MAYAPAABO AND ROGER G. BATES
'106
Deuterium Isotope Effects and the Dissociation of Deuteriophosphoric Acid from 5 to 50" by Maya Paabo and Roger G. Bates National Bureau of Standards, Washinoton D. C.
,90834
(Received Augusl 11, 1969)
The first dissociation constant of deuteriophosphoric acid in deuterium oxide has been determined over the temperature range 5 t o 50" from emf measurements of cells using deuterium gas electrodes arid silver-silver chloride electrodes, with the following results: pK1 = 843,97Q/T - 4.5714 0,0139555T,where T is the thermodynamic temperature. The changes of enthalpy, entropy, and heat capacity characterizing the dissociation process have been derived from pK1 and its temperature coefficient. By comparison of the results with similar data for protiophosphoric acid in water, the deuterium isotope effects on pK1 and the thermodynamic functions have been evaluated. The isotope effect pKl(in DnO) - pKl(in Hn0) is 0.272 unit and almost unchanged over the temperature range studied, This value confirms a linear relationship between the isotope effect for inorganic acids aid pK (in HI@.
+
Introduction A recent paper' has focused attention once again on the relationship between acidic strength and the deuterium isotope effect on the dissociation as measured by pK(in DzO) - pK(in HzO). It was shown that this difference, or ApK, is not uniformly a linear function of pK(in HzO) in spite of earlier evidence supporting a linear variation.2-6 As a group, inorganic acids fell close to a single straight line, especially those with pK greater than 7. However, a critical examination of the data for a considerable number of organic acids, both carboxylic acids and substituted ammonium ions, led to ApK in the vicinity of 0.5 t o 0.6, even though the corresponding pK values in water ranged from 0 t o 6. To make the situation still more complex, the points for the two strongest inorganic acids that have been studied (bisulfate ion and phosphoric acid) fell near, but not exactly on, the straight line fixed by the data for the inorganic acids of pK greater than 7. Because of the high degree of dissociation of these acids, accurate pK values are difficult to obtain. Nevertheless, it has been shown that reliable values for the first dissociation constant of phosphoric acid can be determined by confining the measurements to buffer mixtures in which the concentration of dihydrogen phosphate ion considerably exceeds that of the phosphoric acidae A similar procedure has now been applied to a determination of the p K of deuteriophosphoric acid in deuterium oxide. From the results and those for the pK in water, a new value of ApK which lies almost exactly on the straight line passing through the ApK values for other inorganic acids of higher p K has been obtained. The pK1 for deuteriophosphoric acid in deuterium oxide was determined at ten temperatures from 5 to 50" by the analysis of emf data for cells without liquid junction containing deuterium gas electrodes and silversilver chloride electrodes. The thermodynamic funcThe Journal of Physical Chemistry
tions-enthalpy, entropy, and heat capacity-characteriaing the dissociation of deuteriophosphoric acid in D20 have been derived from pK1 and its temperature coefficient. The deuterium isotope effect, namely pKl(in DzO) - pKl(in HzO), was found to be 0.272 unit and to remain almost unchanged between 5 and 50".
Method The cell can be represented Pt;D2(g, 1 atm), DCl(m), KDzPO,(4nz), AgC1;hg (in DzO) where m is the molality in mol kg-l. The standard emf (E")of this cell being known,' the pK1 for deuteriophosphoric acid in DzO can be related to the emf ( E ) by pK1=
( E - E")F ~~
(RT In 10)
+ log m +
The last term of eq 1 is expected to be small in very dilute solutions and to vary linearly with ionic strength ( I )at higher concentrations. Hence
(1) R. A. Robinson, M. Paabo, and R. G. Bates, J. Res. Nut. Bur. Stand., 73A, 299 (1969). (2) R. P. Bell, "The Proton in Chemistry," Cornel1 University Press, Ithaoa, N. Y.,1959,Chapter 11. (3) P. Ballinger and F. A. Long, J. Amer. Chem. SOC.,82, 795 (1960). (4) N. C. Li, P. Tang, and R. Mathur, J. Phys. Chem., 65, 1074 (1961). (5) See also E.Hogfeldt and J. Bigeleisen, J . Amer. Chem. Soc., 82, 15 (1960). (6) R.G.Bates, J. Res. Nat. BUT.Stand., 47, 127 (1951). (7) R. Gary, R. G. Bates, and R. A. Robinson, J. Phys. Chem., 68, 1186 (1964).
DEUTERIUM ISOTOPE EFFECTS AND
THE
DISSOCIATION OF DEUTERIOPHOSPHORIC ACID
where pK1', the approximate value of p& a t finite I , becomes e q u d to pK1on linear extrapolation to I = 0. The deuteriophosphoric acid is rather extensively ionized, even though the stoichiometric ratio of DzP04to D3P04was kept a t a high value (3) to repress ionization. Consequently, estimation of m D + was necessary in order to calculate the second term on the right of eq 2 with the required accuracy (3)
It was evaluated from the acidity function p(aDYCl), together with values of the mean activity coefficient of deuterium chloride calculated by a Debye-Huckel formula
- log m D +
+
~ ( U D Y C ~ ) 2 log Y& (DCI)
- ( E - E")FR1' In 10
"' + log m - 1 +2A (Id,,) Bd(ldo)"a (4)
I n eq 4, A and B are the Debye-Huckel constants for the appropriate temperature and solvent dielectric constant,7 it is the ion-size parameter, and do is the density of the solvent.8 The ionic strength is given by
I
=
41% f m D i
(5)
It was shown earlierfithat the estimation of mn+ (or mH+) introduces the major uncertainty into the determination of pK values for moderately strong acids by the emf method. I n the present work, a computer program combining eq 2, 3, 4, and 5 was written. Values of the ion-size parameter d were selected, and corresponding values of pK1' were computed, The intercept pKl was obtained by the method of least squares, and the standard deviation of the intercept was derived. The value of the ion-size parameter which yielded the smallest standard deviation for pK1 was judged t o be the best choice. The d values all fell in the range 4.5 t o 5.5 8, showing no dependence on temperature. It is noteworthy that this is of the same order (4.3 t o 5.0 8)found for HC1 in watera and for DC1 in deuterium oxide' over this temperature range. However, in the present procedure, 8 enters only indirectly in a correction term (eq 4).
Experimental Section Materials. Potassium dihydrogen phosphate, NBS Standard Reference Material 21861, was used. It was heated a t 120" for 2 hr. The salt was converted into KDtP04 upon solution in deuterium oxide; the resulting contamination of the solvent (0.4 atom % hydrogen in the most concentrated of the cell solutions) was too slight to be of concern. The deuterium oxide used had an isotopic purity of 99.8% and a conductivity of 9 X lo-' S2-l cm-1 a t
707
25". Deuterium gas, obtained commercially, was analyzed by mass spectrometry a t intervals during the course of the work. The results uniformly indicated a hydrogen content less than 0.6 atom %. The stock solution of deuterium chloride was prepared by diluting a commercial sample of the acid (2001, DC1 in D20) with deuterium oxide. Its molality was determined by coulometric analysis. Cells and Procedures. The cells were of all-glass construction except for the Teflon stopcock plugs. These cells have been described in ref 7. The procedures differed slightly from those followed earlier, in that the vacuum treatment of the electrodes was omitted. Instead, the solutions were saturated with deuterium gas, and the cell vessels were flushed with deuterium before the solutions were admitted. The measurements a t the ten temperatures extended over a period of 3 days. A comparison of the initial and final emf a t 25" indicated excellent stability during this period, the average change for all cells during the course of the run being only 0.05 mV. To conserve deuterium gas, the flow was interrupted during the night when measurements were not being made.
Results The observed emf values recorded a t a total pressure (deuterium gas plus solvent vapor) equal to the atmospheric pressure were corrected in the usual way to a deuterium partial pressure of 1 atm. The corrected values are summarized in Table I . I n the analysis of the data, a range of values of the ion-siEe parameter d was chosen, and the corresponding intercepts ( ~ K I ) were obtained in the manner already described, together with the standard deviations of the intercepts. The best estimates of d and pK1 were assumed to be those for which the standard deviation was a t a minimum. This criterion, adopted in the analysis of the data for the standard emf of the hydrogen-silver chloride cell in ordinary waterI9 has proved useful in many other similar studies in this laboratory. The values of pK1 and the corresponding standard deviations of the intercepts are listed in Table 11. The dissociation constants are based on the molal scale. The experimental values of pK1 can be expressed by the equation pK1
843.979/T - 4.5714
+ 0.0139555T
(6)
+
where T is the thermodynamic temperature, t("C) 273.15. The standard deviation of the ten pK1 values from those calculated by this equation is 0.0003. The constants of eq 6 were found by the method of least squares with the aid of the OMNITAB program. The pK1 in water is listed in the next to the last column and
(8) T.L. Chang and L. H. Tung, Chinese J. Phys., 7, 230 (1949). (9) R. G . Bates and V. E. Bower, J. Res. Nat. Bur. Stand., 53, 283
(1054) I
Volume 74nNumber 4 February 19,1970
MAYAPAABO AND ROGER G. BATES
708
Table I : Electromotive Force of the Cell Pt;D&, 1 atm), DCl(m), KD1P04(4m), AgC1;Ag from 5 to 60" (in V) m
50
10"
15'
200
25"
30'
350
40'
450
50"
0.003416 0.004754 0 1008000 0.01044 0.01448 0 01976 0.02496 0.02946 0.03467 0.03936 0.04487 0 04876
0 62942 0,51898 0 50347 0.49670 0.48642 0 I47793 0.47164 0.46716 0 46279 0.45951 0.45597 0 45368
0 53268 0.52200 0.50629 0.49841 0.48901 0.48039 0.47402 0.46946 0,46506 0.46171 0.45813 0,45580
0.53564 0.52495 0.50904 0 60106 0.49153 0.48280 0 47633 0.47172 0.46724 0.46386 0.46021 0.45787
0.53865 0,52785 0.51164 0 50362 0.49399 0.48520 0 47861 0.47395 0.46940 0.46588 0.46227 0,45990
0.54161 0 . ,53069 0.51431 0,50621 0,49646 0.48755 0 48090 0.47615 0.47155 0.46796 0.46431 0.46191
0.54444 0.53347 0.51700 0 50880 0.49888 0.48987 0.48312 0.47835 0.47365 0 47022 0.46636 0.46395
0.54725 0 53616 0.51957 0.51125 0 . 60125 0.49211 0.48533 0.48046 0.47574 0.47222 0.46835 0.46587
0.54997 0,53887 0.52207 0.51372 0.50355 0.49437 0.48745 0.48258 0.47774 0.47425 0.47026 0.46781
0.55269 0 64149 0.52454 0.51611 0.60585 0.49655 0.48956 0.48462 0.47975 0.47618 0.47216 0.46966
0.55537 0 54408 0.52697 0.51844 0.50811 0.49863 0.49163 0,48662 0.48166 0.47808 0.47403 0.47149
I
I
I
I
I
I
I
I
I
I
Table I1 : Dissociation Constant of Deuteriophosphoric Acid in Deuterium Oxide from 6 to 50" and Related IsoLope Effect PKI, Std devs
(H10)*
APK
5 10 15 20 25 30 35 40 45 50
2.3445 2.3610 2.3788 2.3981 2.4202 2.4437 2 4677 2.4939 2.5211 2.5501
0.0007 0.0007 0.0007 0 IO005 0.0005 0.0007 0,0007 0 0008 0.0007 0 0008
2.071 2.088 2.106 2 127 2,148 2.171 2.196 2 I222 2.249 2.278
0.273 0 273 0.273 0.271 0.272 0.273 0.272 0.272 0.272 0.272
At each temperature, n
t
1
=
12.
I
I
Reference 6.
the isotope effect, ApK = pKl(in D20> - pKl(in HzO),in the last column. To provide a further confirmation of the magnitude of the isotope effect, a series of measurements of pK1 in ordinary water was also made, following exactly the same procedure as for the measurements in deuterium oxide. The values of pKl(in HzO) so obtained agreed very well with those found earlier by BatesJ6the average difference a t the ten temperatures being only 0,0019 unit. The value of pK1 a t 25" given in TabIe I1 (2,420 on the molal scale) can be compared with 2.362 (molar scale) obtained from conductance measurements by Mcnougall and Long.lo Inasmuch as the scalar difference is log do (0.043 at 2 5 " ) , the latter value becomes 2.405 on the molal scale. The agreement is therefore seen to be quite satisfactory, particularly so in view of the difficulties encountered in measuring the pK of an acid as strong as phosphoric. The thermodynamic functions enthalpy, entropy, and heat capacity for the dissociation of deuteriophosphoric acid in heavy water a t 25' have been calculated in the usual way from the temperature coefficient of pK1 using the constants of eq 6. They are given in Table I11 The Journal of Phgsical Chemistry
I
I
Table I11 : Thermodynamic Functions for the Dissociation of Phosphoric Acid in Deuterinm Oxide and Water at 25' In H B O ~
In D20
Dz0, exptl
a
I
PKI
t, e c
I
I
I
I
AH",J mol-1 AS",J K-lmo1-l ACpe, JK-lmol-l
-7593 (- 1815)" -71.8(-17.2) -159 (-38)
a Values in parentheses are in calories (1 cal erence &.
- 7650 (- 1828) -66.8 (-16.0) -154 (-37) =
4.184 J).
Ref-
and compared with the corresponding quantities for the dissociation of phosphoric acid in ordinary watern6 The uncertainties in the thermodynamic functions estimated from the variances of the least-squares constants of eq 6 are as follows: AH", 12 J mol-' (3 cal mol-'); AS", 0.1 J K-I mol-' (0.1 cal deg-l mol-'); AC,', 2 J K-' mol-' (0.5 cal deg-l mol-').
Discussion Six weak acids have now been studied in both water and deuterium oxide by thermodynamically rjgorous methods utilizing cells without liquid junction. The p K values for these acids in the two solvents are compared in Table IV. The isotope effect ApK, that is, pK(in D20)- pK(in HzO),is given in the last column. The acids are arranged in order of decreasing strength in water. It can be seen in Figure 1, where ApK is plotted as a function of pK(in HzO), that ApK for phosphoric acid now lies on the straight line fixed by the isotope effects for the t,hree inorganic acids with pK greater than 7. The point marked "AcOD" represents the ApK value for both acetic acid and acetic acid-da. Its departure from the line is too large to be attributed to experimental error. Taken with other data summarized elsewhere,' it supports the conclusion that the isotope effect is different for organic and inorganic (mineral) acids below pK = 7. It cannot be said, however, that a clear difference between carboxylic acids, phenols, and ariilinium acids exists.' The charge (10) A, 0 . McDougall and F. A. Long, J . Phyd. Chem., 66,429 (1962).
DEUTERIUM ISOTOPE EFFECTS AND
THE
DISSOCIATION OF DEUTERIOPHOSPHORIC ACID
709
Table IV : Deuterium Isotope Effects on pK at 25' PK
a
Acid
(in Dz0)
PK (in HzO)
APK
Phosphoric Acetic Acetic-do Primary phosphate ion Bicarbonate ion Solvent
2.420 5 312 5.326 7 780 11.077 14.955
2.148 4.756 4.771 7 200 10.329 13.997
0.272 0.566 0.553 0 580 0.748 0.958
I
I
I
I
Refa
This work, 6 11,12 13,14 15,16 17,18 19,20
The first reference is to the pK (in DzO),the second to the pK (in HzO).
Table V: Deuterium Isotope Effect on the Thermodynamic Functions for the Dissociation of Weak Acids at 2 5 O 5 AHo(d) -AHo(h),
Add
J mol-1
Phosphoric Acetic Acetic-da Primary phosphate ion Bicarbonate ion Solvent 5
d = in DzO; h = HBO.
b
-
ASo(d) ASo(h); J K-1 mol-'
57 (13)b 1561 (373) 1456 (348) 1628 (389) 2564 (613) 3284 (785)
-5.0 -5.4 -5,4 -5.9 -5.9 -7.3
(-1.2) (-1.3) (-lS3) (-1.4) (-1.4) (-1.7)
ACp"(d)
- ACpo(h)p
J K-1 mol-1
-4 (-1) -8 (-2) - 17 (-4) -18 (-4) 4-15 (+4) -34 (-8)
Values in parentheses are in calories (1 cal = 4.184 J).
1.c
in both water and deuterium oxide to make worthwhile a comparison of the enthalpy, entropy, and heat capacity effects of changing solvent. Such a comparison is made in Table V; the acids are again arranged in order of increasing pK (decreasing strength). The figures given in each column represent the excess of the indicated thermodynamic function (AH" , AS", or AC,") in deuterium oxide over the value in water. The differences in enthalpy and heat capacity are seen to increase as the acid becomes weaker, but the isotope effect on the entropy change appears to be nearly independent of the acidic strength. The very slight decrease in ApK from 5 to 50" apparent in Table I1 is consistent with the small positive
0.E
0.E
A PK 0.4
(11) R. Gary, R. G, Bates, and R. A. Robinson, J . Phys. Chem. 69, 2750 (1965).
0.2
I 2
IO
6 pK IN
14
H,O
Figure 1. Deuterium isotope effects ( A p K ) for some weak acids plotted as a function of pK (in Ht0) at 25".
type thus appears to be of secondary importance, in agreement with the conclusion of Martin.21 The pK values of a sufficient number of weak acids have now been determined over a range of temperatures
(12) H. S. Harned and R. W. Ehlers, J. Amer. Chem. Sac., 5 5 , 652 (1933). (13) M. Paabo, R. G. Bates, and R. A. Robinson, J . Phys. Chern., 70,2073 (1966). (14) M. Paabo, R. G. Bates, and R. A. Robinson, ibid., 70, 540 (1966). (15) R. Gary, R. G. Bates, and R. A. Robinson, (bid.,68,3806 (1964). (16) R. G. Bates and S. F. Acree, J . Res. Nat. Bur. Stand., 34, 373 (1945); F. Ender, W. Teltschik, and K. Schafer, 2. Elektrochem., 61, 775 (1957); A. K. Grzybowski, J. Phys. Chem., 62,555 (1958). (17) M. Paabo and R. G. Bates, ibid., 73, 3014 (1969). (18) H. 5. Harned and S. R. Scholes, Jr., J . Amer. Chem. Sac., 63, 1706 (1941). (19) A. K. Covington, R. A. Robinson, and R. G. Bates, J . Phys. Chem., 70, 3820 (1966). (20) H. S. Harned and R. A. Robinson, Trans. Faraday Soe., 36, 973 (1940). (21) R. B. Martin, Science, 139, 1198 (1963). Volume '7Jt Number 4 February 19, 1970
I. NODA, T. TSUGE, AND M. NAGASAWA
710
values of AH"(d) - AHo(h) given in Table V. The two quantities are related by
- -d(ApK) -
-
dT
AHo(d) - AHo(h)
(1'
RT2 In 10
The enthalpy differences listed in the second column of Table V show that the isotope effect on pK decreases
uniformly with increasing temperature and that the decrease is greater the weaker the acid.
Acknowledgment. The authors are indebted to C. E. Champion and W. D, Dorko of the Analytical Chemistry Division for the coulometric analyses of the DCl solution and the mass spectrometric examination of the deuteriumgas.
The Intrinsic Viscosity of Polyelectrolytes by Ichiro Noda, Takeaki Tsuge, and Mitsuru Nagasawa Department of Xynthetic Chemistry, Nagoya University, Chilcuswlcu, Nagoya, Japan,
(Received July 28, 1969)
The intrinsic viscosity of sodium poly(acry1ate)was determined in sodium bromide solutions of various concentrations C, as a function of molecular weight A4 and degree of ionization i. It was found experimentallythat the electrostatic part of the expansion factor, which is defined as the ratio of the intrinsic viscosity to its value at infinite ionic strength, can be expressed in terms of a reduced parameter (M/C,)'/z at all degrees of neutralization if the molecular weight is lower than IO6. On the other hand, in almost all theories so far published, the electrostatic part of the expansion factor is given as a function of M'/z/C, if the charge density of the polyion is low enough.
Introduction Among various characteristic properties of linear polyelectrolyte solutions, the intrinsic viscosity may be one of the most important problems remaining unsolved. In spite of a vast number of publications on this problem, the following features of the intrinsic viscosity of linear polyelectrolytes have not yet been explained. ( 1 ) Molecular Weight Dependence of Intrinsic Viscosity. The theories on the expansion factor of nonionic polymers can be, roughly speaking, divided into two types. The representative of the first type is the well known theory of Floryl which gives a5
- a3 = 2 . 6 0 ~
(1)
where a2 represents the ratio of the mean-square endto-end distance of polymer perturbed by the excluded volume effect to that of the unperturbed polymer and z is proportional to the square root of molecular weight such as
Here, (h02)is the mean end-to-end distance of unperturbed polymer, M and m, are the molecular weights of the polymer and a segment, p is the excluded volume characterizing the interaction between a pair of segments, and u(r)is the potential of average force between two segments at distance r apart. The other representative is the theory of Stockmayer and Fixman2 which gives aa -
(24
and A'
=
(ho')/M
(2b)
B = P/rnB2 = (4T/mS2)
Im{ 1 - exp[-u(r)/kT]}r2dr
The .Journal of Physical Chemistry
(2c)
28
(3)
The recent experimental results of light scattering from nonionic polymer solutions appear to show that eq 1 agrees with experimental results better than eq 3.314 However, the conclusion cannot always be valid if we compare the theories with experimental data of intrinsic viscosity by assuming the Flory-Fox relationship'that [91/[9ls = a,3
x = (3/2.rr)8'/"BA-3M1/2
1=
=
Q1q3
a3
(4) (5)
(1) P.J. Flory, "Principles of Polymer Chemistry," Cornel1 University Press, Ithaca, N. Y., 1953,Chapter XIV. (2) W. H. Stockmayer and M. Fixman, J. Polym. Sci. Part C , 1, 137 (1963). (3) G. C. Berry, J . Chem. Phys., 44,4550 (1966);46, 1338 (1967). (4) T. Norisuye, K. Kawahara, A. Teramoto, and H. Fujita, ibid., 49,4330 (1968);K.Kawahara, T. Norisuye, and H. Fujita, ibid., 49, 4339 (1968).
