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Ion-pair formation of cyclic phosphate anions
J. im~rg, nucl. Chem., 1972. Vol, 34, pp. 3899-3912.
ION-PAIR
Pergamon Press.
Printed in Great Brit~dn
FORMATION OF CYCLIC ANIONS
PHOSPHATE
G E N 1 C H I R O K U R A and S H I G E R U O H A S H I Department of Chemistry, Faculty of Science, Kyushu University, Hakozaki, Fukuoka, Japan (First received 1 July 1971;in re t.:isedJbrm 9 November 1971)
Abstract--An ion-exchange method, conductivity method and activity measurements have been applied to the investigation of ion-pair formation by cyclic phosphate anions in aqueuos solution. By conductivity measurements on their tetramethylammonium salts, the charges on the cyclic phosphate anions and ion equivalent conductivities at infinite dilution have been determined. From conductivities of sodium tetrameta-, hexameta- and octametaphosphate, association constants of ion pairs containing one sodium ion have been calculated by the method of Davies and Monk. From activity measurements on the sodium ions, an average number of sodium ions bound to each cyclic phosphate anion and association constants of the ion pairs containing one or more sodium ions have been calculated. It was concluded that the following ion pairs are present in dilute aqueous solutions of sodium metaphosphates; N a P 4 0 ~ in sodium tetrametaphosphate solution, NaP~O~, Na,.,P~O~ and Na:~P~;O:~, in sodium hexametaphosphate solution and NaP~O~, NaePsO~, Na:~PxO~] and Na4PsO~r in sodium octametaphosphate solution. INTRODUCTION
CONDENSED phosphates are well known as multivalent inorganic anions. Their
ability to form complexes with various metals has been reported by several investigators[l]. However, relatively few studies have been made on ion association of phosphate anions with alkali metal cations [2-4]. Among the condensed phosphates, there is a group called metaphosphates, which may have cyclic structures. Trimeta- and tetrametaphosphate have been well known for many years and can be easily prepared by conventional methods [5]. Hexameta- and octametaphosphate were recently prepared by Griffith and Buxton[6], and Schiilke [7], respectively. Our previous study [8] of ion-exchange equilibria of cyclic phosphate anions with an anion-exchange resin or dextran gel indicated that the charges of highermembered cyclic phosphate anions than the tetrametaphosphate anion, in the resin phase, are coincident with those in highly dilute solution where no ion-pair formation occurs. However, it was found that in the dextran gel phase these cyclic phosphate anions have lower charges than those in the resin phase. This 1. 2, 3. 4. 5.
J. R. Van Wazer and C. F. Callis, Chem. Rev. 58, 1011 (1958). C. W. Davies and C. B. Monk, J. chem. Soc. 413 (1949). U. Schindewolf and K. F. Bonhoeffer, Z. Elektrochem. 57, 216 (1952). G. L. Gardner and G. H. Nancollas, A nalyt. ('hem. 41,202 (1969). G. Brauer, Handbuch der Priiparativen Anorganischen Chemie, pp. 494-496. Ferdinand Enke Verlag, Stuttgart (1960). 6. E. J, Griflith and R. L. Buxton, lnorg. Chem. 4, 549 (1965). 7. U. Schiilke, Z. anorg, allg. Chem. 360, 231 (1968). 8. O. Kura and S. Ohashi, J. Chromatog. 56, 111 (1971). 3899
3900
G. K U R A and S. O H A S H I
suggests that in the anion-exchange dextran gel phase an interaction of cyclic phosphate anions with the cations takes place. Since the dextran gel is more highly swollen than the resin, an appreciable amount of cations can invade the gel phase. Therefore, multivalent anions such as cyclic phosphate anions adsorbed in the gel phase tend to attract the invading cations to form ion pairs. The purpose of this work is to study interaction of the higher-membered cyclic phosphate anions with alkali metal ions, especially sodium ions. For such a purpose many methods can be used, e.g. an ion-exchange method, conductivity measurements, activity measurements, optical methods and solubility measurements. Dissociation pherrorn~a of sodium trimeta- and tetrametaphosphate and corresponding acids have beer~ ~tudied by Davies and Monk by a conductivity method[2]. They have determined limiting equivalent conductivities of the above salts and acids and obtained/~eir dissociation constants from differences in observed and calculated condu~tivities. They have concluded that ion-pair formation occurs slightly in phosl~ate solution and the logarithm of the association constant of NaP40~-, log/31,'is about 2.04. However if more than one associated species are present as unsymmetrical electrolytes, it is difficult to apply a conductivity method for this purpose, because of the large uncertainty in the limiting equivalent conductivities of the associated species. On the other hand, Gardner and Nancollas [4] recently investigated ion association in solutions of sodium trimeta- and tetrametaphosphate using a sodium ion-selective electrode, and gave a value of 2.12 for 1og/31 for NaP40~. By means of activity measurements free sodium ions can be directly determined and association constants of highly associated species can be calculated. In the present study we employed an ion-exchange method, conductivity measurements and activity measurements in order to obtain information on ion-pair formation by tetrameta-, hexameta- and octametaphosphate anions with sodium ions. EXPERIMEN~TAL Cyclic phosphates The preparation of sodium tetrameta-, hexameta- and octametaphosphate was described in our previous paper [8]. Tetramethylammonium salts of cyclic phosphoric acids were prepared as follows. Tetramethylammonium hydroxide was prepared by passing tetramethylammonium chloride solution through a column of a Dowex IX4 resin in the hydroxide form. Tetrameta-, hexameta- and octametaphosphoric acid were prepared by converting their sodium salts to the respective free acids by treating with a Dowex 50W X8 resin in the hydrogen form. We obtained tetramethylammonium tetrameta-, hexameta- and octametaphosphates by neutralization of the respective free acids with tetramethylammonium hydroxide.
Determination of phosphates Determination of each cyclic phosphate was carried out by converting it to the free acid and titrating the acid with sodium or tetramethylammonium hydroxide, or by the colorimetric method of Lucena-Conde and Prat [9]. Measurement of distribution ratios in ion-exchange equilibria A study of the pH dependence of the distribution ratios was carried out by the method described 9. F. Lucena-Conde and L. Prat, Analytica chirn. Acta 16, 473 (1957).
Ion-pair formation
390 i
in our previous paper[8] using a QAE-Sephadex A-25 gel and 0.20-0.50 M potassium chloride as eluents. In the study of the effect of eluents, distribution ratios of the cyclic phosphates for a QAESephadex A-25 gel were measured by the usual batch method at room temperature. About 0.5 g of the air-dried gel in the chloride form was put into a stoppered Erlenmeyer flask and then 25 ml of an eluent and 2 ml of a solution containing a known amount of a cyclic phosphate (ca. 250 p,g as P) was added. After equilibrium was reached, the gel phase was separated by decantation. The phosphate in the solution phase was determined colorimetrically [9]. Eluent concentrations were varied from 0.10 to 0.30 M.
Conductivity measurements The equipment used for the conductivity measurements was a Yanagimoto Conductivity Outfit, Model MY-7, A cell was designed so as to be adaptable for titration and its electrodes were coated with platinum black. The cell was standardized with 0.01 M potassium chloride and the cell-constant was 0.3447 cm -~ at 25°C. The temperature of the cell was precisely controlled at 25 +_0.1°C. Conductivities of solutions of the following salts were measured; [N(Me)414P40,~, [N(Me)al~P~;O~, IN(Me/.~]sP~Oe~. Na~P,P~2, Na~P,O~s and NasP, Oe~. The pH of the cell solutions was always between 6.5 and 7-0. Activity measurements o f free sodium ions Activity measurements of sodium ions were made at 25 +_0-1 °C in a cell. Glass electrode Isolution under studyl calomel electrode. A Hitachi-Horiba sodium-ion glasselectrode, #2535-05 T and a reference calomel electrode, #4142-05 T were connected with a HitachiHoriba pH meter, model F-Sss. E.m.f. values measured with the above equipment were converted into sodium ion concentration by using a calibration curve obtained for a series of standard solutions of sodium chloride. About 15 rain were needed for each activity measurement until a constant reading was obtained, because of the slow response of the electrode. The precision of the e.m.f, values was +_0.3 mV, if the electrodes were carefully preconditioned. Activities of sodium ions in solutions of the following salts were measured; Na4P4Piz, Na~PrO~8 and NasP~O.,4. The pH of the cell solutions was always between 6.5 and 7.0 during the experiments, considerably larger than pNa, indicating a negligible degree of hydrolysis and hydrogen ion interference. RESULTS AND DISCUSSION
As has been described in the previous paper [8], charges on phosphate anions can be determined by measuring their distribution ratios in ion-exchange equilibria. If cyclic phosphate anions form ion pairs with protons, their degrees of association must depend on the pH of an eluent. The charges of cyclic phosphate anions were calculated from the slope of the plots of log D,, (distribution ratio) against log ICI ] (eluent concentration) at pH 5.2 and 9.8 and are shown in Table 1. The charges on the cyclic phosphate anions are nearly the same at each pH values. Therefor, it can be said that the association of cyclic phosphate anions Table 1. Charges of cyclic phosphate anions in the dextran gel phase at pH 5.2 or 9-8. Eluent: KCI
pH5-2 pH9.8
PITh
Prm
Psm
3.4(4-0)* 3-3(3-7)
4.3(5.0) 4.3(5.0)
5.7(6.5) 5.7(6.2)
*Values in the parentheses are corrected for the invasion in the gel phase.
3902
G. K U R A and S. O H A S H I
with protons does not occur even at pH 5.2. Thus it can be concluded that ionpair formation by cyclic phosphate anions with cations in the eluent causes the decrease in their charges in the gel phase. It is generally accepted that in outer-sphere complexes the complexibility of alkali metals increases in the order of Li < Na < K < Rb -- Cs. In order to examine this, we used alkali metal chlorides, i.e. LiC1, NaCI, KCI and CsC1 as eluents. From the results shown in Table 2, it appears that the charges of the cyclic phosphate anions decrease slightly as the sizes of the hydrated cations of the eluents decrease. However, since the differences in the charges are very small, it seems that there are no marked differences in the ability of alkali metal ions to form ion pairs. We applied a conductivity method to the quantitative study of ion-pair formation. Onsager's limiting law for the equivalent conductivity of a completely dissociated salt can be expressed by Eqns (1), (2) and (3). (1)
A = A°--AX/I
in which I is the ionic strength, and Eqn (1) can be written as equation (1)'. A = A°-A'X/C
(1)'
where C is the equivalent concentration of a phosphate. In Eqn. (1), for a single electrolyte dissociating into ions 1 and 2, A becomes A = 2.801 x 1061Zl. Z21qA° ~ 41"25(IZll + IZ21) (eT)S/2(l + V/q) n ( e T ) lie
(2)
where • and ~ are the dielectric constant and viscosity of the solution, respectively [10]. Here the quantity q is defined by
Iz,. z21 q = IZ, I + [Z2I }Z2JA~+ IZ, IA2
(3)
where ZI(Zz) and A~(A~) are the charge and ion equivalent conductivity of the ion 1(2), respectively. For aqueous solutions at 25°C Eqn (2) reduces to: A = 0.78521Z~. Z2J = -qA° - - - 7 - + 30.32 (IZal + JZ2f)
l+vq
Table 2. Charges of cyclic phosphate anions in the dextran gel phase when various alkali chlorides are used as eluents, pH: 5.2
Prm Psm
LiCI
NaCI
KC1
CsCI
4"7 6"0
4"3 5"7
4"4 5"7
4"4 5"5
10. R. A. Robinson and R. H. Stokes, Electrolyte Solutions. Butterworths, London (1959).
(2)'
Ion-pair formation
3903
It is known that these equations agree with the experimental data for numerous uni-univalent, uni-bivalent and uni-tervalent salts in aqueous dilute solutions at various temperatures. However, when the valency product is four, Onsager's limiting law does not always hold. This may be due to the increase of electronic interaction between cations and highly negatively charged anions. In other words, Bjerrum-type ion association certainly occurs and exerts some effect upon the mobility of the ions. Conductivities of sodium tetrameta-, hexameta- and octametaphosphate, which are 1-4-, 1-6- and 1-8-valent electrolyte, respectively, were measured in dilute solutions and are given in Table 3. When the conductivities of these salts are plotted against the square roots of the equivalent concentrations, the resulting curve has a much steeper slope than that expected from Onsager's limiting law as shown in Fig. 1. This lower conductivity is certainly due to ion pairing. From the difference between the conductivity observed and that calculated from Onsager's limiting equations where full dissociation of salts is assumed, the degree of ion association can be approximately evaluated by the method of Davies and Monk [2]. For sodium trimetaphosphate, they regarded this salt as a mixture of l-3-valent electrolyte, (3Na). (P309) and 1-2-valent electrolyte, (2Na). (NaP30,) and assumed that ion equivalent conductivity of NaP,~O,;'- at infinite dilution is two-thirds of that of P30.,,:~-. They also assumed that Onsager's limiting equations hold individually for both 1-2- and 1-3-valent electrolytes in the solution. Table 3a. Equivalent conductivity of sodium tetrametaphosphate and apparent association constant Kn~o (× 106)
C (× 104)
"~/C ()( 102)
A
~3'1
2-337
0.5035 1'002 1.496 1.985 2.468 2.948 3.423 3.897 4 "347 4.819 5"728 6'626 7.496 8'356 9.200 11.24 13-20 15.07 16-87 20.24 23.36 26-24 28.91
0,7095 1.001 1 '223 1.409 1.571 1.717 1-850 1.973 2.088 2-195 2"393 2.573 2 -738 2.891 3'033 3-353 3"633 3' 882 4-107 ,4,499 4.833 5-122 5-377
141.5 139"5 137.6 137.2 136.1 135- I 134.7 134.4 134- I 133.5 , 132.2 131.2 130.5 129-8 128.9 127.6 126.1 125.0 123.7 122" 1 120,8 119-7 118.4
65-20 187.6 104.1 124-1 134.6 107.4 83.64 66.58 68-04 73"56 77 "43 69-77 65.41 68"56 56"96 56,41 51.63 53-11 45.32 39.23 34.60 34.75
C (× 10')
0-5274 1,049 1"567 2-079 2'585 3"088 3"585 4.077 4.564 5-045 6'000 6.935 7 -851 8"752 9-636 11"78 13.82 15-78 17.67 21-20 24.64 27.49
~,,o (× 10~)
1.942
0.7262 1-024 1"252 1.442 1"068 1"757 1'893 2-019 2" 136 2'247 2-449 2-633 2' 802 2-958 3"104 3'432 3"718 3 "972 4-204 4'604 4"946 5"243
X/C (× 102) 145"1 141 '3 139"6 138"5 135-6 124-6 132"6 131 '2 129"7 128-1 126.2 124.0 122-1 120"5 118.8 115"8 113-2 110.8 109" 1 106.0 103-9 101 "5
A 7163 4676 3273 2461 2506 2135 2127 2030 2004 2037 1887 1892 1897 1883 1940 1921 1967 2099 2098 2264 2243 2699
/3;
Table 3b. Equivalent conductivity of sodium hexametaphosphate and apparent association constant
1-542
~n,O (x 106) 0.5473 1-089 1.626 2.157 2.683 3-204 3'720 4"231 4'737 5-238 6.226 7' 196 8.148 9.083 10.00 12-22 14.34 16 "38 18.34 22 -00 25-39 28.52 31.43 36.66
C (x 10,) 0.7397 1.044 1.275 1.496 1.638 1-790 1.929 2-057 2-176 2.289 2-495 2.684 2-854 3'014 3"162 3-496 3.787 4-047 4.283 4"690 5-039 5-340 5.060 6.005
~C (× 102)
141-3 139.2 135-6 133.0 129.0 125 "6 123"6 122'0 120" 1 118'4 115-4 112"9 110 "4 108"8 107"0 103"5 100-6 98.35 96.07 93 "05 90'43 88-78 87" 18 84-45
A
15560 7393 6348 5591 8027 8420 8796 8944 10300 12120 10110 42660
~
Table 3c. Equivalent conductivity of sodium octametaphosphate and apparent association constant
> rat)
©
>
Ion-pairformation
140
~
'\
,so-
\\
3905
\
:~.3"-.".
u
"~ ILl I10
X\\
I00 90
ko
z.o
s.o
4.o
s.o
~.o
IO0./E
Fig. l. Equivalentconductivityof sodiumtetrameta-,hexameta-and octametaphosphate. (..... , calculatedfrom Onsager's limitinglaw); O O, tetrarnetaphosphate; ID ~, hexametaphosphate;O-----Q, octametaphosphate. In the next step, accurate values of A° are required for the calculation of association constants. In order to estimate A°, conductivities, A, are extrapolated linearly to infinite dilution against the square roots of concentrations. However, as shown in Fig. 1, for sodium hexameta- and octametaphosphate accurate values of A° cannot be obtained by this extrapolation. Tetraalkylammonium ions are considered to have no tendency to form ion pairs with various anions in dilute aqueous solution, because of their large size and symmetrical shape with low charge. It is therefore expected that the conductivities of the tetraalkylammonium salts of cyclic phosphate anions should be a linear function of the square roots of the concentrations in a low concentration range. Conductivities of these tetramethylammonium salts are given in Table 4. As shown in Fig. 2, the plots of A against 100~/C form a good straight line. In this case, a linear extrapolation Of the conductivities to zero concentration could easily be made and gave limiting equivalent conductivities, A °, for tetramethylammonium tetrameta-, hexameta- and octametaphosphate. After deduction of the mobility of the tetramethylammonium ions, 44-9 [10], the mobility of the cyclic phosphate anions was obtained. A° values for tetramethylammonium tetrameta, hexameta- and octametaphosphate and the respective anions are tabulated in Table 5. By inserting experimentally determined A°, A~ and A~ values into Eqn (2), Onsager's limiting slopes for these salts were calculated. On the other hand, the slopes of A vs. V C curves could be obtained from the analysis of Fig. 2. Comparison of these calculated and observed values is shown
3906
G. K U R A and S. O H A S H I Table 4a. Equivalent conductivity of tetramethylammonium tetrametapbosphate at 25°C
~.2o
C
~/C
(× 106)
(× 104)
(× 102)
0.989
0.3913 1-163 1.919 3.025 3'387 3"746 4.452 5.826 6.495 7.151 7.795 9-046 10"26 I 1"71 13' 11 14"45 15"73 16-97 20"40 24'41 29"50
0.6255 1'078 1"385 1.739 1.840 1.935 2-110 2.414 2.594 2.674 2.792 3.008 3.203 3.422 3'621 3.801 3.966 4-119 4"517 4'941 5-431
A 136.5 134-5 133.6 132-1 131 '4 131 "6 131-1 129.9 129"4 128"1 127"9 127"0 126-5 125'7 124"9 124"1 123 "7 122"6 121.6 120"2 117.8
Table 4b. Equivalent conductivity of tetramethylammonium hexametaphosphate at 25°C ~o (× 10~) 1"123
C
~C
(× 104)
(× I0~)
1"142 2-202 3'271 4"319 5"348 6-356 8'318 9"273 10-21 11"13 12-03 12"48 13.35 14-64 16-72 18-72 20"63 22-46 25 "93 20-12 34'85
1"055 1"484 1-809 2"078 2"313 2"521 2"884 3-045 3"195 3"336 3 "468 3-553 3-654 3"826 4 "089 4'327 4 '542 4"739 5 '091 5"396 5"903
A 148-4 146"7 142"8 140"7 139"0 136'8 134"5 133 -5 132'0 131"6 130 "5 130'0 129"2 127-7 126-5 124-9 124-3 122"7 120-5 118.1 116-4
Ion-pair formation
3907
Table 4c. Equivalent conductivity of tetramethylammonium octametaphosphate at 25°C KH~O
C
X/C
(× 10~
(× 10a)
(× l0 s)
A
1.110
0.5318 1.580 2.096 2-607 3.114 3.615 4.111 4.603 5-091 6.057 6.993 7,918 8.827 9.718 10.59 I 1.45 12-29 13.13 13.94 15.92 17.82 19.64 21.38 24.67 33-17
0.7292 1-257 1.448 1.615 1.765 1-901 2-028 2-145 2.256 2-460 2.644 2.814 2-971 3.117 3.254 3-384 3.506 3.624 3.734 3.990 4.221 4.432 4.424 4-967 5.759
149-5 144,3 140.5 140-3 139.6 137.2 136.4 135.5 133.7 131.5 129.6 127.4 126.1 124.1 123.0 121.4 120.5 119.5 118.4 116. I 114.3 112-7 111.3 109.0 103-6
Table 5. A ° values of tetramethylammonium tetrameta-, hexameta-, octametaphosphate and the respective anions A° IN (Me)414P40,.,, [N(Me)414PnO,x [N(Me)4]~P8024
136.9 155.6 156.6
A° P 40,3 4P,~O~ P,O~f
94-0 110.7 111.7
in Table 6. The coincidence of the both values is satisfactory. From these results, it is concluded that in the case where ion pairing does not occur, tetrameta-, hexameta- and octametaphosphate anions are quadrivalent, sexivalent and octavalent, respectively. By adding the sodium-ion mobility, 50.1 [10], to the mobility of the cyclic phosphate anions, "true" limiting equivalent conductivities of the sodium salts can be obtained. For sodium tetrameta-, hexameta- and octametaphosphate, it is assumed that only one sodium ion would associate with each metaphosphate anion, i.e. sodium tetrameta-, hexameta- and octametaphosphate solution is
3908
G. K U R A and S. O H A S H I
,6ol
140
-~ ,30-
IlO
-~ ~ . ~
--
~0 I p-o
I 2.0
I 3.0
I 4.0
I S-o
IO0 ,/'Y Fig. 2. Equivalent conductivity of tetramethylammonium tetrameta-, hexameta- and octametaphosphate. 0 ©, tetrarnetaphosphate; ~) IO, hexametaphosphate; 0 - - - - 0 , octametaphosphate. Table 6. Comparison of slope calculated from Onsager's limiting law with that determined experimentally Theoretical
P4m Psm Psm
411 699 978
Experimental
390 707 1010
Difference
(%) --5 + 1 +3
regarded as a mixture of 4-1-valent, (4Na). (P401~) and 3-1-valent electrolyte, (3Na). (NaP4012), a mixture of 6-1-valent, (6Na). (P6018) and 5-1-valent electrolyte, (5Na). (NaP6Ols) and a mixture of 8-1-valent, (8Na). (P8024) and 7-1valent electrolyte, (7Na). (NaPsOz4), respectively. We also assume that the limiting ion equivalent conductivities of these a- NaPrO~s, and NaP80~4 are three-fourths, fiveassociated species, NaP40~z, sixths and seven-eighths of those of P404~-, P60~6~ and PsO8~, respectively. Equations for the calculation of degrees of ion dissociation, a, from the experimentally measured conductivities can be written as follows. P4m:
4Aexpt = 4a (144.1 -- 260.6V'I) +3(1 --a) (120.6-- 192.2V'I) / = (c~+ 1.5) c
(4) (5)
Ion-pair formation
3909
|--S
fl~
m s (3 + s)
(6)
PG,,,: 6A,,x,,t = 6~ ( 160.8 - 369.3X/I) + 5 ( 1 -- s) ( 142.4 -- 303.7X/I)
(7)
! -- (s + 2.5) C
(8)
1--s
fl~ = m s (5 + s)
(9)
Psm: 8Aex0t ~ 8s(161"8-456"9X/I) + 7 ( 1 - s ) (147"7- 395"0X/I)
(10)
1 =- ( s + 3 " 5 ) C
(1 1)
1--s
fll ~ ms (7 + ~)
(12)
where/31 denotes an apparent association constant. Eqns. (4), (7) and (10) are cubic equations for s and can be solved using a computer. The calculated values o f f l ' 1 are tabulated in the fifth columns of Tables 3a, 3b and 3c. In order to evaluate a true thermodynamic association constant, a correction must be made for the activity coefficient term. If activity coefficients can be calculated using Debye-Hiickel's equation, a straight line should be obtained when log/3'1 is plotted against ~ / / a n d a thermodynamic association constant can be obtained by extrapolating this to zero ionic strength. For sodium tetrametaphosphate, this plot gave a straight line. This treatment was also applied for sodium hexameta- and octametaphosphate, but irregularity was found in these cases. However, below C = 2.2 × 10-4, a straight line was obtained and gave an approximate value of flj. The results for log fll were summarized in Table 7. It is concluded that ion pairs containing only one sodium ion, i.e. NaP40~-, N a P r O ~ and NaPsO274, are present in (0.3-7)?< 10-4M sodium tetrameta-, (0-8--4) × 10-SM sodium hexameta- and ( 0 . 6 - 3 ) × 10-SM sodium octametaphosphate solutions, respectively. The irregularity above C = 2-2 × l0 -4 might be caused by the existence of highly associated species, e.g. Na2PrO4~, Na3PnO]~, Na2PsO~4, Na3PsO~4 , Na4P804~, etc. and also by the uncertainty of the values of the anions mobilities. When more than one associated species are present in solution, it is difficult to estimate their association constants by
Table ]~. Association constants of sodium tetrameta-, hexameta- and octametaphosphates Conductivity measurements log/31 Davies and Monk This work P4,, 2.04 2.15 Pr,, 4.3 P~,,, 4.6
G a r d n e r and Nancollas 2.12
* Corrections for activity coefficients were not tried.
Activity measurements log~fl~ log/~,, log/3:~ This work* 2-35 3.7 4.0
log fl~
This work* 6.0 7.0
9.7 11
15
3910
G. K U R A and S. O H A S H I
the conductivity method. The conductivity method is also inadequate for measurement in the high-concentration region. We applied activity measurements to the calculation of association constants. Since concentrations of free sodium ions can be directly determined by this method, it is possible to estimate association constants of highly associated species without any unreliable assumptions. However, at extremely high dilution the precision of the activity measurement is worse than that of the conductivity measurement. It is convenient to express the extent of ion-pair formation in terms of an average number of sodium ions, h, bound to each cyclic phosphate anion. If we denote molar concentrations of total sodium, free sodium and total cyclic phosphate as [Na]t, [Na] and [P]~, respectively, h is represented by h = [ N a ] t - [Na] [P]t
(13)
An apparent over-all association constant for the reaction, nNa + P ~ N a , P is [NanP] /3n = [Na]n[P]"
(14)
fl~[Na] + 2fl2[Na] 2+ . . . . fi --- 1 + fll[Na] +fl2[Na]2+ . . . . "
(15)
Combining Eqns. (13) and (14),
This equation becomes t~+ (~-1)fl~[Na] + (n-2)fl~[Na]2+ (n-3)B3[Na]3 + . . . . . Table 8a. Sodium t e t r a m e t a p h o s p h a t e ion-pair formation at 25°C [P]t (x 104)
[Na]t (X 104)
[Na] (X 104)
[Na]bouna (X 104)
0"82 1"60 2'32 3"00 3-65 4"25 4'83 5.38 6"38 7.28 8.13 8.88 9.58 10.2 11.6 12-8
3-28 6"38 9-27 12.0 14.6 17"0 19.3 21.5 25.5 29.1 32-5 35.5 38.3 40.8 46.4 51.0
3"55 6'31 8'92 11-5 13-8 15.9 18'2 19.5 22-9 26"3 28.8 32.4 33.9 36.3 40-8 42-7
0'07 0.35 0-50 0.77 1' 10 1.10 2.00 2.60 2"80 3-70 3' 10 4.40 4-50 5.80 8-30
0.044 0.15 0.17 0.21 0-259 0.228 0-372 0-408 0.385 0.455 0.349 0.459 0.441 0.500 0-651
0.
(16)
Ion-pair formation
391 I
Table 8b. Sodium hexametaphosphate ion-pair formation at 25°C [Plt (X 104)
[Na]t (X 10')
[Na] (X 104)
[Na]bound
0.547 1.06 1.55 2.00 2"43 2.83 3"22 4.25 4-85 5"40 5-92 6.38 6.80 7.73 8-50
3.28 6.38 9.27 12.0 14.6 17.0 19.3 25 "5 29-1 32.4 35.5 38-3 40'8 46.4 51-0
2-89 5.37 7-59 9.34 11-0 12"9 14,5 18-2 20.0 22.4 24-0 25.7 26-9 30.9 33.1
0-39 1.01 1.68 2.66 3"60 4.10 4.80 7.30 9.10 10-0 11-5 12.6 13.9 15.5 17,9
(X 104)
n 0.713 0.950 1-09 1-33 1.48 1.45 1.49 1.72 1.88 1-85 1.94 1.97 2.04 2"00 2" I 1
Table 8c. Sodium octametaphosphate ion-pair formation at 25°C [Pit (X 104)
[Na]t (X 104)
[Na] (X 10')
[Na]bound
0.426 0-829 1.21 1-56 1-89 2-21 2.50 2-79 3.31 3.79 4.21 4-61 4.98 5.30 6.03 6.63
3.41 6.63 9.64 12.5 15.1 17-7 20.0 22-3 26.5 30.3 33.7 36.9 39.8 42-4 48-2 53.0
3.34 3-93 6-03 7.95 9.12 10.7 11.5 12,9 15.5 17-4 20-9 21.9 22-4 24.5 26.3 28.2
1.I)7 2.65 3.61 4-55 5.98 7.00 8.50 9.40 l 1-0 12.9 12.8 15.0 17.4 17-9 21-9 24.8
(× 1114)
/~ 2-15 3.20 3.00 2.91 3.17 3-16 3'40 3.37 3-32 3-41 3.04 3.25 3.50 3.38 3.73 3.74
As a set of h and [Na] can be determined experimentally the above equation can be solved by a least-square procedure. The values of [Na] and ~ are shown in Tables 8a, 8b, and 8c. However, since errors in [Na]-measurements were large, only approximate values for log/3, were obtained, which are given in Table 7. In these calculations, a correction for the activity coefficient was not tried. For sodium tetrametaphosphate, we can reasonably conclude that only NaP40:~ exists as an associated species in 6 x 10-4-5 x 10-:~ M solution of sodium tetrametaphosphate. For sodium hexameta- and octametaphosphate, it can be said that N a P 6 0 ~ , Na2P60~- and Na.~P~O~7 and NaP~O~V~-, Na2P~O~-, Na:~P~O~-
3912
G. K U R A and S. O H A S H I
and Na4PsO~- are present as associated species in 3 x 10-4-5 x 10-3 M solution of sodium hexameta- and octametaphosphate, respectively. For sodium tetrametaphosphate the conductivity method and activity method gave nearly equal values of log fll. However, for sodium hexameta- and octametaphosphate, the 131 values obtained by the two methods were somewhat different. This discrepancy may result from the uncertainty of the values of the associated-ion mobility in the calculation employed in the conductivity method and from that in potential measurements with the sodium ion-selective glass electrode, and from omitting the correction for activity coefficients in the latter method.
ION-PAIR
Pergamon Press.
Printed in Great Brit~dn
FORMATION OF CYCLIC ANIONS
PHOSPHATE
G E N 1 C H I R O K U R A and S H I G E R U O H A S H I Department of Chemistry, Faculty of Science, Kyushu University, Hakozaki, Fukuoka, Japan (First received 1 July 1971;in re t.:isedJbrm 9 November 1971)
Abstract--An ion-exchange method, conductivity method and activity measurements have been applied to the investigation of ion-pair formation by cyclic phosphate anions in aqueuos solution. By conductivity measurements on their tetramethylammonium salts, the charges on the cyclic phosphate anions and ion equivalent conductivities at infinite dilution have been determined. From conductivities of sodium tetrameta-, hexameta- and octametaphosphate, association constants of ion pairs containing one sodium ion have been calculated by the method of Davies and Monk. From activity measurements on the sodium ions, an average number of sodium ions bound to each cyclic phosphate anion and association constants of the ion pairs containing one or more sodium ions have been calculated. It was concluded that the following ion pairs are present in dilute aqueous solutions of sodium metaphosphates; N a P 4 0 ~ in sodium tetrametaphosphate solution, NaP~O~, Na,.,P~O~ and Na:~P~;O:~, in sodium hexametaphosphate solution and NaP~O~, NaePsO~, Na:~PxO~] and Na4PsO~r in sodium octametaphosphate solution. INTRODUCTION
CONDENSED phosphates are well known as multivalent inorganic anions. Their
ability to form complexes with various metals has been reported by several investigators[l]. However, relatively few studies have been made on ion association of phosphate anions with alkali metal cations [2-4]. Among the condensed phosphates, there is a group called metaphosphates, which may have cyclic structures. Trimeta- and tetrametaphosphate have been well known for many years and can be easily prepared by conventional methods [5]. Hexameta- and octametaphosphate were recently prepared by Griffith and Buxton[6], and Schiilke [7], respectively. Our previous study [8] of ion-exchange equilibria of cyclic phosphate anions with an anion-exchange resin or dextran gel indicated that the charges of highermembered cyclic phosphate anions than the tetrametaphosphate anion, in the resin phase, are coincident with those in highly dilute solution where no ion-pair formation occurs. However, it was found that in the dextran gel phase these cyclic phosphate anions have lower charges than those in the resin phase. This 1. 2, 3. 4. 5.
J. R. Van Wazer and C. F. Callis, Chem. Rev. 58, 1011 (1958). C. W. Davies and C. B. Monk, J. chem. Soc. 413 (1949). U. Schindewolf and K. F. Bonhoeffer, Z. Elektrochem. 57, 216 (1952). G. L. Gardner and G. H. Nancollas, A nalyt. ('hem. 41,202 (1969). G. Brauer, Handbuch der Priiparativen Anorganischen Chemie, pp. 494-496. Ferdinand Enke Verlag, Stuttgart (1960). 6. E. J, Griflith and R. L. Buxton, lnorg. Chem. 4, 549 (1965). 7. U. Schiilke, Z. anorg, allg. Chem. 360, 231 (1968). 8. O. Kura and S. Ohashi, J. Chromatog. 56, 111 (1971). 3899
3900
G. K U R A and S. O H A S H I
suggests that in the anion-exchange dextran gel phase an interaction of cyclic phosphate anions with the cations takes place. Since the dextran gel is more highly swollen than the resin, an appreciable amount of cations can invade the gel phase. Therefore, multivalent anions such as cyclic phosphate anions adsorbed in the gel phase tend to attract the invading cations to form ion pairs. The purpose of this work is to study interaction of the higher-membered cyclic phosphate anions with alkali metal ions, especially sodium ions. For such a purpose many methods can be used, e.g. an ion-exchange method, conductivity measurements, activity measurements, optical methods and solubility measurements. Dissociation pherrorn~a of sodium trimeta- and tetrametaphosphate and corresponding acids have beer~ ~tudied by Davies and Monk by a conductivity method[2]. They have determined limiting equivalent conductivities of the above salts and acids and obtained/~eir dissociation constants from differences in observed and calculated condu~tivities. They have concluded that ion-pair formation occurs slightly in phosl~ate solution and the logarithm of the association constant of NaP40~-, log/31,'is about 2.04. However if more than one associated species are present as unsymmetrical electrolytes, it is difficult to apply a conductivity method for this purpose, because of the large uncertainty in the limiting equivalent conductivities of the associated species. On the other hand, Gardner and Nancollas [4] recently investigated ion association in solutions of sodium trimeta- and tetrametaphosphate using a sodium ion-selective electrode, and gave a value of 2.12 for 1og/31 for NaP40~. By means of activity measurements free sodium ions can be directly determined and association constants of highly associated species can be calculated. In the present study we employed an ion-exchange method, conductivity measurements and activity measurements in order to obtain information on ion-pair formation by tetrameta-, hexameta- and octametaphosphate anions with sodium ions. EXPERIMEN~TAL Cyclic phosphates The preparation of sodium tetrameta-, hexameta- and octametaphosphate was described in our previous paper [8]. Tetramethylammonium salts of cyclic phosphoric acids were prepared as follows. Tetramethylammonium hydroxide was prepared by passing tetramethylammonium chloride solution through a column of a Dowex IX4 resin in the hydroxide form. Tetrameta-, hexameta- and octametaphosphoric acid were prepared by converting their sodium salts to the respective free acids by treating with a Dowex 50W X8 resin in the hydrogen form. We obtained tetramethylammonium tetrameta-, hexameta- and octametaphosphates by neutralization of the respective free acids with tetramethylammonium hydroxide.
Determination of phosphates Determination of each cyclic phosphate was carried out by converting it to the free acid and titrating the acid with sodium or tetramethylammonium hydroxide, or by the colorimetric method of Lucena-Conde and Prat [9]. Measurement of distribution ratios in ion-exchange equilibria A study of the pH dependence of the distribution ratios was carried out by the method described 9. F. Lucena-Conde and L. Prat, Analytica chirn. Acta 16, 473 (1957).
Ion-pair formation
390 i
in our previous paper[8] using a QAE-Sephadex A-25 gel and 0.20-0.50 M potassium chloride as eluents. In the study of the effect of eluents, distribution ratios of the cyclic phosphates for a QAESephadex A-25 gel were measured by the usual batch method at room temperature. About 0.5 g of the air-dried gel in the chloride form was put into a stoppered Erlenmeyer flask and then 25 ml of an eluent and 2 ml of a solution containing a known amount of a cyclic phosphate (ca. 250 p,g as P) was added. After equilibrium was reached, the gel phase was separated by decantation. The phosphate in the solution phase was determined colorimetrically [9]. Eluent concentrations were varied from 0.10 to 0.30 M.
Conductivity measurements The equipment used for the conductivity measurements was a Yanagimoto Conductivity Outfit, Model MY-7, A cell was designed so as to be adaptable for titration and its electrodes were coated with platinum black. The cell was standardized with 0.01 M potassium chloride and the cell-constant was 0.3447 cm -~ at 25°C. The temperature of the cell was precisely controlled at 25 +_0.1°C. Conductivities of solutions of the following salts were measured; [N(Me)414P40,~, [N(Me)al~P~;O~, IN(Me/.~]sP~Oe~. Na~P,P~2, Na~P,O~s and NasP, Oe~. The pH of the cell solutions was always between 6.5 and 7-0. Activity measurements o f free sodium ions Activity measurements of sodium ions were made at 25 +_0-1 °C in a cell. Glass electrode Isolution under studyl calomel electrode. A Hitachi-Horiba sodium-ion glasselectrode, #2535-05 T and a reference calomel electrode, #4142-05 T were connected with a HitachiHoriba pH meter, model F-Sss. E.m.f. values measured with the above equipment were converted into sodium ion concentration by using a calibration curve obtained for a series of standard solutions of sodium chloride. About 15 rain were needed for each activity measurement until a constant reading was obtained, because of the slow response of the electrode. The precision of the e.m.f, values was +_0.3 mV, if the electrodes were carefully preconditioned. Activities of sodium ions in solutions of the following salts were measured; Na4P4Piz, Na~PrO~8 and NasP~O.,4. The pH of the cell solutions was always between 6.5 and 7.0 during the experiments, considerably larger than pNa, indicating a negligible degree of hydrolysis and hydrogen ion interference. RESULTS AND DISCUSSION
As has been described in the previous paper [8], charges on phosphate anions can be determined by measuring their distribution ratios in ion-exchange equilibria. If cyclic phosphate anions form ion pairs with protons, their degrees of association must depend on the pH of an eluent. The charges of cyclic phosphate anions were calculated from the slope of the plots of log D,, (distribution ratio) against log ICI ] (eluent concentration) at pH 5.2 and 9.8 and are shown in Table 1. The charges on the cyclic phosphate anions are nearly the same at each pH values. Therefor, it can be said that the association of cyclic phosphate anions Table 1. Charges of cyclic phosphate anions in the dextran gel phase at pH 5.2 or 9-8. Eluent: KCI
pH5-2 pH9.8
PITh
Prm
Psm
3.4(4-0)* 3-3(3-7)
4.3(5.0) 4.3(5.0)
5.7(6.5) 5.7(6.2)
*Values in the parentheses are corrected for the invasion in the gel phase.
3902
G. K U R A and S. O H A S H I
with protons does not occur even at pH 5.2. Thus it can be concluded that ionpair formation by cyclic phosphate anions with cations in the eluent causes the decrease in their charges in the gel phase. It is generally accepted that in outer-sphere complexes the complexibility of alkali metals increases in the order of Li < Na < K < Rb -- Cs. In order to examine this, we used alkali metal chlorides, i.e. LiC1, NaCI, KCI and CsC1 as eluents. From the results shown in Table 2, it appears that the charges of the cyclic phosphate anions decrease slightly as the sizes of the hydrated cations of the eluents decrease. However, since the differences in the charges are very small, it seems that there are no marked differences in the ability of alkali metal ions to form ion pairs. We applied a conductivity method to the quantitative study of ion-pair formation. Onsager's limiting law for the equivalent conductivity of a completely dissociated salt can be expressed by Eqns (1), (2) and (3). (1)
A = A°--AX/I
in which I is the ionic strength, and Eqn (1) can be written as equation (1)'. A = A°-A'X/C
(1)'
where C is the equivalent concentration of a phosphate. In Eqn. (1), for a single electrolyte dissociating into ions 1 and 2, A becomes A = 2.801 x 1061Zl. Z21qA° ~ 41"25(IZll + IZ21) (eT)S/2(l + V/q) n ( e T ) lie
(2)
where • and ~ are the dielectric constant and viscosity of the solution, respectively [10]. Here the quantity q is defined by
Iz,. z21 q = IZ, I + [Z2I }Z2JA~+ IZ, IA2
(3)
where ZI(Zz) and A~(A~) are the charge and ion equivalent conductivity of the ion 1(2), respectively. For aqueous solutions at 25°C Eqn (2) reduces to: A = 0.78521Z~. Z2J = -qA° - - - 7 - + 30.32 (IZal + JZ2f)
l+vq
Table 2. Charges of cyclic phosphate anions in the dextran gel phase when various alkali chlorides are used as eluents, pH: 5.2
Prm Psm
LiCI
NaCI
KC1
CsCI
4"7 6"0
4"3 5"7
4"4 5"7
4"4 5"5
10. R. A. Robinson and R. H. Stokes, Electrolyte Solutions. Butterworths, London (1959).
(2)'
Ion-pair formation
3903
It is known that these equations agree with the experimental data for numerous uni-univalent, uni-bivalent and uni-tervalent salts in aqueous dilute solutions at various temperatures. However, when the valency product is four, Onsager's limiting law does not always hold. This may be due to the increase of electronic interaction between cations and highly negatively charged anions. In other words, Bjerrum-type ion association certainly occurs and exerts some effect upon the mobility of the ions. Conductivities of sodium tetrameta-, hexameta- and octametaphosphate, which are 1-4-, 1-6- and 1-8-valent electrolyte, respectively, were measured in dilute solutions and are given in Table 3. When the conductivities of these salts are plotted against the square roots of the equivalent concentrations, the resulting curve has a much steeper slope than that expected from Onsager's limiting law as shown in Fig. 1. This lower conductivity is certainly due to ion pairing. From the difference between the conductivity observed and that calculated from Onsager's limiting equations where full dissociation of salts is assumed, the degree of ion association can be approximately evaluated by the method of Davies and Monk [2]. For sodium trimetaphosphate, they regarded this salt as a mixture of l-3-valent electrolyte, (3Na). (P309) and 1-2-valent electrolyte, (2Na). (NaP30,) and assumed that ion equivalent conductivity of NaP,~O,;'- at infinite dilution is two-thirds of that of P30.,,:~-. They also assumed that Onsager's limiting equations hold individually for both 1-2- and 1-3-valent electrolytes in the solution. Table 3a. Equivalent conductivity of sodium tetrametaphosphate and apparent association constant Kn~o (× 106)
C (× 104)
"~/C ()( 102)
A
~3'1
2-337
0.5035 1'002 1.496 1.985 2.468 2.948 3.423 3.897 4 "347 4.819 5"728 6'626 7.496 8'356 9.200 11.24 13-20 15.07 16-87 20.24 23.36 26-24 28.91
0,7095 1.001 1 '223 1.409 1.571 1.717 1-850 1.973 2.088 2-195 2"393 2.573 2 -738 2.891 3'033 3-353 3"633 3' 882 4-107 ,4,499 4.833 5-122 5-377
141.5 139"5 137.6 137.2 136.1 135- I 134.7 134.4 134- I 133.5 , 132.2 131.2 130.5 129-8 128.9 127.6 126.1 125.0 123.7 122" 1 120,8 119-7 118.4
65-20 187.6 104.1 124-1 134.6 107.4 83.64 66.58 68-04 73"56 77 "43 69-77 65.41 68"56 56"96 56,41 51.63 53-11 45.32 39.23 34.60 34.75
C (× 10')
0-5274 1,049 1"567 2-079 2'585 3"088 3"585 4.077 4.564 5-045 6'000 6.935 7 -851 8"752 9-636 11"78 13.82 15-78 17.67 21-20 24.64 27.49
~,,o (× 10~)
1.942
0.7262 1-024 1"252 1.442 1"068 1"757 1'893 2-019 2" 136 2'247 2-449 2-633 2' 802 2-958 3"104 3'432 3"718 3 "972 4-204 4'604 4"946 5"243
X/C (× 102) 145"1 141 '3 139"6 138"5 135-6 124-6 132"6 131 '2 129"7 128-1 126.2 124.0 122-1 120"5 118.8 115"8 113-2 110.8 109" 1 106.0 103-9 101 "5
A 7163 4676 3273 2461 2506 2135 2127 2030 2004 2037 1887 1892 1897 1883 1940 1921 1967 2099 2098 2264 2243 2699
/3;
Table 3b. Equivalent conductivity of sodium hexametaphosphate and apparent association constant
1-542
~n,O (x 106) 0.5473 1-089 1.626 2.157 2.683 3-204 3'720 4"231 4'737 5-238 6.226 7' 196 8.148 9.083 10.00 12-22 14.34 16 "38 18.34 22 -00 25-39 28.52 31.43 36.66
C (x 10,) 0.7397 1.044 1.275 1.496 1.638 1-790 1.929 2-057 2-176 2.289 2-495 2.684 2-854 3'014 3"162 3-496 3.787 4-047 4.283 4"690 5-039 5-340 5.060 6.005
~C (× 102)
141-3 139.2 135-6 133.0 129.0 125 "6 123"6 122'0 120" 1 118'4 115-4 112"9 110 "4 108"8 107"0 103"5 100-6 98.35 96.07 93 "05 90'43 88-78 87" 18 84-45
A
15560 7393 6348 5591 8027 8420 8796 8944 10300 12120 10110 42660
~
Table 3c. Equivalent conductivity of sodium octametaphosphate and apparent association constant
> rat)
©
>
Ion-pairformation
140
~
'\
,so-
\\
3905
\
:~.3"-.".
u
"~ ILl I10
X\\
I00 90
ko
z.o
s.o
4.o
s.o
~.o
IO0./E
Fig. l. Equivalentconductivityof sodiumtetrameta-,hexameta-and octametaphosphate. (..... , calculatedfrom Onsager's limitinglaw); O O, tetrarnetaphosphate; ID ~, hexametaphosphate;O-----Q, octametaphosphate. In the next step, accurate values of A° are required for the calculation of association constants. In order to estimate A°, conductivities, A, are extrapolated linearly to infinite dilution against the square roots of concentrations. However, as shown in Fig. 1, for sodium hexameta- and octametaphosphate accurate values of A° cannot be obtained by this extrapolation. Tetraalkylammonium ions are considered to have no tendency to form ion pairs with various anions in dilute aqueous solution, because of their large size and symmetrical shape with low charge. It is therefore expected that the conductivities of the tetraalkylammonium salts of cyclic phosphate anions should be a linear function of the square roots of the concentrations in a low concentration range. Conductivities of these tetramethylammonium salts are given in Table 4. As shown in Fig. 2, the plots of A against 100~/C form a good straight line. In this case, a linear extrapolation Of the conductivities to zero concentration could easily be made and gave limiting equivalent conductivities, A °, for tetramethylammonium tetrameta-, hexameta- and octametaphosphate. After deduction of the mobility of the tetramethylammonium ions, 44-9 [10], the mobility of the cyclic phosphate anions was obtained. A° values for tetramethylammonium tetrameta, hexameta- and octametaphosphate and the respective anions are tabulated in Table 5. By inserting experimentally determined A°, A~ and A~ values into Eqn (2), Onsager's limiting slopes for these salts were calculated. On the other hand, the slopes of A vs. V C curves could be obtained from the analysis of Fig. 2. Comparison of these calculated and observed values is shown
3906
G. K U R A and S. O H A S H I Table 4a. Equivalent conductivity of tetramethylammonium tetrametapbosphate at 25°C
~.2o
C
~/C
(× 106)
(× 104)
(× 102)
0.989
0.3913 1-163 1.919 3.025 3'387 3"746 4.452 5.826 6.495 7.151 7.795 9-046 10"26 I 1"71 13' 11 14"45 15"73 16-97 20"40 24'41 29"50
0.6255 1'078 1"385 1.739 1.840 1.935 2-110 2.414 2.594 2.674 2.792 3.008 3.203 3.422 3'621 3.801 3.966 4-119 4"517 4'941 5-431
A 136.5 134-5 133.6 132-1 131 '4 131 "6 131-1 129.9 129"4 128"1 127"9 127"0 126-5 125'7 124"9 124"1 123 "7 122"6 121.6 120"2 117.8
Table 4b. Equivalent conductivity of tetramethylammonium hexametaphosphate at 25°C ~o (× 10~) 1"123
C
~C
(× 104)
(× I0~)
1"142 2-202 3'271 4"319 5"348 6-356 8'318 9"273 10-21 11"13 12-03 12"48 13.35 14-64 16-72 18-72 20"63 22-46 25 "93 20-12 34'85
1"055 1"484 1-809 2"078 2"313 2"521 2"884 3-045 3"195 3"336 3 "468 3-553 3-654 3"826 4 "089 4'327 4 '542 4"739 5 '091 5"396 5"903
A 148-4 146"7 142"8 140"7 139"0 136'8 134"5 133 -5 132'0 131"6 130 "5 130'0 129"2 127-7 126-5 124-9 124-3 122"7 120-5 118.1 116-4
Ion-pair formation
3907
Table 4c. Equivalent conductivity of tetramethylammonium octametaphosphate at 25°C KH~O
C
X/C
(× 10~
(× 10a)
(× l0 s)
A
1.110
0.5318 1.580 2.096 2-607 3.114 3.615 4.111 4.603 5-091 6.057 6.993 7,918 8.827 9.718 10.59 I 1.45 12-29 13.13 13.94 15.92 17.82 19.64 21.38 24.67 33-17
0.7292 1-257 1.448 1.615 1.765 1-901 2-028 2-145 2.256 2-460 2.644 2.814 2-971 3.117 3.254 3-384 3.506 3.624 3.734 3.990 4.221 4.432 4.424 4-967 5.759
149-5 144,3 140.5 140-3 139.6 137.2 136.4 135.5 133.7 131.5 129.6 127.4 126.1 124.1 123.0 121.4 120.5 119.5 118.4 116. I 114.3 112-7 111.3 109.0 103-6
Table 5. A ° values of tetramethylammonium tetrameta-, hexameta-, octametaphosphate and the respective anions A° IN (Me)414P40,.,, [N(Me)414PnO,x [N(Me)4]~P8024
136.9 155.6 156.6
A° P 40,3 4P,~O~ P,O~f
94-0 110.7 111.7
in Table 6. The coincidence of the both values is satisfactory. From these results, it is concluded that in the case where ion pairing does not occur, tetrameta-, hexameta- and octametaphosphate anions are quadrivalent, sexivalent and octavalent, respectively. By adding the sodium-ion mobility, 50.1 [10], to the mobility of the cyclic phosphate anions, "true" limiting equivalent conductivities of the sodium salts can be obtained. For sodium tetrameta-, hexameta- and octametaphosphate, it is assumed that only one sodium ion would associate with each metaphosphate anion, i.e. sodium tetrameta-, hexameta- and octametaphosphate solution is
3908
G. K U R A and S. O H A S H I
,6ol
140
-~ ,30-
IlO
-~ ~ . ~
--
~0 I p-o
I 2.0
I 3.0
I 4.0
I S-o
IO0 ,/'Y Fig. 2. Equivalent conductivity of tetramethylammonium tetrameta-, hexameta- and octametaphosphate. 0 ©, tetrarnetaphosphate; ~) IO, hexametaphosphate; 0 - - - - 0 , octametaphosphate. Table 6. Comparison of slope calculated from Onsager's limiting law with that determined experimentally Theoretical
P4m Psm Psm
411 699 978
Experimental
390 707 1010
Difference
(%) --5 + 1 +3
regarded as a mixture of 4-1-valent, (4Na). (P401~) and 3-1-valent electrolyte, (3Na). (NaP4012), a mixture of 6-1-valent, (6Na). (P6018) and 5-1-valent electrolyte, (5Na). (NaP6Ols) and a mixture of 8-1-valent, (8Na). (P8024) and 7-1valent electrolyte, (7Na). (NaPsOz4), respectively. We also assume that the limiting ion equivalent conductivities of these a- NaPrO~s, and NaP80~4 are three-fourths, fiveassociated species, NaP40~z, sixths and seven-eighths of those of P404~-, P60~6~ and PsO8~, respectively. Equations for the calculation of degrees of ion dissociation, a, from the experimentally measured conductivities can be written as follows. P4m:
4Aexpt = 4a (144.1 -- 260.6V'I) +3(1 --a) (120.6-- 192.2V'I) / = (c~+ 1.5) c
(4) (5)
Ion-pair formation
3909
|--S
fl~
m s (3 + s)
(6)
PG,,,: 6A,,x,,t = 6~ ( 160.8 - 369.3X/I) + 5 ( 1 -- s) ( 142.4 -- 303.7X/I)
(7)
! -- (s + 2.5) C
(8)
1--s
fl~ = m s (5 + s)
(9)
Psm: 8Aex0t ~ 8s(161"8-456"9X/I) + 7 ( 1 - s ) (147"7- 395"0X/I)
(10)
1 =- ( s + 3 " 5 ) C
(1 1)
1--s
fll ~ ms (7 + ~)
(12)
where/31 denotes an apparent association constant. Eqns. (4), (7) and (10) are cubic equations for s and can be solved using a computer. The calculated values o f f l ' 1 are tabulated in the fifth columns of Tables 3a, 3b and 3c. In order to evaluate a true thermodynamic association constant, a correction must be made for the activity coefficient term. If activity coefficients can be calculated using Debye-Hiickel's equation, a straight line should be obtained when log/3'1 is plotted against ~ / / a n d a thermodynamic association constant can be obtained by extrapolating this to zero ionic strength. For sodium tetrametaphosphate, this plot gave a straight line. This treatment was also applied for sodium hexameta- and octametaphosphate, but irregularity was found in these cases. However, below C = 2.2 × 10-4, a straight line was obtained and gave an approximate value of flj. The results for log fll were summarized in Table 7. It is concluded that ion pairs containing only one sodium ion, i.e. NaP40~-, N a P r O ~ and NaPsO274, are present in (0.3-7)?< 10-4M sodium tetrameta-, (0-8--4) × 10-SM sodium hexameta- and ( 0 . 6 - 3 ) × 10-SM sodium octametaphosphate solutions, respectively. The irregularity above C = 2-2 × l0 -4 might be caused by the existence of highly associated species, e.g. Na2PrO4~, Na3PnO]~, Na2PsO~4, Na3PsO~4 , Na4P804~, etc. and also by the uncertainty of the values of the anions mobilities. When more than one associated species are present in solution, it is difficult to estimate their association constants by
Table ]~. Association constants of sodium tetrameta-, hexameta- and octametaphosphates Conductivity measurements log/31 Davies and Monk This work P4,, 2.04 2.15 Pr,, 4.3 P~,,, 4.6
G a r d n e r and Nancollas 2.12
* Corrections for activity coefficients were not tried.
Activity measurements log~fl~ log/~,, log/3:~ This work* 2-35 3.7 4.0
log fl~
This work* 6.0 7.0
9.7 11
15
3910
G. K U R A and S. O H A S H I
the conductivity method. The conductivity method is also inadequate for measurement in the high-concentration region. We applied activity measurements to the calculation of association constants. Since concentrations of free sodium ions can be directly determined by this method, it is possible to estimate association constants of highly associated species without any unreliable assumptions. However, at extremely high dilution the precision of the activity measurement is worse than that of the conductivity measurement. It is convenient to express the extent of ion-pair formation in terms of an average number of sodium ions, h, bound to each cyclic phosphate anion. If we denote molar concentrations of total sodium, free sodium and total cyclic phosphate as [Na]t, [Na] and [P]~, respectively, h is represented by h = [ N a ] t - [Na] [P]t
(13)
An apparent over-all association constant for the reaction, nNa + P ~ N a , P is [NanP] /3n = [Na]n[P]"
(14)
fl~[Na] + 2fl2[Na] 2+ . . . . fi --- 1 + fll[Na] +fl2[Na]2+ . . . . "
(15)
Combining Eqns. (13) and (14),
This equation becomes t~+ (~-1)fl~[Na] + (n-2)fl~[Na]2+ (n-3)B3[Na]3 + . . . . . Table 8a. Sodium t e t r a m e t a p h o s p h a t e ion-pair formation at 25°C [P]t (x 104)
[Na]t (X 104)
[Na] (X 104)
[Na]bouna (X 104)
0"82 1"60 2'32 3"00 3-65 4"25 4'83 5.38 6"38 7.28 8.13 8.88 9.58 10.2 11.6 12-8
3-28 6"38 9-27 12.0 14.6 17"0 19.3 21.5 25.5 29.1 32-5 35.5 38.3 40.8 46.4 51.0
3"55 6'31 8'92 11-5 13-8 15.9 18'2 19.5 22-9 26"3 28.8 32.4 33.9 36.3 40-8 42-7
0'07 0.35 0-50 0.77 1' 10 1.10 2.00 2.60 2"80 3-70 3' 10 4.40 4-50 5.80 8-30
0.044 0.15 0.17 0.21 0-259 0.228 0-372 0-408 0.385 0.455 0.349 0.459 0.441 0.500 0-651
0.
(16)
Ion-pair formation
391 I
Table 8b. Sodium hexametaphosphate ion-pair formation at 25°C [Plt (X 104)
[Na]t (X 10')
[Na] (X 104)
[Na]bound
0.547 1.06 1.55 2.00 2"43 2.83 3"22 4.25 4-85 5"40 5-92 6.38 6.80 7.73 8-50
3.28 6.38 9.27 12.0 14.6 17.0 19.3 25 "5 29-1 32.4 35.5 38-3 40'8 46.4 51-0
2-89 5.37 7-59 9.34 11-0 12"9 14,5 18-2 20.0 22.4 24-0 25.7 26-9 30.9 33.1
0-39 1.01 1.68 2.66 3"60 4.10 4.80 7.30 9.10 10-0 11-5 12.6 13.9 15.5 17,9
(X 104)
n 0.713 0.950 1-09 1-33 1.48 1.45 1.49 1.72 1.88 1-85 1.94 1.97 2.04 2"00 2" I 1
Table 8c. Sodium octametaphosphate ion-pair formation at 25°C [Pit (X 104)
[Na]t (X 104)
[Na] (X 10')
[Na]bound
0.426 0-829 1.21 1-56 1-89 2-21 2.50 2-79 3.31 3.79 4.21 4-61 4.98 5.30 6.03 6.63
3.41 6.63 9.64 12.5 15.1 17-7 20.0 22-3 26.5 30.3 33.7 36.9 39.8 42-4 48-2 53.0
3.34 3-93 6-03 7.95 9.12 10.7 11.5 12,9 15.5 17-4 20-9 21.9 22-4 24.5 26.3 28.2
1.I)7 2.65 3.61 4-55 5.98 7.00 8.50 9.40 l 1-0 12.9 12.8 15.0 17.4 17-9 21-9 24.8
(× 1114)
/~ 2-15 3.20 3.00 2.91 3.17 3-16 3'40 3.37 3-32 3-41 3.04 3.25 3.50 3.38 3.73 3.74
As a set of h and [Na] can be determined experimentally the above equation can be solved by a least-square procedure. The values of [Na] and ~ are shown in Tables 8a, 8b, and 8c. However, since errors in [Na]-measurements were large, only approximate values for log/3, were obtained, which are given in Table 7. In these calculations, a correction for the activity coefficient was not tried. For sodium tetrametaphosphate, we can reasonably conclude that only NaP40:~ exists as an associated species in 6 x 10-4-5 x 10-:~ M solution of sodium tetrametaphosphate. For sodium hexameta- and octametaphosphate, it can be said that N a P 6 0 ~ , Na2P60~- and Na.~P~O~7 and NaP~O~V~-, Na2P~O~-, Na:~P~O~-
3912
G. K U R A and S. O H A S H I
and Na4PsO~- are present as associated species in 3 x 10-4-5 x 10-3 M solution of sodium hexameta- and octametaphosphate, respectively. For sodium tetrametaphosphate the conductivity method and activity method gave nearly equal values of log fll. However, for sodium hexameta- and octametaphosphate, the 131 values obtained by the two methods were somewhat different. This discrepancy may result from the uncertainty of the values of the associated-ion mobility in the calculation employed in the conductivity method and from that in potential measurements with the sodium ion-selective glass electrode, and from omitting the correction for activity coefficients in the latter method.