This comparison of the ~ u Hassigned to the solution of tricine (0.06m) and sodium tricinate (0.02m) with the primary pH standards was made as follows. The emf of a cell with hydrogen gas electrode, 3.5M calomel reference electrode, and liquid junction formed in a 1-mm vertical capillary tube was determined when the cell contained the 1:3.5 phosphate standard composed of KHzPO4 ( m = 0.008695) and NazHP04 ( m = 0.03043). The phosphate standard was then replaced by the tricine buffer and the measurement repeated. The barometric pressure and the temperature (26.3 "C) were the same for the two measurements, and the emf for the tricine buffer was higher than that for the phosphate standard by 0.01056 V. At 26.3 "C,
therefore, the pH of the tricine buffer was 0.178 unit higher than that of the phosphate standard [7.409 a t this temperature (IO)]or 7.587. This value is 0.009 unit lower than 7.596 calculated by Equation 4 of this paper. Thus, the residual liquid-junction potential is probably not so large as to exclude tricine from consideration as a primary standard for pH measurements. Received for review February 16, 1973. Accepted April 6, 1973. Work supported in part by the National Science Foundation under Grant GP-14538 and by the donors of the Petroleum Research Fund, administered by the American Chemical Society.
Ionic Hydration and Single Ion Activities in Mixtures of Electrolytes with a Common Unhydrated Anion R. A. Robinson and Roger G. Bates Department of Chemistry, University of Florida, Garnesville, Fla. 32607
In earlier papers, a hydration convention permitting single ion activities to be derived from mean ionic activities of unassociated electrolytes has been outlined. This convention has now been combined with the thermodynamic theory of electrolyte mixtures to obtain single ion activity coefficients for the three ions in mixtures of two electrolytes with a common unhydrated anion. The method is illustrated with data for mixtures of potassium chloride and sodium chloride at a total molality of 4 mol kg-' and for hydrochloric acid and sodium chloride at a total molality of 3 mol kg-l.
Ion-selective electrodes responsive to a considerable variety of ions are now available. Thermodynamics leads one to expect that the potentials of these electrodes are a function of ion activities instead of concentrations, and this conclusion is supported by experimental evidence (1-4). Ion-selective electrodes are therefore capable of measuring ionic activity relative to a standard in which the activity of the ion is known or assigned. Inasmuch as thermodynamic theory can offer no unique definition of the activity of any single ionic species, no consistent standard scales of ionic activity yet exist. The problem is compounded by the common use of ion-selective electrodes at high ionic strengths, where marked specific differences among the activity coefficients of ions of like charge become apparent. As a result, a chaotic situation is rapidly developing. A conventional scale of hydrogen ion activity ( 5 ) has proved very satisfactory for the standardization of glass pH electrodes. The convention on which it is based (6) re-
lates the activity coefficient. of chloride ion to ionic strength (0 in the region 0 < I < 0.1. Other simple conventions would have served equally well in this dilute range. The problem of establishing scales for a variety of ions a t higher ionic strengths is, however, of considerably greater complexity. The scales that will eventually be adopted for single ion species must be consistent with the measurable thermodynamic constants for pairs or other combinations of ions. Accordingly, they must take account of specific ionic properties of which differences in the interaction of ions with the solvent are probably the most important. In earlier papers (7, 8) we have suggested that the hydration number h can be used to characterize ionic specificity of the latter type. In 1948, Stokes and Robinson (9) put forth a hydration theory combining ion-ion interaction with ion-solvent interaction and showed that the mean activity coefficients (r*)of strong electrolytes can be expressed accurately by an electrostatic (DebyeHuckel) term dependent on the ion size, together with a hydration number and the solvent activity. We have introduced (7) a conventional method (based on hcl- = 0) to derive ionic hydration numbers from the hydration number for the electrolyte and have shown that thermodynamics then leads to a formula for separating the mean activity coefficient into the activity coefficients of the component ions. For a uni-univalent electrolyte ( u = 2) at molality m , the equations for the cation (+) and anion ( - ) are logy+ = logy, 0.00782(h+ - h-)m$ (1)
+
and logy- = logy,
(1) p. Schindler and E. Walti, Helv. Chim. Acta, 51, 539 (1968). (2) A. Shatkayand A. Lerman,AnaL Chem., 41, 514 (1969). (3) J. N. Butler, in "Ion-Selective Electrodes," R. A. Durst, Ed., Nat. Bur. Stand. Spec. Pub., 314, Washington, D.C., 1969, Chap. 5. (4) R. G. Bates and M. Alfenaar, Nat. Bur. Stand. Spec. Pub/., 314, Chap. 6. (5) R. G. Bates, J. Res. Nat. Bur. Stand., 66A, 179 (1962). (6) R . G. Bates and E. A. Guggenheim. Pure Appl. Chem., 1, 163 (7960).
1666
+
0.00782(h- - h+)m$
(2)
where 4 is the osmotic coefficient, (-ln a,)/0.018 u r n . (7) R. G. Bates, B. R. Staples, and R. A. Robinson. Anal. Chem., 42, 867 (1970). (8) R. A. Robinson, W. C. Duer, and R. G. Bates, Anal. Chem., 43, 1862 (1971). 70, 1870 (9) R. H. Stokes and R. A. Robinson, J. Amer. Chem. SOC.. (1948).
ANALYTICAL CHEMISTRY, VOL. 45, NO. 9, AUGUST 1973
This approach to activity scales for ions in solutions of a single electrolyte appears to be a promising way to establish consistent useful standards for ion-selective electrodes, Furthermore, the ionic activities derived by this method are in acceptable agreement with those obtained from the potentials of cells with ion-selective electrodes, corrected for the residual liquid-junction potential (7,IO). MIXTURES OF ELECTROLYTES The increasing use of ion-selective electrodes in mixtures of electrolytes such as blood, sea water, and the like, introduces special problems. Neff (II), assuming that a properly standardized electrode yields accurate ionic activities in blood (I2),has discussed means by which these activities can be converted to concentrations. In two investigations (13, I4), the lanthanum fluoride electrode was used to measure the activity coefficient of fluoride ion in mixtures of sodium or potassium fluoride with other alkali halides. Leyendekkers interpreted the fluoride ion activity in these mixtures in terms of ion-ion interaction coefficients which were, in turn, shown to be consistent with Harned rule coefficients. Bagg and Rechnitz demonstrated that the single ion activities in the mixtures of sodium salts were close to those in pure sodium fluoride up to a total ionic strength of 1.0. In mixtures of potassium salts, slightly better agreement was obtained by introduction of Harned's rule for the mean activity coefficient in the mixtures. It is the purpose of our present contribution to show that the thermodynamic treatment of binary mixed electrolytes with a common ion leads to a simple extension of our earlier ion hydration convention, permitting the individual ionic activities in these mixtures to be derived. The problem can be illustrated by reference to the KC1 (A)-NaC1 (B) system a t 25 "C and a total molality (m = mA m g ) of 4.0. It is known (15)that the activity coefficient of potassium chloride in its own solution a t m = 4.0 mol kg-1 is given by log ~ A O= -0.2390, the superscript being used to denote that the solution of potassium chloride contains no second electrolyte. Similarly for sodium chloride in its own solution a t this molality, log y B 0 = -0.1061. The work of Robinson (16)has shown that the activity coefficient of potassium chloride in mixtures of this salt with sodium chloride a t constant total molality decreases linearly with the amount of sodium chloride in the solution,
O
t
6'
KCI
" yo CI- in NaCl
I
I
0 KCI
0.5
1.0 NaCl
YE
Mean molal activity coefficients of potassium chloride and sodium chloride in a mixed solution at a total molality of 4.0 mol kg-'. The possible variation of t h e three single ion activity coefficients is indicated by means of dashed lines Figure 1.
(3)
a solution containing only potassium chloride is given by log yBtr = -0.1061 - 4.0 X 0.0246 = -0.2045. I t will be noted that the two trace activity coefficients are almost equal. The behavior of the activity coefficients of potassium chloride and of sodium chloride in these mixed solutions is shown by the full lines in Figure 1. The hydration convention outlined in our earlier papers (7, 8) permits one to derive the activity coefficients of potassium and chloride ions in solutions containing only KC1, as well as those of sodium and chloride ions in solutions containing only NaC1. In KC1 (rn = 4), log yKO = -0.1811 and log yc10 = -0.2958, whereas in NaCl (m = 4), log y~~~ = 0.0162 and log ycl0 = -0.2284. The superscript zero has been used to emphasize that the activity coefficients refer to solutions containing a single electrolyte. Thus, the convention for single ion activity coefficients being accepted, one knows the activity coefficient of chloride ion in a solution of potassium chloride and also in a solution of sodium chloride, but the manner in which this activity coefficient varies with composition of a mixture of the two electrolytes is unknown (see Figure 1). In a similar way, the activity coefficient of potassium ion in KCl and the activity coefficient of sodium ion in NaCl are known, but the value of neither activity coefficient in a mixture of KC1 and NaCl is as yet obtainable.
with CXA = -0.0090 a t a total molalitv of 4.0 mol kiz-1. In the limit, when the solution contains only sodium chloride, the trace activity coefficient of potassium chloride is 4.0 x 0.0090 = -0.2030. given by log y A t r = -0.2390 On the other hand, Robinson found that the activity coefficient of sodium chloride does not vary linearly with the concentration of potassium chloride; instead
HYDRATION EQUATION FOR A MIXED ELECTROLYTE SOLUTION Consider a solution containing two electrolytes, each of the 1:l charge type, A with cation M and anion X and B with cation N and anion X. Let the solution contain S moles of water, Y A moles of A and YB moles of B (yA y e = 1).Then the total Gibbs energy of the solution is
+
log?,
= lOgY.4'
-
aAmB
-
+
log?,
=
logy:
-
aRm.4
-
PBm.4'
+
(4)
The p coefficient, however, is small and can be ignored for the present purpose of illustration. Then with C X B = 0.0246, the trace activity coefficient of sodium chloride in
This, however, can equally well be expressed in terms of the partial molal free energies of the hydrated electrolytes,
(10) J. Bagg and G. A. Rechnitz, Anal. Chem., 45, 271 (1973). (11) C. W. Neff.Anal. Chem., 42, 1579 (1970). (12) C. W. Neff. W. A. Radke, C. J. Sarnbucetti, and G. M. Widdowson, Clin. Chem., 16, 566 (1970). (13) J. V. Leyendekkers,Anal. Chem., 43, 1835 (1971). (14) J. Bagg and G. A. Rechnitz, Anal. Chem., 45, 1069 (1973). (15) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2nd revised ed., Butterworths, London, 1970, p 441 (Eq. 15.9) and Appendix 8.10. (16) R. A. Robinson, J. Phys. Chern., 65, 662 (1961).
where the primes indicate that the quantities refer to the h o B , h A and he hydrated electrolytes and h = h o A being the hydration numbers of electrolytes A and B, respectively. Then,
G
= (S
- h)GL +
y,\GA'
+
YBG~,'
(6)
+
hGw + Y . ~ G A +
Y B G B = YAG.:
+
YBG;
ANALYTICAL CHEMISTRY, VOL. 45, NO. 9, A U G U S T 1973
(7) 1667
HYDRATION EQUATION FOR THE CATIONS We now consider the mixture of MX (electrolyte A) and NX (electrolyte B) where X is an unhydrated anion, such as the chloride ion, and hM, hN are the hydration numbers of the cations M and N, respectively. If there are mA moles of A in 55.51 moles of “total” water, there are mA moles of A in (55.51 - hwA m - hNYn m ) = (55.51 - h m ) moles of “free” water. Thus, mA‘, the molality of A expressed as moles per kilogram of “free” solvent, is given by m,’ = m . 4 / ( l
-
0.018hm)
(18)
with similar equations for mR‘, and the total molality m‘ = (mA’ m”). From the Gibbs-Duhem equation, one obtains -(55.51/m)d lna, = yAd lnyM+.m
+
1
I.o Na CI
0.5
0
+
I
I
KCI
Ye Figure 2. Osmotic coefficients of potassium chloride-sodium chloride mixtures at rn = 4 mol k g - I
Expressing the partial molal free energies in terms of activity coefficients (ti)on the mole fraction scale,
+ ~ ~ ( +c .2RT ~ ’lnf,N,A) + 2RT In fBNB) = YA(G,%O’ + yB(G2 2RT In f.A’NAf + yB(CBs‘ + 2RT In f:N
