6038
J. Phys. Chem. 1994, 98, 6038-6045
Ising Models of Polyprotic Acids and Bases Michal Borkovec' Federal Institute of Technology, ETH-ITO, Grabenstrasse 3, 8952 Schlieren, Switzerland
Ger J. M. Koper Department of Physical and Macromolecular Chemistry, Leiden University, Gorlaeus Laboratories, P.O. Box 9502, 2300 RA Leiden, The Netherlands Received: January 27, 1994"
Ising models, which have received much attention in statistical mechanics, are used to predict acidity constants (pK values) of polyprotic acids or bases. The present model circumvents the calculation of microscopic pK values, which is necessary in the classical approach, and leads directly to all experimentally observable macroscopic pK values. The Ising model allows a transparent parametrization of the problem where the basic parameters are the microscopic pKvalues of the acid-base groups (given all other groups in the molecule are deprotonated) and interaction energies between pairs and triplets of protonated sites. The classical description is recovered if only pair interactions are considered while the model can be systematically improved by including triplet (and higher order) interactions. As an illustration, the model is applied to linear aliphatic amines and polyphosphoric acids.
Introduction The prediction of acid-base properties of small organic molecules has reached a respectable sophistication on the phenomenologica11.2and ab initio level.3 The popular empirical approach reviewed by Perrin et al.1 allows the estimation of pK values (logarithms of the acidity constants) from the molecular structure with reasonable accuracy. The approach rests on the assumption that the pKvalue of an individual group can be written as the sum of empirical group contributions (either obtained from tables or calculated from Hammett or Taft equations) and the pK value of the isolated acid-base group in consideration. While this phenomenological approach was developed mainly for monoprotic acids and bases, in principle, it can be used for thecalculation of all pKvaluesof an arbitrarily complexpolyprotic molecule. One then has to estimate the pK values of all acidbase groups on the molecule for a given protonation state of all other groups. The procedure must be repeated for all different protonation states of the molecule, since the pKvalues of individual groups often depend on whether the neighboring groups are protonated or deprotonated. Proceeding this way, one obtains the so-called microscopic pKvalues, which must be converted to the macroscopic pK values observed in potentiometric titration experiments.4-6 For typically more than three acid-base groups in a molecule, the above approach becomes very tedious and the results often cease to be reliable. The reasons for this are as follows: (1) The number of microscopic pK values, which must be estimated, soon becomes prohibitively large (for a molecule with N different acid-base sites one must calculate N2N-1pK values). ( 2 ) With an increasing number of sites, the pK values no longer differ substantially and the common conversion procedure from the microscopic to the macroscopic pK values by selecting the proper reaction pathway134 is no longer applicable. (3) As we shall demonstrate below, the pair additivity of the pK increments may no longer apply. Acid-base behavior of polyprotic molecules with a large number of sites is commonly explained in quite a different fashion in the case of polyelectrolytes7-9 and proteins.10-12 The free energy of the molecule is written as the sum of contributions from individual acid-base groups, their pairs, and sometimes their triplets. The
* Author to whom correspondence should be addressed. 0
Abstract published in Advance ACS Abstracts, May 15, 1994.
titration curve is evaluated from the partition function of such a model, which is nothing but the much studied Ising model in statistical mechanics.13J4 In the case of proteins, this approach has evolved from the early work on acids with a small number of protonsl5J6 while in the case of polyelectrolytes the relation to small molecules has received only marginal attention.' The aim of thepresent article is to show how such king models can be applied to the calculation of pK values of polyprotic acids or bases. The common approach of Perrin et al.,l which is based on microscopic constants, is recovered as a special case of an Ising model with pair interactions. The description can be systematically improved by including triplet (and higher order) interactions. The Ising model leads to a systematic parametrization of the problem and permits a straightforward evaluation of macroscopic pKvalues for polyprotic molecules with many acidbase sites. The microscopic pK values can also be determined, if necessary. We shall exemplify the applicability of the present approach by calculating pK values of a homologous series of linear polyamines which can be predicted quite accurately with a minimumof adjustable parameters. These result scan bedirectly compared with the recent Ising model of the potentiometric titration data of linear poly(ethylenimine).* The applicability of the model to inorganic molecules (phosphorus acids) is also demonstrated.
General Concepts Picture a polyprotic acid or base as a collection of N distinct acid-base sites which can be either protonated or deprotonated. The protonation state of an individual site i (i = 1, 2, ...,N ) can be described by a state variable si such that if si = 0 the site is deprotonated whileif si = 1 thesite is protonated. The protonation state of the entire polyprotic molecule, on the other hand, is characterized by the set of values sl, s2, ... S N . The free energy of a molecule in such a given protonation state can be generally written as
ij,k
For short-range interactions this type of expansion ought to
0022-3654/94/2098-6038%04.50/0 0 1994 American Chemical Society
Ising Models of Polyprotic Acids and Bases
The Journal of Physical Chemistry, Vol. 98, No. 23, 1994 6039
converge quite rapidly, and we shall neglect in the present discussion all contributions beyond three-body interactions. Because the statevariables are discrete (i.e. si = 0, l ) , the function f;.(l)(si)can be written as a linear combination of si and 1 - si, fii(2)(si,sj) as a linear combination of sisj,(1 - si)sj, si( 1 - sj), and (1 -si)( 1 -si), and similarly forfjk(3)(si, sj, sk). Collecting terms up to third order allows eq 1 to be simplified to
Without loss of generality we have chosen the fully deprotonated molecule as the free energy zero and have assumed that the coefficients Eij and Lijk vanish if any two indices are equal. Furthermore, we observe that the coefficient matrices are symmetrical, i.e. Eij = Eji and Lijk = Ljik = L i k j = Ljki. This type of free energy is well-known in the statistical mechanics literature as the Ising Hamiltonian in an inhomogeneous field including arbitrary pair and triplet interactions. The frequently investigated king model is recovered by neglecting all interactions except for nearest neighbor pairs.13J4 Potentiometric titration curves of polyelectrolytes7-9 and proteinslGl2 were interpreted on the basis of the free energy equation, eq 2. Let us now use this free energy of a polyprotic molecule to evaluate its macroscopic and microscopic pKvalues. Let us first focus on the macroscopic description, which is essential for the understanding of potentiometric titrations. The protonation equilibria to consider are BH,,
+ H + BH,
(3) where B is the fully deprotonated basic form, while BH, ( n = 1, 2, ...,N) represent the succession of protonated ampholytic forms which terminates with the fully protonated acidic form BHN. The protonation reaction is characterized by the acidity constant6
functions13
Qn K, = -
Q O G
where QH and Qm( m = 0 , 1,2, ...,N) are the partition functions of the proton and the species BH,, respectively. Such a macroscopic description does not distinguish between different microscopic states of the molecule (as defined by the set of state variables, SI, s2, ..., s N ) as long as they lead to the same number of bound protons n = Eisi. Therefore, given the free energy equation, eq 1, the partition function of the species BH, can be expressed as
SI.sZv-r”
where l l p = kgT is the thermal energy and 6 , j is the Kronecker 6. The partition function Qnis given by the sum over all Boltzmann factors of all protonation states of the molecule compatible with the constraint that n sites are protonated. The above macroscopic description averages over all possible states of the molecule which lead to a given degree of protonation. Sometimes, it may appear more useful to consider microscopic equilibria between particular protonation states of the molecu1e.u These equilibria arecharacterized by microscopicpKvalues which can also be measured experimentally under special circumstances.5 Let us denote the molecule in the configuration SI,s2, ..,,S N by B(sl, s2, ..., S N ) and thereby omit the protons from the notation since their number n = zisi is already uniquely specified. For the sake of simplicity let us only consider microscopicequilibria where only one particular site k changes its protonation state. Thus for the deprotonated form we have s k = 0, and therefore the configuration of the protonated molecule s’~,s$, ...,s;V is given by The corresponding chemical equilibrium
(4) has an equilibrium constant where square brackets denote concentrations in moles per liter. The familiar pK values are given by pK, = log K,
(5) The present numbering scheme implies that ~ K characterizes I the first protonation step of the fully deprotonated base while PKNdescribes the first deprotonation step of the fully protonated acid (last protonation step). For later convenience we shall focus on the chemical reactions
B + n H 6 BH,
(6)
with the corresponding formation constants6
wkich is related to the corpmonly used microscopic pK value by pK(s1, s2, ..., Sdk) = log K(sl, s2, ..., s d k ) . Note that there are N2N-I such microscopic constants. The acidity constant of eq 14 is given by
where the free energy change is
A ~ ( s , ~,
..., sJ k )
2 ,
= Z ( S ’ ~s;,,
...,sh) - SI, ~ 2 ..., , s
~ )
(7) Comparing with eq 4,we observe that the formation constants can be expressed as products of the acidity constants
and, reversely, acidity constants are given by ratios of formation constants K, = K,/K,+,
(9)
where = 1. Neglecting intermolecular interactions, we can express the formation constants in terms of the corresponding partition
where the last equation follows from eqs 2 and 12. If we apply eq 16 to the fully deprotonated state (where si = 0 for all i ) , one can elimin_ate th_e chemical protential p k in favor of the acidity constants Kk = K(0, 0, ..., Olk) of the individual acid-base sites in the fully deprotonated molecule. The resulting relation
allows us to rewrite eq 16 as
6040
Borkovec and Koper
The Journal of Physical Chemistry, Vol. 98, No. 23, 1994
where pitk = pK(0, 0, ..., Olk) = log it& is the microscopic pK value of site k, given that all other sites are deprotonated. The pair and triplet interaction parameters tij
A,,
= PEij/ln 10 = PLijk/ln 10
are related to the excess pair and triplet energies, respectively. Equation 18 simplifies for those microscopic protonation steps which result in the fully protonated molecule. If only the site k is deprotonated but all other sites protonated, the state variables are given by si = 1 for i # k. Inserting this relation into eq 18, one obtains
Neglecting triplet interactions (xijk = 0) allows eq 18 to be reduced to expressions for the microscopic pK values reported previously.lJ8 Perrin et al.1 express these pK values as a sum of group contributions of neighboring groups which are added to the pK value of the individual acid-base group. Since the contributions from neighboring acid-base groups depend on the protonation state of these groups, they can be written as linear functions in the occupation number si (the difference between the contributions of the protonated and deprotonated group being tij). All contributions from the deprotonated and inert groups are collected in the constant term which corresponds to the microscopic pK value of this group? given that all other groups are deprotonated (their sum being pKk). The present treatment demonstrates how to obtain systematic improvement of the description by including triplet (or even higher order) interactions. The king model formulation shows also unequivocally that the matrix til must be symmetrical. This constraint is not easily recognized on the level of microscopic pK values and has not been taken into account previous1y.I Now we are in the position to eliminate the chemical potential terms and the partition function of the protons from eq 10 in favor of the microscopic pK values of the first protonation step and express the formation constants which are directly related to the experimentally observable macroscopic pK values of a polyprotic acid (cf. eq 9) as
where
Very similar relations were already proposed many decades ag0.16J’ Given the pK values for the individual groups in the deprotonated molecule p& ( i = 1 , ...,N) and the matrix elements tij and Xijk, recursive application of eqs 21 and 7 for n = 1 , 2, ..., N leads to all pK values of a given polyprotic base or acid. In two special cases, namely for n = 1 and n = N , eq 21 can be evaluated directly. The results for the macroscopic acidity constants read
lead to the generalization of the relations between macroscopic and microscopic pK value^.^ As long as the pK values are sufficiently far apart, one can also search for the correct path in the microscopic reaction scheme and include the necessary combinatorial factors to obtain the macroscopic pK values.’ Particularly, with an increasing number of sites (already more than three), this approach becomes prohibitively complicated. However, eq 21 can be evaluated in a straightforward fashion either analytically or numerically for a much larger number of sites (see next section), and one directly obtains all macroscopic pK values which include the proper combinatorial factors. Equation 18 can be used to evaluate the microscopic pK values, if required. In the following, we shall discuss several applications of this development. The basic parameters of the model are (1) the microscopic pK values of the individual sites p&, given all other sites are deprotonated, (2) the symmetrical pair interaction parameter matrix til, and (3) if triplet interactions are included, the set of coefficients Xijk. The main problem is to estimate the values of these parameters, which can be done within the established framework for the prediction of pKvalues as discussed by Perrin et al.1 The number of unknown interaction parameters can be reduced by exploiting the symmetries of the molecule. Also, one expects that the absolute values of the interaction parameters decrease with increasing number of atoms between the acid-base groups. Thus, in practice, most of the elements ti, and Aijk will be zero, and as a first approximation,one can consider nearest neighbor pair interactions only. A more detailed description can be achieved by including more distant neighbor pair interactions and triplet interactions between neighboring groups. Weshallsee furtherbelow that thismethodcan beapplied to linear aliphatic amines in quite a convincing fashion.
Special Cases In this section we show that the king model not only can be used to derive systematically several known expressions for pK values of polyprotic molecules but also leads to new analytical results. First, we focus on the simplest case of a polyprotic molecule without interactions between the individual acid-base groups. Physically speaking, acid-base groups on the molecule are situated far apart and do not influence each other. The microscopic pK values of the isolated groups are, however, not identical to the macroscopic pK values of the polyprotic molecule. The proper relation can be recovered by inserting tij = Aijk = 0 into eq 21, which simplifies to6J6J7
The structure of eq 25 becomes more transparent by evaluating the expression for different N . For N = 1 we have K1 = b, for N = 2,
K,= b,+ b2 K2 = b,b2
N
K, = ybj Y ’ i= 1
and for N = 3,
L,= b, + R2 + R3
and 1 --
KN -
N 1
2%
where log k: = p& are the microscopic pK values of the last protonation steps, given all other sites are protonated (cf. eq 20). Within the approximation of triplet interactions, eqs 18 and 21
K3= blb2b3 In the case o,f identical groups, all pK values are equal and p k = pK, = log K = log Ki. One finds that the formation constants
The Journal of Physical Chemistry, Vol. 98, No. 23, 1994 6041
Ising Models of Polyprotic Acids and Bases are related to binomial coefficients6
Kn = k"(
HzN
y)
NH2
and using eq 9, one obtains pK, = p k
+ log N + nl - n
H H ~) N LN
The macroscopicpKvalues of a polyprotic molecule which carries identical and noniteracting groups are symmetrically distributed around the single microscopic pK value of the groups. Let us now focus on the more realistic situation of interacting acid-base groups. In the simplest case of a diprotic molecule ( N = 2 ) , from eqs 7 and 21 one ~ b t a i n s ~ ~ J ~
+ loPG)
pK, = log(lOpkl
+
pK, = -log(lO-pkl
- q2
e; *NH~
Figure 1. Definition of the Ising model for linear amines. Chemical formula of the molecule (left) and the corresponding Ising model (right) with acid-base sites (circles), pair interactions (line), and triplet intercations (curly triangles). Basic model parameters as microscopic pK-values of the isolated groups, pair interaction parameters e, and triplet interaction parameters X are indicated.
is given by eq 30, and the final expression reads
(31)
Specializing to a symmetrical molecule allows eq 3 1 to be reduced to15-17 pK, = p k
+ log 2
pK, = p k - log 2 - q2
(32)
The difference between both pK values is related to the pair interaction parameter 612 between the groups. Analytical expressions for the pK values in the presence of interactions and for a general number of sites N are available in a few special cases only. As a first example, let us focus on a polyprotic molecule where the free energy equation, eq 1, can be decomposed into two parts, namely 3(s,,
...,sN)= 3(O)(sl, ..., sN)+ $mfi(n)
(33)
zi
where n = si. The second contribution to the free energy depends only on the overall protonation state of the molecule irrespective of the protonation states of the individual groups. In other words, this part of the free energy depends on the average protonation state of all other groups and corresponds to a mean field contrib~tion.1~The mean field contribution to the macroscopic pK values can be evaluated by inserting eq 33 into eq 1 1 , and from eqs 9 and 10, one finds
PK, = PK,(O)
+ $"fl(n
- 1) - 4("fl(n)
(34)
where pKnO are the pKvalues evaluated without the mean field contribution and 4("fi(n) = p3(mfl(n)/ln 10 (35) is proportional to the mean field free energy. In the case of pair interactions, the interaction parameter matrix qj contains an additive contribution dmfl which is equal for all pairs of sites. Similarly, the triplet interaction parameters have a contribution A(mfl which is equal for all triplets of sites. Inserted into eq 2, the mean field contribution becomes d(mfi(n)= t(mfln(n- 1)/2
+ X("h(n
- l ) ( n - 2)/6
(36)
which, inserted into eq 34, leads to the result
PK, = PK,") - &mfl(n- 1) - A(mfl(n- l ) ( n - 2 ) / 2
-tLY PKc
A(mfl(n- l ) ( n - 2)/2 (38) The above situation for a diprotic molecule (cf. eq 3 2 ) is a trivial example of such mean field behavior ( N = 2 ) but becomes an interesting model in the case of highly symmetrical polyprotic molecules, such as for an equilateral triangle ( N = 3) or a tetrahedron ( N = 4). Another interesting observation is that, for A(mfl= 0, eq 37 has the same structure as the Debye-Hiickel correction due to finite salt concentration of the pK values extrapolated to infinite dilution.6 The mean field acting on the molecule is now the difference between the screened and bare electrostatic charging energy and gives rise to the Debye-Hiickel contribution to the pair interaction parameter dmfl = -B~e2/(47rD)(where 1 / is~ the Debye length, e the elementary charge, and D the dielectric permitivity of the medium). A second example, which permits an analytical treatment in the presence of interactions, is a linear polyprotic chain molecule with N sites with identical nearest neighbor pair and triplet interactions (i.e. q, = tbj,i+l and Aijk = A6j,i+lbk,j+2fori < j < k). As we shall later apply this model to treat linear-chain amines, we shall also allow for different microscopicpKvalues for terminal groups pK, and all other groups in between pK, (see Figure 1). For the derivation we refer to the Appendix. The final expression for the formation constants reads
Kn = K,"
X
i=max(0,2n-N-l) j=max(O,Zi-n)
n-i
c
ni,
where the nonexisting binomial coefficients must be replaced by unity, log K, = pK,,
c,
= [N-n+(a-l)(n-i)][N-n+ [N-n][N-n+
l+(a-l)(n-i)] 11
Ia2
for n < N
for n = N (40)
(37)
In order to illustrate the use of this equation, let us consider the situation where the interactions between the otherwise identical groups are of mean field type only. Since in this case cij = dmfl and Ajjk = A(mfJ, the free energy contribution 3 ( O ) describes noninteracting and identical acid-base groups Therefore, pK,m
(39)
and a is defined by the relation log CY = pK, - PK,
(41)
Neglecting triplet interactions (A = 0), we can use eq A.4 from
6042 The Journal of Physical Chemistry, Vol. 98, No. 23, 1994
Borkovec and Koper
TABLE 1: Experimental pK Values of Linear Aminesz0Compared with Calculated pK Values Using Parameters in Table 2 I=O.lM Z=lM experiment model 1 model 2 experiment model 1 model 2 ethylenediamine(en) 9.89 9.77 9.72 10.20 10.07 10.02 7.49 7.67 H2N(CH2)2NH2 7.08 6.97 7.15 7.71 10.09 9.92 1,4,7-triazaheptane(dien) 9.84 9.79 9.74 10.04 H2N(CHdzNH(CHz)zNHz 9.02 9.15 9.10 9.24 9.40 9.46 3.98 4.73 4.64 4.23 4.08 4.71 1,4,7,lO-tetrazadecane(trien) 9.74 9.80 10.09 9.76 10.10 10.05 9.16 H~N[(CHZ)~NHIZ(CHZ)~NHZ 9.07 9.20 9.50 9.50 9.45 6.59 6.42 7.15 6.69 6.87 7.11 3.27 3.95 4.34 3.68 3.36 3.99 1.4.7.10.1 3-oentazatridecane 9.74 9.82 9.78 10.11 10.07 9.14 9.24 9.21 H2N [(CH2
