On intramolecular and intermolecular hydrogen bonding

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Ž. Fluid Phase Equilibria 156 1999 51 56 On intramolecular and intermolecular hydrogen bonding Doukeni Missopolinou, Costas Panayiotou ) Department of Chemical Engineering, UniÕersity of

biological interest. Of interest is the case of very dilute systems in inert solvents where intermolecular hydrogen bonding is absent and the only observed hydrogen bonds are the intramolecular ones.

1 Ž. Fluid Phase Equilibria On intramolecular and intermolecular hydrogen bonding Doukeni Missopolinou, Costas Panayiotou ) Department of Chemical Engineering, UniÕersity of Thessaloniki, Thessaloniki, Greece Abstract Intramolecular hydrogen bonding is often an important contribution to the overall hydrogen bonding in fluid systems, especially in systems of biological interest. Of interest is the case of very dilute systems in inert solvents where intermolecular hydrogen bonding is absent and the only observed hydrogen bonds are the intramolecular ones. The objective of this note is to present the necessary formalism in the frame of the Ž. 1 lattice-fluidrhydrogen-bonding LF model for studying fluid systems with both intermolecular and intramolecular hydrogen bonds. For simplicity, we will confine ourselves to the case of molecules with one donor group Ž such as O. and x equivalent proton acceptor sites Ž such as ether oxygen O. per molecule. This type of system Ž polyethoxyalcohols. is of direct interest to the IUPAC project on alkanol alkane ether systems as they involve the same type of interactions, and is of key importance in the study of non-ionic surfactants. The formalism can easily be extended to more complex cases. q 1999 Elsevier Science.V. All rights reserved. Keywords: Statistical mechanics; Chemical potential; LF model 1. The model The presentation in this paragraph will heavily be based on our previous work wx 1 where the details of the original lattice-fluidrhydrogen-bonding Ž LF. theory is presented. As in the LF model, the full partition function Q of our system in the N, P, T ensemble in its maximum-term approximation is factorized into a physical term, Q P, a chemical or hydrogen-bonding term, Q, and the classical exponential volumetric term, or QsQ Q expž ypvrkt. Ž 1. P ) Corresponding author. Tel.: q ; fax: q ; cpanayio@mailhost.ccf.auth.gr r99r$ - see front matter q 1999 Elsevier Science.V. All rights reserved. Ž. PII: S

2 52 D. Missopolinou, C. PanayiotourFluid Phase Equilibria where V is the total volume of the system. We will focus attention on the chemical factor. According to the LF model, this term may, in general, be written as N 0 r ÝN ij G / ij rn kt Q s Vexp y Ž 2. ž where N is the total number of molecules in the system, Nij is the number of hydrogen bonds of type i j characterized by a free energy change G 0 ij and N sýnij is the total number of intermolecular hydrogen bonds. The term rrr is the ratio of the reduced density over the number of segments per molecule as calculated by the LF model wx 1, but for the purposes of this note it may be considered a constant. The preexponential factor V is the number of different ways of distributing the hydrogen bonds in the system without requiring that donor and acceptor groups be neighbors. This requirement of donorracceptor proximity is taken into account by the first term in the rhs of the above equation wx 1. The focus now is on the statistical derivation of V. The method will be explained by applying it to a rather classical case Case: Polyethoxyalcohol PEA with x ether oxygen sites Let us consider a system with N1 PEA molecules and N2 molecules of an inert solvent Ž NsN qn.. The number of proton donors of type 1 Ž O. 1 2 is N 1, of proton acceptors of type 1 Ž O. is N, and of proton acceptors of type 2 Ž O. 1 is xn 1. Let there be N11 hydrogen bonds O O, N12 intermolecular bonds O O, and intramolecular bonds O O in the system. The number of free proton donors are N10sN1yN11yN 12y Ž 3. The number of different ways of distributing the above hydrogen bonds in the system can be found by applying the rationale of the LF model wx 1. According to this rationale, in order to find the different number of isoenergetic configurations of our system we have to do the following: Ž a. Find the number of different ways of selecting the associated donor sites out of the donor population. Ž b. Find the number of different ways of selecting the associated acceptor sites out of the acceptor population. Ž c. Find the number of different ways of making hydrogen bonds between the selected donor and acceptor sites. The number of configurations of the system is the product of these three terms. Let us apply the above procedure to our case. We have first to select the N, N,, and N donors out of the N1 donor population. From simple combinatories, this can be done in N!rw!N!N!N! x ways. In a second step, we have to select the N acceptors 1 out of the N acceptor population. This can be done in N!rw N! Ž N yn.! x ways. In a third step, we have to select the acceptors 2 out of the xn1 acceptor population. owever, once we have selected the proton donors that participate in intramolecular bonds, we have also selected the molecules with the acceptor 2 sites that participate in the intramolecular bonds. We will assume for simplicity that all x acceptor sites are equivalent for the intramolecular bonds. In each of these molecules, we must

3 D. Missopolinou, C. PanayiotourFluid Phase Equilibria now select the acceptor 2 site for the intramolecular bond out of the x acceptor 2 population. For each molecule, this can be done in x!rw1! Ž xy1.! x ways. Thus, for the molecules it can be done in x!rw1! Ž xy1.! x4 sx ways. aving selected the acceptor 2 sites we must now select, out of the remaining Ž xn y. 1 acceptor 2 population, the N12 which will participate in the intermolecular bonds. This can be done in Ž xn y.!rwž xn yyn.!n! x ways. The N11 and N12 bonds can be done in N 11!N 12! ways while the bonds in only one way after we have selected, both, the donor and the acceptor site in each molecule. Thus, the number of configurations in the hydrogen bonded system is: N 1! N 1! x! Ž xn1y.! Vs N 11!N 12!!N!N!N! N! Ž N yn.! Ž xy1.! Ž xn yyn.!n! x Ž N 1!. Ž xn1y.! s Ž 4.!N!N!N! Ž N yn.! Ž xn yyn.! The Gibbs free energy equation Let us now return to Eq. Ž 2. and find the expression for the free energy G for the above case of a hydrogen-bonded system with both intermolecular and intramolecular hydrogen bonds. In order to apply Eq. Ž 2., we have to find the probability factor for the close proximity of each pair of donor and acceptor sites that hydrogen bond. As far as the intermolecular bonds are concerned, this factor is, as 1 before, equal to Ž rrrn. N 11qN 12. The corresponding probability for the proximity of the pair of the intramolecular hydrogen bond is primarily a characteristic property of the molecule and it will be considered here as being a constant c. Thus, the total probability factor becomes equal to Ž. N 11qN 12 rrrn c. In our case, there are three types of hydrogen bonds: N bonds with free energy of bond formation G 0, N 1 2 intermolecular bonds with free energy of bond formation G , and 1 2 intramolecular bonds with free energy of bond formation G 0. The free energy of the i j bond formation can be resolved as following: G 0 se 0 qpv 0 yts 0 ij ij ij ij Ž 5. E 0, V 0, S 0 ij ij ij being the energy, volume, and entropy change of i j bond formation, respectively. The hydrogen bonding term Q of the partition function in Eq. Ž 2. can then be written as following: N 11 qn / N G 0 qn G 0 qng = ž kt / r x Ž N!. Ž xn y.! Q s c ž rn!n!n!n! Ž N yn.! Ž xn yyn.! exp y Ž 6. The hydrogen-bonding part of the free energy of the system is obtained from the equation G sykt lnq Ž 7.

4 54 D. Missopolinou, C. PanayiotourFluid Phase Equilibria y minimizing this equation with respect to the unknowns N, N, and we obtain the following coupled equations: Ž xn y. G 0 1 scexp y sk Ž 8. Ž xn yyn. N x kt N r G K11 s exp y s Ž 9. Ž N yn. N rn kt N N r G K12 s exp y s Ž 10. Ž xn yyn. N rn kt N Ž. Ž. The coupled Eqs must be solved simultaneously by an appropriate iteration scheme. After some algebra, the above three equations lead to the following equations K K Nx N s Ž xn y. Ž 11. K K NxqŽ K11yK12. N s N Ž 12. Ž NxK 1 yk12. K12 xn1 y K11 s N1yy yn1 Ž 13. N xk N xn K qž K yk The last equation contains only the unknown and it can be solved numerically by successive substitutions. The solution for can then be replaced in Eqs. Ž 11. and Ž 12. in order to obtain N12 and N 11, respectively. y taking into account the above minimization conditions, the equation for the Gibbs free energy becomes: G N1m,1 Nq N11 N12 s snqn1ln 1y qn1ln 1y qxn1ln 1y Ž 14. kt kt N N xn y where, m,1 is the hydrogen bonding contribution to the chemical potential of the associating component 1. In the limiting case of highly dilute systems, we have N s N s 0 and, consequently, Eq. Ž becomes K s K or s Ž very dilute system. Ž 15. Ž N y. x N 1qxK 1 1 This is a useful equation which can be used for determining K from experimental Ž such as spectroscopic. information on the degree of hydrogen bonding rn. 1

5 D. Missopolinou, C. PanayiotourFluid Phase Equilibria Discussion The above formalism can be integrated to any equation-of-state framework leading to an equationwx 1 can be of-state theory of hydrogen bonded systems. The procedure used in the LF model applied directly here. The formalism can also be extended in a straightforward manner to the case of more complex systems with more than one proton donor group per molecule. owever, we must keep in mind that, due to steric and other interactions, the acceptor sites may not be equivalent wx 2. This is important when estimating equilibrium constants and hydrogen bonding energies. In the case of PEA molecules, the strength of the intermolecular O O bond is expected to be close to the correspondwx 2. In ing O O intramolecular bond only when the two interacting groups are sufficiently far apart this case, and for concentrated systems, one is justified to neglect the difference between intermolecuw3,4 x. lar and intramolecular hydrogen bonds 4. List of symbols Number of intramolecular bonds c Probability constant E Potential energy G Gibbs free enthalpy k oltzmann constant K Equilibrium constant M Molecular weight N Number of molecules Nr Total number of lattice sites P Pressure Q Partition function r Number of segments per molecule R Gas constant S Entropy T Temperature V Total volume x Proton acceptor sites per molecule Greek Intersegmental interaction parameter m Chemical potential r Density V Combinatorial term in Eq. Ž 2. Subscripts Intramolecular ydrogen bonding i Property of component i

6 56 D. Missopolinou, C. PanayiotourFluid Phase Equilibria ij Property of the pair i j P Physical Superscripts ; Reduced quantity 0 Reference state References wx 1 C. Panayiotou, I.C. Sanchez, J. Phys. Chem. 95 Ž wx 2 M.D. Joesten, L.J. Schaad, ydrogen onding, Chap. 5, Marcel Dekker, New York, wx 3 D. Missopolinou, C. Panayiotou, Fluid Phase Equilib. 110 Ž wx 4 D. Missopolinou, C. Panayiotou, J. Phys. Chem. A 102 Ž

Intramolecular Hydrogen bonding for Lattice fluid EOS. J.H.Lee and C.S.Lee Chemical and Biological engineering in Korea Univ.

Intramolecular Hydrogen bonding for Lattice fluid EOS J.H.Lee and C.S.Lee Chemical and Biological engineering in Korea Univ. Scope Intramolecular Hydrogen bonding Models for intramolecular HB Proposed