Ion Specific Membranes as Electrodes in. Determination of Activity of Calcium Adam Shatkay Isotope Department, The Weizmann Institute of Science, Rehouoth, Israel The suitability of paraffin membranes, solid polymeric membranes, and Orion liquid ion exchange electrodes for the measurement of Ca+z activity is investigated. A convenient method for assessing the specificity of membranes is presented and applied to the above membranes. It appears that the paraffin membranes are not completely permselective and probably are completely nonspecific. The other two electrodes appear to be suitable for the measurement of CaiZ activity in biological systems and have about equal performance.
THE IMPORTANCE of accurate determination of the activity of calcium ions is apparent in the recent research of the metabolism of calcium ( I ) . Successful attempts to determine the activity of calcium potentiometrically were carried out in 1924 by Lucasse ( 2 ) , using Ca amalgam. Lucasse’s method was laborious and unsuitable to biological systems. In 1928 Corten and Estermann (3) reported successful use of a n electrode system consisting of metall cationl anionl (insoluble)cationz anionl (insoluble)-cation2 anionz (soluble) for the determination of Ca+2in blood. (These so-called “electrodes of the third kind” are illustrated by the lead-lead oxalatecalcium oxalate-calcium chloride system.) Their results were disputed in 1933 by LeBlanc and Harnapp (4, who reported a successful electrode of their own, also of the third kind. Unfortunately, this electrode also was unreliable in biological systems. In 1938 Joseph ( 5 ) improved the use of electrodes of the third kind so that the activity of Ca+Z in the presence of amino acids could be determined. The preparation of such electrodes remains difficult, and their use is cumbersome. A new approach to the measurement of Ca+z activity was made with the introduction of membrane electrodes (6-10). Some of these electrodes are reported to offer a convenient and a reliable way to measure the activity of calcium. To fulfill such a claim certain requirements appear to be necessary. The electrode should be either commercially available or easily constructed and handled. It should be robust and durable. The electrode should give reproducible and meaningful results within the range of calcium activities of (1) W. F. Neumann and M. W. Neumann, “The Chemical Dynamics of Bone Mineral,” University of Chicago Press, Chicago, 1958. (2) W. W. Lucasse, J . Am. Chem. Soc., 41, 743 (1925). (3) M. H. Corten and I. Estermann, Z . Physik. Chem., 136, 228 (1928). (4) M. LeBlanc and 0. Harnapp, Ibid., A166, 321 (1933). (5) N. R. Joseph, J . Biol. Chem., 130, 202 (1939). (6) G. J. Hills, “Reference Electrodes,” D. J. G. Ives and G. J. Janz, Eds., Chap. 9, Academic Press, New York, 1961. (7) N . Lakshminarayanaiah, Chem. Rec., 65 ( 5 ) (Oct. 1965). (8) K. Sollner, et a/., “Ion Transport Across Membranes,” H. T. Clarke and D . Nachmansohn, Eds., Chap. 5 , Academic Press, New York, 1954. (9) K. S . Spiegler and M. R. J. Wyllie, “Physical Techniques in Biological Research,” Vol. 11, A. W. Pollister and G. Oster Eds., Academic Press, New York, 1956. (10) T. Teorell, Discussions Faraday Soc., No. 21, 9 (1956).
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interest. In biological systems this means activities of Ca+2 of approximately to 10-2M. The electrode should be ion specific-i.e., it should measure the correct activity of Ca+2in the presence of all other species contained in the biological system, such as Naf (?lO-lM), K+ (“10-2M), Mg+z (-lO-$M), C1- (=lO-lM), POa-3 (=lO-zM), and various macro ions (11). The time of response should be reasonable, with constant readings reached preferably in minutes. It would be helpful (though not essential) if within the above range the relation of emf to activity is Nernstian. Three types of electrodes which claim to fulfill the above requirements have been investigated in this paper. These are : paraffin membranes ; polyvinylchloride (PVC) membranes incorporating tributylphosphate (TBP) and theonyltrifluoroacetone (TTA); and Orion commercial liquid ionexchange membranes (Orion Research, Inc., Cambridge, Mass.). All of the above electrodes satisfy the first requirement. The multilayer membrane electrodes have not been included in this study because they have already been extensively described (12), and d o not claim as yet to satisfy the first requirement. EXPERIMENTAL Preparation of Membrane Electrodes. Orion electrode was prepared for use according to the operating instructions supplied by the firm. PVC, TTA, and TBP membranes were prepared as described in Israel Patent 21709 (13). The paraffin membranes on gauze were prepared from paraffin wax (mp 55-7” C) on surgical cotton gauze (14). To obtain thin uniform paraffin membranes of practically unlimited area we employed a method suggested to us by S. Szapiro of the Isotope Department of the Weizmann Institute of Science. A suitable quantity of clean mercury was heated to about 70” C in a flat dish. The desired quantity of paraffin (or paraffin-oxalate mixture) was placed on top of the mercury. The paraffin melts and spreads in a thin layer over the mercury. After cooling, the uniform thin layer was lifted off the mercury. By this method uniform and tough films of paraffin 0.1 mm thick were easily obtained. For light membranes water may be used instead of mercury. Standard Solutions. The standard solutions of CaCI, used were analyzed for C a f 2 and for C1-, and their concentration can be taken as accurate to within f1 %. Measurements. T h e availability of mechanically strong paraffin membranes allowed us to use (in addition to the glass tubes of Tendeloo and Kripps) the convenient perspex cells, shown schematically in Figure 1.
(11) W. S . Spector, “Handbook of Biological Data,” Saunders and Co., New York, 1956. (12) H. P. Gregor, et a / . , J . Am. Chem. Soc., 85, 3926 (1963). (13) R. Bloch, A. Shatkay, and H. A. Saroff, Weizmann Institute of Science, Rehovoth, Israel, unpublished work, 1966. (14) H. J. C. Tendeloo and A. Kripps, Rec. Trac. Chim., 76, 703, 946 (1 957).
RESULTS AND DISCUSSION MEM3RANE
Measurements in Pure Solutions of CaClz Using Paraffin Membranes. Membranes consisting of paraffin incorporating
-”ELECTROO€
precipitates have been described by Tendeloo and Kripps (14) and by Fischer and Babcock (15). According to Hills (6) “although these membranes were not subjected to a
PERSPEX .-CELL
.
Figure 1. Cell employed for measuring the EMF of membrane electrodes
As reference electrodes we used the Radiometer calomel electrodes, which were tested and found to leak KCl at a rate less than lop6mole of KC1 per hour (as compared with about lov4 mole per hour for the Metrohm calomel electrodes). The emf was measured on a Metrohm null pH meter compensator E 388, with a precision of 1 0 . 1 mV. All measurements were taken at 20.5” + 0.5” C.
L
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thorough test, they gave a specific and theoretical response. . . ” It will be shown that this claim is not justified. The membranes of Tendeloo and Kripps consist of pure paraffin or of paraffin mixed with detergent and calcium oxalate. The melted paraffin (or mixture) is made into membrane on tightly stretched cotton gauze. We have investigated the behavior of membranes consisting of pure paraffin on gauze, and of membranes consisting of a mixture of paraffin and calcium oxalate on gauze. The addition of calcium oxalate was “to increase the selectivity for calcium ions by incorporating in the membranes calcium salts of low solubility in water” (14). The inner (constant concentration) solution was 4.56 X 10-3M CaC12, while the outer solutions were of varying concentrations of CaCI2 (Figure 2). For comparison we give the results of Tendeloo and Kripps for a pure paraffin membrane [ref. (14, p. 706, Table 1, electrode OxPA21 with an inner solution of 10-3M CaC12, and a membrane of paraffin, detergent, and calcium oxalate (p. 948, Table 1, electrode OxPA4) with an inner solution of 10-2MCaC12. T o have better theoretical significance, the graphs in Figure 2 should represent the change of emf with the log of activity and not of concentration. Assuming that acs-2 is directly (15) R. B. Fisher and F. Babcock, ANAL.CHEM., 30, 1732 (1958).
I
-3 10s
c,,
I
I
-2
-I
J
0
(10s ? * C,,)
Figure 2. EMF of a cell employing paraffin-on-gauze membrane Open figures-concentrations; closed figures-concentrations multiplied by mean activity coefficients of CaClz 0 Tendeloo and Kripps, pure paraffin 0 Tendeloo and Kripps, paraffin and calcium oxalate A Our results, standard solution of CaC124.56 X 10-3M, membrane of pure paraffin I3 Our results, standard solution of CaCIz 4.56 X lOP3M,membrane of paraffin and calcium oxalate VOL. 39, NO. 10, AUGUST 1967
1057
108 C C a t a
Figure 3. EMF as function of the concentration of CaClz in a cell employing a paraffin-on-gauze membrane Concentrations of the reference (constant) solutions are : (1) 1.00M (3) 1.10 (5) 4.56
x x
10-1M 10-3~
proportional to yf X CCaC12, where y f is the mean activity coefficient on the molarity scale, we have calculated and plotted log (yf C c a c l J also in Figure 2 using y f from the literature (16,17). The results summarized in Figure 2 indicate that in pure solutions of CaCla there is no difference in the behavior of membranes consisting of paraffin alone, or of paraffincalcium oxalate mixture. The electrodes give fairly reproducible results. The slope of d emf/d log aca+2changes from approximately 15 mV/log a at concentrations of CaCh of about 0.1M to approximately 20 mV/log a at concentrations of about 10-2M, and to 15 mV/log a at concentrations (16) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” Butterworths, London, 1965, p. 478. (17) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” Reinhold, New York, 1964, pp. 252, 550, 735, 738.
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(2) 5.00 X 10-lM (4) 2.27 X 10-zM (6) 7.56 X 10-4M
of about lO-3M. (The Nernst slope at the experimentaI temperature of 20’ C is 29.0 mV/log a.) These results indicate that the membrane is not completely permselectivei.e., it allows both cations and anions to pass, although a t different rates. The ion specificity towards specific cationse.g., towards Caf2-will be discussed below. At concentrations of CaCla below 2 X lO-3M the slope of d emf/d log a falls to 0, and the electrode ceases to be functional. Thus at the interesting concentrations of Ca+2 in biological systems the paraffin electrode does not appear to be useful. The effect of the concentration of the inner (reference) solution on the range of functionality of paraffin membranes has been investigated, and our results are presented in Figure 3. It is apparent that the increase in concentration of the inner solution somewhat extends the range of functionality
4 Figure 4. EMF as function of the concentration of CaClz in a cell employing a paraffin membrane with a high content of calcium oxalate, not incorporating gauze
b
v
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Y
b Figure 5. Change of EMF with time for a paraffin membrane Standard solution: CaClz 6.80
x 10-2M
a: CaClz 8.00 X 10-6M
b: CaC12 6.07 X lO-3M a’ and b’: corresponding slopes of curves a and b
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VOL. 39, NO. 10, AUGUST 1967
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Figure 6. Types of electrodes
v
(1) PVC, TTA, TBP membrane; standard (constant) solution CaClz 6.07 X 10-3M (2) PVC, TTA, TBP membrane; standard (constant) solution - CaCI2 1.10 X 10-'M (3) Orion electrode
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of the paraffin membranes. However, this effect is slight, and even with an inner solution of 1M CaCl2, the electrode ceases to be useful at about 10-3Mconcentrations of CaC12. Having a method of preparation of robust paraffin membranes without the need to support them on gauze, we have investigated such pure paraffin membranes and membranes of paraffin mixed with calcium oxalate. We have found that pure paraffin membranes, not incorporating gauze, form an open circuit, as opposed to pure paraffin membranes incorporating gauze. Addition of small amounts of calcium oxalate, as carried out by Tendeloo and Kripps, has a negligible effect on the conductivity of the paraffin membrane. Only on the addition of very large quantities of calcium oxalate (approaching 1 gram of oxalate per gram of paraffin), as practiced by Fisher and Babcock (1.3, does the membrane become conductive. At such concentrations of oxalate the behavior of the membrane resembles that of a gauze-con1060
ANALYTICAL CHEMISTRY
taining membrane, as illustrated by Figure 4. The gauze employed by Tendeloo and Kripps is not merely an inert supporting structure for the membrane, but takes part in the permselectivity of the membrane. We have mentioned that the markedly non-Nernstian behavior of the paraffin electrodes suggested that the paraffin membranes were not fully permselective, but merely affected to a different extent the mobility of cations and anions. This suggests in turn that the cell utilizing a paraffin membrane is not at equilibrium, and perhaps not even in a steady state, during the measurements. This would make the measured emf difficult to interpret, If our suggestion is correct, the changes of emf with time should be investigated, and the time dependence might be of considerable importance in the assignment of some emf to a cell in which the initial concentrations of CaClz o n both sides of the membrane are known. As the papers of Tendeloo and Kripps (14) do not mention
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Figure 7. Change of EMF with varying concentrations of MgClz in a cell employing paraffin-on-gauze membrane Pure paraffin (experimental) A Paraffin with calcium oxalate (experimental)
Curve : Theoretical
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A 0
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b Figure 8. Change of EMF with varying concentrations of NaCl in a cell employing paraffinon-gauze membrane Standard (constant) solution: CaClz 6.80 X 10-zM. CaClz solution containing the foreign salt: 1.52 x 1 0 - 3 ~ 0 Pure paraffin (experimental) A Paraffin with calcium oxalate (experimental) Curve : theoretical
0
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VOL. 39, NO. 10, AUGUST 1967
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Table I. Effect of Foreign Salt on the EMF of 5 X 10-3M CaClz Solution Measured by Paraffin Electrode Foreign salt Molarity EMF, mV MgCh 10-3 96 MgClz 5 x 10-3 96 10-2 98 MgCh 10-3 97 KC1 KCl 5 x 10-3 98 10-2 98 KC1
the time dependence of the emf (which is not necessary in a permselective membrane), we summarize in Figure 5 two such observations. The emf of a dilute solution of CaCh (8 X 10-5M) changes at the rate of some 5mV/hour at the start of the measurements, and the rate falls to about 0.5 mV/hour after 10 hours. Thus the emf might be taken to be about 40 mV after the instrument has apparently settled down after an hour, or 34 mV after 10 hours, when the reading appears to be quite steady. For the more concentrated solutions, such as 6 X lO-3M in Figure 5, the constant readings appear almost immediately, giving the wrong impression of equilibrium. On such semiconstant and fairly reproducible readings are based Figures 2-4, and their values correspond to those of Tendeloo and Kripps. It should be emphasized, however, that the readings are not constant, and that the solutions continue mixing through the membrane, until the concentration is the same on both sides, and the emf is 0. PVC, TTA, TBP Membranes and Orion Electrodes. The PVC, TTA, TBP membrane has been described in reference (13). A summary of its behavior, including some new data, is presented in Figure 6, curves 1 and 2. 1
I
I
This membrane gives results which are linear within the range of concentrations of 2 X 10-4M to 10-IM; within this range the slope is 27 to 28 mV/log a-Le., almost Nernstian. The curve may be used to determine the activity of Caf2, even where it is not linear, in the range of 5 x 10-6M to 1M. The concentration of the standard (constant) solution has little effect, although a change from 6.07 X 10-3M to 1.10 X 10-IM improves slightly the linearity and the reproducibility. The Orion calcium activity electrode, Model 92-20, is based on a liquid ion exchanger membrane. The summary of measurements carried out on such an electrode is given in Figure 6, curve 3 ; 80 mV have been subtracted from each emf reading to facilitate graphic comparison. The electrode gives results which are linear only within the range of 5 X 10-4M to 10-IM where the slope is 26.5 mV/log a. Thus it appears that the membrane is not ideally permselective. However, the electrode gives reproducible and useful results within the range of 1 M to 5 x lO-5M. Thus in unmixed solutions of Ca2+ all three electrodes (paraffin, solid polymer, and Orion) may be used to determine the activity of Ca+2. The Tendeloo-Kripps electrode reacts slowly, does not reach equilibrium, and is not functional at concentrations lower than 2 x lO-3M; the Orion electrode and the PVC, TTA, TBP membrane may be conveniently used down to 5 X lO-SM. Measurements in Mixed Solutions of Caf2. To measure accurately the activity of Ca+2in the presence of other cations, such as Mg+2 and Na+ which are present in considerable quantities in biological systems, it is necessary that the membrane electrode be specific toward the calcium ions. The assessment of the specificity of a given membrane is, however, unfortunately still ambiguous, as will be shown in evaluating the claims of some electrodes. 0I
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Figure 9. Change of EMF with varying concentrations of foreign salt in a cell employing PVC, TTA, TBP membrane 0 MgClz in 3.80 X 10-SM CaC12 (experimental) 0 NaCl in 1.52 X lOP3MCaC12 (experimental) Curves: theoretical
1062
ANALYTICAL CHEMISTRY
-%
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Figure 10. Change of EMF with varying concentrations of foreign salt for the Orion electrode Q MgC12 in 3.80 X 10-3M CaClz (experimental) 0 NaCl in 1.52 X l O P 3 MCaC12 (experimental) Curves: theoretical
Tendeloo and Kripps (14) tested their paraffin membranes in solutions of CaCla in the presence of MgC12 or KC1. In a typical test a pure solution of 5 x 10-3M CaC12 yielded 100 mV. In the presence of MgClz and KCl the emf was as shown in Table I. On the strength of such results the authors claim that “there is a fairly acceptable specificity for calcium ions.” The implication seems to be that because an addition of MgC12 or KCl changes the emf only slightly, the membrane is specific to calcium. Such a statement does not appear to be correct. A membrane specific to calcium should show the correct calcium ion activity in any solution; if the presence of foreign salts in the solution changes the activity of calcium (as it generally does), then the emf should change on the addition of the foreign salts, and not remain constant. The problem is to evaluate what change of emf is to be expected. If such an expected change occurs, the membrane may be considered specific to calcium. Corning (18) states regardifig its commercial electrode (which has not yet been evaluated by us): “The electrode has high selectivity for Ca+2 over other monovalent and divalent cations.” To support this statement results are given as presented in Table 11. Again, the figures in Table I1 do not support any definite conclusions about the specificity of the electrode. The instructions accompanying the Orion electrode give detailed values for the effect of NaCl on the emf read by the electrode. For a solution of 10-3M CaClz one should subtract 11 mV in a solution of 1M NaCl, 1 mV in a solution of 10-’M NaCl; no correction is necessary in a solution of 10-2M NaCl or less. These data appear to us doubtful, as will be shown below. The manufacturers do not give (18) Corning Glass Works, Corning, N.Y., Product Information
detailed assessment of bivalent ions, but claim that the electrode “is approximately 50 times more sensitive to Ca+2 than Mg+2.” Thus, if we interpret the above statement correctly, a 1 0 - 3 ~solution of CaCh in a 1 0 - 3 ~solution of MgC12 should behave as 1.02 x lO-3M solution of C a C h If the behavior of the Orion electrode were approximately Nernstian-Le., d emf/d log a = f 2 9 mV/log a-the mixed solution should read by 0.3 mV more than the pure solution of CaC12. Similarly, in a solution of 10-2M MgC12 the emf should read about 2 mV high. These conclusions appear to be incorrect, as will be shown below. It appears worthwhile to find a method which would enable one to describe fairly easily, clearly, and unambiguously the specificity of a membrane. The following method seems to be reasonably satisfactory. We assume that the presence of foreign salts changes the activity coefficient ylt of Ca+2 through the change in ionic strength only. This assumption is justified in dilute solutions (4, 12), although in concentrated solutions it certainly is an oversimplification. We calculate the ionic strength of any mixed solution, and estimate the concentration of CaCla in a pure solution of CaClz which would have the same ionic strength as that of the mixed solution. This enables us to
Table 11. ElTect of Foreign Salt on the EMF of 10-3M CaClz Solution Measured by Corning Electrode Molarity of Foreign salt foreign salt EMF, mV ...
...
NaCl NaCl
10-2 lo-’
MgCI,
10-2
MgC12
10-1
...
-96.2 -96.2 -94.0 -96.5 -93.8 -88.1
02052 (1966). VOL. 39, NO. 10, AUGUST 1967
1063
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Figure 11. Change of EMF with varying ionic strength for the Orion electrode 0 Foreign salt: MgCI2 only (experimental) 0 Foreign salt: NaCl only (experimental) 0 Foreign salt: Mixture of MgClz and NaCl (experimental) Curve: -theoretical find the ~ = of k Ca+2 from the values of activity coefficients of CaClz in pure solutions, listed in the literature (16, 17). A theoretical curve for the change of emf of a given solution of CaClz in varying concentrations of foreign salts can now be drawn, (using curves such as in Figures 2-6), and compared with experimental results. This procedure is illustrated in Figures 7-11. For low concentrations of foreign salts the emf is not necessarily constant, yet follows the theoretical curve fairly closely. This indicates that at those concentrations the membrane is specific to calcium. At some concentration of the foreign salt there occurs a more or less abrupt break of the experimental results from the theoretical curves. It is assumed that this break indicates the concentration of the foreign salt at which the membrane ceases to be specific toward Ca+,. The above method has been applied to measure the specificity of the three membranes investigated in this paper. Specificity of Paraffin Membranes. The behavior of a solution of 3.80 X lO-3M CaClz in the presence of MgC12, measured with a paraffin membrane supported on gauze, is summarized in Figure 7. The standard (constant) solution was 6.07 X 10-3M CaCI2. The theoretical curve appears almost trivial; the emf remains constant to a MgClz concentration of lO-SM, and increases only by some 3 mV when the concentration of MgCla becomes 1M. This is due to the fact that the emf of CaClz = lO-3M is very slightly dependent on changes of concentration, as seen in Figures 2 and 3. The presence of 1M MgClz changes the y=k from 0.8 to 0.5, thus reducing the effective concentration of CaCla by about 30z. This, however, has almost no effect on the emf because of the poor performance of the paraffin membrane at this concentration. Figure 7 shows that a t a MgClz concentration of Z ~ O - ~ M , 1064
ANALYTICAL CHEMISTRY
the experimental points start to deviate from the theoretical curve-Le., the membrane is not ion specific for Ca+2 at concentrations of MgClz higher than 10-3M. This does not mean that the membrane is ion specific for lower concentrations of MgC12. It is possible that a membrane completely nonspecific to Cae2 yields a curve following the theoretical curve of Figure 7 for low concentrations of foreign salts. However, the behavior of nonspecific membranes requires special treatment (8), and we shall not discuss it here. Preliminary investigations seem to indicate, however, that the paraffin membranes are completely nonspecific even a t the lowest concentrations of foreign salts. The behavior of paraffin electrodes for a 1.52 X 10-3M CaClz solution, in the presence of varying concentrations of NaC1, is described in Figure 8. Again, the theoretical emf is almost constant, and the membrane shows nonspecificity at NaCl concentrations higher than lO-4M. It is also probable that the membrane is completely nonspecific to Ca+2 against Na+ at any concentration. The presence of calcium oxalate does not appear to improve the specificity of the paraffin membranes. Specificity of PVC, TTA, TBP Membranes. Figure 9 describes the behavior of a 1.52 x lO-3M CaCh solution at various concentrations of MgC1, and of NaC1, as measured by the PVC, TTA, TPB membrane. In both cases the standard (constant) solution was 0.1 10M CaClZ. The experimental results follow the theoretical curve fairly closely until the concentration of the foreign salt reaches the value of about 10-'M. In the case of NaCl the emf decreases by 4 mV, and in the case of MgClz it decreases by 8 mV when the concentration of the foreign salt reaches 10-lM, yet we still consider the membrane specific to Ca+2. On the other hand, at the concentration of MgClz of about 3 X lO-'M, the emf is
equal t o that of pure CaCla solution, yet we consider that a t this concentration the membrane has already lost its specificity. Specificity of the Orion Electrode. Figure 10 represents the behavior of the systems described in the last paragraph as measured with the Orion electrode. The theoretical curves differ slightly from those of Figure 9, as they were calculated from the curves of emf/log aca+2 obtained from the Orion electrode. The general pattern is, however, similar t o that of the PVC, TTA, TBP membrane: the electrodes appear t o be selective towards Ca+2 up to a concentration of 10-lM of the foreign salt. Specificity of Membranes in Solutions Containing More Than One Foreign Salt. The method for assessing Ca+2 specificity described above can be easily extended t o deal with mixed foreign salts. Instead of plotting the theoretical curve and the experimental results as a function of log C of one foreign salt, they may be plotted as a function of the log of the ionic strength I of the mixture. Such a procedure can also serve as a test of the method, for if the method is sound the experimental points obtained by measurements in CaCla solutions, in MgCl, solutions, or in mixed solutions, should all fall on the same line. Figure 11 illustrates such a procedure in the case of the Orion electrode. A solution of 1.52 X 10-3M CaCla was
tested in varying concentrations of NaCl alone, in varying concentrations of MgClz alone, and in varying concentrations of a mixture of MgCla and NaCl, in the proportion of 1 mole of Mg to 10 moles of Na (similar to the proportion encountered in biological systems). U p to a n ionic strength of about 3 X lO-lM, the membrane is specific t o Ca+2. Similar results are obtained for the PVC, TBP, TTA membranes. The agreement between the theoretical curve and the experimental points seems fair. The agreement between the different kinds of experimental points (Na, Mg, mixture) seems to support the method described above. ACKNOWLEDGMENT We are greatly indebted to M. Anbar of the Isotope Department of the Weizmann Institute of Science, under whose direction this investigation was carried out, for his constant interest and advice. We are grateful to H. A. Saroff of the National Institutes of Health and S. Szapiro of the Weizmann Institute of Science for their helpful suggestions.
RECEIVED for review November 30, 1966. Accepted March 6, 1967. This paper is based on work performed under grant No. 5x5121 of the National Institutes of Health.
Quasi-Reversible and lrreversible Charge Transfer at the Tubular Electrode L. N. Klatt a n d W. J. Blaedel Department of Chemistry, Unicersity of Wisconsin, Madison, Wis. Current-potential equations for the quasi-reversible and totally irreversible heterogeneous charge transfer reactions at a tubular electrode have been theoretically derived and experimentally verified. The dependences of the current-potential curve upon the standard rate constant, transfer coefficient, the volume flow rate are shown graphically. A procedure for determining the standard rate constant and the transfer coefficient of a chemical system is presented. Magnitudes of rate constants determinable by this technique are about the same as those at the dropping mercury electrode.
HETEROGENEOUS ELECTRON TRANSFER reactions have received considerable attention since Eyring (I) presented the equations relating the rates of electrochemical reactions t o the electric field present a t the electrode-solution interface. Even though experimental investigation of these reactions would be greatly simplified if concentration polarization of the electrode could be eliminated, such a condition is almost impossible to achieve. As a result, quantitative theories concerned with electrode reactions have had to consider the rates of both mass and electron transfer. Early work in this direction was attempted by Eyring and coworkers (2), who applied the absolute rate theory and the Nernst diffusion layer concept t o the problem of ( 1 ) S. Glasstone, K. J. Laidler, and H. Eyring, “The Theory of Rate Processes,” McGraw-Hill, New York, 1941, pp. 575-7. (2) H. Eyring, L. Marker, and T. C. Kwoh, J . Phys. Colloid Chem., 53, 187 (1949).
polarographic current-potential curves. Several groups of investigators (3-5) developed independently the rigorous solution corresponding to the semi-infinite linear diffusion case. Koutecky (6) solved this problem for the polarographic case. The last decade has seen a tremendous development of new techniques such as stationary electrode polarography (7), fast rise potentiostatic techniques (8), relaxation methods (9), and double pulse galvanostatic methods (10-12), each capable of studying rapid heterogeneous electron transfer reactions. All of these theories and techniques dealt with time-dependent transient phenomena. Studies of heterogenous electron transfer reactions in hydrodynamic systems have been noticeably rare, partly because of the success of the time-dependent techniques, and perhaps partly because of difficulties associated with the mathematical treatment and experimental measurements in hydrodynamic systems. Delahay (13) gave a treatment based upon applica(3) M. Srnutek, Col(ection Czech. Chem. Commun., 18, 171 (1953). (4) P. Delahay, J . Am. Chem. Soc., 73,4944 (1951). (5) M. G. Evans and N. S. Hush, J . Chim. Phys., 49, C159 (1952). (6) J. Koutecky, Chem. Lisfy, 47, 323 (1953). (7) R. S. Nicholson and I. Shain, ANAL.CHEM.,36, 706 (1964). (8) S. P. Perone, ANAL.CHEM.,38, 1158 (1966). (9) P. Delahay and T. Berzins, J . Am. Chem. SOC.,77, 6448 (1955). (10) H. Z. Gerischer, Physik Chem., 10,264 (1957). (11) H. Z. Gerischer, Physik Chem., 14, 184 (1958). (12) P. Delahay, J . Am. Chem. SOC.,81, 5077 (1959). (13) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, 1954, pp. 222-6. VOL. 39, NO. 10, AUGUST 1967
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