NOTES approximately linear up to Na+/A13+ = 2.5, at which point the hydroxyls are almost evenly distributed between the 2.85-p and the 3.6-11 bands. Within the limits of experimental error, this also corresponds to the ratio a t which Eisenman notes a change from K + to Na+ selectivity. It is tempting therefore, to ascribe the ion selectivity of glass electrodes to the superimposition of two types of sites in the glass. The one which is Na+ selective (2.85 p ) can constitute 100% of the sites, whereas the other (3.6 p ) which is K + selective, can apparently constitute no more than about 70% of the sites. This accounts for the highly selective Na+ electrodes that can be obtained and the poor selectivity of the available K + glass electrodes. Nicolskii’ has reported that in electrodes containing only small quantities of alumina (less than 7%), a distinct step occurs when the electrode potentials of a series of sodium aluminosilicate glasses are plotted against pH. These results were taken as indicating contributions to the electrode potentials by (Si0)- and (Al-O-Si)- sites. The role of the hydrated layer which must be formed before glasses become K + selective is still vague. The present correlation suggests that if the layer is directly important in ordering the selectivity, then the underlying glass structure directly controls the leaching and structure of the gelatinous layer which in turn controls the ion selectivity. The present model differs from the Eisenman model in that the K + selectivity is associated with the phase that gives rise to the 3.6-p band-Le., thes odium silicate phase or the Si-O--Na+ structure. Sodium silicate glasses are known to be leachable to give molecular sieve type materials which have pore sizes between 2 and 7 in diameter.8 This is the same order of magnitude as the “pores” observed in sintered porous glasses which gave rise to similar K+-IVa+ selectivityg and suggests that a pore size effect could be important in determining ion selectivity. The role of the alumina in the glass is seen to be of less importance. An alkali silicate glass, rich in soda, would be predicted to give high K+-Na+ selectivity on the basis of relative hydroxyl concentrations. Such glasses are notoriously soluble in water (as distinct from leachable) and the role of the alumina may be primarily to provide stability to the glass network in high-alkali glasses. (Alumina is renowned in the glass industry as the great homogenizer.) Similarly, a pure silica would be predicted to give high Na+-K+ selectivity. Such a membrane, however, would have very high resistance and, in this case, the role of the alumina could be the purely practical 0n.e of allowing the addition of NazO to reduce the resistance, without causing formation of the 3.6-p phase.
1147
Addendum A reviewer has commented on the emphasis placed on the correlation between the composition a t which the selectivity changes from potassium to sodium and the distribution of the hydroxyls. Quite rightly, he points out that this composition would be completely different had I chosen K+-Li+ or K+-Rb+ and suggests that some insight into the question might be obtained by looking a t lithium aluminosilicate or potassium alumjnosilicate glass compositions rather than the sodium aluminosilicates. Unfortunately, the small amount of data in the literature does not permit the corresponding plots for lithium and potassium aluminosilicate glasses to be drawn. However, the data do show that lithium glasses lie well above and potassium glasses lie well below the curve drawn for sodium glasses in Figure 2-just as they do in Eisenman’s selectivity plot.2 The general trend is thus established but more data are needed to confirm an exact relationship. (7) B. E’. Nicolskii, M. M. Shul’ts, E. A. Materova, and A. A. Belijusten, Dokl. Akad. Nauk SSSR, 140,641 (1961). (8) L. S. Yastrebova, A. A. Bessonov, S. S. Khvaschev, Taeolity, I k h Sin., Svoistva Primen., Mater. Vses. Soveshch. Tseolitam, 2nd, 1964, 229 (1965). See Chem. Abstr. 64, 19149~(1966). (9) I. Altug and M. L. Hair, J. Phys. Chem., 72,2976 (1968).
The Anomalous Frequency Effect in Conductometric Measurements at High Dilution
by Estella K. Mysels,’ Chemistry Department, University of Southern California, Los Angeles, California 90007
Piet C. Scholten, Philips Research Laboratories, Eindhouen, The Netherlands
and Karol J. Mysels R. J . Reynolds Tobacco Company, Winston-Salem, North Carolina (Received September, 82, 1969)
8‘7208
The precise and accurate measurement of conductivity of dilute solutions is an important tool in the study of both aqueous and nonaqueous electrolytes. The necessity of a solvent correction makes it generally desirable to use dilution cells capable of determining the resistance of both the solvent and the solution and therefore requires the measurement of a wide range of resistances extending to high values. Alternating current is usually used in these measurements and this can lead, and has led, to significant errors. Jones and (1) To whom correspondence should be sent at Salem College, Winston-Salem, N. C. 27108.
Volume 74, Number 6 March 6 , 19’70
1148
NOTES
his students2have analyzed most of these, particularly the “capacitative shunt” effect, and have shown that they can always be detected by a decreasing measured resistance at increasing frequencies. Hence, a resistance independent of frequency has been accepted as a criterion of accurate measurement. The literature reports, however, instances of measured resistances increasing with frequency. This anomalous effect has been variously explained without p o l I , , 630 detailed analysis. Thus, Nichol and FUOSS~ ascribed it to higher order terms in the capacitative shunt effect, Hawes and Kay4 to the polarizability of glass, and 0 I 2 3 4 5 6 7 8 9 IO FREQUENCY, k H Z Mukerjee, Mysels, and Dulin5 to the presence of inducFigure 1. Typical variation of the measured resistance with tion in windings of an electric motor. In the two cases frequency when cell resistance is about 0.5 megohm. where equivalent conductivity values were i n v ~ l v e d , ~ ~ ~ extrapolations to zero frequency were used although the exact method is not reported. and that linear extrapolation along either the highWe have again encountered this anomalous frefrequency or the low-frequency part can lead to quite quency effect, and as the previous explanations did not different results than the one shown. seem pertinent or sufficient, a more detailed study was We attribute this effect to a leakage to the ground undertaken. Our results indicate that the effect is along a resistance in series with a capacitance. The due to a capacitive leakage to the ground and that principal capacitance is believed to be between the extrapolation to zero frequency is indeed correct and bottom of the erlenmeyer part of the cell and the magshould be done on the basis of frequency squared from netic stirrer mechanism. The resistance is that of the quite low frequencies. electrolyte within the cell, particularly in the channels Experimental Section between the measuring bulb and the erlenmeyer flask. Figure 2 shows the essential elements of the system Resistances were measured on a Jones-Dyke bridge6 including the equal fixed arms of the bridge, the true with a General Radio Type 1232-A null detector. Rt, the measuring resistance E,, and the cell resistance Auxiliary high-quality resistors (General Radio Co. resistance to the ground Rg,along with its series capacType 500) served either to shunt the measured cell itance C,. For simplicity, the latter is assumed to be or to extend the measuring arm of the bridge. Both connected to the middle of Rt which corresponds to the procedures gave equivalent results. The Wagner position of the narrow channels between the bulb and ground of the bridge was balanced for each measurethe erlenmeyer. Ct is the true parallel capacitance of ment. The cell was of the type described by Daggett, the cell and C, is the measuring capacitance. Other Bair, and Kraus’ with the configuration of the lead capacitances are omitted since they do not enter the arms somewhat modified to reduce the Jones shunt calculation. Ct includes two items: the true parallel effect. This basically is a 1000-ml erlenmeyer flask capacitance of the cell due mainly to its parallel disk with a small bulb containing the electrodes connected electrodes acting as condenser plates and the effective to the side of this flask by two channels, one very near series capacitance due to electrode polarization. The the bottom and the other vertically above it. The lead impedance of the latter is negligible when the electrodes arms extend sideways from the bulb. The cell conare even lightly platinized and especially when the stant was 0.2254. The cell was immersed in a thermoCt is then essentially a measured resistance is large, stated oil bath. Within the bath were a number of K and for aqueous solutions function of the cell constant metallic accessories, such as a cooling coil, a heater, 1.1 X 10-l2 X 8 0 / ( K X 4 ~ F)so that in our given by a temperature sensor, and, particularly, a riser enclosing measurements Ct is about 31 pF. When the bridge the motor of the magnetic stirrer.6 These, as well as and the Wagner ground are both balanced, the two the steel thermostat vessel, were grounded. The data reported are for distilled water containing three different (2) (a) G. Jones and G. M. Bollinger, J. Amer. Chem. Soc., 53, levels of dissolved COz.
,
Results and Interpretation
A typical variation of measured resistance as a function of fequency is given in Figure 1. The line drawn from the intercept through the first two points is a parabola corresponding to a linear extrapolation on an u2plot. It may be seen that the effect is significant The Journal of Physical Chemistry
I,,
411 (1931); (b) G.Jones and S. M. Christian, {bid., 57,272 (1935). (3) J. C.Nichol and R. M. Fuoss, J.Phgs. Chem., 5 8 , 696 (1954). (4) 5. L. Hawes and R . L. Kay, ibid., 69, 2420 (1965); see p 2423 top. (5) P. Mukerjee; K. J. Mysels, and C. I. Dulin, ibid., 62, 1390 (1968). (6) P.H.Dyke, Rev. Sci. Instr., 2, 379 (1931). (7) H. M. Daggett, E. J. Bair, and C. A. Kraus, J. Amer. Chem. Soc., 73, 799 (1951).
1149
NOTES
This difference is indeed positive and for low frequencies approaches zero proportionately to the square of the frequency. This explains then qualitatively our observations and permits the determination of the true resistance by such an extrapolation. Further support for this explanation can be obtained by evaluating the characteristics of the postulated leakage to the ground; i.e., p and C, should both be constant. To simplify, we neglect 1 in comparison with 4p and y and 1 in comparison with p2y2. Equation 6 can then be rewritten as
Figure 2. Schematic diagram of the Wheatstone bridge with Wagner ground a t the bottom and with the error-causing Rg - C , bypass to the ground from the middle of the measured resistance Et.
terminals of the detector are at ground potential. Hence, the system must satisfy the conditions
where Z denotes the complex impedance of the element indicated by the subscript and i the corresponding current with the exception that i t is the current in only the left part of Rt. Solution of these equations is simplified if one takes into account the postulated nature of the stray resistance and capacitance. C, is constant if the position of the cell remains constant in the apparatus and Ct is also constant as explained above. Hence, we can write c g
= rCt
(4)
where y is a constant. R, and R t are both due to the same electrolyte filling the cell, the former to the portion in the channels and the latter to that between the electrodes. Hence, they must be proportional to each other and we can write
where p is a constant >1. The exact solution of eq 1-3 gives for the error A in the measured resistance
As the product Rt2Ct2is of the order of the first term on the right is essentially constant and a plot of eq 7 on a u2scale should reduce all data to a set of parallel lines having different intercepts. These intercepts can thus be obtained but are very small and of the order of the scatter of the data. Figure 3 therefore shows our data according to eq 7 after subtracting these intercepts. It may be seen that, up to a frequency of about 7 kHz, the points lie close to a single straight line passing through the origin. At the highest frequency (10 kHz) there are marked methodical deviations increasing with the resistance of the cell. These are of little importance in normal measurements, and we attribute them to the emergence of residual shunt effects. The slope of this line is equal to p and gives the value of 25 for the ratio of R, to R t which is reasonable. The low-frequency limit of eq 7 can be written as:
Since Rt, A, and p have been evaluated, the value of C, may be obtained from the slope of a plot of this relation. The points scatter considerably since they depend on the small values of A at low frequencies but give a value of about 10 p F for C, which is reasonable. The simplification introduced in obtaining eq 7 and 8 is therefore consistent with the values obtained, i.e., p = 25 and y = 0.3, which gives p 2 y 2 = 56. The same approximations seem to remain valid over the probable range of conditions of interest. Equation 8 can be rewritten as A =
W2RtapCg2 W2Rm'pCg2 4 4
(9)
which shows that for any experimental arrangement, Volume 74,Number 6 March 6, 1970
1150
NOTES small terms in A and in RtCt. Hence, the error in the measured capacitance is negative and very small. /A;--
Conclusion
/’:;
//9
0
@”
Figure 3. Agreement of data with eq 8. Additional 10 points not shown are located in the small dashed recta.ngle near the origin. The ordinate shows R t p / ~ minus the small intercepts (see text).
once the two constants have been evaluated, the correction can be readily applied to any measurement at a low enough frequency (500 Hz or below). It may be noted that the calculated measuring capacitance is
Ct
- c,/4
Acknowledgment. This work was supported in part by PHS Research Grant GM 10961-03 from the Division of General Medical Services, Public Health Service.
(10)
The approximation involves the neglect of the very
The Journal of Physical Chemistry
Because of the very different grounding system used by Hawes and Kay4it is possible that our analysis does not apply to their system. It certainly can account, however, for the anomalies observed in the more conventional systems of Kichol and Fuoss3 and of Mukerjee, et aL6 The effect can, of course, be minimized by careful design of the constant-temperature bath and especially of the stirring arrangement to avoid any leakage of the current from the bridge circuit against which Jones has warned already.s Any residual effects can be precisely corrected for by proper extrapolation or by eq 9. Thus, the range of accurate cell resistance measurements can be extended to at least 500 kilohms from the 10 kilohms recommended by Jones1 and the 50 kilohms shown to be possible with dip cells by Nichol and FUOSS.~ Hence, the range of concentrations over which a solvent correction measured in the same cell can be applied to give an accurate conductance value can be correspondingly increased.
(8) G. JoneR and R. C. Josephs, J . Amer. Chem. Sac., 50,1049 (1928).
