Some remarks on the titration equation of weak polyacids

Electroanalytical Chemistry and Interracial Electrochemistry Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

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S O M E R E M A R K S ON T H E T I T R A T I O N E Q U A T I O N OF W E A K POLYACIDS* M. MANDEL and J. C. LEYTE Gorlaeus Laboratorium der Rijksuniversiteit, Afdelin9 Fysische Chemie 111, Leiden (The Netherlands) (Received5th October 1971)

INTRODUCTION In 1948 Overbeek 1 derived the following equation to account for the dissociation equilibrium of a simple weak polyacid: pH = p K o + l o g ~

0.434 ( O F ~ + ~\~-/

(1)

where ~ is the degree of dissociation, K0 the dissociation constant of an acidic group of the polyacid and Z the total number of charged groups along the chain. The term involving the electric free energy Fe takes into account the extra amount of free energy per acidic group necessary to remove, at a given value of ~, a proton of the polyacid. This equation was based on the assumption that it is possible to split up the total change in free energy into two independent contributions, one involving the free energy of the macromolecule and determined only by its conformation and another arising from the electrostatic forces, an assumption which was later questioned by Lifson 2. It is the purpose of the present note to show that (1) can be derived in a straightforward manner using the principles of statistical mechanics, without the above assumption. A STATISTICALDERIVATION OF THE TITRATION EQUATION Consider a very dilute polyelectrolyte solution containing also salt. If the dilution is high enough we may, as a first approximation, neglect interactions between the macromolecules. This can be done, as shown by Marcus 3, by dividing the total volume of the solution into electrically neutral subvolumes Vp, each of which contains a polymeric ion (Vp= total volume divided by the number of polyacids present) and is assumed to be large enough to be treated macroscopically. W will also contain N1 solvent molecules, several kinds of small ions from the salt present (No in total) and NH + hydrogen ions regarded as the counter ions of the polymeric ion (H ÷ and O H ions from the dissociation equilibrium of water will be neglected). The polyacid is assumed to consist of n units, Z of which are in a dissociated form A - (bearing one elementary charge) and (n-Z) in art undissociated form AH. The degree of dissociation * Dedicated to ProfessorJ. Th. G. Overbeekon the occasion of the 25th anniversary of his appointment as a Professor of Physical Chemistry. J. Electroanal. Chem., 37 (1972)

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M. M A N D E L , J. C. L E Y T E

is defined by the fraction Z/n. The centre of mass of the macromolecule is assumed to be fixed and will be the origin of the coordinate system used to describe the configuration of all molecules within Vp. At constant Vp, temperature T and composition (constant N 1,No,Z) the total free energy F of the subvolume can be written as F = F (0) +

(2)

Here F(0) is the free energy of volume Vpin the absence of charge interactions and F e is the free energy associated with these interactions. It will be assumed that F(0) represents the free energy of an ideal system, i.e. all interactions other than charge interactions will be neglected. It is easy to show that (2) follows directly from the statistical mechanical definition of the free energy, which is proportional to the logarithm of the canonical partition function Q(Vp, T, N1, No, Z) of the system. Thus F(0) is related to the partition function Qo for the ideal system. Therefore it can bewritten as a product of independent contributions of all the particles present. If exp ( - f f / k T ) represents the molecular partition function of a particle of species u, F(0) can be put into the following form e(o) = N l f ° ( O ) + f ° ( 0 ) + ~

(3)

N,f~°(O)- TSs(O) (v)

Here the subscripts 1, p and v refer to the solvent, the polyelectrolyte and the different species of small ions respectively. ( y n v = No). (v)

The entropy Ss (0) represents the ideal entropy of distribution of all the small ions and solvent molecules : S~(O) -- - kN1 (1n •N1 - 1) - k r N~ ( l n -Xv --

1)

(4)

Note that fo and N,+ will depend on Z, the latter through the condition Z = N,+ according to the assumption given above. If the macromolecule is considered to be large enough to be treated macroscopicallyJp° (0) can also be split up into a contribution depending on the chemical constitution and an entropy contribution - TSp(0). The latter depends as much on the distribution of the units AH and A - along the macromolecular chain as on the conformational degrees of freedom. j;o (0) = Z f ° - (0)+ (n - Z) fOH(0) -- rSp(0).

(5)

In order to derive an expression for Spa molecular model will be used. The macromolecule will be represenfed as a collection of point masses, each of which is located, for a given conformation, at a position R k (k varying from 1 to n). The point masses are not independent, each point mass at Rk being connected to the point mass R k_ ~ and at Rk+ 1. The centre of mass, origin of the coordinate system, is determined by the condition : ,Y___,mk Rk/,Y__,'nk = 0 (k)

(k)

d. Electro(real. Chem., 37 (1972)

(6)

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where ln~ represents the mass at position R k. Each conformation C i of the macromolecule is defined by the set coordinates R1, R 2 . . . . . R, = {Rk}i. The total configuration of the macromolecule (configuration being used here in the statistical and not the stereochemical sense) is not only determined by its conformation but also by the distribution of the undissociated and dissociated groups respectively. The latter can be defined by characterizing each unit (or point mass) by a charge parameter (k which is, by definition, zero for an AH unit and unity for an A - unit. A given charge distribution ?i is completely fixed by the set of values ~1, ~2, "" ", ~ n = {~k}i" Note that Z =Z(k)~ k. Thus each configuration of the macromolecule is defined in terms of the conformation Ci = [Rk}i and the charge distribution ),i = {(k}~. It should be observed that all R k are not continuously changing variables because of the conditions establishing the connection between the point masses and the condition (6). Each configuration, out of a total G, is characterized by a normalized probability Pi depending, for the ideal system considered, only on the potential energy of the macromolecule in this particular configuration. P~ = Pi(C~, 7~)

(7)

E Pi -~" 1 fi)

(8)

In (8) the summation includes all of the G possible configurations. The total entropy of the macromolecule can be represented in the following way : Sp(O) = --

k Z P ~ In P~

(9)

0)

In general the potential energy of a configuration will depend on the conformation as well as the charge distribution. The probability Qi= Q~(C~) of finding a given conformation {Rk }~independently of the charge distribution, can also be defined however. Qi(Ci) = Z Pi(Ci, Yl)

(10)

Here the summation is over all possible charge distributions within a given conformation. The probability Q~ is also normalized to unity:

E ~i(Ci)= 1 (Ci)

As charge interactions are not included inJp° (0), all configurations corresponding to a given conformation but differing only by the distribution of the charges along the chain will have the same probability (the small structural change occurring in a unit by removal of a proton being assumed to have no effect on the potential energy of the conformation). Thus, according to (10) Qi(Ci) = Z P~(C~ ~i) = Z Pi(C~)

(11)

where Pi (C~) is the identical probability of each charge distribution for a given C, and Z the total number of charge distributions that can occur. The latter is independent of the conformation and given by the total number of ways in which Z charged and (n-Z) uncharged groups can be distributed along n units. J. Electroanal. Chem., 37 (1972)

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z = n!/z!(,,-z)!

(12)

Combining (9) and (12) yields the following expression for Sp: S,(0)=-k

E [-ZP, l n e ] = - k ~ Q ~ l n Q ~ + k l n z (Ci) [_(7i)

J

(13)

(CO

This equation establishes that, in the absence of charge interactions, and neglecting all other interactions between macromolecule and small ions, the entropy of the macromolecule may be split up into a conformational entropy S o = - k Z(ci)Q~ In Q~ which will not depend on Z, and an entropy of ideal mixing Sm of charged and uncharged groups along the chain. Combining (2), (3) and (5), the free energy of the system is now given by F = Nlf°(0) + Z UJ°(O) + Z f °- (0) + (n--Z)fOH(0) (~)

(14)

- TSs(0)- TSp(O)+F¢ where S~(0) and Sp(0) are given by (4) and (13) respectively. As already pointed out, the free energy depends only on the state variables W, T, N 1, N~ and Z (with the restriction that Nn+ = Z because of the condition of electroneutrality). No conformational parameter, connected with the average dimensions appears in this expression explicitly. The equilibrium value of F corresponding to a given value of Z is thus found by minimizing F with respect to Z, at constant Vp, T, N1 and Nc (expect NH+ ). +

= 0

(15)

Observing that all f°(0) as well as all Q1 are by definition independent of Z, this yields: Nn+ Z [fo+ (O)+fo_ (O)_fOH(o)] + kT In ~ + kT In ~

+

i ~ J v.,ul,uo¢.uH +)= 0 (16) where use has been made of the Stirling approximation lor the factorials. Rearanging (16) and introducing pH-= -log(Nn +/V=) and ~ = Z/n, the titration equation is obtained : 0.434 0 c~ 0.434 (0Fe~ _ [fo (0)+fo+ (0)_f~H(0) ] +log~z~_~ + k T - \~Z] V.,N.U°(*U.+) p H - _- _kT (17) This expression is equal to (1) if pK 0 is identified with the first term on the r.h.s, of (17) and proves the validity of the former within the framework of the model adopted. CONCLUSIONS

The Overbeek equation for the titration of a weak polyacid has thus been J. Electroanal. Chem., 37 (1972)

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shown to be valid if it is assumed that all interactions except charge interactions can be neglected and the confo~mational degrees of freedom of the macromolecules are taken into account. It should be noted that the correction for the electrostatic interactions appearing in (17) must be calculated by taking the derivative of the total electrostatic free energy, as defined by (2) with respect to Z, and constant Vp, T, N 1 and Nc ( ~ NH+). The identification of this correction with the derivative of only a part of F e, as done by Harris and Rice 4 is in general incorrect. This will be discussed in more detail in a subsequent paper. SUMMARY

The titration equation for weak polyacids, derived by Overbeek in 1948 using a specific model for a polyion and making some assumptions on the addivity of conformational and electrostatic free energy, is shown to be generally valid for polyelectrolyte systems in which all intermolecular interactions except electrostatic ones are neglected. The proof is based on a statistical mechanical evaluation of the change in total free energy with degree of ionization. REFERENCES 1 2 3 4

J. Th. G. Overbeek, Bull. Soc. Chim. Belg., 57 (1948) 252. S. Lifson, J. Polym. Sci., 23 (1957) 431. R.A. Marcus, J. Chem. Phys., 23 (1955) 1057. F. E. Harris and S. A. Rice, J. Phys. Chem., 58 (1954) 725.

J. Electroanal. Chem., 37 (1972)