108
`Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
`y = vertical direction, em
`Ye = position where a light ray enters the electrolyte, em
`Y = dimensionless vertical distance, y I o
`z = distance from electrode leading edge in the flow direction,
`em
`Greek Letters
`{3 = interfacial velocity gradients, s-1
`f3oo = interfacial velocity gradient, hlw--+ 0, s-1
`'Y = interfacial concentration gradient, Ml em
`o = boundary layer thickness, em
`oN = Nernst boundary layer thickness, em
`t.C = concentration difference, Cb - C., M
`iln = refractive-index difference nb - n.
`17 = similarity variable; see eq 5
`e = dimensionless concentration, ( c - c.) 1 ( cb - c.)
`/.. = light wavelength, nm
`v = electrolyte kinematic viscosity, cm2 Is
`T = dimensionless time; see eq 29
`</> = constant; see Table II
`Literature Cited
`Abramowitz, M., Stegun, 1., Ed., "Handbook of Mathematical Functions", pp
`255-262, 320, National Bureau of Standards, Washington, D.C., 1964.
`Beach, K. W., Muller, R. H., Tobias, C. W., Rev. Sci. Instrum., 40, 1248 (1969).
`Beach, K. W., Ph.D. Thesis, UCRL-20324, University of California, Berkeley,
`1971.
`Beach, K. W., Muller, R. H., Tobias, C. W., J. Opt. Soc. Am., 63, 559 (1973).
`Bird, R. B., Stewart, W. E., Lightfoot, E., "Transport Phenomena", p 354, Wiley,
`New York, N.Y., 1960.
`Chapman, T. W., Newman, J. S., "A Compilation of Selected Thermodynamic
`and Transport Properties of Binary Electrolytes in Aqueous Solution",
`UCRL-17767 (1968).
`Durou, C., Giraudou, J. C., Moutou, C., J. Chern. Eng. Data, 18, 289 (1973).
`Eversole, W. G., Kindsvater, H. M., Peterson, J.D., J. Phys. Chern., 46, 370
`(1942).
`
`Fritz, J. J., Fuget, C. R., J. Phys. Chern., 62, 303 (1958).
`Haul, W., Grigull, U., Adv. Heat Transfer, 6, 133 (1970).
`Howes, W. L., Buchele, D. R., J. Opt. Soc. Am., 56, 1517 (1966).
`Lapidus, L., "Digital Computation for Chemical Engineers", p 51, McGraw-Hill,
`New York, N.Y., 1962.
`Lin, C. S., Ph.D. Thesis, Department of Chemical Engineering, University of
`Washington, 1952.
`Lin, C. S., Moulton, R. W., Putnam, G. L., Ind. Eng. Chern., 45, 640 (1953).
`Love, A. E. H., "Treatise on Mathematical Theory of Elasticity", p 129, Cambridge
`University Press, 1927.
`Mclarnon, F. R., Ph.D. Thesis, LBL-3500, University of California, Berkeley, 1975.
`McLarnon, F. R., Muller, R. H., Tobias, C. W., App/. Opt., 14, 2468 (1975a).
`McLarnon, F. R., Muller, R. H., Tobias, C. W., J. E/ectrochem. Soc., 122, 59
`(1975b).
`Mclarnon, F. R., Muller, R. H., Tobias, C. W., J. Opt. Soc. Am., 65, 1011 (1975c).
`Mclarnon, F. R., Muller, R. H., Tobias, C. W., Electrochim. Acta, 21, 101 (1976).
`Muller, R. H., Adv. Electrochem. Electrochem. Eng., 9, 326-353 (1973).
`Newman, J. S., "Electrochemical Systems", pp 225, 322, 318, 331, 342,
`Prentice-Hall, Englewood Cliffs, N.J., 1973.
`Newman, J. S., private communication, 1974.
`Norris, R. H., Streid, D. D., Trans. ASME, 62, 525 (1940).
`Rousar, !., Hostomsky, J., Cezner, V., J. Electrochem. Soc., 118, 881 (1971).
`Sand, H. J. S., Phil. Mag., 1(6), 45 (1901).
`Schlichting, H., "Boundary Layer Theory", p 291, McGraw-Hill, New York, N.Y.,
`1968.
`Selman, J. R., Ph.D. Thesis, UCRL-20557, University of California, Berkeley,
`1971; also: Tobias, C. W., Selman, J. R., Adv. Chern. Eng., 10,211-318
`(1978).
`Simon, H. A., Eckert, E. R. G., Int. J. Heat Mass Transfer, 6, 681 (1963).
`Tobias, C. W., Hickman, R. G., Z. Phys. Chern., 229, 145 (1965).
`Wagner, C., J. Electrochem. Soc., 98, 116 (1951).
`Whitaker, S., "Fundamental Principles of Heat Transfer", 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 Materials Sciences,
`Office of Basic Energy Sciences, U.S. Department of Energy.
`
`Estimation of Entropies of Fusion of Organic Compounds
`
`Samuel H. Yalkowsky
`
`The Upjohn Company, Kalamazoo, Michigan 49001
`
`The entropy of fusion for many drugs and molecules of intermediate size can be estimated in the following manner:
`(1) for rigid molecules, ASf ~ 13.5 eu; (2) for long chain derivatives of such molecules, ~Sf ~ 13.5 ± 2.5(n
`- 5) eu, where n is the number of flexible links in the chain.
`In most cases, these simple rules will provide an
`estimate of ~Sf which is sufficiently accurate to obtain reasonable estimates of ideal solubility.
`
`According to Hildebrand (1950, 1962), the ideal solu(cid:173)
`bility of a crystalline substance can be calculated from a
`knowledge (or an estimation) of either T m and C.Hr or T m
`and f1Sr. Because melting points are easily determined,
`it is only necessary to estimate f1Hr or C.Sr 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 parameter
`directly from considerations of chemical structure (Bondi,
`1968). There are, however, 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 important of
`these is the Walden Rule (which is analogous to Troutons
`Rule for entropy of vaporization). Walden (1908) observed
`that the entropies of fusion for most organic compounds
`fall in a fairly narrow range about 13 eu. The data of
`
`Tsonopoulos and Prausnitz (1971) show, in agreement with
`Walden, 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 molecular shape and t:..Sr, with spherical molecules
`having the lowest values and highly elongated molecules
`having the highest values. More recently, Bondi (1968)
`attempted to calculate f1Sr from molecular moments of
`inertia and empirical corrections for hydrogen bonding
`groups.
`The entropy of fusion of long-chain molecules has been
`discussed by several workers (Bondi, 1968; Pirsch, 1937,
`1956; Aranow et al., 1958; Garner et al., 1926; Bunn, 1955)
`who have shown that there is a regular increase in f1Sr with
`increasing chain length. The above relationships will
`provide much of the basis for the calculations offered in
`this report.
`
`0019-7874/79/1018-0108$01.00/0
`
`© 1979 American Chemical Society
`
`

`
`CRYSTAL
`
`HYPOTHETICAL
`PARTIAL MELTS
`
`LIQUID
`
`Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 109
`
`--
`"''-'~
`/ - ........ / -
`--
`t-
`-
`'\ \
`
`/
`
`-::..
`
`-
`
`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 between a crystal and its
`melt is the difference in their degrees of geometric order.
`The separation distance, packing arrangement, orientation,
`and conformation of molecules in a crystal are fixed within
`narrow limits, whereas in the liquid these parameters can
`vary over a much wider range of values.
`For purposes of visualization, the melting process can
`(1)
`be divided into four independent subprocesses:
`expansional-the change in the average distance between
`molecules that usually occurs on melting 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 change
`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,
`however, for long-chain molecules.) This strictly geometric
`interpretation of fusion provides an intuitive means of
`understanding the process in terms of molecular size, shape
`and interactions.
`The first three subprocesses are illustrated schematically
`in Figure 1 and the fourth in Figure 2. Each of these
`submelting processes has associated with it a probability
`of occurrence and thus an entropy of occurrence. Since
`the probabilities are multiplicative, the entropies are
`additive
`
`t.Sr = t.Sexp + t.Spos + t.Srot t.Sint
`
`(1)
`
`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 magnitude than the entropy of vaporization.
`When a crystal melts there is usually, but not always, a
`slight increase in volume. 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 which the centers of gravity of
`
`~ ~
`~ y
`~ r
`
`~ ~
`
`~ ~
`b
`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(fhq
`Pvol = V(f)
`solid
`
`V(f)liq
`.
`Sr = R ln -V(f) .
`solid
`
`(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
`between 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 allow for
`positional randomization of its molecules. Highly eccentric
`compounds, because of their greater space requirements
`for rotation, show much greater increases in volume on
`melting and consequently have higher expansional en(cid:173)
`tropies of melting. Similarly, compounds, 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
`molecules in the positions that are within the crystal
`lattice. This is analogous to the probability of finding 64
`l-in. diameter checkers on an 8 X 8 in. checker board
`arranged so that the centers of all the checkers fall within
`a different square. The 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 around 2 to 3 eu (Hirshfelder et al., 1937; Lennard-Janes
`and Devonshire, 1939).
`The 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,
`methane, etc. For these substances t.Sr invariably falls
`between 2 and 4 eu. Since expansion on melting is minimal
`for these substances, the observed tl.Sr values can be re(cid:173)
`garded as being composed of primary t.Spos·
`Rotational Entropy
`In considering t:.Spos for spherical molecules it is not
`necessary to be concerned with molecular orientation (since
`
`

`
`110
`
`Ind. Eng. Chern. Fundam., Vol. 18, No. 2, 1979
`
`Table I. Component Entropies of Fusion
`
`type of entropy
`expansional
`positional
`rotational
`total (rigid molecules)
`internal
`total (flexible molecules)
`
`most likely values, eu
`2
`2.5
`9
`13.5
`2.5 (n- 5)
`13.5 + 2.5(n- 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. Within the
`crystal, 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).
`However, in the case of nonspherical molecules (which
`includes nearly all drugs) the entropy associated with the
`change from a fixed orientation with respect to near
`neighbors in the crystal to the nearly random orientation
`of the liquid t:.Srot 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 7-11eu, we would expect the total en(cid:173)
`tropy of fusion to fall between 10 and 17 eu. (See Table
`1.) 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 D..Sc 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 following two
`assumptions. (1) In the crystal, the molecules with their
`centers of mass fixed (and accounted for by D..Sexp and
`D..S""") can "wobble" or librate to only a certain extent (say
`about 10° in the spherical coordinates cp and 8 from 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 8.
`The probability difference between these two different
`degrees of orientational freedom can be calculated by
`tracing the allowable positions of any point on the mo(cid:173)
`lecular surface. In a liquid molecule the chosen point will
`trace out a sphere about the center of mass, whereas in a
`crystal molecule it will 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 8. (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 8 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 0.00754n and the entropy contribution is -k ln 0.00754n
`or -R ln 0.00754 or about 10 eu. Similarly, the entropy
`associated with 8 = cp = 20° is 7 eu. Although the actual
`values of 8 and ¢ very likely will depend on the overall
`geometry of the molecules and their degree of interaction,
`the relative constancy of D..Sc for rigid molecules suggests
`
`Table II. Entropies of Fusion of Some
`Disubstituted Benzenesa
`Cl
`CH,
`{ ortho 13.2
`CH, meta
`12.4
`para
`14.0
`
`Br NO,
`
`11.4
`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
`
`{ ortho
`Cl meta
`para
`
`{ ortho
`Br meta
`para
`
`{ ortho
`NO, meta
`para
`
`{ortho
`NH, meta
`para
`
`13.8
`
`NH, OH
`12.4
`9.0
`9.8
`9.1
`9.2
`13.8 11.1
`9.7
`
`10.5
`11.2 11.7
`12.0 13.8
`14.6 15.0
`
`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
`11.8
`11.5
`10.8
`11.6
`
`16.7
`
`{ ortho
`OH meta
`para
`a All entropy values expressed in eu.
`
`14.3
`13.3
`14.5
`
`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 molecules and thus restrict free rotation of the liquid
`and that this results in a reduction in D..Sr over the ho(cid:173)
`momorphic CH3 containing molecules. 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 two (lower right)
`hydrogen bonding groups. (Specifically o-, m-, and p(cid:173)
`xylenes have nearly the same values as catechol, resorcinol,
`and hydroquinone, respectively.)
`It is also evident from Table II that there is little
`systematic difference among ortho, meta, and para isomers.
`Evidently the increased symmetry of para isomers which
`would tend to decrease D..Srot is offset by their greater
`packing efficiency which tends to increase D..Sew
`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
`In a crystal, a
`conformational freedom of the liquid.
`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 some 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 e-c-c-e twist angles (this is equal to the number
`of bonds about which there is free rotation of nonhydrogen
`
`

`
`atoms). The bond lengths and bond angles are not affected
`to any great extent by melting, but the twist angles are.
`In 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 assumed to be equally probable, then the
`probability of finding a fully outstretched chain in (1 / 3)n-3.
`This corresponds to an idealized internal entropy of fusion
`of
`~Sint = R In (;,"s)n-3 = R(n- 3) In (%) = -2.3(n - 3)
`(4)
`Heteroatoms in the chain, e.g., amide nitrogens, ether
`oxygens, and ester oxygens, are included in the value of
`n.
`Actually, the values of a.J.S /an most often observed
`experimentally are 2.3 eujCH 2 for homologous series of
`orthorhombic crystal forming compounds and 2.7 eu/CH 2
`for series that form monoclinic crystals. In the absence
`of specific information about the type of crystal formed,
`a value of 2.5 eujCH 2 can safely be used for purposes of
`estimation.
`It has been observed for a number of different series
`containing alkyl groups attached to large rigid moieties
`(Breusch, 1969; Ubbelohde, 1965; Yalkowsky et al., 1972)
`that the melting behavior characteristic of aliphatic
`compounds is not observed until there are at least 4 to 6
`atoms in the chain. This is in agreement with our ob(cid:173)
`servation that short chains (n ::5 5), which are configu(cid:173)
`rationally constrained by the rigid portion of the molecule,
`do not contribute appreciably to .J.Sint· Therefore, mol(cid:173)
`ecules with less than 5 chain atoms, as a first approxi(cid:173)
`mation, can be treated as rigid molecules. (The actual
`chain length required for the aliphatic chain to dominate
`the crystal forming properties is dependent upon the size
`and interaction ability of the nonhydrogen portion of the
`molecule. In alkylbenzenes, benzoates, and parabens, 5
`chain atoms are requ:ired whereas in alkylnaphthalenes,
`6 chain atoms are required.) For longer chains we can
`estimate the contribution to internal entropy by adding
`2.5 (n - 5) eu, where n is the total number of chain atoms
`(exclusive of protons).
`The total entropy of fusion of a flexible or semiflexible
`molecule is calculated from the sum 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 provide more
`sophisticated and/ or more accurate estimates of entropy
`of fusion than the above variation of Walden's Rule tend
`to be more cumbersome and not consistently more accurate
`than the following
`.J.Sr = 13.5 eu for rigid molecules
`
`(5)
`
`and
`.J.Sr = 13.5 + 2.5(n - .5) eu for flexible molecules
`
`(6)
`
`Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 111
`
`Table III. Calculated Entropies of Fusion of Some
`Alkyl-p-aminobenzoates at 37 o C
`
`ester
`methyl
`ethyl
`propyl
`butyl
`pentyl
`hexyl
`heptyl
`octyl
`nonyl
`dodecyl
`hexadecyl
`
`mp,
`oc
`112
`89
`74
`56
`52
`61
`75
`71
`69
`82
`87
`
`Ll.Sf,
`obsd
`15.1 a
`13.1
`14.6
`17.8
`17.8
`25.2
`18.1
`28.3
`31.4
`41.5
`55.5
`
`13.5 +
`2.5
`(n- 5)
`13.5a
`13.5
`13.5
`16.0
`18.5
`21.0
`23.5
`26.0
`28.5
`36.0
`46.0
`
`n-5
`0
`0
`0
`1
`2
`3
`4
`5
`6
`9
`13
`
`a All entropy values are expn•ssed in eu.
`
`The ideal solubility of many crystalline compounds can
`be estimated from the melting point and entropy of fusion
`as given by eq 5 and 6.
`Because the above treatment is based on many as(cid:173)
`sumptions and approximations, it cannot be expected to
`provide highly accurate solubility estimates for all com(cid:173)
`pounds. It does, however, provide a very simple means of
`obtaining a reasonable estimate of ideal solubility from
`nothing more than the structure and melting point of the
`compound in question. If means were available for pre(cid:173)
`dicting melting point from chemical structure, this type
`of approach could be used for the design of compounds
`having desired solubility properties.
`
`Literature Cited
`
`Aranow, R. H.; Witten, L.; Andrews, D. H. J. Phys. Chern. 1958, 62, 812.
`Bondi, A ... Physical Properties of Molecular Crystals, Liquids and Glasses .. , Wiley:
`New York, 1968.
`Breusch, F. L. Fortschr. Chern. Forsch. 1969, 12, 119.
`Bunn, C. W. J. Polyrn. Sci. 1955 323.
`Garner, W. E.; Madden, C. F.; Rushbrooke, J. E. J. Chern. Soc. 1926,2491.
`Hildebrand, J. H.; Scott, R. L ... Regular Solutions .. , Prentice-Hall: Englewood
`Cliffs, N.J., 1962.
`Hildebrand, J. H.; Scott, R. L ... The Solubility of Nonelectrolytes .. , Reinhold: New
`York, 1950.
`Hirschlelder, J. 0.; Stevenson, D.P.; Eyring, H. J. Chern. Phys. 1937, 5, 896.
`Lennard-Jones, J. E.; Devonshire, A. F. Proc. R. Soc. London, Ser. A 1939,
`170, 464.
`Luttinghaus, A:, Vierk, G. Ber. 1949, 82, 376.
`Pirsch, J. Ber., 1937, 12; Mikrochirn. Acta 1956, 992.
`Tsonopoulos, C.; Prausnitz, J. M. Ind. Eng. Chern. Fundarn. 1971, 10, 593.
`Ubbelohde, A. R ... Melting and Crystal Structure .. , Oxford: London, 1965.
`Walden, P. Z. Elektrochern. 1908, 14, 713.
`Yalkowsky, S. H.; Flynn, G. L.; Slunick, T. G. J. Pharrn. Sci. 1972 61, 852 .
`
`Received for revieu: December 19, 1977
`Accepted November 27, 1978