33) D. D. Wilkey, H. W. Fenrick, and J. E. Willard, J . Phys. Chem., 61, 220 (1977).
+
-
(34) T. Kimura, N. Bremer, and J. E. Willard, J . Chem. Phys., 66, 1127 (1977). (35) M. Iwasaki, K. Toriyama, K. Nunome, M. Fukaya, and H. Muto, J . Phys. Chem., 81, 1410 (1977). (36) At present, formation of RH', is not clear, though H. Yoshida, T.Shiga, and M. Irie ( J . Chem. Phys., 52, 4906 (1970)) have reported ESR observation of trapped positive holes in y-irradiated 3-MHX glasses without additives. (37) K. D. Asmus, J. M. Warman, and R. H . Schuler, J . Phys. Chem., 74, 246 (1970); K. M. Bansai and R. H. Schuler, ibid., 74, 3924 (1970).
Forces between Molecules in Liquids. 1. Pure Nonpolar Liquids R. Thomas Myers Depaftment of Chemistry, Kent State University, Kent, Ohio 44242 (Received February 9, 1978; Revised Manuscript Received October IO, 1978)
The London equation for dispersion energy between nonpolar molecules is used to calculate boiling point as a function of polarizability, ionization energy, and size, provided the shape of the molecule is taken into account. For spherical molecules, density is proportional to molecular weight M divided by the cube of the radius. For the zero-point energy hvo,the ionization potential I is used. For polarizability, the molar refraction RM is used and the boiling point Tb is assumed proportional to the energy of vaporization (Trouton's rule). The result is that Tbllzis proportional to RM1112/vb, where v b is the molar volume at Tb. The data for rare gases and for group 4A tetrahalides give good straight lines. The method is then extended to cylindrical and flat molecules, in which case T212is proportional to R M I ~ ~Vb3l2, ~ Lwhere ~ ~ L~ is/ the length of the molecule, and RMI1I2A3/ v b 3 , where A is the area of the molecule. Good straight lines are obtained for normal hydrocarbons (cylindrical) and aromatic compounds (flat).
The purpose of this paper is to use the London equation for dispersion energy to calculate the normal boiling point of nonpolar molecules. This can be done provided the shape of the molecules is considered in the calculations. Equations are obtained for spherical, tetrahedral, cylindrical, and flat molecules. The other independent variables are polarizability, ionization energy, and size. The data fit the derived equations closely. The anomalous behavior of SiC14, as compared to CCl,, is explained. Theory The energy involved in the attractive London, or dispersion, forces between two molecules is given by eq 1.l
E = 3a2(hv0)/(4r6) (1) If the molecules are spherical, then the density is proportional to molecular weight divided by some function of R3, where R is the radius (eq 2 ) . This radius R will d
a
M/R3
(2)
determine the closeness of approach of the molecules. Squaring both sides and rearranging we obtain eq 3, where R6 a r6 0: @/db2 = v b 2 (3) db and v b are the density and molar volume at the boiling temperature. The quantum-mechanical vibrational energy hvoof single atoms is approximated very closely1 by the ionization energy I. For consistency, an attempt is made in this article to always use the vertical ionization potential. The values used are given, since it is not always clear as to which ionization energies were given in the literature source. The molar polarizability RM is used for the D line2 of sodium. Thus we obtain eq 4. The energy of vaporization 0022-3654/79/2083-0294$01.00/0
E 0: RM21db2/w (4) divided by the boiling temperature is a constant (Trouton's law), which leads to the final relationship in eq 5. If the Tb112 a RMpI2/vb = s (5) molecules are spherical (or nearly so) then a graph of TJ12 vs. RMFl2/ Vb should be a straight line. (The boiling point is used because accurate data are easy to find, as compared with the case of heats of vaporization.) If the molecules are cylindrical, e.g., the normal alkanes, then the density is given by eq 6, where L is the overall
d
a
M/nR2L
(6)
length and R is the cylindrical radius (the latter determining the distance of approach). Carrying out the same operations as before, we obtain eq 7. Tbl/2 a RM L3/211/2/Vb3/2 = C (7)
For flat molecules, such as aromatic compounds, the volume is given by AR, where A is the cross-sectional area and R is now the thickness of the molecules. Carrying out the same operations as before, the final result is eq 8. Tb1I2
R M A ~ I = ~F ~ ~ / ~ (8) ~ ~
It can be seen that a fourth variable has arisen: the shape of the molecules. Application of t h e Theory The only substances with spherical or nearly spherical symmetry for which sufficient reliable data are available to properly test the validity of eq 5 are the inert gases and the group 4A tetrahalides. (A few other substances are included, although data are skimpy and, in many cases, of low reliability.) The data from the literature are dis0 1979 American Chemical Society
The Journal of Physical Chemistry, Vol. 83, No. 2, 1979 295
Forces between Molecules in Liquids
TABLE)I: D a t a for "Spherical" Molecules
TABLE 11: - D a t a for Cylindrical Molecules
_ I _ -
substancle 3He ,He Ne
-.___
R h f , mL
0.5216 0.5216 1.002 Ax 4.22 Kr 6.38 10.49 Xe Rn 16.2) 6.82 CH 4 12.64 SiH, 13.35 GeH, SnH, 17.3) 6.08 CF, SIF, 8.40 GeF, 10.34 26.6 cc1, SiCl, 28.7 31.5 GeC1, SnC1, 34.94 37.8 CBr, SiBr, 40.8 GeBr, 44,65 CMe, 2530 SiMe, 30.33 GeMe, 32.26 SnMe, 36.42 PbMe, 42.4 TiCI, 38.0 TiBr, (50.3)
vb, mL
I, eV
51.75 24.58 26.92 24.58 16.760 21.56 28.456 15.76 14.00 33.9 42.866 12.13 54.7 10.75 12.7 34.4 12.6 55.0 12.2 55.68 63.25 (12.0) 15.8 54.63 15.8 64.18 15.7 62.4 103.8 11.48 121.1 11.9 12.0 124.3 131.7 12.0 123.04 10.75 143.26 10.9 144 10.8 119.7 10.35 'L36.2 9.99 140.3 9.56 145 9.1 151.0 (8.8) 124.70 11.70 146 10.55
Tb, K 3.19 4.22 27.17 87.0 121.33 166.2 212 111.6 161.3 184.8 221.1 145 187 239 350 331 357 387 463 427 460 283 299.6 316.5 351.1 383 409.5 506.5
S
0,0500 0.0961 0.2776 0.589 0.704 0.8523 0.981 0.702 0.813 0.838 0.95 0.442 0.520 0.656 0.868 0.818 0.878 0.919 l.007 0.940 1.018 0.679 0.704 0.711 0.756 0.833 1.043 1.12
compound
RM,
Vb,mL L , cm
mL _-.
n-C,H, 11.44 n-C3H, 16.05 n-C4Hlo 20.67 n-CSH1225.29 n-C,H,, 29.91 n-C,H,, 34.53 n-C,H,, 39.18
I,eV
Tb,K
C
56.36 4.48 11.5 75.75 5.81 11.1 93.86 7.13 10.6 118.18 8.46 10.35 140.54 9.78 10.1 163.42 11.11 9.9 186.68 12.43 (9.7)
184.6 231.0 272.6 309.2 341.8 371.5 398.8
0.869 1.186 1.4019 1.558 1.745 1.926 2.096
-
TABLE 111: D a t a for F l a t Molecules €EM,
compound
mL
C6H6 26.19 1,4-C6H,Me, 36.00 1,3,5-C,H3Me, 40.69 1,4-C6H,C1, 36.45 1,4-C6H,Br, 42.7 C6F6 26.2 ClOH8 44.35
13
14
A, Vb,mL cml I , e V 96.08 140.6 161.8 132.2 143.7 126.00 150.5
15
36.5 46.6 53.2 46.8 49.0 39.7 54.5
16
T F OKK"
17
9.25 8.45 8.40 9.17 8.97 9.9 8.12
18
Th,K
F
353.2 411 437.8 447.27 493.5 353.41 491.1
4.37 3.81 4.19 4.90 5.07 2.58 6.00
19
1
20
Figure 2. Test of the derived function for cylindrical molecules. ---,
Tb'
I
'
1
'
I
'
I
(OK?
Figure 1. Test of the derived function for spherical, or nearly spherical, molecules. The points are: 1, 3He; 2, ,He; 3, Ne; 4, Ar; 5, Kr; 6, Xe; 7, CH,; 8, SiH,; 9 GeH,; IO, SnH,; 11, CF,; 12, SiF,; 13, GeF,; 14, SiCI,; 15, CCI,; '16, GeCi,; 17, SnCI,; 18, SiBr,; 19, GeBr,; 20, CBr,; 21, CMe,; 22, SiMe,; 23, GleMe,; 24, SnMe,; 25, TiCI,; 26, TiBr,; 27, PbMe,.
played in Table I. Some of the values for density a t the boiling point were obtained by linear extrapolation over a wide temperature range and therefore are not highly accurate. The estimated accuracy of all data is indicated by the significant figures given; estimated values are in parentheses. The results are graphed in Figure 1 with least-squares lines drawn through the points for inert gases, the tetrahalides, and the group 4A tetramethyl compounds. The only molecules of cylindrical symmetry for which data are available are the normal alkanes. The data are given in Table I1 and graphed in Figure 2, again with a least-squares line. The length of the molecules was determined by stretching Fisher-Hirschfelder-Taylor molecular mlodels out to maximum length and measuring with a ruler. These numbers were then fitted to a least-squares straight line, asl a function of the number of carbon atoms, and then the length for each alkane was calculated from
h - - - - . ; o " " "
21
22
23
Ti4 oK14
Figure 3. Test of the derived function for fiat molecules.
this line. This procedure minimized errors of individual measurements. The only flat molecules for which data are available are aromatic compounds. Their area was determined by laying Fisher-Hirschfelder-Taylor molecular models on paper, tracing out the arz?aby straight lines along the outermost atoms, and deterrnining the area by weighing the paper cutouts. A special difficulty here is that the polarizability used is the threle-dimensional average, although the molecules are aligned in essentially only one direction.
296
The Journal of Physical Chemistry, Val. 83, No. 2, 1979
MOLECULAR WEIGHT
Figure 4. Molar refraction and density of group 4A as a function of maleciilar weight,
Also, in order to get a sufficient number of compounds to test the equation, it was necessary to use some molecules with sizeable bond dipoles. The density (molar volume) and the area are cubed in the final equation; therefore errors are magnified greatly. The data are in Table 611 and graphed in Figure 3.
Discussion As can be seen from the graphs, the equations derived represent the data quite well for the spherical, tetrahedral, and cylindrical molecules. The equation for the inert-gas line has a correlation coefficient b of 0.9998. The deviation of helium is probably due to its “quantum” behavior. Use of the line yields 16.34 mL for the molar polarizability of radon. This agrees with the value of 6.3 f 0.6 A3 for the atomic polarizability estimated by The equation representing the group 4 tetrahalides has p equal to 0.9968. The equation to fit the group 4A tetramethyl compounds has p equal to 0.9729. If all the molecules were actually spherical, then the slopes of the three lines should be the same; they are probably the same, within experimental error. The line for alkanes has p equal to 0.9976. Much of the deviation in this case is due to an odd-even alternation:
R. Thomas Myers
all but one of the points show such an effect. (There appears to be some tendency for the longer alkanes to turn up in a curve. This may be due to a systematic error in length which is amplified by the three-halves power.) The polarizabilities of the tetrahalides behave in a regular manner, Figure 4. It can be seen from the data in Table J, and the graph, Figure 4, that the “anomalous” behavior of SiCl,, SiBr,, and GeRr, is due to low densities of the liquids. The reason for the low density of silicon chlorides and bromides is clear. The increase in the molecular weight from CC14 to SiCI4 is small (153.8 to 169.9). The volume, however, increases as the cube of the radius, and the much larger covalent bond length of silicon (1.17 as compared with 0.77) more than compensates for the increase in mass, giving a lower density. GeF4 can maintain a “normal” position in the line, as compared with SiF4,because of its high density. This is due to a large increase in molecular weight (104.1 to 148.6) accompanied by a small increase in covalent bond length of Ge (1.22 as compared with 1.17). The group 4 tetramethyl compounds, with about the same geometry as the tetrahalides, lie very close to the line for the latter, but much of the data is not very reliable. This is true also for TiCl, and TiHr4. The group 4 tetrahydrides, although the data are scarce and not too accurate, also appear to give a straight line, but this is close to that for the inert gases. The data for flat molecules are given in Table TI1 and graphed in Figure 3. The range of boiling points is not very large, which does not permit a good test of the derived function. However, most of the points are approximately on a straight line. As pointed out before, errors are magnified by the necessity of the third power in eq 8. The equation for the leasbsquares line through the data points (excluding C&,) has a b of 0.955.
References and Notes (1) S. Glasstone, “Theoretical Chemistry”, van Nostrand, New York, 1944. (2) This IS felicitous choice of frequency. I f we set hvo = I = 10 eV and then calculate the wavelength of light corresponding to uo, we obtain k = 120 nm, which is not an order of magnitude removed from the D line of sodium, 589.3 nm. Differences in dispersion should then not cause significant errors. I n any event, measurement of polarizability at lower frequencies, such as in dielectric studies, shouM not be used. (3) D. G. Tuck, J . Phys. Cfiem., 64, 1775-6 (1960).
