each element or group of elements. is now in progress in our laboratory.
Received for review February 11,1970. Accepted April 21, 1970. This work was supported in part by U.S.A.E.C. Contract AT(11-1)-1705, U.S.P.H.S. Grant AP-00585, N.S.F. Grant GA-811, and the University of Michigan. Contribution No. 174 from the Department of Meteorology and Oceanography and No. 127 from the Great Lakes Research Division, University of Michigan.
Such an investigation
ACKNOWLEDGMENT We are grateful to S. S. Brar of the Argonne National Laboratory and to personnel of the division of air pollution control of the city of East Chicago, Indiana, for collection of the sample described in Table IV.
Ionic Hydration and Single Ion Activities in Unassociated Chlorides at High Ionic Strengths Roger G. Bates University of Florida, Gainesville, Fla.
Bert R. Staples National Bureau of Standards, Washington, D. C.
R. A. Robinson State University
of New York at Binghamton, Binghamton, N.
Although the convention on which standard reference values of pH are based Is suitable at Ionic strengths below 0.1, an extension of this formula Is required to provide the single Ionic activities needed for the standardization of ¡ -selectlve electrodes at high Ionic strengths. This procedure must take into account the specific differences among activity coefficients of Ions of the same charge, apparent at elevated concentrations but negligible at low Ionic strengths. It is shown that a reasonable method for deriving single Ionic activities can be based on the StokesRobinson hydration theory. The assumption Is made that the chloride Ion Is not hydrated. Individual Ionic activity coefficients In solutions of seven unassociated uni-univalent chlorides and four alkaline earth chlorides have been calculated. Advantages and limitations of the hydration treatment are discussed. Recently it was proposed (7) that the convention on which standards for ion-selective electrodes are based be made consistent with the existing pH convention. The convention used to establish standard pH values is as follows (2,3): log Tci-
=
-Al1,2 ! + i.57t
2
d)
where A is the Debye-Hiickel slope constant and 7 is the ionic strength, both on the molality scale. This convention was intended to apply to aqueous solutions at any temperature (with appropriate change in the value of A) but only at ionic strengths of 0.1 or less. The activity coefficient of chloride ion so defined is nearly the same as the mean activity coefficient of sodium chloride. In the region of ionic strengths to which the pH convention applies, activity coefficients display a considerable degree of (1) R. G. Bates and M. Alfenaar in “Ion-Selective Electrodes,” R. A. Durst, Ed., Chap. 6., NBS Special Publication 314, U. S. Government Printing Office, Washington, D. C., 1969. (2) R. G. Bates and E. A. Guggenheim, Pure Appl. Chem., 1, 163 (1960). (3) R. G. Bates, J. Res. Nat. Bur. Stand., Sect. A, 66, 179 (1962).
Y.
Consequently, a convention leading to the same numerical value for the activity coefficient of chloride ion in different buffer solutions of the same low ionic strength is not unreasonable. In concentrated solutions, however, the differences among the mean activity coefficients of the uniunivalent chlorides become pronounced, and it seems likely that extension of the pH convention in its simple form has limited justification. At these higher ionic strengths, the mean activity coefficients of the alkali chlorides not only differ from one another but also display a characteristic minimum. Moreover, it appears that ion-selective electrodes also indicate a minimum in the sodium and chloride ion activity coefficients of concentrated sodium chloride solutions, both before and after liquid-junction potential corrections have been applied (4). These observations constitute further evidence that the extension to high ionic strengths of Equation 1, which does not account for this minimum, is not to be recommended. It now seems highly likely that the differences among the activity coefficients of unassociated electrolytes of the same charge type are due in part to ion-solvent interactions (J). Furthermore, the minima in the activity coefficient curves can be accounted for rather successfully in terms of changes in the “real" ionic strength resulting from the removal from the bulk solvent of water bound to the ions. The purpose of this paper is to suggest an approach to the evaluation of the individual activity coefficients of the ions in concentrated solutions of unassociated uni-univalent and bi-univalent chlorides. This procedure is based on considerations of ionic hydration (6). Though conventional in nature, it constitutes, in our opinion, the most plausible solution of this difficult problem that has yet been proposed.
uniformity.
(4) A. (5) R. 2nd (6) R.
Shatkay and A. Lerman, Anal. Chem., 41, 514 (1969). A. Robinson and R. H. Stokes, “Electrolyte Solutions," Ed., Chap. 9, Butterworths, London, 1959. H. Stokes and R. A. Robinson, J. Amer. Chem. Soc., 70,
1870 (1948).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
·
867
In Equation 3, A and B are constants of the Debye-Hiickel theory (5), á is an ion-size parameter, and I is the ionic strength in molarity units (mol l.~ *). Equation 2 is remarkably successful in expressing the mean activity coefficients of unassociated electrolytes in concentrated aqueous solutions, and the ion sizes á obtained for a considerable number of electrolytes were consistent with the likely dimensions of the solvated ions. The values of A and á found by Stokes and Robinson for a number of uni-univalent and bi-univalent chlorides are as follows: á
(Á) 4.47 4.32
A
HC1
8.0
LiCl
7.1 3.5 1.9 1.2 1.6 13.7 12.0 10.7 7.7
NaCl KC1
RbCl NH»C1 MgCl2 CaCl2 SrCl2 BaCl2
3.97 3.63 3.49 3.75 5.02 4.73 4.61 4.45
The agreement between the mean activity coefficients of sodium and calcium chlorides and the values calculated by the hydration equation is apparent in Table I. Figure 1. Hydration numbers (A) of alkali chlorides plotted as function of crystallographic radii of cations
Table I. Comparison of Mean Activity Coefficients for Sodium Chloride and Calcium Chloride with Values Calculated Equation 2 NaCl CaCla Caled Obsd / Obsd Caled 0.616 0.614 0.778 0.775 0.1 0.550 0.731 0.552 0.735 0.2 0.518 0.516 0.710 0.705 0.3 0.482 0.480 0.681 0.678 0.5 0.453 0.449 0.657 0.656 1.0 0.450 0.447 2.0 0.668 0.670 0.500 0.498 0.714 0.715 3.0 0.567 0.570 0.783 0.780 4.0 0.670 0.666 0.874 0.868 5.0 0.792 0.814 0.986 0.981 6.0
HYDRATION THEORY Stokes and Robinson (6) derived the following equation for the mean molal activity coefficient y± of an electrolyte in a solution of molality m :
In ±
=
i
A
i
|z+z_[ In fDH
-
-
In
aw
—
In
[1
+
0.018 (v
-
h)m]
In fcH 868
·
1
+ Bá/1/2
We now describe how the hydration theory can be extended to provide a formula for dividing the mean activity coefficients of certain unassociated chlorides into their ionic components. In order to accomplish this, we assume that the chloride ion is not hydrated. This is a non-thermodynamic assumption, but, unlike many conventions which are highly arbitrary in nature, there is considerable evidence to support this assumption. In Figure 1, the hydration numbers A for four alkali chlorides as found by Stokes and Robinson (6) are plotted as a function of the crystal radii of the cations. A hydration number for cesium chloride has not been obtained, but the plot makes it appear highly likely that A is about 0 for this salt. Moreover, a somewhat different treatment by Glueckauf (7) led to the conclusion that the hydration number of the chloride ion is small, about 0.9. Further support for the assumption that A for the chloride ion is small may be found in a recent publication by Maksimova and Zatsepina (8). It is therefore reasonable to regard this ion as unhydrated for our present purposes. Uni-univalent Chlorides. For a solution of the electrolyte MCI of molality m, the Gibbs-Duhem equation gives (4) dln(7M + ni) + din a ci—(55.51/m) din aw =
If the cation is hydrated with A molecules of water and m' is its molality in moles per kilogram of unbound water, m'
(2)
where fD¡¡ is the electrostatic contribution expressed as an activity coefficient on the mole fraction scale, aK is the water activity, v is the number of ions produced by a mole of the electrolyte, and A is the hydration number (number of moles of water bound to one mole of electrolyte irrespective of how it is distributed among cations and anions). This equation rests on two assumptions: that water bound to one or both species of ions is no longer a part of the bulk solvent, and that the Debye-Hiickel expression correctly gives the true activity coefficient (mole fraction scale) of the solvated ions:
—AIUi
SINGLE IONIC ACTIVITY COEFFICIENTS
(3)
ANALYTICAL CHEMISTRY, VOL. 42, NO. 8, JULY 1970
(5)
=
1
-
0.018Azn
and hence
—(55.51/m') din
—(55.51 ¡m) din aa + Adln aw din (ym+z m') + din « (6) The activity of chloride ion is the same on the two scales: m'yci-'. From Equations 4 and 6, myc\din (7m+z «') din (ym+ ni) + Adln aw (7) =
aw
—
=
=
or
In 7m+
=
In 7m +
z
—
Ain aw
—
In
(1
—
0.018Am)
(8)
(7) E. Glueckauf, Trans. Faraday Soc., 51, 1235 (1955). (8) I. M. Maksimova and N. N. Zatsepina, Russ. J. Phys. Chem., 43, 570 (1969).
It follows from the postulates of the Stokes-Robinson hydration treatment that the activity coefficient of the hydrated species i on the mole fraction scale is correctly given in terms of Equation 3 by In /< Hence, since z
