THE INTERFACE SYMPOSIUM
PAUL D. CRATIN
Partitioning at the Liquid- Liquid Interface he importance of the distribution coefficient is freTquently overlooked in the undergraduate physical chemistry curriculum. Sometimes it is mentioned in connection with “systems which lend themselves nicely to straightforward applications of the phase rule.” At other times, the distribution coefficient will appear in one form or another when zone refining, partition chromatography, or solvent extraction is discussed from a mathematical standpoint (27). Even in many texts on chemical thermodynamics, the “Nernst distribution law’’ is often relegated to a position of secondary importance. This situation is a regrettable one, for herein lies an opportunity to inter-relate such significant and often abstruse concepts as fugacity, activity coefficient, chemical potential, Henry’s law, and ficticious standard states (9, 70, 72, 79). For those embarking on a study of complex and competing equilibria, the MoelwynHughes treatment of distribution phenomena gives an outstanding introduction (22). Indeed, one might say that virtually every physicochemical equilibrium may be regarded as a particular case of molecular distribution. Likewise, in the field of surface chemistry, there exist numerous examples where some form of the distribution coefficient can be used to represent various equilibria. For example, consider the apportioning of molecules between a gas phase and an adsorbing solid surface ( I 7 ) , the equilibrating of surfactant molecules in solution with those adsorbed at a liquid surface or interface (24, 26), or the partitioning of solubilizate between micelles and the bulk phase (3, 27). In this paper, we shall be concerned only with the distribution coefficient which describes the partitioning of a nonionic amphipathic solute between two immiscible solvents. Accordingly, we shall define the distribution coefficient by Equation 1 :
where C ( W ) and C(o) refer to the equilibrium molar concentrations of the monomer in the aqueous and nonaqueous phases, respectively. By defining K D in this manner, we restrict ourselves to ideal solutionsthat is, solutions in which micelles are not present, and 14
INDUSTRIAL A N D ENGINEERING CHEMISTRY
in which neither phase contains solubilized or dissolved material of the other phase. Relationship between Distribution Coefficient a n d HLB One would expect, intuitively, that a relationship would exist between the distribution coefficient, which is firmly based upon thermodynamics, and the hydrophilelipophile balance (HLB) of a molecule, which was established on a purely empirical basis (75). Early attempts to relate HLB to the distribution of nonionic surfactants between water and oil met with litttle success (76), but no doubt much of this was due to experimental difficulties. The first breakthrough was scored by Davies (7, 8),who derived a quantitative relationship between HLB and K , from a consideration of the kinetics of coalescence in emulsion systems. According to Davies, “the empirical HLB values have a fundamental significance in terms of free energies, and should be related to the distribution of surface-active agents between oil and water under certain conditions” (7). Davies’ equation in its simplest form is
(HLB - 7)
( 21 Once an HLB-K, relationship was established, Davies proceeded to give thermodynamic significance to the ability of a molecule to determine emulsion type and stability and to its capabilities as a wetting agent, detergent, or defoamer ( I , 2). Assuming that it is possible to obtain valid distribution coefficient data, one would no longer need to rely upon empirically determined group numbers to obtain a correlation between HLB and surfactant properties. I n principle, one could assign to each group constituting the surfactant molecule standard thermodynamic values which describe quantitatively its transfer from one phase into another. Unfortunately, there are many experimental difficulties associated with this type of measurement, and these probably account for the scarcity of published data on nonionic surfactants. For example, isolation and purification techniques must be developed to separate isomers and/or homologs From the reaction potpourri; a suitable immiscible solvent pair must be =
0.36 In C(w)/C(o)
The distribution coefficient, long relegated to a position of secondary importance, assumes new prominence in providing themdynamic signifiance to surface chemical phenomena
-1
V O L 6 0 NO. 9 S E P T E M B E R 1 9 6 8
I5
found; experimental methods to determine accurately the (wide range of) surfactant concentrations must be available; and corrections for solubilization must be made. Because agitation of the system frequently results in the formation of stable emulsions, one must rely solely upon the extremely slow diffusion processes to establish equilibrium between the two phases (4). I n spite of these difficulties, some meaningful data have been published in this area (5, 6, 73, 74, 20); the recent work by Crook, Fordyce, and Trebbi is especially noteworthy (5, 6 ) . Crook and co-workers studied the distribution of a series of p,t-octylphenoxyethoxyethanols (OPE’S) between iso-octane and water a t several temperatures. From their data they were able to calculate the changes in standard free energy, enthalpy, and entropy of transfer from one phase into the other of the ethylene oxide moiety, The derivations given in this paper form a logical extension of Crook’s work by providing a method whereby thermodynamic properties may be assigned to all groups-even, perhaps, to atoms themselves. Choice of Standard Reference States
+ R T In X t
(3)
where p i e is the chemical potential of pure “ 2 ’ in the solution at a specified temperature and pressure, and X i is its mole fraction. In the event that the solution becomes nonideal before X i = 1, F~ is not the actual chemical potential of pure “i” (sometimes denoted by uo) but the value it would have if the solution remained ideal up to X , = 1. I n other words, it is a limiting form of Henry’s law. As a consequence, it can be shown that the thermodynamic distribution coefficient, KD’, based upon ideal solutions, should have the form :
KD’
=
X(w)/X(o)
(4)
in which X ( w ) and X ( o ) refer to the mole fractions of solute in the aqueous and nonaqueous phases, respectively. Hence, it will be necessary to convert from one system of concentration units to the other. The mathematical development which follows is patterned after Everett ( I 7). The molar concentration of the ith component, c,, is defined as
c,
= ni V
(5)
where n, is the number of moles of component i, and V is the volume of solution. If the solution is sufficiently dilute,
v 16
Ns17so
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Furthermore,
since N , >> Xi. By substituting X i for N i / N , in Equation 7 we obtain (9) Rearranging Equation 9 and taking logarithms will give In X, = In C,
(6)
+ In VsO
(10)
Finally, Equation 3 may be written for component i in the following manner :
w,(T, P, X) = p : ( T , P)
We shall first discuss the standard reference states chosen in our mathematical development so there will be no ambiguity as to what the thermodynamic changes refer. An ideal solution may be defined (77, 23) as one in which each component follows the equation :
d T , P, X ) = ~ d s ( TP, )
vso
in which N , and are the number of moles and molar volume, respectively, of solvent in the solution. Thus,
+ R T In
V80
+ R T In C, (1 1)
Equation 11 shows that the chemical potential based upon mole fractions is larger than that based upon the molar concentration by R T In pso. This means that the value of the chemical potential if expressed on the molar concentration scale, even in ideal solutions, depends upon the molar volume of the solvent! Having defined the standard reference state for our system, we may now proceed with our derivation, Derivation of Free-Energy Equations
Let us assume that the total free energy of a molecule, of a lipophilic group ( L ) and “n” hydrophilic groups ( H ) may be represented by the equation ~.l~, comprised
P t b ) =
PLb)
+ nPH(w)
(12)
+
(13)
and Pt(0)
= PA01
nPH(0)
where (w)and ( 0 ) again refer to the aqueous and nonaqueous phases. If we are dealing with sufficiently dilute solutions so that they behave ideally, we may write ~ . t ( w )=
I*L’(~)
and
PAO) =
+
+ R T In X ( W ) + nPfle(o) + R T In X(o) WH’(~)
(14)
(15 ) Introducing Equation 10 into Equations 14 and 15, we obtain ~ ~ ‘ ( 0 )
+ npHe(w)+ R T In vo(w)+ R T In C(w)
ht(w) = p L o ( w )
(16)
D.Cratin is a Research Associate in the Surface Chemistry Group at St. Regis Paper Co., Technical Center, West iiyack, hi. Y .
AUTHOR Paul
Figure 1 is a graph of Crook's data (5) on OPE1-lo. The curve is linear for four orders of magnitude above n = 3. Below n = 3, some nonlinearity develops, but this can be explained on the basis ofinteractions between the phenolic group and the small ethylene oxide side chains. Crook gives the equation for this line as
100
10
log KD = 0.4420 12 - 3.836
(20)
This corresponds to a standard free energy change (at 25 "C) of transfer per mole of ethylene oxide (0 + w ) of -0.602 kcal. The intercept, -3.836, may be used with the values of the molar volumes of the two phases to give the standard free energy charge (o + w ) of the p,t-octylphenoxyethoxy group, f6.52 kcal mol-'. Note that the free energy change for the transfer of the hydrophilic group from nonaqueous to aqueous phase is negative; this indicates a spontaneous process. O n the other hand, the free energy change for the lipophile is positive-which means that this group prefers to be in an oil-like environment. This is precisely what one would predict.
10-
Y
lo-'
Determination of Group Free Energies 10-
101
0
2
3
4
5
7
6
8
9
10
n(CH&.O-)
Figure 7 . Logarithm of the distribution coejicient us. the number of ethylene oxide adducts in p,t-octylphenol, After Crook (5)
and pt(o) = pL'(o)
+ npHe(o) + R T In vo(o)+ R T In C(o) (17)
When equilibrium is established between phases, so, we may equate Equations 16 and 17 and collect terms:
pLt(w)= p t ( o ) ,
- NL'(0)
IPL%J>
12 [ P H ' ( ( ~ >
I
+ R T In [ V"w>/ PHe(o)
I
I+
V0(0)
= --RT In [C(w)/C(o)]
(18)
To simplify Equation 18, let us replace C(w)/C(o) by K D and put Ape = p'(o)
The extension of Equation 20 for determining other group (and perhaps atomic) free energies is carried out in a straightforward manner. As an illustration, let us suppose that we wish to determine Ape for the methylene group, CH2. There are many ways in which this can be accomplished, but only two will be taken up. These can serve to check the validity of the theory. Method A. Let us assume that we have, for example, two homologous series of compounds, both having the same lipophile, but different hydrophiles. For the sake of convenience, let us assume one contains propylene oxide (CH2-CH2-CH2-0-) adducts, and the other, ethylene oxide (CH2-CH2-0-) groups. The intercepts of log K D us. n for both of these materials will be the same, since the lipophiles are identical. The propylene oxide materials will exhibit a less positive slope because the extra CH2- tends to make the molecule more lipophilic. Indeed, based upon HLB group numbers ( 8 ) )one would expect a slightly negative slope. Figure 2 is an idealized sketch of log KD us. n for both homologous series. The slope of B is given by Ape(CH2-CH*-CH$-R T , whereas the slope of A is Ape(CH20-)/2.3 R T . If the free energies are additive, CHz-0-)/2.3 Ap'(CH2-CH2-CH2-0-)
- pe(w)
=
3Ape(CH2-)
+ A$(-0-)
(21)
and
Hence,
ApLB(CH2-CH2-O-)
= 2Ap'(CHz-)
+ Ape(-0-) (22)
(19)
If Equation 19 is valid, a plot of log KD us. n will be linear with a slope equal to A p ~ ' / 2 * 3R T , and with an log [ P o ( o ) / v o ( w ) ] . intercept of ApL'/2.3 RT
+
so by subtracting Equation 22 from Equation 21 we obtain
Ape(CH~--CHz-CHz-O-)
-
ApB(CH2-CH2-0-) VOL. 6 0 NO. 9
=
Ap'(CH2)
SEPTEMBER
1968
(23) 17
Similarly, since both Ap'(CH-)
CH+)
and Ap'(CH-
are known,
-
A~*(CH-CH+)
2 A/.t'(CH-)
=
Ap'(U)
(24)
Hence, a table of parachor-like values, not unlike the thermodynamic group contributions given by Janz (78), can be compiled. Method B. In the original derivation, we had assumed a molecule comprised of one lipophilic group and W' hydrophilic units. Since this designation is entirely arbitrary, we can speak of a molecule containing a single hydrophilic unit (H) and "n" lipophilic units (L). In our example, the lipophilic unit would be the CH- group. It is not unreasonable to expect a log K , us. n for this homologous series to be similar to the idealized curve in Figure 3. Should the two values of Ar$(CH,) as determined by the independent methods differ, additional experimentation would be required, and suitable corrections for deviations from ideality would have to be made, as is often done in thermodynamics (78).
lo-'
9
10-1
0
1
2
3
4
5
6
7
I
Determination of Group Enthalpies and Entropies
To determine the group contribution to the molecular enthalpy, one needs to know the dependence of K, upon temperature. From thermodynamic considerations we can show that
Figure 2. Idealized curut to show the relationship bctwcm log KO and fhe number of odducrs of (A) ethyl- oxide and ( E ) propylene oxide inp,t-octylphenol. Calculated uring D a v i d cquaiion andHLB group numbers (7, 8 ) . Circles are Crook's expnimmral data pOints (5)
10-3
where &" is the standard molar change in enthalpy associated with the transfer of solute from the nonaqueous to the aqueous phase. We are neglecting in Equation 25 the effect of temperature on the molar volumes of the two liquid phases. This effect will be small compared to the magnitude of heat changes. If &" is independent of the temperature in the interval studied, Equation 25 may be integrated to yield
In K , =
- -A@ -+ constant RT
--.(26)
To remain consistent in our symbolism, let AR" -AI?, where ARB refers to the enthalpy change accompanying the transfer of a mole of solute from the aqueous phase to the nonaqueous phase. We may write ARBas the difference between two states-namely,
AI? =
R,b(o)- R,b(w)
(27)
Io-
10-7
If the group enthalpies, like group free energies, are assumed to be additive, Equations 28 and 29 follow:
RP(w)
=
R,'(w)
+ nR,B(w)
(28)
and
0
R,B(o)= R,B(o)+ nR,b(o)
(29) Subtracting Equation 28 from Equation 29 we obtain
AI? 18
=
+ n&,b
(30)
INDUSTRIAL AND ENGINEERING CHEMISTRY
1
2
3
4
5
6
wr) Figure 3. Idealized curve to show the relationship betwm the logarithm of the dism'bution coc&eni and number of methyIem adducts in p,t-octylphmol. After DaUies (7,s)
To evaluate both terms on the right-hand side of Equation 30, we need to know the quantitative dependence of K , upon temperature for two members of the homologous series. This is best illustrated by another example. Let us assume we have two members of a homologous series, one containing n hydrophilic units, the other 1) units. containing ( n Substitution of Equation 30 into Equation 26 for these two homologs gives
+
Subtracting Equation 32 from Equation 31 yields
where CB = C1 - Cz. Thus, a graph of In [K, ( n l ) / K D ( n ) ] us. 1 / T yields a straight line of slope equal to Ai?,’/R. Since ARHecan be determined in this manner, and A R B is known for a given material of “n” hydrophilic units, the contribution of the lipophilic group can be calculated by difference :
+
AH -Le - AB’
-
to the nonaqueous phase. They represent endothermal, athermal, and exothermal changes, respectively. Entropies can now be calculated using Equation 35 :
A$ = - TAP (35) I n conclusion, we should comment on the assumption made and the conditions which need to be met for the method to be valid. 1. The total free energy of a molecule is given by the sum of the free energy contributions of the hydrophile and lipophile units constituting the molecule-Le., there must be no interactions between or among the groups which comprise the molecule. For example, once an ethylene oxide side chain in the OPE’S gets much larger than n = 10, interactions begin playing a significant role, so that the thermodynamic properties are no longer additive. 2. The molecular species in both phases is the same. Should ionization and/or association of the solute occur, the distribution coefficient will show a rather marked dependence upon concentration as well as temperature (and pressure). Additional equations describing these equilibria must be written, and the effects accounted for. The inclusion of these equations complicates the mathematics considerably (22). 3. The concentration is below the critical micelle concentration. Above the cmc, KO varies greatly with concentration and the solutions no longer behave ideally. 4. The enthalpies of transfer are constant in the temperature range studied. Should the Ai?“s vary with temperature, one would need to know the_ heat capacities to establish a functional relation between AHe and temperature, for
(34)
A graphical schematic is depicted in Figure 4. Curves A , B , and C refer to enthalpy changes which accompany the transfer of hydrophilic groups from the aqueous
By extending this treatment to many homologous series, we would accumulate a table of group thermodynamic properties which should prove to be a worthwhile addition to our technology. REFERENCES
1/T
Figure 4. Idealized curves to show the relationship between log [KD(n 7 ) / K ~ ( n )and ] 7 / T . The slopes of A, B, and C are related to the sign and magnitude of the enthaLpy changes accompanying the transfer of hydrophiles from the aqueous to the nonaqueousphase
+-
(1) Becher, P., “Emulsions, Theory and Practice,” Reinhold, New York, 1966, p. 232 ff. (2) Becher, P., J . Sot. Cosmetic Chemistr, 11, 325 (1960). (3) Cratin, P.D., and Robertson, B. K., J . Phys. Chem., 69,1087 (1965). (4) Cratin, P. D., and Zetlmeisl, M. J., unpublished. (5) Crook, E. H., Fordyce, D. B., and Trebbi, G. F., J.ColloidSci., 20,191 (1965). (6) Crook, E. H., Fordyce, D. B., and Trebbi, G. F., J . Phys. Chem., 67, 1987 (1963). (7) Davies, J. T., Proc. 2ndIntern. Congr. Surfoce Activity, 1,476 (1957). (8) Davies, J. T., and Rideal, E. K.,“InterfaciaI Phenomena,” p 343 ff, Academic Press, New York, 1961. (9) Denhigh, K “ T h e Principles of Chemical Equilibrium,” p 270 ff, Cambridge, London, 1966: (10) Ibid., Chapters 9 and 10. (11) Everett, D. H,“Chcmical Thermodynamics,” p 54 ff and p 90 ff, Longmans, 1959. (12) Ibid., Chapters 7 and 8. (13) Grcnwald, H. L., Kice, E. B., Kenly, M., and Kelly, J., Anal. Chem., 33, 465 (1961). (14) Grieger, P. F., and Kraus, C. A , , J.Amer. Chem. Soc., 71,1455 (1949). (15) Griffin, W. C., J . Sot. Cosmetic Chem., 1, 311 (1949). (16) X d . , 5 , 249 (1954). (17) Hudson, J. B., and Ross, S . , “Chemistry and Physics of Interfaces,” p 103 ff, ACS Publications, Washington, 1965. (18) Janz G. J. “Estimation of Thermodynamic Properties of Organic Compounds,” Chapter 5, Academic Press, New Y a k , 1953. (19) Klotz, I. M., “Chemical Thermodynamics,” Chapters 18 and 19, PrenticeHall, New York, 1950. (20) Lawrence, A. S. C., and Stenson, R., Proc. 2nd Intern. Congr. Surface Activity, 1, 388 (1957). (21) McB+ M. E. L., and Hutchinson, E., “Solubilization and Related Phenomena, p 139, Academic Press, New York, 1955. (22) Moelwyn-Hughes, E. A., “Physical Chemistry,” p 1077 ff, Pergamon, New York, 1961. (23) Prigogine, I., and Defay, R., “Chemical Thermodynamics,” p 78 ff, Wiley, 1954. (24) Ross, S., and Chen, E . S., “Chemistry and Physics of Interfaces,” p 43 ff, ACS Publications, Washington, 1965. (25) Ross, S . , Chen, E. S., Becher, P., and Ranauto, H., J . Phys. Chem., 69,1681 (1959).
(26) Ross, S., and Cratin, P. D., unpublished. (27) Sheehan, W. F., “Physical Chemistry,” p 319 ff, Allyn and Bacon, Boston, 1961.
VOL. 6 0
NO.
9
SEPTEMBER 1968
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