108
`
`Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
`
`y = vertical direction, cm
`Ye = position where a light ray enters the electrolyte, cm
`Y = dimensionless vertical distance, y / 0
`z = distance from electrode leading edge in the flow direction,
`cm
`Greek Letters
`{3 = interfacial velocity gradients, s-1
`{3 .. = interfacial velocity gradient, h/w-+ 0, s-1
`y = interfacial concentration gradient, M/cm
`0 = boundary layer thickness, c1n
`Or-; = Nernst boundary layer thickness, cm
`UC = concentration difference, Cb - C,, M
`Un ::; refractive-index difference nb -
`tJ 6
`11 = similarity variable; see eq 5
`0 =dimensionless concentration, (C - C1)/(Cb - Cs)
`;\ = light wavelength, nn1
`v =electrolyte kinematic viscosity, cm2/s
`T = dimensionless titne; see cq 29
`¢ = constant; see Table II
`Literature Cited
`Abramowitz, M., Stegun, !., Ed., "Handbook: ol Mathematica\ Functions", pp
`255-262, 320, NaUonal Bureau ol Standards, Wasn~ton, D.C., 1984.
`Beach, K. W., Mufer, R.H., Tobtas, C. W., Rev. Sci. Jnstrum., 40, 1248 (1969).
`Beach, K. \'I., Ph.D. Thesis, UCRL-20324, University of Ga~fornla, Berkeley,
`1971.
`Beach, K. W., Munar, A.H., Tobias, C. W., J. Opt. Soc. Am., 63, 559 (1973).
`mrd, R. 8 .. Stewart, W. E., Llghtloo!, E .. "Transport Phenomena", p 354, W~ey,
`New York, N.Y., 1960.
`CN!pman, T. \V., Newman, J, $., "A CompDatlon ol Selected TherffiOOynamlc
`and Transport Properties of Binary Electrolytes In Aqueous So!u1ion",
`UCAL-17-787 (1968).
`Durou, C., Giraudoo, J.C., Moutou, C .. J. Chem. Eng. Data, 18, 289 (1973).
`Eversole,\"/. G., Kindsvater, H. M., Peterson, J, D., J, Phys. Chem., 46, 370
`(1942).
`
`Frltz, J. J., Fugel, G. A., J. Phys. CMm .. 62, 303 (1958).
`Haul, VI., Gdgu\I, U .. Adv. /"ff>af Transfer, 6, 133 (1970).
`Howes, W. l., Buchele, D. A .. J. Op/. Soc. Am., 56, 1517 (1966).
`Lapldus, l., "D;gital Computation for Chemlcal Englnee<s'', p 51, McGtaw-H!ll,
`New YOJk, N.Y., 1962.
`Un, C, $., Ph.O. Thesis, Oepartmen\ of Chemlcal Engineering, University of
`Washington, 1952.
`Lin, C. S., Moulton, R. W., Putnam, G. L., Ind. Eng. Chllm .. 45, 640 (1953),
`Love, A. E. H .• "Treatise on Mathematica I Theory of Elasticity", p 129, Carl"tb0.1ge
`University Press, 1927.
`lvklarnon, F. R., Ph.D. Thesis, LBL--3500, Unlvarsity of Cal'forM, Be<keley, 1975.
`Mclarnon, F. R., Muller, R.H., Tobias, C. \'/.,Appl. Opt., 14, 2468 (1975a),
`Mclarnon, F. R., Mut!er, R.H., Tob!as, C. W., J. Electrochem. Soc,, 122, 59
`(1975b).
`Mda1non, F. R., J..\iler, A.H., Tob:as, C. W .. J. Opt. Soc. Am., 65, 1011 (1975c).
`1'.'d..arnon, F. R., IJA>'let', R.H., Tobias, C. W., ElGctrochlm. Acts, 21, 10111976).
`Mu!!er, R. H., Adv. E'6ctrochem. Electrochem. Eng., 9, 326-353 (1973).
`Newman, J. S., "Electrochemical Systems", pp 225, 322, 318, 331, 342,
`Prentice·Hall, Englewood Clills, N.J., 1973.
`Newman, J. S., p<ivate communication, 1974.
`Norri$, R.H., Streld, D. D., Trans. ASME, 82, 525 (1940).
`Rousar, l .. Hostomsky, J., CezMr, V., J. Electrochom. Soc., 118, 881 (1971).
`Sand. H.J. s .. Phil. Mag., 1(6), 45 (1901).
`Scr.lk:htlng, H., "Boun-d.ary layer Theory", p 291, McG'aw-H~I. New York, N.Y.,
`1968.
`Selman, J. R .. Ph.D. Thesis, UCRL-20557, University ol Ca!ilornla, Berke:ey,
`1971; also: Tobias, C. \'I., Selman, J, A., Adv. Cham. Eng., 10, 211-318
`(1978).
`Simon, H. A., Eckert, E. A. G., Int. J. Heat Mass Transfer, 8, 681 (1983).
`Tobias, C. VI .. Hickman, A.G., Z. Phys. Chem .. 229. 145 (1965).
`Wagner, C., J. E!ectrochem. Soc., 98, 118 {1951).
`Wh~aker, S .. "Fundamental Principles of Heat Transrer", p 155 Pergamon,
`Elmsford, N.Y., 1977.
`
`Received for review August 16, 1977
`Accepted December 4, 1978
`
`This work was supported by the Division of I\'faterials Sciences,
`Office of Basic Energy Sciences, U.S. Dcpart?nent of Energy.
`
`Estimation of Entropies of Fusion of Organic Compounds
`
`Samuel H. Yalkowsky
`
`Th8 Upfohn Company, Ka/af'TlfjZOo, A1/chlgan 49001
`
`The entropy of fusion for many drugs and molecules of Intermediate slze can be estimated In the following manner;
`(1) for rigid molecules, t>.S 1 "" 13.5 eu; (2) for long chain derivatives of such molecules, i'lS 1 "" 13.5 ± 2.5(n
`- 5) eu, where n is the number of flexible links in the chain. In most cases, these slmple rules will provide an
`estimate of 6.8 1 which is suffic!ently accurate to obtain reasonable estimates of ideal solublll\y.
`
`According to Hildebrand (1950, 1962), the ideal solu(cid:173)
`bility of a crystalline substance can be calculated from a
`kno\vledge (or an esthnation) of either Tm and b.Hr or Tm
`and b.Sr. Because inelting points are easily determined,
`it is only necessary to estimate Af/f or USr in order to
`estimate the ideal solubility of existing compounds.
`Although heat of fusion can be measured experimentally,
`it has not been found possible to estimate this para1neter
`directly from considerations of chemical structure (Bondi,
`1968). There are, ho\vever, several empirical relationships
`between entropy of fusion and structure in the literature
`(Bondi, 1968; Walden, 1908; Pirsch, 1937, 1956; Luttin(cid:173)
`ghaus and Vierk, 1949). The first and most itnportant of
`these is the \Vaiden Rule (\\1hich is analogous to Troutons
`Rule for entropy of vaporization). \Vaiden (1908) observed
`that the entropies of fusion for n1ost organic compounds
`fall in a fairly narro\v range about 13 eu. The data of
`
`Tsonopoulos and Prausnitz (1971) show, in agree1nent with
`\V alden, that entropy of fusion tends to be nearly constant
`but 13.5 appears to be a better average value.
`Pirsch (1937, 1956) observed a relationship between
`overall tnolecular shape and 6.Sr, \vit.h spherical molecules
`having the lowest values and highly elongated molecules
`having the highest values. More recently, Bondi (1968)
`atteinpted to calculate osf from inolecular 1non1ents of
`inertia and empirical corrections for hydrogen bonding
`groups.
`The entropy of fusion of long-chain molecules has been
`discussed by several v.•orkers (Bondi, 1968; Pirsch, 1937,
`1956; Arano\v et al., 1958; Garner et al., 1926; Bunn, 1955)
`\\'ho have shO\'lll that there is a regular increase in D.Sr \vith
`increasing chain length. The above relationships \vill
`provide 1nuch of the basis for the calculations offered in
`this report.
`
`0019-7874/79/1018-0108$01.00/0
`
`© 1979 American Chemical Society
`Breckenridge Exhibit 1007
`Breckenridge v. Novartis AG
`
`

`
`CRYSTAL
`
`HYPOTHETICAL
`PARTIAL MELTS
`
`LIOU10
`
`Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
`
`109
`
`~ ~
`~ y
`
`~ ~
`
`~ (\_
`
`~ ~
`
`'
`Figure 2. Internal melting of flexible molecules: a, crystal; b, melt.
`
`the molecules are free to move. It has also been called the
`fluctuation volume (Bondi, 1968). Therefore
`
`and
`
`V(f)li,
`P,.01 = -V(f) .
`solid
`
`V(f)liq
`.
`Sr = R ln V---(f) .
`sohd
`
`(2)
`
`(3)
`
`Although the amount of expansion that occurs on
`melting and thus the entropy of expansion is largely
`dependent on molecular shape, it is usually found to lie
`bet\veen 1 and 3 eu. Compounds which are spherical or
`nearly spherical require very little expansion to attain free
`rotation and thus need expand only enough to allo\v for
`positional randomization of its 1nolecules. Highly eccentric
`compounds, because of their greater space requirements
`for rotation, sho\v much greater increases in volume on
`melting and consequently have higher expansional en·
`tropics of melting. Similarly, con1pounds, which, because
`of their shape, have high packing densities in the crystal,
`will have high entropies of expansion associated with
`melting.
`Positional Entropy
`The positional entropy of fusion as stated above is
`related to the probability of finding a collection of n
`1nolecules in the positions that are \Vit.hin the crystal
`lattice. This is analogous to the probability of finding 64
`1-in. diameter checkers on an 8 X 8 in. checker board
`arranged so that the centers of all the checkers fall \\•it.bin
`a different square.
`'fhe requirement that there are 64
`checkers on a 64 in, 2 surface and that there is no overlap
`of the checkers has been taken care of by the separation
`probability term. More sophisticated theoretical calcu(cid:173)
`lations of this type lead to calculated positional entropies
`of arowld 2 to 3 eu (Hirshfelder et al., 1937; Lennard-Jones
`and Devonshire 1 1939).
`'l'he combined effects of separational and positional
`disorder can best be illustrated by the entropies of fusion
`observed for spherical molecules such as the inert gases,
`1nethane, etc. For these substances ~Sr invariably falls
`bet\veen 2 and 4 eu. Since expansion on melting is 1ninhna1
`for these substances, the observed <.\Sr values can be re(cid:173)
`garded as being composed of prilnary LlSp<:w
`Rotational Entropy
`In considering D.Spv, for spherical molecules it is not
`necessary to be concerned \vith 1nolecular orientation (since
`
`-= ---
`
`Figure l. Schematic illustration of melting process: a, crystal; b,
`rotational melting; c, expansional melting; d, positional melting; e,
`liquid (complete melting).
`
`Entropy of Fusion
`The most obvious difference bet\veen a crystal and its
`melt is the difference in their degrees of geometric order.
`'fhe separation distance, packing arrangement, orientation,
`and conformation of molecules in a crystal are fixed \vi thin
`narrow limits, \vhereas in the liquid these parameters can
`vary over a much wider range of values.
`For purposes of visualization, the inelting process can
`(1)
`be divided into four independent subprocesses:
`expansional-the change in the average distance bet\veen
`molecules that usually occurs on melt.ing and is evidenced
`by an increase in volume; (2) positional-the change from
`the ordered arrangement of molecular centers of gravity
`in the crystal to the randomized arrangement in the liquid;
`(3) rotational~the change from the ordered arrangement
`of the major axes of crystalline molecules to the randomly
`oriented arrangement in the liquid (This process is not
`applicable to spherical molecules); (4) internal-the chm1ge
`from the uniform conformation of flexible molecules of the
`crystal to the random conformation of such molecules in
`the liquid. (This process is not applicable to·rigid mol(cid:173)
`ecules and thus to most drugs. It does become important,
`ho\vever, for long-chain molecules.) This strictly geometric
`interpretation of fusion provides an intuitive means of
`w1derstanding the process in terms of molecular size1 shape
`and interactions.
`The first three subprocesses are illustrated schematically
`in Figure l and the fourth in Figure 2. Each of these
`subtnelting processes has associated \vith it a probability
`of occurrence and thus an entropy of occurrence. Since
`the probabilities are multiplicative, the entropies are
`additive
`
`(1)
`
`6S1 = 6Se1 p + llSpo! + D.S1-0t ilSint
`it is possible to estimate the entropy of fusion from a
`consideration of the probabilities of the various processes.
`Entropy of Expansion
`The entropy of separation is similar in nature but
`smaller in n1agnitude than the entropy of vaporization.
`When a crystal melts there is usually, but not ahvays, a
`slight increase in volun1e. The contribution to the entropy
`of fusion resulting from the change in free volume is
`calculated from the probabilities of finding a collection of
`liquid molecules in the crystal density. The volumetric
`probability is equal to the ratio of free volumes of the
`liquid and solid. The term free volume V(f) as used here
`refers to the volume into \\•hich the centers of gravity of
`
`

`
`110
`
`Ind. Eng. Chem, Fundam., Vol. 18, No. 2, 1979
`
`Table I. Component Entropies of Fusion
`
`type of entropy
`expansional
`positional
`rotational
`total (rigid nlolecules)
`internal
`total (flexible molecules)
`
`most likely values, eu
`
`2
`2.5
`9
`13.5
`2.5 (11- 5)
`13.5 + 2.5(11- 5)
`
`normal range of values, eu
`high
`
`low
`
`1
`2
`7
`10
`(2.3-2.7)
`
`3
`3
`11
`17
`[(n - 3)-(n - 6)]
`
`Figure 3, Rotational freedom of crystalline molecule. \Vithin the
`crysta1, the rotation (libration) of a molecule is restricted by its nearest
`neighbors, whereas in the liquid it can rotate much more freely.
`
`all orientations are equivalent for spherical molecules).
`Ho,vever, in the case of nonspherical molecules {which
`includes nearly all drugs) the entropy associated with the
`change from a fixed orientation \Vith respect to near
`neighbors in the crystal to the nearly random orientation
`of the liquid "13"" is a major factor in determining the total
`entropy of fusion.
`If the rotational entropy of fusion of all rigid molecules
`is assumed to be 7-1 f eu, \Ve \vould expect the total en·
`tropy of fusion to fall between 10 and 17 eu. (See Table
`I.) That this is the case for most compounds having
`melting points above 25 'C is obvious from the data in the
`literature (Aranow et al., 1958; Garner et al., 1926; Bunn,
`1955). The near constancy of 118, has been noted, but not
`explained by several early workers (Walden, 1908; Pirsch,
`1937; Luttinghaus, 1949).
`An intuitive justification of the nearly constant rota(cid:173)
`tional entropy of fusion is based upon the follo\\'ing t\vo
`assu1nptions. (1) In the crystal, the molecules \\•ith their
`centers of mass fixed (and accounted for by .6.Seip and
`.6.Sp:J can '\vobble" or librate to only a certain extent (say
`about 10° in the spherical coordinates¢ and 0 fro1n their
`most stable position, after averaging over all axes). (2) In
`the liquid the individual molecules have nearly total
`orientational freedom and thus rotate freely in ¢ and 0,
`The probability difference bet\veen these t\vo different
`degrees of orientational freedom can be calculated by
`tracing the allo\'.iable positions of any point on the mo(cid:173)
`lecular surface. In a liquid molecule the chosen point \vill
`trace out a sphere about the center of n1ass, whereas in a
`crystal molecule it \vill only describe a segment of a sphere
`(see Figure 3). The size of the spherical segment with
`respect to the sphere is dependent only on the average
`values of¢ and 0. (Free rotation in the liquid is assumed
`for mathematical convenience. It is not necessary for the
`applicability of the above approach.)
`The area of a spherical segment obtained by. a ±10°
`variation in 0 and ¢ is 0.00754 times that of a sphere of
`the same radius. Thus, the probability of n molecules
`being oriented within the allowed limits for crystallinity
`is 0.00754" and the entropy contribution is -k In 0.00754"
`or -R In 0.00754 or about 10 eu. Similarly, the entropy
`associated with 0 = </> = 20' is 7 eu. Although the actual
`values of 0 and </> very likely will depend on the overall
`geotnetry of the inolecules and their degree of interaction,
`the relative constancy of O.St for rigid molecules suggests
`
`Table II. Entropies of Fuston of Some
`Disubstituted Benzenes0
`
`Cl
`
`Br N0 1
`
`I 1.4
`11.9 12.6
`12.7
`12.1
`14.6
`12.3
`13.1 13.3 14.0
`11.0 16.3
`12.5 13.7
`13.5
`
`13.9
`11.5
`15.1
`
`13.8
`
`NH, OH
`12.4
`9.0
`9.8
`9.1
`9.2
`13.8 11.l
`9.7
`
`10.5
`11.2 11. 7
`12.0 13.8
`14.6 15.0
`
`CH,
`{ ortho 13.2
`CH, meta
`12.4
`para
`14.0
`
`Cl
`
`&
`
`{ ortho
`1neta
`para
`
`{ ortho
`meta
`para
`
`{ ortho
`N0 1 meta
`para
`
`{ortho
`NH 1 meta
`para
`
`{ ortho
`Oif meta
`para
`0 All entropy values expressed in eu.
`
`14.3
`13.3
`14.5
`
`COOH
`12.8
`9.8
`12.0
`14.9
`13.3
`15.0
`14.9
`13.3
`15.0
`15.9
`11.1
`17.2
`I 1.8
`11.5
`10.8
`11.6
`
`16.7
`
`that the variation is not too large or that factors which
`inhibit rotation in the liquid also inhibit rotation in the
`crystal.
`It has been proposed (Bondi, 1968) that hydrogen
`bonding groups such as OH and NH2 allow association of
`liquid inolecules and thus restrict free rotation of the liquid
`and that this results in a reduction in OSr over the ho·
`momorphic CH3 containing n1olecules. Analysis of the data
`in Table II suggests that this is not the case. The entropies
`of fusion of compounds which have no hydrogen bonding
`groups (left) are not significantly different from those of
`compounds having one (upper right) or t\VO (lo\ver right)
`(Specifically o-, m-, and p·
`hydrogen bonding groups.
`xylenes have nearly the same values as catechol, resorcinol,
`and hydroquinone, respectively.)
`It is also evident from Table II that there is little
`systetnatic difference among ortho, meta, and para isotners.
`Evidently the increased symmetry of para isomers \Vhich
`would tend to decrease O.Srvt is offset by their greater
`packing efficiency \Vhich tends to increase D.Sup·
`Internal Entropy
`If the compound under consideration is not a rigid
`molecule as discussed above, it becomes necessary to
`account for the entropy that results from the greater
`conformational freedom of the liquid.
`In a crystal, a
`molecule is not only fixed in its position and orientation
`but is also fixed in its conformation. Fatty acids and other
`long-chain compounds, for example, are fully outstretched
`in the crystal but may be coiled to so1ne extent in the
`liquid.
`A compound having a long chain of n carbons will have
`n - 1 carbon carbon bonds, n - 2 C-C-C bond angles, and
`n - 3 C-·C-C-C t\vist angles (this is equal to the number
`of bonds about \vhich there is free rotation of nonhydrogen
`
`

`
`atoms). The bond lengths and bond angles are not affected
`to any great extent by melting, but the t\\1ist angles are.
`ln the fully stretched conformations of most crystals these
`angles are invariably 180°. In the liquid state, other angles,
`especially 60 and 300°, are likely to be observed. If these
`three angles are assun1cd to be equally probable, then the
`probability of finding n fully outstretched chain in (l/3)ri-3,
`'I'his corresponds to an idealized internal entropy of fusion
`of
`"S'"' = R In (Y,) 0 -s = R(n - 3) In 1%l = -2.3(ri - 3)
`(4)
`Heteroatoms in the chain, e.g., amide nitrogens, ether
`oxygens, and ester oxygens, are included in the value of
`ll.
`Actually, the values of 11D.S /an n1ost often observed
`experimentally are 2.3 eu/CH2 for homologous series of
`orthorhomhic crystal forming compounds and 2.7 eu/CH2
`for series that form n1onoclinic crystals. In the absence
`of specific information about the type of crystal fonned;
`a value of 2.5 eu/CH:: can safely be used for purposes of
`estiination.
`It has been obserV(!d for a number of different series
`containing alkyl groups attached to large rigid tnoieties
`(Breusch, 1969; Ubbelohde, 1965; Yalkowsky et al., 1972)
`that the n1elting behavior characteristic of aliphatic
`coinpounds is not observed until there are at least 4 to 6
`ato1ns in the chain. This is in agreen1ent \Vith our ob·
`servation that short chains (n .:5 5), \Vhich are configu(cid:173)
`rationally constrained by the rigid portion of the molecule,
`do not contribute appreciably to ..\Sirit· Therefore, mol·
`ecules \Vith less than 5 chain ato1ns, as a first approxi(cid:173)
`mation, can be treated as rigid tnolecules. ('fhe actual
`chain length required for the aliphatic chain to doininate
`the crystal forming properties is dependent upon the size
`and interaction ability of the nonhydrogen portion of the
`n1olecule. In alkylbenzenes, benzoates, and parabens, 5
`chain ato1ns are required \vhereas in alkylnaphthalenes,
`6 chain aton1s are required.) For longer chains \Ve can
`estimate the contribution to internal entropy by adding
`2.5 (n - 5} eui \vhere 11 is the total nutnber of chain aton1s
`(exclusive of protons).
`The total entropy of fusion of a flexible or se1niflexible
`n1olecule is calculated fro1n the sun1 of the four partial
`entropies described above. Table III gives calculated and
`observed entropies of fusion for some nonrigid molecules.
`Attempts (Bondi, 1968; Pirsch, 1937, 1956) to pro\'ide more
`sophisticated and/or n1ore accurate estimates of entropy
`of fusion than the above variation of \Valden's Rule tend
`to be 1nore cun1berso1ne and not consistently inore accurate
`than the follo\ving
`.
`.).Sf = 13 .. ) eu for rigid inolecules
`
`(5)
`
`and
`.:\Sr:::: 13.5 + 2.5(n - .)) eu for flexible 1nolecules
`
`(6)
`
`Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 111
`
`Table III. Calculated Entropies of Fusion of So1nc
`Alkyl-p-aminobenzoatcs at 37 ° C
`
`1np,
`6.St,
`,c
`ester
`obsd
`112
`n1ethyl
`15. 1°
`13.1
`89
`ethyl
`74
`14.6
`propyl
`17 .8
`56
`butyl
`17.8
`52
`pentyl
`61
`25.2
`hexyl
`18.1
`75
`heptyl
`28.3
`71
`octyl
`31.4
`69
`nonyl
`41.5
`82
`dodecyl
`87
`55.5
`hexadecyl
`a All entropy values are exprt'\SSed in eu.
`
`n-5
`0
`0
`0
`1
`2
`3
`4
`5
`6
`9
`13
`
`13.5 +
`2.5
`(n - 5)
`
`13.5°
`13.5
`13.5
`16.0
`18.5
`21.0
`23.5
`26.0
`28.5
`36.0
`46.0
`
`The ideal solubility of 1nany crystalline co1npounds can
`be estimated from the n1elting point and entropy of fusion
`as given by eq 5 and 6.
`Because the above treat1nent is based on tnanv as(cid:173)
`sun1ptions and approxin1ations, it cannot be expected to
`provide highly accurate solubility estin1ates for all com(cid:173)
`pounds, It does, however, provide a very siinple n1eans of
`obtaining a reasonable estimate of ideal solubility fron1
`nothing tnore than the structure and 1nelting point of the
`con1pound in question. If 1neans \Vere available for pre·
`dieting n1elting point from chen1ical structure, this type
`of approach could be used for the design of con1pounds
`having desired solubility properties.
`
`Literatu1·e Cited
`
`Aranow, R.H.; Witten, L.; Andrews. D. H.J. Phys. Chem. 1958, 62, 812.
`BOl'ldi, A. "Physical Properties of .VO'ecutar Crystals, Llqu'ds and G!asses", \V1'ey:
`New York, 1968.
`Breusch, F. l. Fortschr. Chem. Forsch. 1969, 12, 119.
`Bunn, C. W. J. Po/ym. Sci. 1955 323.
`Garner, W. E.; MaCden, C. F.; Rushbrooke, J.E. J. Chem. Soc. 1926, 2491.
`Hi'debrand, J. H.; Sco11, R. L. "Regular So!ullons", Pren\lce·Hall: Englewood
`Cliffs, N.J., 1962.
`H!debrand, J. H.; Scott, R. L. "The So!ubi~1y of None:ectrolytes", Reinhold: New
`York, 1950.
`H'rschfe!der, J. Q.: Stevenson, D. P.; Eyrlng, H.J. Chem. Phys. 1937, 5. 896.
`Lennard-Jones, J.E.; Devonsh're, A. F. Proc. A. Soc. London, Ser. A 1939,
`170, 464.
`Lu\linghaus, A.: Vie1k, G. Ber. 1949, 82, 376.
`Pirsch, J. Ber., 1937, 12; Mikrochim. Acta 1956, 992.
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`
`ReceiL•ed for reL'i!?lr Dece1nber 19, 1977
`Accepted Noven1ber 27, 1978