the potentiometric titration of weak polyacids

European Polymer Journal. 1970, Vol. 6, pp. $07-822. Pergamon Press. Printed in England.

THE POTENTIOMETRIC TITRATION OF WEAK POLYACIDS i~[. MANDEL LaboratoD" for Physical Chemistry Ill, The University, Leiden, The Netherlands

(Receit'ed 14 June 1969) Abstract--It is proposed to represent the increase with degree of dissociation (a) of pK (K = apparent dissociation constant) for weak polyacids, not exhibiting a conformational transition during titration, by a series expansion in a; pK = pKo + ~ -- ¢~:ct2 - . . . . Experimental results for poly(acrylic acid) show that a second degree polynomial in a represents the dependence of pK on the degree of dissociation within experimental accuracy for aqueous solutions containing an excess of sodium salt. In the absence of such an excess, this second degree polynomial remains a fairly good approximation. The three coefficients pro, ¢1 and ¢2, are shown to depend significantly on the salt concentration (C,). The proposed empirical relationship is dicussed in terms of the expression derived for the titration equilibrium of polyacids by Marcus "3) from thermodynamics a~'~dthe PoissonBoltzmann equation, not taking into account specific interactions along the macromolecular chain (non-random distribution of charged and uncharged groups) and formation of ion-pairs between charged sites on the polyelectrolyte and small ions. The dependence of pro on salt concentration is partially attributed to medium effects. The positive coefficient ¢'t is shown to be related to the mean distribution of macromolecular groups Vs (an electrically neutral sub-volume assigned to each macromolecule in a dilute solution) for the uncharged polyelectrolyte and the increase with a of the mean electric potential around the poly-ion near a = 0. Values of this coefficient are compared to corresponding values derived from model calculations using a fixed mean distribution of polyion charges. The coefficient ¢~,, negative at low salt concentrations and positive at higher ones, is shown to depend also on the perturbation (near a = 0) of the mean distribution of the macromolecular groups in Vs on dissociation, due to chain-expansion. Finally it is emphasized that the proposed representation for the pK-dependence on a is not inconsistent with the empirical Henderson-Hasselbalch equation for the titration curve of polyacids valid for a limited region of a-values near a = 0" 5. INTRODUCTION IN Trm last 25 years, the potentiometric titration of weak polyacids in a c q u e o u s solutions has been extensively studied by investigators of the behaviour of polyelectrolyte systems both from an experimental a n d a theoretical point of view. One of the most striking features of the dissociation e q u i l i b r i u m of these systems is the c o n t i n u o u s decrease of the a p p a r e n t ionization c o n s t a n t (K) with increasing degree of dissociation (a). In 1948 Overbeek ~I) pointed out that this effect should be connected with the increasing difficulty to remove protons from polyions with increasing charge Attempts to derive theoretical expressions, based u p o n model calculations for the relation between K and a, have only been successful qualitatively and no good q u a n t i tative agreement between theoretical and experimental curves has been achieved. This is primarily due to the crudeness of the models used in these calculations. I n m a n y cases, the model is based on a m e a n distribution of p o l y i o n charges of simple symmetry which remains unaltered during the titration (thus H e r m a n s and Overbeek ~2) used a spherical charge d i s t r i b u t i o n whereas Katchalsky a n d co-workers c3' '*) or K o t i n a n d Nagasawa cs) started from a u n i f o r m charge distribution on a rigid cylinder). In other calculations the real polymer chain is replaced by a K u h n - l i k e chain o f r.v.J. 6,'6---t,

807

808

M. MANDEL

"statistical elements" of constant length but with variable charge. (~. 7) Furthermore, some details have been overlooked when fitting experimental data to theoretical curves. Thus the occurrence of a conformational transition in a limited range of avalues [as has been found for poly-(methacrylic acid) (8)] for which no allowance is made in the model-calculations, gives rise to very significant deviations. Also the common implicit assumption of an "intrinsic dissociation constant" pKo for the acidic group on the chain, independent of the concentration of added neutral salt, will be shown below not to be consistent with experimental data, as expected from theory. On the other hand, a number of empirical relations between K, pH and a have been proposed (9-~°~ to represent the titration curve of weak acids in terms of experimentally available parameters. It has been emphasized by Nagasawa and Rice (~2) that these empirical equations are all equivalent and may be derived from the one most commonly used, the so-called extended Henderson-Hasselbalch equation originally proposed by Katchalsky and Spitnik. " t ) pH = pK, -- n log(l -- a)/a.

(1)

Here pKo and n are constants for a given titration but depend on the nature of the polyacids and the ionic strength of the solution. Unfortunately Eqn. (I) is applicable only for a limited range of a-values around a = 0.5 and cannot represent the entire titration curve. In many investigations where the potentiometric titration of the polyacids is used to study the behaviour of polyelectrolyte systems under different conditions, it is desirable to represent these titration curves by a simple analytical function. As for a polyacid not exhibiting a conformational transition in course of the titration, --log K - pK increases monotonously with a, it was thought possible to represent this functional dependence by a converging series expansion of pK around e = 0. It is the purpose of this paper to show that in the case of poly-(acrylic acid) (PAA) of molecular weight below l0 s for which no evidence for the occurrence of a conformational transition with dissociation is known, the titration curve in the presence of monomonovalent salt can be characterized by a simple polynomial dependence of pK on a. In general three coefficients will be necessary for such an empirical representation to be valid over the whole range of a-values. These coefficients, dependent on the ionic strength of the solution, will be discussed in terms of a generally valid expression for the potentiometric behaviour of polyacids derived by Marcus (:3~ from thermodynamics and the Poisson-Boltzmann equation without using any particular model. EXPERIMENTAL (a) Materials

Conventional PAA was prepared by polymerization of acrylic acid at 80" in water in the presence of HzO~."'~ Crude fractionation was performed by precipitation of the polymer from a methanol solution with ethyl-ether. The mean viscosity molecular weight M, of the fractions obtained after freeze-drying was determined according to Newman et al. "5~ Two different fractions were used in the present investigation, fraction A with [M~I = 790,000 and fraction B with Mn = 120,000. All aqueous solutions were prepared with demineralized water obtained from a mixed-bed exchange column and by dilution from a concentrated stock solution stored frozen in a polyethyleneflask. The concentration of the polyelectrolytesolutions was determined by potentiometric titration. Titrations were performed with 0" 1 M NaOH obtained by dilution from Merck's Tiwisol ampoules. The sodium salts were of different origin but of analytical grade.

The Potentiometric Titration of Weak Potyacids

809

(b) Potentiometric titrations T h e titrations ,,,,ere performed with a R a d i o m e t e r A u t o m a t i c Titration E q u i p m e n t ( T T I ' I in conj u n c t i o n v,'ith a SBR2 titrigraph a n d a SBU1 syringe burette) in a titration vessel at c o n s t a n t temperature (20~). R a d i o m e t e r glass (G 202 B) a n d calomel (K 401) electrodes were used. C o n t a m i n a t i o n o f the solutions with C O : was avoided by working in a nitrogen a t m o s p h e r e . Each system was titrated at least twice a n d only differences between duplicates of 0 . 0 4 p H units or less were tolerated. T h e degree of dissociation c, was calculated from the electroneutrality condition,

= [C,

+

C.+

-

Co.-]iC.,,

(2)

where Ca, CH ÷ and Co.- represent the concentration (in ion-g, l- z) of the sodium ions due to added N a O H , o f the protons a n d hydroxyl-ions respectively a n d Cm s t a n d s for the concentration of potyelectrolyte in (monomeric) equivalents I- t. Ca+ a n d C o ~ - were calculated from m e a s u r e d pH-values neglecting corrections for the activity coefficient. In general, the difference between a a n d the degree of neutralization (=' = C./C..) was negligible for =' > 0" 3 at all concentrations of polyelectrolyte a n d added salt. T h e apparent dissociation c o n s t a n t was calculated according to Eqn. (3). o K = p H + log(1 - -

a)/a.

(3)

( c) Investigated solutions T h e investigated solutions are s u m m a r i z e d in Table 1. Solutions prepared from fraction A c o n t a i n e d N a N O ~ as a d d e d salt; solutions from fraction B c o n t a i n e d NaCI. T h e concentration o f salt is represented by C,. T h e polyelectrolyte concentration (C.~) was kept low (C.~ < 10 -3 equiv. 1-~) in order to avoid interactions between macromolecules. TABLE 1. POLYELECTROLY'rE AND SALT CONCENTRATION OF THE DIFFERENT SOLUTIONS INVESTIGATED ,~l, = 790,000 ; N a N O ~ Code

A, .4., As

C~, (equiv. 1-~)

4'71 :,: 10 -3

.ql, = 120,000 ; NaCI

Cs (equiv. 1-~)

Code

-3"33 ;< 10 -5 10 -2

Bt

A, .-15

2 × 10 -2 4 x 10 -2

A6 Av

6 X 10 -2 10 - t

.4s

2 x 10-*

.4~

Ato Art At: At3 At.. Als ,4t6

1"79 x 10 -3

C.u (equiv. 1-~)

6"04 × 10 -3

B,_ B3 B. Bs B6 B-I

Cs (equiv. l - t )

-5 × 10 - s 10 =2 2 × 10 -2 5 × 10 -2 10- t 2 × 10 -1

3"33 x 10 - t -I0 -a

2 ' 7 5 × 10 -3

5'41 X 10 -3 2"09 x 10 -3

2 x 10 -3 5 × 10 -3 10 - t 2 × 10 - t --

(d) Fitting procedure for the experimental curves T h e experimental pK-values for a given s y s t e m at different values o f cx were fitted to polynomials of different degree in a by a least-squares procedure. This fitting procedure was performed with the help of an Olivetti Programma 101 for the polynomials of the second degree in a a n d with a n IBM 360/50 c o m p u t e r for the polynomials o f higher degrees. F o r each titration at least 14 points were used. T h e s t a n d a r d deviation S/z~ of the experimental points y, = p K ( " 3 with respect to the calculated polynomial o f degree p, y," = y/(ct), was determined according to the definition

S (,, = [ q -- (p1 + 1) ~ O't - - y , ) 2 ]* where q represents the n u m b e r o f points 0 ' 3 used in the least-squares procedure.

(4)

810

M. MANDEL

The polynomial of lowest degree in ~z for which S~(v) was equal to or lower than the estimated experimental uncertainty in pK (of the order of 0-03) and no systematic distribution of experimental points around the calculated curve was observed, was chosen as the empirical representation of pK VS. a .

RESULTS AND DISCUSSION F o r most of the investigated systems, the second degree polynomial represented the change of p K with ct within experimental accuracy. Figure 1 represents some calculated curves with the experimental points and in Table 2 the coefficients of the second degree polynomials p K = PKo ,'-- ¢~1 ~2) ~ - - ¢~2 c2~ ~2

(5)

are tabulated together with the standard deviations Sy c:~. Only for the solutions At and A~6 containing no added salt, was this standard deviation larger than the estimated uncertainty on p K ( 0 . 0 3 ) . F o r A : and At3 containing no real excess o f salt, the standard deviation was close to 0.03. F o r these four systems, the coefficients

8.5--

f

I

7.~

6'0

5,9

0

0"2

0"4

0"6

0'8

L'O

C~

FIG. 1. The dependence of pK on the degree of dissociation (a) for different PAA solutions of Table 1. Experimental points are calculated from titration curves according to (3). Drawn curves represent the least-squares second degree polynomials (5).

The Potentiometric Titration of Weak Polyacids

811

TABLE 2. COEFFICIEN-r$OF LEAST-SQUARESSECOND DEGREE POLYNO.~M.S

Code

pKo

¢q

¢':

S, ¢'-t

Code

pKo

¢~t

¢~z

S, ~'~

At* A:" A~ A.L A5 A6 A~ As A9

5"14 4.94 4.77 4'70 4"58 4'57 4'50 4"39 4"33 5"50 4"61 5'03 4"87 4'57 4'43 5"08

4'3 3-5 3.0 2"5 2'2 1'9 1"7 1"3 1"0 4"-I. 1"4 3"9 3"6 1'5 1'2 4"6

--1"8 -l'1 --0.8 --0"5 --0"2 --0-I 0-0 0"3 0"5 --2"0 0"4 --1"5 --1'2 0"3 0"3 --1'9

0-045 0'026 0.014 0"008 0"008 0"007 0'015 0"012 0"019 0'020 0"016 0'022 0"026 0'015 0"009 0"039

Bt B., B3 B~ B5 B6 B7

4"80 4'68 4'72 4"63 4"61 4"53 4"50

3"8 3"0 2"5 2"2 1'5 1 '4 0"9

-1.5 -0.8 -0-5 -0'2 0"3 0"2 0-6

0"017 0.017 0"007 0.009 0.006 0.008 0-007

Ato Art Azz A~3* At,

Ats AI~*

*The following coefficients for the least-squares third degree polynomials are obtained: A1 : p K = 5"05 + 5.7 a -- 5"8 ~2 + 3 ' 0 ~3, Sy~3~ = 0"014;A., : p K = 4"87 + 4"3 a -- 3"2 a2 + 1"4 a 3 S~ ~3~ = 0'007; At3 : p K = 4"81 + 4"4 a - 3"2 a 2 + 1"4 a 3, Sy c3~ --- 0 ' 0 1 5 ; A,n : p K = 4 ' 8 8 6"3 a - 5"7 a: + 2"3 a s , $7 c3~ = 0"033.

of the third degree polynomials and S/3) are also mentioned in Table 2. It should be noted that pKo, Oh, and ¢~2 do not differ either in order of ma~itude or in sign for the second and the third degree polynomials. It can be concluded that for PAA a polynomial of the second degree in ~ represents the change of pK with de~ee of dissociation for solutions containing an excess of added salt and that in the absence of such an excess this representation remains a good approximation. The dependence of pKo, ~ - ' ~ and ¢~,~2~ with concentration of added salt (ionic strength) is shown in Figs. 2--4. The values of the last two coefficients fall on two different curves corresponding to systems with different molecular weights. No systematic effect of the polyelectrolyte concentration could be detected, which is probably due to the absence of interaction effects between polyions. In order to interpret the physical meaning of the three coefficients and their salt concentration dependence, reference should be made to the titration equation for polyelectrolytes as derived by Marcus ¢~3~ from thermodynamics and the Poisson-Boltzmann equation for the electrostatic mean potential determining all charge interactions in a dilute polyelectrolyte solution. This equation reads as follows: pH = log

a

-i-,

0"4343 ( ~ o + + ~,o_ _ ~ , o )

Vs

+ 0.4343 m~k-T f

¢ (R) p~ (R) dR -9 constant

(6)

Here p~ represents the mean volume charge density due to macromolecular charges and m the total number of acidic groups (both dissociated and undissociated) on the

812

M. M A N D E L

,.ol "--

7"O

5.0

e) 4-

5-0

-

B 4-

I

I

I

!..

I

0

I

I

5

x I0-' FIG. 2. The dependence of pKo on salt concentration (C,). The left-hand ordinate scale refers to PAA solutions A of Table I. [ O : AI-Ag; .-7 : A~o, A1,; 0 : AI:-A,4; V : A:s, A~6]. The right-hand ordinate scale refers to PAA solutions B of Table 1. ( A : B~-Bv). Also represented is the dependence of p r o for acetic acid ( + ) according to Harned eta/. ( ~ .

2--

I 0

! 2

I

! 4

I

I 6

x 10-'"

FIG. 3. The dependence of¢~ [O:

on

salt concentration (C,) for different PAA solutions of Table ].

AL-Ag; ~ : Aio,-.411; ~ : Al2-Al4; ~7 : A,s, A16; ~.: BI-B~].

The Poten:iometricTitration of Weak Polyacids

t

~

C

I 2

1

I

I ,

4

813

I 6

x l O -t

FIG. 4. The dependence of ~2 on salt concentration (6",) for different PAA solutions of Table 1 (same symbols as in Fig. 3). polymer chain. The integration extends over the total electrically neutral sub-volume (Vs) assigned to each polymeric ion (total volume divided by number of such ions). Furthermore ~o+ is the standard free energy (per ion) of the protons in solution (defined on the molarity scale). The definition of the standard free energy ~ o and ~° H of the dissociated (A-) and undissociated (AH) acidic groups on the macromolecule, which are not specified in Marcus' paper, needs some consideration. As will be shown in the appendix, ~o and ~° H represents the free energy, per group, of the completely dissociated or undissociated polyacid respectively in the sub-volume containing the same amount of solvent and added salt. Therefore the "intrinsic" dissociation constant K£ defined by pK'o --

0.4343 (F°÷ -1- F ° - -- /~°H) kT

(7)

still depends on the total salt concentration of the solution. The constant appearing in the r.h.s, of (6) is due to contributions arising from the fact that the experimentally measured pH with glass and calomel electrode, using buffer solutions as references, is not equal to the negative value of the logarithm of the proton activity (thermodynamically a non-measurable quantity). The most important of these is due to the noncompensated liquid-junction potentials. This contribution may be assumed, in a first approximation, to remain constant with ~ but may still be dependent on the nature of the polyelectrolyte and the concentration of added salt. The term log =/(l--a) in (6) is only valid for a random distribution of charged and uncharged groups along the macromolecular chain (i.e. when specific group interactions are neglected along this chain). Furthermore no allowance is made in (6) for the formation of ion-pairs between counterions and charged groups on the chain, in

814

M. MANDEL

which case not all the dissociated groups, according to the proton dissociation equilibrium, should be counted as charged. CtT) Under these circumstances pp may be expressed in terms of ct and the local concentration C a of polymeric groups (dissociated and undissociated) in V s, using the simple relation p, = ,, e, C,

(8)

where ep is the charge of a dissociated group, equal to one negative elementary charge for a carboxylate group as in PAA. As ep and ~b will always have the same algebraic sign, the following expression may be derived from (6), (7) and (8) __ _ , 0"4343 l e, I ( vs pK - pH + log l -- a pXo -v} ~b(R) I C, (R) dR. m kT .J '

(9)

Here pKo is the sum of pKo and the constant appearing in (6) due to non-compensated liquid-junction potentials. The absolute values of the charge !ep[ and potential [~bl have been introduced. For a simple polyelectrolyte not undergoing a conformational transition over a small range of ~-values in course of the titration, i~bl can be assumed to be a monotonously increasing function of ~ which can be expanded into a Taylor's series around

(1 -~--0. I ~, (R) [ = a ~b, (R) + c~'- ¢2 (R) + . . .

(10)

Here $~ stands for (i[)-i(~li~b[/~ al),=o . No constant term appears in (i0) because ([~bD,=o = 0, the electric potential around the polymer being assumed to disappear for the completely uncharged macromolecule (effects due to the interactions between the small ions in the presence of the uncharged macromolecule are neglected, as usual). It is to be expected that ~bl should be positive, the absolute value of the potential increasing always when the charge is increased near a = 0. Likewise, an analogous series expansion can be written for Cp('R)

C, (R) -- Cp.o (R) + ~ Co. 1 (R) + o 2 Co. 2 (R) + ...

(11)

Now G. o

= [Co (R)J,= o

represents the distribution of the groups on the uncharged macromolecule within the sub-volume. Furthermore Co. t = (i!) -1 (3~Co/O a'),= o. In general Cp, C~.o and C~.~ will be different from zero only in limited fraction of the total space occupied by the sub-volume Vs for any location of the centre of gravity of the macromolecule. As in general the actual volume occupied by the polyelectrolyte will increase with increasing charge near ct = 0, Co. t may be expected to be negative. Substituting (10) arid (1 I) into (9) yields the following series expansion for pK Vs

pK----- p.Ko + 0"4343 le o I ~'af m kT l d

.q- ¢12; s

¢I (R) C~,o (R) dR

[$z (R) C~.o (R) + St (R) Cp.t (R) dR

+ ,~3; s

(12)

[~, (R) Cp.o (R) + ¢2 (R) C..z (R) + 6, (R) G., (R) dR + ... 3

The PotentiometricTitration of Weak Polyacids

815

Comparison of (12) and (5) shows that pKo has the same meaning in both equations and that qSt and ~ , are given by

.vs 4,1(R) C o.o(R) dR

(13)

[¢: (R) Co. o (R) + ~1. (R) Cp. ~ (R)] dR.

(14)

qbt --0"4343mkTle'!)

vs ~

_ 0.4343mkTlep ! f

It should therefore be possible to understand, at least qualitatively, the dependence of the three coefficients on the concentration of added salt. The systematic decrease of pKo with ionic strength is in accordance with the definition of pK~ directly related to this quantity. This medium effect is of the same order of magnitude as for acetic acid in water (16~ for which the dependence of pK o upon ionic strength is also represented in Fig. 2. However, at low salt concentrations (C.~t < 10-2 equiv. 1-~) a systematic deviation of pKo for fraction A (high molecular weight sample) with respect to acetic acid and fraction B is observed. This discrepancy is somewhat reduced if pKo values, as obtained from a third degree pol~-nomial fitting are used at low C~ but, even in that case, it remains significant. Several effects may be thought to be responsible for this: (a) the pK o of PAA at infinite salt dilution (C, = 0) is not necessarily equal to the pK o of ordinary carboxylic acids. The standard values of ~° H and ~o_ for the polyelectrolyte are defined by extrapolation of the properties of an infinitely diluted solution of a macromolecular chain of only AH and A - groups respectively. This means that both standard chemical potentials contain contributions of the intramolecular binding along the chain. As in the case of fraction B (IVI, = 120,000), pKo at infinite salt dilution does not differ very significantly from the pK o of acetic acid, it is not to be expected that for fraction A (El, = 790,000) the deviation of more than 0.2 pK-units could be accounted for only by this effect; (b) the contribution of the non-compensated liquid-junction potentials to pKo, depending on the transference number of the diffusible ions, may become more important for high molecular weights and low salt concentration than for low molecular weights (and even be dependent on the actual glass and calomel electrode used). Although it is very difficult to estimate the order of magnitude for this effect, contributions to pKo are not to be excluded. In any case, it is clear from this discussion that in view of these two effects, it is extremely hazardous to predict pKo-values for polyelectrolyte titrations on a priori grounds. From (13) it is seen that q~ depends on the distribution of macromolecular groups in Vs for the uncharged polyelectrolyte Cp,o and [0 ~t4,'/0a]~=o- As pointed out before, 4,~ can be assumed to be positive so that q~ is expected to be positive also. This agrees with the experimental results over the whole range of salt concentrations, both for A and B solutions. It is also to be expected that ~ will decrease with C, as the slope of the ~b = 4,(a) curves at a --+ 0 can be assumed to decrease with increasing ionic strength. As ¢'~ depends on the mean conformation of the uncharged macromolecules and the change of 4' near a = 0, this coefficient may be compared to theoretical values obtained with models using afixed charge distribution. Two such comparisons have been performed. Alexandrowicz and Katchalsky ('L~ have derived the

816

M. M A N D E L

following equation for the change of pK with a in the case of a uniformily charged rigid-rod model in the presence of added salt, valid at low values of ApK

c~

= (pK -- pKo)/a = 0 . 8 6 8 6 . ,V.

Ko (~a)

~:aKl (Ka)

(15)

where Ko and KI are modified Bessel functions of order zero and one respectively, a is the thickness of the cylindrical rod representative for the polyelectrolyte, and ~ is the Debye-HiJckel screening factor depending on the ionic strength and defined for mono-mono-valent salts as

~¢= {8~re2 N°~÷Cs *. G0 ; kr/

(16)

Here e is the elementary charge, No Avogadro's number, • the dielectric constant of water and k Boltzmann's constant. The charge parameter A' is given by m e2 A' = - (17)

Ehk~"

with h standing for the effective cylindrical length of the macromolecule. In Fig. 5, values o f ~ t as function of v/C,, obtained in this work, are compared with calculated ApK/c~ curves, according to (15), using a = 5.5 A ° and ,V = 1.80 and ,V = 1.45 for solution A and B respectively. The agreement is very satisfactoD'. Surprisingly enough, these values for A' are nearly a factor 0.5 smaller than the value A' = 3 estimated by Alexandrowicz and Katchalsky from the length of the polyelectrolyte,

6

4

(~1

0

0

2

-

I

t 2

t

I 4

T Xa

f 6

I

f 8

I

1

xlO -I

FIG. 5. C o m p a r i s o n between ~ t values at different C, a n d calculated ApK/a curves according to Eqns. (15) a n d (16) due to Alexandrowicz a n d K a t c h a l s k y . (4) Curve I: ,V = 1 "80; a = 5"5 A ° ; values o f ~ 1 ( O ) refer to solutions A2-A9 o f Table 1. C u r v e II: A' = 1-45; a = 5 . 5 A°; values of ~ ( r ] ) refer to solutions B2-B7 o f Table 1.

The Potentiometric Titration of Weak Pol~acids

~

Io

S17

.~'rr[

~'

4

6

x IO "~

FIo. 6. Comparison between • t values (O) for different PAA-solutions (At-Ag) at different C, and calculated ~pK/a curves according to Eqn. (18) due to Hermans and Overbeek.~=~ Draa-n curves correspond to different values of R,r: I: 260 A°; II: 240 A°; III: 220 A ' ; IV: 200 A°; V: 180 A ° (degree of polymerization : 11.000).

15

• = Io

I 2

4

6

~ 1 0 -I

FIG. 7. Comparison between ¢,~ values ( A ) for different PAA-solutions (B,-B-t) at different 6", and calculated ApK/a curves according to Eqn. (18) due to Hermans and Overbeek. (2) Drawn curves correspond to different values of R.•: I: 120 A=; II: 110 A°; III: 100 A°; IV: 90 A°; V: 80 A = (degree of polymerization: 1670).

818

M. MANDEL

h m 1, where 1 is the contribution of a monomer unit to the len~h of the fully extended chain. At present, no explanation for this small value nor for the molecularweight dependence of ,V can be put forward. Values of¢)~ were also compared with the expression for kpK:'c~derived by Hermans and Overbeek (2~ using a model where the macromolecular charge density pp is constant throughout a spherical volume of radius R u (see Figs. 6 and 7). =

,.XpK _ 0.4343 "

" 7-~

[RM (1 + 0"6~ R.~, ~' 0"4K ~ R.~)] -~ .

(18)

Curves of :XpK/cLas function of v/C, were calculated for several reasonable values R.~ (for solutions A with m = 11,000, 180 < R.u < 260 A ° and for solutions B with m = 1670, 8 0 < R ~ < 120 A°). From Figs. 6 and 7, it can be concluded that no good agreement can be realized over the entire range of C, values between ¢'t and ApK/a according to (18). This is not surprising in view of the generally formulated criticism of the Hermans-Overbeek theory which is based upon a linearized Poisson-Boltzmann equation, an approximation which can only become reasonable at large excess of added salt even within the framework of the spherical model (see Wall and Berkowitz ¢~8) and Lifson('9)). The coetficients ¢~2, according to (14) are determined by two contributions, one of which is presumably negative because Cp., < 0. From Table 2 and Fig. 4 ¢~2 appears to be negative at low C, values, turning positive if C, > 0.1 equiv. I - ' for sample A and C, > 0-04 equiv. 1-' for sample B. This implies that, above these concentrations, ~b2 = 0"5 (~-'l~bi/O~)~=o should be positive which is probably also the case at lower concentrations. The negative value of q~2 at low salt concentrations is then, apparently, due to the predominance of the integral gs

Va

f ~1 C,.l dR over f @2Cp.odR. The difference between both integrals decreases with increasing salt concentration, finally becoming negative for an excess of added salt. A possible explanation is that the two factors in the integrand of the first integral decrease with increasing salt concentration (near a = 0 boththe increase of~b with ct and the expansion of the chain responsible for C,.1 strongly depend on the ionic strength) whereas in the second integral Cp.0 should be only very slightly dependent on C,. In this discussion of e t and ¢2, the effect of non-random distribution of AH and A - groups along the macromolecular chain and the association between counterions and A - groups have been systematically neglected. Harris and Rice ( : ) have pointed out that in the absence of added salt the distribution of charged and uncharged groups on the polyelectrolyte chain is probably far from random. This could explain the somewhat exceptional behaviour of solutions At, A2, Al3 and A16 for which the third degree polynomial seemed to be a better representation for p K vs. ct than the second degree polynomial in contrast to all other systems. However from the present results it is not to be expected that the deviations from random distribution should represent a large effect, as predicted by Rice and Harris, especially not for the low molecular weight sample. On the other hand, association effects should only be taken explicitly into account if the interaction between counterions and A groups reduces the real charge on the polyelectrolyte chain (chemical association)


819

and cannot be accounted for through the electrostatic potential ~, appearing in (6) or (9) (see Mandel(19~). Such evidence has not yet been given in the case of P A A and N a + ions and cannot be derived from the titration curves discussed here, which remain explicable in terms o f electrostatic effects only. This does not mean that the possibility for strong association should be ruled out altogether. Finally it is necessary to point out that the empirical representation for p K = pK(a) according to (5) is not inconsistent with the empirical representation of the pKvariation by means of the extended H e n d e r s o n - H a s s e l b a l c h Eqn. (I). In fact, this equation can be derived in a completely analogous way by expanding p K into a power series in the variable x - log("-'~/ct around x = 0 (corresponding to a = 0.5). In as far as the successive derivatives ( ~ t p K / ~ x % = o are finite, one can ~Tite p K = (pK).~=o + ¢~ x -- ¢2 x z q- . . .

(19)

where ¢, -----(i!)-*(a'pK/ax')~=o. Equation (19) can be transformed into an explicit function of = by using 3/3x = 0"4343 (e2-ct) a/?= p K = (pK)~=o.s -- ~

\ ~a I~=O.S

o1__-o a

+ (0.~43)-" (a~pK 1 ( 1--=)z - log \ ~ - /~=o.s ct

+

...

(20)

In a limited range of a-values around ~ = 0.5, the third and higher terms on the r.h.s. of (20) can be neglected so that, substitution of (20) into (3) yields the H e n d e r s o n Hasselbalch equation, pH = (pK)~=o.s-

I 1 --

------~-

]

1-

Both coefficients (pK)==o.5 = pK~ and I I + 0"4343 ( ° p K / ~ c 0 " = ° " ] can be easily expressed in terms of pK,, ~ , and 02 as defined by (5). Acknowledgements--The author thanks Miss C. M. K6rmeling and Mr. Th. Vreugdenhil for help

with the titrations and calculations of least-squares polynomials and Dr. J. C. Leyte for discussing critically the manuscript. REFERENCES (I) J. Th. Overbeek, Bull. Soc. Chim. Beiges 57, 252 (1948). (2) J. J. Hermans and J. Th. G. Overbeek, Rec. Tray. Chim. 67, 762 (1948); Bull. Soc. Clam. Beiges 57, 154 (1948). (3) A. Katchalsky, N. Shavit and H. Eisenberg, J. Polym Sci. 13, 69 (1954). (4) Z. Alexandrowicz and A. Katchalsky, J. Polym. Sci. A1, 3231 (1963). (5) L. Kotin and M. Nagasawa, J. chem. Phys. 36, 873 (1962). (6) A. Katchalsky and J. Gillis, J. Polym. Sci. 13, 43 (1954). (7) F. E. Harris and S. A. Rice, J. Phys. Chem. 58, 733 (1954). (8) J. C. Leyte and M. Mandel, J. Polym. Sci. A2, 1879 (1964). (9) W. Kern, Z. Phys. Chem. A181, 249 (1938). (10) I. Kagama and K. Tsumura, J. Polym. Sci. 7, 89 (1951).

820 (I1) (12) (13) (14) (15) (16) (17) (18) (19) (20)

M. M A N D E L A. Katchalsky and P. Spimik, J. Polym. Sci. 2, 432 (1947). M. Nagasawa and S. A. Rice, J. Am. chem. Soc. 82, 5070 (1960). R. A. Marcus. J. Chem. Phys. 23, 1057 (1955). P. Seller, Thesis, Leiden (1966). S. Newman, W. R. Krigbaum, C. Langier and P. g. Flory, J. Polym. Sci. 14, 451 (1954). H. S. Harned and B. B. Owen, The Physical Chemistry o f Electrolytic Solutions. Reinhold, New York (1958). F. E. Harris and S. A. Rice, J. phys. Chem. 58, 725 (1954). F. T. Wall and J. Berkowitz, J. chem. Phys. 26, 114 (1957). S. Lifson, J. chem. Phys. 27, 700 (1957). M. Mandel, J. Polym. Sci. C16, 2955 (1967). APPENDIX

The free energy of the sub-volume Vs of the polyelectrolyte system containing one macromolecular particle may be written in the following way: F = F(O) + F,

(A. 1)

Here F(O) stands for the free energy of the sub-volume in the absence of charge interactions and F, is the free energy associated with the reversible charging process establishing these interactions. If F(o) represents the free energy of an ideal system, i.e. if all interactions besides charge interactions between polyelectrolyte and small ions are neglected and the standard chemical potentials for solvent (~L°), small ions (m °) and macromolecular particle (v..~°) are introduced, then F(O) = nt t~L* + 57 n~ m ° + t~.~* -- k T l n Vs -- T Sc4 I

(A.2)

where nt and nc stand for the number of molecules of solvent in Vs and number of small ions of the ith kind respectively and Std represents the ideal entropy of mixing of the small ions with the solvent. Note that, according to the definition of the standard chemical potential on the molarity scale, ~.,¢o will be defined by extrapolation of the properties of an infinitely siluted solution of macromolecules in the given salt solution acting as solvent! Formally ~ . o can be split up into a part depending only on the constitution of the polymer and a part depending on the conformational entropy S o of the chain, depending also on the degree of dissociation a. /zu ° = m [(1 -- a) ~,,,, (0) + a ~.,t (0)] -- T S , (a)

(A.3)

where ~,~a and ~,,- are the free energy contributions of a AH and A - group respectively along the macromolecular chain. Assuming random mixing of AH and A - groups and defining the free energy contribution of the completely undissociated or completely dissociated polyacid by [m,A. °-(TS~)~ ~ O] and [m/~, - ° -- (TSp) a . :] respectively, ~ . o may be written as ~.~° = m [(1 -- a) re, H* + (I -- a) k T l n (1 -- a) + a t~A-* + a k T l n a] -- TSp (a).

(A.4)

Combining (A.1), (A.2)and (A.3), the expression for the total free energy of the sub-volume is given by: F = (nl , l * -- k T l n Vs) + X nL I~fl -- TSt~ - TSp + m [(1 -- a)/~,r~* + a ~,-] I

+ mkT(l

- a) In (1 - a) + a l n a] + F,.

(A.5)

It follows from this derivation that ~A, ° and ~,~o represent the free energy per monomeric group of a completely undissociated or completely dissociated polyacid respectively in the same amount of solvent and added salt as in the sub-volume Vs in the polymeric system, contributions of the conformational entropy being excluded and all charge interactions neglected. Equation (A.5) is practically equal to the expression used by Marcus (see Eqns. (3) and (5) of Ref. 13) of which Eqn. (6) was derived. The constant contribution n, ~ o _ k T In Vs omited in Marcus' Eqn. (3) is of no influence on the relation between pH and a. ROsum~---Dans le cas des polyacides faibles ne prgsentant pas de transition de conformation durant la titration, on propose un d~veloppement en sgrie du pK (K = constante de dissociation apparente) en fonction du coefficient de dissociation a; pK = pKo + @ta + ~2a a + . . . . Pour le poly-(acide aerylique) les resultats exp~rimentaux montrent qu'un polyarme du second degr~ en a represente la variation de pK avec le degr6 de dissociation dans les limites des erreurs exp~rimentales pour les solutions aqueuses contenant un exces de sel de sodium. En absence d'un tel exces, ce polynrme du


:321

second degr6 demeure une approximation satisfaisante. On a montr~ que les trois coefficients Ko, ¢)t et ¢'2 varient de fa¢on significative avec la concentration en sel (C,). La relation empirique proposee est discutse en tenant compte de I'expression dSriv~e par Marcus de dormees thermodynamiques et de l'~quation de Poisson-Boltzmann pour les ~quilibres de titrations des polyacides, sans tenir compte des interactions slz~cifiques se produisant le long de la chaine macromol&:ulaire (distribution nonstatistique des groupements charges et non charges) ni de la formation de paires d'ions entre les sites charges du poly-~lectrolyte et de petits ions. La variation de pKo avec la concentration en sel est partiellement attribute a. des effets de milieu. On montre que le coefficient positif O1 est relie ~- la distribution moyenne des groupements macromol&:ulaires darts Vs (un subvolume electriquement neutre attribu~ h chaque macromol~cule en solution diluee) pour le poly-81ectrolyte non-charg~ et ~. l'accroissement avec a, au voisinage de a = 0, du potentiet ~:lectrique mo.~en autour du poly-ion. On compare les valeurs de ce coefficient b. des valeurs correspondantes obtenues par le catcul ~. partir de modules utilisant une distribution moyenne determin~e des charges du poly-ion. Le coefficient ¢):, n~gatif pour de faibles concentrations en sel et positif au-delb., d~pend ~galement de la perturbation (au voisinage de a = 0) de la distribution moyerme dans Vs des groupements macromol~ulaires au cours de la dissociation, ph~nom(~ne dfi b. l'expansion de la chaine. En fin de compte, on met en relief que la representation propos~e de la variation de pK avec a e s t coh(~rente avec I'~quation empirique propos(~e par Henderson-Hasselbalch pour la courbe de titration des poly-acides valable dans un domaine limit8 de valeurs de a aux environs de a = 0,5. Sommario--Si propone di rappresentare l'incremento del pK (K -~- costante di dissociazione apparente) con il grado di dissociazione ((~) per poliacidi deboli che non esibiscano una tramizione conformazionale durante la titolazione, mediante una espansione in serie in a; pK = pKo ~- ¢'1 + ¢)2a2 -- . . . . I risultati sperimentali per l'acido poliacrilico mostrano che un polinomio di secondo grado in ,x rappresenta la dipendenza di pK dal grado di dissociazione entro l'accuratezza sperimentale per soluzioni acquose contenenti un eccesso di sale sodico. In assenza di un tale eccesso, questo polinomio di secondo grado rimane una approssimazione abbastanza buona. Si mostra che i tre coefficienti pKo, ¢'t e ¢'_, dipendono in modo significativo dalla concentrazione di sale (C,). Si discute la relazio ne empirica proposta, in termini della espressione derivata da .%larcus~3) per l'equilibrio della titolazione di poliacidi dalla termodinamica e dall'equazione di PoissonBoltzmann, non prendendo in considerazione le interazioni specifiche lungo la catena delle macromole'cole (distribuzione non casuale dei gruppi con carica e senza carica) e la formazione di accoppiamenti fra siti carichi sul polielettrolita e piccoli ioni. La dipendenza di pKo sulla concentrazione del sale 8 attribuita parzialmente agli effetti del mezzo. Si mostra che il coefficiente positivo @~ correlato alia distribuzione media di gruppi macromolecolari in Vs (un sottovolume elettricamente neutro assegnato a ciascuna macromolecola in soluzione diluita) per il polielettrolita sen:a carica e all'incremento con (x det potenziale elettrico medio intorno at poll-lone vicino ad a = 0. I valori di questo coefficiente sono paragonati ai corrispondenti valori derivati per via di calcolo su modelli, usando una distribuzione media fissa della cariche dei poli-ioni. Si mostra che il coefficiente ¢):, negativo a basse concentrazioni di sale e positivo a pit) alte concentrazioni di sale, dipende anche dalla perturbazione (vicino ad a = 0) della distribuzione media dei gruppi macromolecolari in Vs sulla dissociazione, dovuta ad espansione della catena. Infine si mette in evidenza chela rappresentazione proposta per la dipendenza del pK d a a non e inconsistente con la equazione empirica di Henderson-Hasselbalch per la curva di titolazione dei poliacidi, valida per una limitata regione di valori d i a vicino ad a = 0,5. Zusammenfassung--FiJr schwache Polys~.uren, die w~.hrend der Titration keinen Konformationstibergang zeigen, wird vorgeschlagen, den Anstieg des pK-Wertes (K = scheinbare Dissoziationskonstante) mit dem Dissoziationsgrad (a) darzustellen dutch eine Reihenentwicklung von ,z; pK = pKo + ¢q,~ ÷ ¢'z,x2 + . . . . Experimentelle Ergebnisse ftir Polyacrs'ls~.ure zeigen, dab ein Pol.vnom 2. Ordnung t'iir ,x die Abh/ingigkeit yon pK vom Dissoziationsgrad fi.ir w~.ssrige LOsungen mit einem Oberschul3 an Natriumsatz innerhalb der experimentellen Fehlergrenze wiedergibt. Wena kein CrberschuI3 an Natriumsalz vorliegt, bleibt dieses Polynom 2. Ordnung eine recht gute N~.herung. Es wird gezeigt, dab die drei Koeffizienten pKo, ¢'1 und ¢'2 deutlich von der Salzkonzentration (C,) abh~ingen. Die hier vorgeschlagene empirische Beziehung wird diskutiert im Hinblick auf die Beziehung von Marcus "3~ die fi.ir das Titrationsgleichgewicht von Polys/iuren aus thermodynamischen Daten im Zuzammenhang mit der Poisson-Boltzmarm Gleichung abgeleitet ist. Dabei werden spezifische Wechselwirkungen entlang der makromolekularen Kette (nicht statistische Verteilung tier geladenen und ungeladenen Gruppen und die Bildung yon Ionenpaaren zwischen den Ladungszentren des Polyelektrolyten und kleinen lonen nicht beriicksichtigt. Die Abh~.ngigkeit yon PKo yon der Salzkonzentration wird teilweise auf Einfltisse des Mediums zu~ckgefi~hrt. Es wird gezeigt, dab der positive Koeffizient 01 in Beziehung steht mit der durchschnittlichen Verteilung der makromolekularen Gruppen in Vs (ein elektrisch neutrales Subvolumen) dasjedem Makromoleki.il in verdi~nnter

822

M. M A N D E L

LSsung zugeordnet wird fiir den ungeladenen PolyeIetctrolyten und die Zunahme des durchschnittlichen elektrischen Potentials in der Umgebung des Polyions nahe ~ = 0 mit ~. Die Werte fOx diesen Koeffizienten werden verg/ichen mit entsprechenden Werten aus Modellberechnungen, wobei eine festgelegt mittlere, Verteilung der Ladungen des Polyions angenornmen wird. Der KoeIT-~.ient ¢~, negativ bei rdedrigen Salzkonzentrationen und positiv bei h6heren, er~eist sich auch als abh~ngig yon der St~Srung (nahe a = 0) der durchschnittlichen Verteilung der makxomolekulaxen Gruppen in Its bei der Dissoziation auf Grund der Expansion der Kette. Zum AbshcluB wird betont, daft die hier vorgeschlagene Darstellung der pK-Abh~.ngigkeit von a nlcht im Widerspruch steht zu der empirischen Gleichung yon Hcnderson-Hasselbalch for die Titrationslcurv¢ von Polys.~uren, g~ltig fiir einen be~'enzten Bereich yon a-Werten nahe a = 0"5.