4102
NORIO ISE AND TSUNEO OKUBO
Mean Activity Coefficient of Polyelectrolytes.
I.
Measurements of
Sodium Polyacrylates’
by Norio Ise and Tsuneo Okubo Department of Polyner Chemistry, Kyoto University, Kyoto, Japan
(Received May 11, 1966)
Using a concentration cell with transference, the mean activity coefficient of a polyelectrolyte was directly measured for the first time. Control experiments were carried out with sodium chloride: The observed mean activity coefficient agreed with the literature value. Those of the polyelectrolyte were found to decrease with increasing concentration. By comparison of the observed mean activity coefficient of the polyelectrolyte with the observed single-ion activity coefficient of gegenions, it was found that these two coefficients were not generally equal. This indicates that the contribution of macroions to the thermodynamic properties of the solutions is not negligible a t all but is very influential. The mean activity coefficient observed was successfully compared with that computed by a previous theory.
Introduction In a great number of papers2 concerning the solute activity of polyelectrolyte solutions, emphasis has been put on the single-ion activities of gegenions or of simple electrolytes coexisting. This situation is unsatisfactory if one admits a point of view proposed by Guggenheim3 that single-ion activities are inaccessible to exact thermodynamics, and only mean activities are physically significant. This situation, however, appears to be inevitable for the following reasons. The first reason comes from a consideration of the existing theories of polyelectrolyte dilute solutions. As was pointed out already,4 it has been assumed in most theories that macroion-macroion interactions were not as important as small ion-macroion interactions. Thus, only properties at infinite dilution are considered. It is rather dif€icult to derive expressions for the mean activity coefficient on the basis of this theoretical framework. The second reason is related to the experimental technique of activity measurements. Usually, in the study of polyelectrolytes, electrochemical methods, e.g., e.m.f. measurements o f cells, have been employed to derive the activity coefficient. According to electrochemical theory, the question as to what kind of activity has been measured is answered by finding with respect to which species of ions the electrodes were reversible. Electrodes so far known are reversible with respect to small ions such as H+, C1-, and so on, but The Journal of Physical Chemistry
not to macroions. Therefore, if one of these electrodes was used in a cell, together with a reference electrode, only the single-ion activity of the relevant small ions could be obtained, provided the liquid junction potential could be evaluated. This unsatisfactory situation can be improved by direct or indirect measurements of the mean activity of polyelectrolyte^.^ There are two well-known direct ways for electrochemical measurement of mean activity: e.m.f. measurement of cells with transference and of cells without transference. If the latter type of cell is employed, however, it is required to have electrodes reversible with respect to each of the ions of the polyelectrolyte, gegenions, and macroions. (l! Presented at the 14th Annual Meeting of the Society of Polymer Science, Tokyo, Japm, May 1965. (2) See S. A. Rice and M. Nagasawa, “Polyelectrolyte Solutions,” 1st Ed., Academic Press Inc., New York, N. Y., 1961, Chapter 8. (3) E. A. Guggenheim, J . Phys. Chem., 33, 842 (1929). (4) (a) N. Ise and M. Hosono, J. Polymer Sci., 39, 389 (1959); (b) N. Ise, J . Chem. Phys., 36, 3248 (1962); J. Phys. Chem., 67, 382 (1963); N. Ise and P. Ander, J. Chem. Phys., 39, 592 (1963). (5) It should be mentioned that what is described as the observed value of the mean activity coefficient of the polyelectrolyte in the present paper is the stoichiometric one, which will be denoted by y*. It is possible to define another mean activity coefficient for polyelectrolytes on the basis of the number of free ions, which will be denoted by y. If confusion can be avoided, however, as in the case of low molecular weight strong electrolytes in which no ion association can be supposed to occur, the asterisk will be omitted. See Results and Discussion for the difference between y* and y.
MEANACTIVITYCOEB'FICIENTS OF POLYELECTROLYTES
4103
a stable e.m.f. Before and after e.m.f. measurement of one pair of solutions, the e.m.f. of cell I1 was measNa-glass electrodeINaC1 (m)lcalomel electrode (11)
Na-glass polyelectrolyte j polyelectrolyte Na-glass electrode Na salt (m2) Na salt (m1) electrode solution 5; i solution 1
where R is the gas constant, T temperature in OR., and F Faraday's constant. We shall use eq. 1 to obtain a from E, hp, and a. Apparatus. (a) Cell Design. The cell design employed here is the one used by Stokes and Levied to obtain transference numbers from e.m.f. data of a concentration cell. En order to avoid contamination due to carbon dioxide, the air inside the half-cells was replaced with nitrogen before the cell was filled with the solution, and the volume of the gas contacting the solution was diminished as much as possible. The Naglass electrodes were supported in silicon rubber stoppers which fit tightly into the tops of the two halfcells. Contamination was thus reduced to a minimal amount. (b) Electrodes. The Na-glass electrodes,S products of Horiba Manufacturing Co., Kyoto, were stored in an NaCl aqueous solution (m = 0.100) when not in use since it was found that dry electrodes failed to give
ured at m = 0.100 to standardize the glass electrode. (c) Electric Circuits. The e.m.f. values were measured with a precision potentiometer, Type K-2, of Shimazu Manufacturing Co., Kyoto, in conjunction with a vibrating-reed electrometer, TR-85, manufactured by Takeda Riken Industry Co., Tokyo, as a null detector. The electrometer has an input impedance of more than lo1*ohms, and the maximum sensitivity was 50 pv. A standard cell was purchased from Yanagimoto Manufacturing Co., Kyoto, and had been certified by the Electrochemical Laboratory, Agency of Industrial Science and Technology, Tokyo. The concentration cell was immersed in a liquid paraffin thermostat maintained at 25 =!= 0.02'. The thermostat with its accessories was put in a metal box, which was grounded, to avoid outer disturbance, and the high-impedance side of the circuit was shielded. Materials. Sodium chloride, reagent grade, was used for the experiments without further purification. Sodium polyacrylate was kindly furnished by Prof. M. Nagasawa, Nagoya University. It was used without fractionation for this experiment of an exploratory nature, and the weight-average degree of polymerization was 1640. Conductivity water was used in preparing all of the solutions. As collected from the delivery tip of columns of ion-exchange resins it had a specific conductance of lo-' ohm-l cm.-l. The solutions of sodium chloride were made up by dilution of a 0.1 m stock solution, which was prepared in measuring flasks. I n order to obtain molalities, densities were measured. The polymer solutions were prepared from a stock solution of sodium polyacrylate; the stock solution was diluted and treated with ion-exchange resins. The polyacid thus obtained was quantitatively reconverted into sodium salt by means of potentiometric titration. Polymer concentration was determined from this titration.
Results and Discussion Calibration of Nu-Glass Electrode. Using a cell shown by (11), calibration of the Na-glass electrode was undertaken. For this type of cell, if the liquid (6) A. 8. Brown and D. A. MaoInnes, J. Am. Chem. SOC.,57, 1356 (1936). (7) R. H. Stokes and B. J. Levien, {bid., 68, 333 (1946). (8) For a discussion of the general properties of Na-glass electrodes, see R. G. Bates, "Determination of pH, Theory and Practice," 1st Ed., John Wiley and Sons, Inc., New York, N. Y., 1964, Chapter 10.
Volume 69, Number 19 December 1966
4104
NORIOISEAND TSUNEO OKUBO
junction potential can be assumed to be negligible, the e.m.f. (E)is given by
E = Eo
+ RT -In F
aNa+
(2)
where EOis the standard value of the e.m.f. and aNa+ is the single-ion activity of Na+. With the value of aNa+ determined by using MacInnes' convention concerning ion activity coefficient^,^ eq. 2 was tested. As is clear from Figure 1, a linearity between E and log a N a + was obtained over a wide range of aNa+. Between a N a + = 1 and 0.01, the slope was 59.0 mv., which is in good agreement with the theoretical value, 59.167 mv., a t 25". A departure from this linearity appears at aNa+ = 0.001. For comparison, a response curve of a sodium amalgam electrode used by Nagasawa and KagawalO is shown by a dotted curve. Although Activity of sodium ion their electrode displayed better performance than is Figure 1. Calibration curve of the Na-glass electrode: -, usually expected,ll it is clear that the Na-glass electheoretical line; - X - * - observed curve; ------ , calibration trodes are reversible in a much wider range of activity. curve of sodium amalgam electrode. (The original curve was vertically shifted.) See ref. 10. The e.m.f. reading varied appreciably a t first after the electrodes were inserted into solutions. They reached a limiting value which was constant within Table I : Computation of Mean Activity AO.1 mv. over a 30-min. period. The e.m.f. value Coefficient of Sodium Chloride given in this paper is this limiting value. Measurements of the E.mf. of Concentration Cells of MolalE.m.f., lo@: ity mv. ta (YdYl) Yobsd Ylit Sodium Chloride. It is interesting to measure the e.m.f. of the same type of cell of sodium chloride as 0.010 135.0 0.608" 0,1448 0.92 0.903' 0.88 0.873O 0.020 115.0 0.610" 0,1257 that of polymer samples and to compare the activity 0,612" 0.0947 0.82 0.822' 0.050 87.9 coefficient thus obtained with the literature value. 0. 615a 0.78 0.77gd 68.2 0.0705 0 * 100 The cell was 0,
Na-glass NaCl 1 NaCl Na-glass electrode (m2) i (ml = 1.021) electrode
(111)
The e.m.f. of the concentration cell was measured twice for each pair of solutions. I n the second measurement, the positions of the two electrodes were reversed. The difference between these two measurements was always smaller than 0.5 mv. The results which were obtained by averaging these two values are given in the second column of Table I. The first column of the table gives the concentration m2. It should be remembered that the concentrations studied were in the range (1.0 to 0.003) where a straight line was obtained for the plot of e.m.f. vs. a N a + as was shown in Figure 1. The slope agreed with the theoretical value. Needless to say, this means that the galvanic cell was reversible, which is a basic requirement for electrochemical determination of activity coefficients. The observed e.m.f. was constant within *0.6 mv. for 2 days.12 It was also observed that the additivity of the e.m.f. holds for three pairs of three different concentrations. This additivity, together with The Journal of Physical Chemistrv
0.201 0.506 0.762 1.021
a
49.2 21.5 8.9
0
0.618" 0. 623b 0 . 626b 0 . 62gb
0.0308 0,0105 0.0050 0
0.71 0.67 0.66 (0.657)
0.735d 0.681d 0.665d 0 . 657d
G. L. Longsworth, J. Am. Chem. Xoc., 54, 2741 (1932).
* Determined by the use of the data at 18 and 30°, "Inbmational
G. Soatchard and Critical Tables,',' Vol. 6, 1929, p. 310. S. S. Prentiss, J. Am. Chem. Xoc., 55,4355 (1933). R. H. Stokes and B. J. Levien, ibid., 68, 333 (1946).
the constancy, indicates that concentration difference and diffusion at the junction do not have a serious effect upon our e.m.f. measurements. The Na-glass electrode, as is readily seen from the present description, is not quite as accurate as the (9) See ref. 8, Chapter 3. (10) M. Nagasawa and I. Kagawa, J. Polymer Sci., 25, 61 (1957). (11) It is believed that reliable measurements with amalgam electrodes cannot be carried out for solutions more dilute than about 0.1 M . See S. Glasstone, "An Introduction to Electrochemistry," D. Van Nostrand Co., Inc., Princeton, N. J., 1964, p. 198. (12) This conclusion is right if the asymmetry potential changes slowly with time, which is believed to be the case.
4105
MEANACTIVITYCOEFFICIENTS OF POLYELECTROLYTES
silv er-silver chloride electrodes used by Brown and MacInnes.G For example, these authors reported a value of 0.01 mv. for the reproducibility of their measurements. I n view of the performance of the commercially available glass electrode, an accuracy better than h0.5 mv. cannot be claimed. The reproducibility was about f2%, Even if it were possible to improve the electrode and set up special measuring circuits, it would be difficult to obtain transference number data of macroions accurately enough. Therefore, in the present work, efforts have not been made to aim at high precision. I n the light of such accuracy, impurity problems are relatively unimportant. Brown and MacInness have shown that the correction AE, to be applied to observed e.m.f. reading (due to minute traces of salts) is given by
(3) where K~ is the specific conductance due to the contamination, AEi is expressed in mv., and C is the electrolyte concentration in normality. For a value of K( = ohm-' cm.-', aE, is 0.047 mv. for the cell having the concentration range 1.0 to m. This assumed value for K $ is certainly large compared to the actual K value of water used. Therefore, the correction due to contamination is well within our experimental error and is neglected in this paper. The e.m.f. of the cell shown in (111) is given by
E
= (2RT/F)J*'ta it2
d In a
(4)
Using the observed value of E and the literature value for t,, the mean activity coefficient y13 was calculated using the method described by Glasstone." Equation 4 was rearranged to
FE 4.606ta,RT where 6 = (l/ts) - (l/ta,) and tal is the transference number of the anion at some reference concentration ml; in this case ml = 1.021, and y1 is the activity coefficient at ml. The e.m.f. measurements allow us to evaluate the first two terms on the right-hand side of eq. 5. The third term was obtained by graphical i n t e gration of 6 against E since 6 can be derived from the concentration dependence of the transference number. Thus log (y2/y1) was determined and is given in the fourth column of Table I. Usually, in order to determine activity coefficients, use is made of the familiar equation of the DebyeHuckel theory. I n the present paper, an expedient was adopted instead; y1 was as-
sumed equal to the literature value of the activity coefficient at ml = 1.021. Then it is possible to derive log y for any concentration from the values of log (yz/ yl) obtained previously. The activity coefficient data thus obtained (Yobsd) are shown in the fifth column of Table I and can be compared with literature values of the activity coefficient (y1it) shown in the sixth column. The absolute values of the first and second term on the right-hand side of eq. 5 are close to each other, the third term making only a small contribution. Therefore log (y2/y1) is most sensitive to the e.m.f. data. Taking into consideration the reproducibility and accuracy of the measurements, we have an error which does not exceed f5% for Yobsd values at lower concentrations (0.020 to 0.010 m) and *3% between 0.201 and 0.050 m. It can thus be said that there is good agreement between Yobsd and ylit. Though the agreement is less than that reported by Brown and MacInnes, it is satisfactory for our purpose. Measurement of the E.m.f. of Concentralion Cells of Sodium Polyacrylate. Though experimental procedures and the treatment of data are basically the same as those employed in the case of sodium chloride, the study of polyelectrolytes brings forth a special problem. As was mentioned already14 electrostatic interaction in polyelectrolyte solutions does not disappear at zero polymer concentration since the electric charges on the polymer chain cannot be separated from each other unless the polymer chain is broken into the corresponding monomer units. Therefore, if the adopted standard state of the free energy is the same as that chosen in the case of simple electrolytes, the activity coefficient (of ionized groups) of the polyelectrolytes converges to a value other than unity at the zero polymer concentration. This value clearly depends on the electrostatic interaction between electric charges. This interaction, by various theories, has been related to the intrinsic properties of the polyelectrolyte molecule such as the dimensions of the macroion, the number of electric charges on it, and so on. In the absence of a complete theory, however, the basis of the activity coefficient cannot be provided at zero polymer concentration. I n order to avoid this difficulty, we have used the following convention. By definition, the observed mean activity coefficient y* is written as y + l =
*,*a2gY
(6)
*2P
where y*zg and y*Zp represent the single-ion activity coefficients of gegenions and macroions, respectively. (13) In the case of sodium chloride, y* is equal to dissociation can be assumed.
y
since complete
Volume 69,Number 18 December 1966
NORIOISEAND TSUNEO OKUBO
4106
Since a is usually large compared to unity as far as typical polyelectrolytes are concerned, we obtain an approximate relation y* =
Y*2B
(7)
on the condition that y*zP i s not extremely small or large. While this condition cannot yet be proved to be true, we expediently determine y*1 value on the basis of eq. 7, using a y*zg value (at a reference concentration of 0.01 m) determined independently by measuring the e.m.f. of a cell shown by (II).14 We then proceed to evaluate y* values for other concentrations using the y * ~value. The essential data for the computation of the mean activity coefficient from e.m.f. data are gathered in Table 11. I n the first column, the monomolal concentgtion is given. The e.m.f. values of the cells shown by (I) are given in the second column. The transference numbers (third column) a t each concentration were obtained from the data of Okubo, Nishizaki, and Ise,14 who carried out the transference experiment following the method developed by Huizenga, Grieger, and Wal1.16 I n column 4 the observed mean activity coefficients are given. Column 5 gives the single-ion activity coefficients of gegenions of the same sodium polyacrylate, which were determined independently. l 4 I n the case of the polyelectrolyte solutions, the accuracy of the e.m.f. measurement was 1 0 . 5 mv., and the reproducibility was =!=2%. The accuracy of transference number data is estimated to be =k2%. Taking these factors into consideration, the limit of error associated with the y * o ~ s dvalues was found to increase with decreasing concentration and hp value; the highest limit of error obtained was about =k7% for hP = 0.48 and m = 0.00435. Except for this extreme case, a value of 15% was the limit of experimental error. For convenience, the experimental data are also presented in Table 111, together with theoretical values for activity coefficients. From Table I1 (the fifth column), it is seen that y*pg is about 0.28, almost independent of the polymer concentration in the concentration range studied. This independence is also demonstrated in the fifth column of Table 111, which shows that the ratio of y*zg to that at the reference molality (0.01) is about unity. On the other hand, the observed value of y * decreases with increasing concentration, as is seen in the fourth column of Table I1 and the second column of Table 111. Thus it is evident that y* is not always equal to y*2g; in other words, ep. 7 does not hold generally. It is then interesting to estimate the relative change of y*zP. The eighth column of Table I11 gives the relative change of y*zP obtained from the observed y* and y*2g using eq. 6. The Journal of Physical Chemistry
Table 11: Mean Activity Coefficient of Sodium Polyacrylate Molality
0.00435 (0.01) 0.0116 0.0166 0.0331 0.0663 0.0995 0.167 0.227
E.m.f., mv.'
t2pb
34.6
0.480
27.5 24.0 16.0 7.8 5.7
0.480 0.480 0.481 0.485 0.494 0.522 0.522
...
0.0 -2.5
...
Y*obsd'
0.32 (0.29) 0.28 0.26 0.25 0.24 0.19 0.18 0.15
Y*Z2
0.29 (0.29) 0.29 0.28
...
0.28 0.28 0.28
*..
The e.m.f. measurements were carried out with ml = 0.167, whereas the Y*obsd values were determined using a y*zg value at 0.01. From the measurements in the work cited in ref. 14.
The enormous change of y*zPwith concentration should be noted and will be discussed later. It would be useful to mention the single-ion activity coefficients of macroions and gegenions (Y**~). However, as explained in the Introduction to this paper, the single-ion activities are inaccessible to exact thermodynamics. It should also be noted that a meaningful discussion of y*zP cannot be given because of experimental difficulties as well as of the basic inaccessibility alleged by Guggenheim. Since an electrode, reversible with respect to macroions, is not available, we are forced to estimate y*zp indirectly by measuring y* and eq. 6. Usually, when polyelectrolytes are concerned, a is very large compared to unity. Therefore, it is clear from eq. 6 that y* must be known to a high degree of accuracy in order to obtain a reasonably accurate value of y*zP. For example, if a! is assumed to be 1O00, which appears to be a reasonable estimate for ordinary polyelectrolytes, an error of y* of 0.1% corresponds to an error in y*2pof 100%. Evidently, such high accuracy for y* cannot be expected. Thus, it is difficult to compute exsct values of y*2p from data of y*.la On the other hand, eq. 6 indicates that, however large a may be, the magnification of error does not matter when the aim is to estimate y*Zg. This is one reason why y*2g could often be used for discussion of thermodynamic properties of polyelectrolyte solutions. It should be carefully noted, however, that this usefulness relies ultimately on the well-known assump(14)T. Okubo, Y . Nishizaki, and N. Ise, J . Phys. Chem., 69, 3690 (1965). (15) J. R. Huizenga, P. F. Grieger, and F. T. Wall, J . Am. Chem. Soc., 72, 2636 (1950). (16) Note that the enormous change of r*zp mentioned in the preceding paragraph exceeds largely the limit of error associated with Y*ZP.
4107
MEANACTIVITYCOEFFICIENTS OF POLYELECTROLYTES
Table 111: Relative Changes of
--Molality
0.001 0.01
0.03 0.1 a
Obsd.
... 1.00 0.86 0.66
y*, Y * ~ and ~ , y*zPwith Concentration and Comparison with Theory. ?‘*2d?’*2gQ.Ql
?‘*/-f*0.01--
-Calcd.R = 100 A.
R = 120 A.
2.59 1.00 0.63 0.53
1.31 1.00 0.86 0.80
Obsd.
... 1.00
0.97 0.97
-Calcd.R = 100 A.
1.21 1.00 0.89 0.82
-?‘*~P/?’*2IIQ.U
R
120 A.
1.21 1.00 0.89 0.82
Obsd.
-------Calcd. R = 100 A.
...
102as
1.00 10-88 10 -120
1.00 10-106 10-186
*
R = 120 A.
1028 1.00 10-10 10-18
The reference molality is 0.01.
tion pertaining to the liquid junction potentials which has not yet been rigorously established. I n view of ambiguities associated with the singleion activity coefficients discussed briefly in the preceding paragraph, it is desirable to discuss the solution properties of polyelectrolytes in terms of the mean activity coefficient. It is also thermodynamically significant. Furthermore, it would be interesting to show how useful the mean activity coefficient is. First of all, we have to mention a test of the GibbsDuhem equation in polyelectrolyte solutions. l7 The test of this equation, as is well-known, had been impossible because of the lack of mean activity coefficient data. However, using the data presented in this paper and the osmometrically determined activity coefficient of solvent, the equation can now be checked. I n other words, the reliability of thermodynamic solution data can be conclusively discussed. It would also be useful to recall recent observation by Nagasawa and Fujital* that, when the mean activity coefficient in the Gibbs-Duheni equation is replaced by the single-ion activity coefficient, the modified equation gives single-ion activity coefficient values, which not only disagree numerically with values determined by other direct methods but also show a concentration dependence quite different from the experimental one. It is plausible that the origin of this contradiction is the replacement of y* by y*zg in the equation, Le., the adoption of eq. 7. At3 a matter of fact, y* values calculated by the (unmodified) GibbsDuhem equation agreed with observed values of y* in a high concentration range.” Here asgain, the inaccuracy of eq. 7 is clearly demonstrated. The next step to be taken is comparison of the experimental data with theory. As may be understood from the discussion given later, in order that the experimental data of mean activity Coefficient can be accounted for by any theory, the macroion-macroion interaction has to be taken into consideration in that theory. Regrettably, there are only a few theories: a theory by Harris and Ricelg and our previous theory4&
appear to meet this requirement. The former theory, however, led us to a complicated expression for y* so that the comparison using this theory will be reported in a subsequent paper. I n the present paper we confine ourselves to our theory. As is mentioned in another paper,14 mean activity coefficient and single-ion activity coefficients of macroions and gegenions formulated in our theory are denoted by y, yZp,and yZg,respectively, and are based on the assumption that the electrolyte is partially dissociated; that is, the macroion is not 2 valent but a valent.20 These quantities should not be confused with the experimentally found activity coefficients, which are “stoichiometric” ones determined on the assumption that the electrolyte is fully dissociated and which have been and will be expressed by y* terms in this paper. If the fraction of free gegenions is denoted by 0, we obtain
Y* = 67, y*zg = Pyzg, and y*zP = PyzP
(8)21
Furthermore, thermodynamics gives (9)
If we assume that is equal to the polymer charge fraction (i/s), which was determined by transference experiment, l4 observed values of mean activity coefficient, y*, can be compared with the theoretical value by means of eq. 8. The expressions for ypp (17) N. Ise and T. Okubo, publication in preparation. (18) M. Nagasawa and H. Fujita, J . Am. Chem. Soc., 86, 3005 (1964). (19) F.E.Harris and S. A. Rice, J . Chem. Phys., 25, 955 (1956). (20) The macroion is assumed to have 2 ionizable groups and A bound gegenions, and o = 2 - A. (21) It should be recalled that the same standard state of chemical potential of polyelectrolyte as that of the 1-1 type electrolyte was adopted. To be correct, consequently, yzP must be called the “singleion activity coefficient of cy ionizable groups of the macroion,” whereas 7zP must be called the “single-ion activity coefficient of cy ionized groups of the macroion,” though the term “single-ion activity coefficient of the macroion” does not result in confusion. In the light of auch a physical meaning‘of r*zp,the reason for the difference between y*zp and yzP, and between y* and y would be understood.
Volume 69, Number Id
December 1966
4 108
NORIOISEAND TSUNEO OIWBO
and yzgottained in the previous treatment4areduce to the following in salt-free polyelectrolyte solutions
R= 120; 0.7 azKeo2
kT In yzP = --{T(KR) 3e
+
0.d
\
%
I
Polymer concentrotion (molality) Figure 2. Concentration dependence of the mean activity ,theory; x, observed. coefficient:
-
with
where eo is the absolute value of the elementary charge, I% the Boltzmann constant, E the dielectric constant of solvent, C the mean radius of an ionized group and a gegenion, nz the number of macroions per unit volume of solution, T the radius of macroion sphere, mr the exclusion volume parameter of ion i (& = 1 %vzP for macroions), and v1 the exclusion volume of ion i (v2 = 4 r r a / 3 for macroions). The subscript 2 refers to polyelectrolyte. u and r are familiar functions which appear in the Debye-Huckel theory.22 Using eq. 8-11, y* was calculated with the values of /3 = 0.44, Z = 1640, X = 920, a = 720,25b = 4 8., and appropriate values of T . I n Table 111, the theoretical values of y*/y*0.01, y*zg/y*2m.a, and y*@/ y*2p0.01 are presented. It is clear that the theoretical value of y*/y*o.Ol decreases with concentration in accordance with the observed tendency. On the other hand, the calculated value of y * ~ g l y * 2 is ~ ~much . ~ more insensitive to the concentration than that of y*/y*o.~. This is also a fact experimentally found (compare columns two and five). This difference can be attributed to the contribution of y*zP, as was mentioned already. I n the ninth and tenth columns of Table 111,the theoretical values of y * ~ ~ / y * mare ~ . ~presented. , Taking into account the fact that the calculated values are very sensitive to the assumed value of T , we note that the agreement between theoretical and observed (eighth column) values is satisfactory. It should be further noted that y~r,consists of contributions of macroion-macroion interaction and intramacroion interaction. The former is represented by the K-containing first term24 on the right-hand side of eq. 11, and the latter by the rest. According to our nuThe Journal of Physieol Chemistry
merical calculations, the intermacroion interaction is as strong as the intramacroion interaction. Therefore, the observed enormous change of y*zp with concentration should be attributed to these two kinds of interactions and indicates that the contribution of macroion-macroion interaction in the thermodynamic properties of dilute polyelectrolyte solutions is not negligible at all, but strikingly influential. This is in contrast with what has been often assumed in many theoretical treatments of polyelectrolytes. The important role of the macroion-macroion interaction has been pointed out in a series of our previous papers4 and is now again demonstrated by measurements of a thermodynamically significant quantity, e.g., the mean activity Coefficient. In Figure 2, the theoretical and observed values of y * / ~ * ~are . ~ lcompared. Three curves represent theoretical ones obtained with T values indicated in the figure, whereas the crosses denote experimental data. Figure 2 shows that a theoretical curve obtained with T = 120 8.is in excellent agreement with experimental data between 0.01 and 0.07 m, and observed values at high concentrations deviate from this curve, appr2aching a theoretical curve obtained with T = 100 A. as the concentration increases. This is quite understand.~
(22) R. H. Fowler and E. A. Guggenheim, "Statistical Thermodynamics," 1st Ed., Cambridge University Press, London, 1939, Chapter 9. (23) The transference experiment in the work cited in ref. 14 shows thst the polymer charge fraction i/s ( = 8) varies only slightly with concentration. The value adopted here, Le., 0.44, is the average one. The value of a wm obtained by a = pZ, where Z, the total number of ionizable groups of a macroion, happens t o be equal t o the degree of polymerization under our experimental condition. A. kwasdetarmined by a = Z (24) This term contains also macroion-gegenion interaction. However, this interaction would be much smaller than the macroionmacroion interaction, judged from the nature of Coulombic interaction.
-
IONIZATION CONSTANTS OF SULFONIC ACIDSBY RAMAN MEASUREMENTS
able since the radius of the polymer sphere would decrease with increasing polymer concentration. Finally, mention should be made of t,he value of T found above. Figure 2 shows that 120 A. for r gives a very satisfactory agreement between theory and experiment in the dilute region. Since information on this quantity a t finite concentrat,ions from other sources is not available, no convincing conclusion can be drawn. It should be noted, however, that, if this value of r is used for computation of yzg (by means of eq. lo),
4109
an excellent agreement between theory and experiment can also be obtained as is shown in Table 111. Acknowledgments. The authors gratefully acknowledge the encouragement and useful criticisms received from Professor Ichiro Sakurada. N. I. expresses sincere thanks to Professor U. P. Strauss for having directed attention to the ?;a-glass electrode and to Professors F. T. Wall and J. T. Edsall for comments a t the early stage of the experiments, and Professors M. Nagasawa and H. Fujita for valuable discussion.
The Determination of the Ionization Constants of Some Sulfonic Acids by Raman Measurements
by 0. D. Bonner and Arnold L. Torres Departmeni of Chemistry, University of South Carolina, Columbia, South Carolina (Received M a y 12, 1966)
The degrees of ionization of p-toluenesulfonic acid and 2,5-dimethylbenzenesulfonic acid in aqueous solutions have been determined by comparison of the ratio of intensities of the Raman lines arising from the C-H stretch in the CH3 group and the S-0 stretch for each sulfonate ion. These data have been combined with activity coefficient data to yield ionization constants of 11.6 -f 0.5 and 2.7 f 0.5 for the two acids. A comparison of infrared spectra for samples of an ion-exchanger film indicates a degree of ionization comparable to that of the two “model” compounds.
The introduction of the sulfonic acid type of cationexchange resins, consisting of sulfonate-exchange sites on polystyrene-divinylbenzene matrices, has resulted in the accumulation of very precise equilibrium data. The structures of these resins are reasonably well known, and resin samples from the same batch are quite uniform. There have been many attempts to interpret these data for various pairs of ions. One of the most interesting phenomena is the reversal of selectivity shown in exchanges involving an alkali metal ion, e.g., sodium ion, and hydrogen ion. The resin exhibits the expected preference for sodium ion except in cases where the sodium ion loading exceeds
about 90%, in which case hydrogen ion is then preferred. This behavior has led to considerable speculation as to the strengths of sulfonic acids. Because of the insolubility of the ion-exchange resins, many studies have been made on solutions of L‘modelJ1compounds similar in structure to the repeating units of the ion exchanger. Mock and l\/Iarshalll have investigated a sulfonated 1:l copolymer of styrene and vinyltoluene having a molecular weight of approximately 224,000 and report a degree of ionization, a, of 0.38 as determined from (1) R. A. Mock and C. A. Marshall, J. Polymer Sci., 13, 263 (1954).
Volume 69, Number 18 December 1966
