J. Phys. Chem. 1993,97, 7144-7146
7144
Thermodynamics of Hydrogen-Bonded Fluids: Effects of Bond Cooperativity Boris A. Veytsmant Polymer Science Program, 320 Steidle Building, Pennsylvania State University, University Park, Pennsylvania I6802 Received: May 20, I993
The thermodynamic theory of hydrogen-bonded networks is extended to fluids with interacting bonds. I t is shown that the results are consistent with those obtained in the framework of association models. The thermodynamic theory, however, can be applied to systems with more complicated networks of H bonds, where association theory fails.
Introduction There are two approaches in the theory of H-bonded liquids. The first one (association theory) treats liquid as an equilibrium mixture of monomers, H-bonded dimers, trimers, etc. The stoichiometric equations are used to obtain concentrations of the species, and the free energy of the fluid is calculated using some approximation for the free energy of mixing the monomers, dimers, trimers, etc. A good review of this approach can be found in ref 1. The second approach (contact point or thermodynamic theory)ZJ is based on the equilibrium between the formation and capture of hydrogen bonds (Le., treats contacts rather than species). At a given number of hydrogen bonds one calculates the excess free energy due to their formation. Minimizing it with respect to the number of hydrogen bonds, one obtains both stoichiometric equations and the free energy. In the recent study of Sanchez and P a n a y i o t o ~ , ~it. ~was shown that both these approaches predict similar behavior if both are applicable. However, there are a number of advantages of the second approach. First, it is more straightforward, and some things that are rather obscure in the association theory become clear in the thermodynamic theory. As an example, one can mention the discussion of the proper choice of association model, whether it should be the Mecke-Kempter or Kretschmer-Wiebe one (see ref 5 , Chapter 4 for details). In the thermodynamic theory, the proper choice becomes self-evident. Second, the thermodynamic theory can be applied to hydrogenbonded networks-where the association theory cannot be readily applied. Consider, for example, a mixture of A and B molecules each having a proton-donor and a proton-acceptor group. In the framework of the association theory we must consider equilibrium mixture of complexes like A, B, AB, AB, BB, AAA, AAB, ABA, ABB, BBB, etc. Some ingenious play with recursive equations can give results even here, but if the molecules have two or more donor or acceptor groups, the complexes become branched and we can only quote ref 3: this problem “has resisted our attempts of solution.” The calculations of the thermodynamic theory approach are in this case very straightforward, and the results are in a good agreement with experiments.3~4 However, the thermodynamic theory has a serious drawback, preventing its application to a number of systems. Namely, in this theory all hydrogen bonds of the same kind must have the same equilibrium constant. In other words, the formation of hydrogen bonds must be independent of whether other donor or acceptor groups of the molecules are H-bonded. This is true for some systems (e.g., containing carboxylic acid groups5), but it does not hold for molecules with certain types of functional groups. For example, for phenols the equilibrium t On leave from Physics-Chemical Institute, 86 Chernomorskaya Doroga,
Odessa 270080, Ukraine.
constant of dimer formation is several times less than the equilibrium constant describing subsequent h-mer formation.$ This effect is called hydrogen-bond cooperativity and may be of considerable importance for many systems (see ref 6 and references therein). It is evident why this effect is easily described by association models and is difficult to account for in the framework of the thermodynamic theory. The association models are “global”; they take the H-bonded “polymer“ as a whole. The thermodynamic theory is “local” and deals with one H-bond at a time. The cooperativity of H bonds is not a local effect, so it is difficult to describe it from the viewpoint of a local theory. However, this cooperativity is apparently not too “global”, and it affects only neighboring H bonds, so one may hope that extending a local theory a little will solve this problem. In this paper we shall see that the thermodynamic theory can be extended to account for the interaction of hydrogen bonds. Thus, one can use the thermodynamic theory for all cases. The general case of the system we are discussing is the solution of k kinds of molecules with d kinds of proton4onor and a kinds of proton-acceptor groups, each molecule of kind i having d i donor groups and a; acceptor groups of kinds k and m. If there were no interaction, the equilibrium constants would be Kkm, but d u e t o interaction they a r e something like @d ,...Sol,. . . , P d l ~ d a ~ ~ ~ ~where ~ ~ ~ ~ ~symbols ~ , . . . ) s and p are 0 or 1, km’ depending on whether the proton active groups of the molecules forming the (k,m) bond are engaged in other bonds. The number of sub- and superscripts in the subsequent equations becomes formidable, and the equations themselves are quite unreadable. To make the typist’s life easier, we shall not discuss this case here. Instead, we shall illustrate how this method works on a very simple, almost toy model. After this the generalization of the method to a more complex case is a straightforward task. Moreover, for this case the results of the association theory are available, and we can compare the two models. A Simple Model and Calculations
Let us discuss the simplest possible case: a fluid of molecules (A) each having one donor group and one acceptor group. The constant of dimer formation is Kz,and that of the subsequent polymer’s formation is K p . In a chain k-mer A-A- ...& there are (k - 1) H bonds, ( k 2) of them are polymerlike and one is dimerlike (the primal one). It is not important which one is dimerlike, so we suppose that it is the hydrogen bond formed by the acceptor group of the molecule having a free donor group. We shall call the acceptor group of the molecule weak if the donor group of the same molecule is free and strongotherwise. If a neighboring acceptor group and donor group form an H bond, the change of free energy would be AF, for a strong acceptor and AF, for a weak acceptor.
0022-3 654/93/2097-7 1443Q4.QQ/Q 0 1993 American Chemical Society
Letters
The Journal of Physical Chemistry, Vol. 97, No. 28, 1993 7145
Let us consider N molecules that formed M bonds, N? of them are dimerlike and ( M - N2) are polymerlike. As in ref 2, the excess free energy due to H bonds in the mean field approximation would be
FH = AF,M
+ (AF,- AFs)N2- k T In E
(1) where Z is the number of ways of distributing M bonds between neighboring donor/acceptor groups, but now we have a constraint: N2 of the bonds must be dimerlike. The mean field approximation for Z is (see ref 1)
E =EopM
(2) where EO is the number of ways of distributing H bonds disregarding that acceptor and donor groups must be neighbors, and p is the probability that the given donor/acceptor pair are neighbors. It was shown in ref 1 that
p = c/N (3) where c is a constant depending on the geometry of the molecules. Now let us calculate Eo = Z0(M,N2). Let us enumerate the H bonds, in such a way that the first N2 of them are dimerlike and the subsequent ( M - N2) are polymerlike. First let us choose donor groups participating in the H bonds. It can be done N / ( N - M)! ways. After that, we must choose acceptor groups. There are N - M weak acceptors, and only N2 of them participate in the H bonds, so the number of ways to choose themis ( N - M ) ! / ( N - M - N 2 ) ! Analogously, thenumber of way to choose strong acceptors is M l N z ! , and in a final touch, we must recall that the hydrogen bonds are indistinguishable. However, one can distinguish dimerlike bonds from polymerlike ones, so the result must be divided by N2!(M - Nz)! Thus 3
Lo=
--
N! (N-M)! M! 1 ( N - M)! ( N - M - N2)! N2! ( M - N2)!
N,!
N! M. (N-M-N,)!
(4)
[N2!I2(M-N2)!
Using Stirling’s approximation N! = N In N / e , one has from eqs 1-4
FH = - M l n kT M In
K p - N2In r
e N ( M - N2) M(N - M - N2)
+ N l n (1 --
N
N2)
(6)
r =K2/Kp
Minimizing eqs 5 and 6 with respect to M and N2, we obtain both the free energy and stoichiometric relations. It is more convenient to switch now to concentrations: if Vis the volume of the system, then let n = N / V , m = M / V , n2 = Nz/V, and fH = FH/V. Then we obtain the following result:
(7)
where
n(m - n2) = Kpm(n- m - n2)
n,? = r(n - m - n2)(m- n2)
(9)
in agreement with the previous results.233 For the case T # 1 the association model results are at hand. So one can recall thems fH -- n In 41- n-rl
kT r2 where 41 is the volume fraction of monomers, and r,=1-r+r2=1-r+
7
1 - KP41 t
(1 - Kp41)2 The stoichiometry equation is Now let us determine 41from our calculations. In fact 41is the probability for a molecule to be free. The probability of a donor group to be free is (n - m)/n. If a donor group is free, the corresponding acceptor group is weak, and a probability for a weak group to be free is ( n - m - n 2 ) / ( n - m). Thus
n-m-n, n It is easy to see from eq 8 that
41 =
r2= n - mn- n ,
(5)
Kp = c exp(-AFJkT)
_-
mn = Kp(n- m ) 2
(13)
rl = n -nm- -mn ,
where
fH
First, let us check the results for their consistency with those obtained for noninteracting bonds. If bonds are not interacting then T = 1, and it immediately follows from eq 8 that nz = m(n - m)/n. Substituting this value of nz, we obtain
+
22 + N, In (N - M -NN&(M - N2)
K2 = c exp[(-AF,/kT)
Are Our Results Consistent?
(8)
(14)
and eqs 10-12 are valid. One can say that our paper is just a circumvented way of obtaining eqs 10-12, but this is not exactly so. First, this method can be easily extended to more complex systems with a number of active groups. Second, the derivation of eqs 7 and 8 is much more clear this way. Third, eqs 7 and 8 are not only nicer than eqs 10-12 but also more meaningful. Indeed, it is very difficult to measure 41 in an experiment. What can be readily measured in I R experimentsSis the number of hydrogen bonds ( m )and the fraction of dimerlike ones (nzlm). So eqs 7 and 8 are more handy for the processing of experimental data. Last, but not least, it is easy to expand this approach for the complex systems where association models fail.
Conclusion We showed that the thermodynamic theory can cope with problems that up to now were themonopoly of association models. It should be interesting to expand the thermodynamic theory to other problems: e.g., the formation of cyclic species, complexes, and chain-ring transitions. The theory of more complex cases than the one described above (such as mixtures or highly bonded liquids) demands a priori
7146 The Journal of Physical Chemistry, Vol. 97, No. 28, 1993
knowledge of the interaction Darameters describing H bonds. This can-be achieved either b; quantum mechanicscalculation or by group contribution methods or just by lucky insight. The data in this field are far from complete, so there is much work to be done.
Acknowledgment. This paper is a side result of long and fruitful discussions with Professor Paul Painter. These discussions were financed by the NSF (Grant DMR-9017387).
Letters
References and Notes (1) March, K.; Kohler, F. Mol. Uq.1985, 30, 13. (2) Veytsman, B. A. J . Phys. Chem. 1990, 94, 8499. (3) Sanchez, I. C.; Panayiotou, C. G. In Thermodynamic Modeling,
Sandler, S.,Ed.; Marcel Delcker: New York, in press. (4) Panayiotou, C.; Sanchez, I. C. J. Phys. Chem. 1991, 95, 10090. ( 5 ) Coleman, M.M.; Graf, J. F.; Painter, P. C. SpecIflc Interactionsand the Miscibility of Polymer Blends; Technomic: Lancaster, PA, 1992. (6) Maes, G.; Smets, J. J . Phys. Chem. 1993, 97, 1818.
