Table I. Comparison of Methods of Data Treatment Relative standard deviation Equation Minimum Maximum Average 2 0.06 0.7 0.27 5, modified Guggenheim 0.7 7.0 2.7 5, standard Guggenheim 2.0 3.9 3.0
(z)
The magnitude of the error involved in neglecting the final term in Equation 5 may be examined by considering three readings at times tl, r2, and f3. For the interval tl, t2 let all quantities be defined as above; for the interval t2, r3 let Tl* = r3 - f 2 , A* = ( z ; - z2)/T2*. Further let t2 t3, T2* Aln R = In(-R*) - In(-R), A(T42) = (Tl* - T,)/2 and AT2 = T2* - T2. When Equation 5 is written for the interval f 2 , r3 t o give In( -R*), solution for Aln R and rearrangement yields
+
(7) which defines the slope given by these three measurements. As in the Guggenheim method, when equal time intervals T2 are used, no error results from the term in AT2 in Equation 7 . When unequal time intervals are used with a n average aT5of 0.02 or less, values of aAT2of the order 0.01 may occur. The error in slope of the plot, with neglect of the last term in Equation 5 , would then be of the order 0.1 Z. Because AT2 would typically have both positive and negative values over a series of measurements, such errors would tend t o cancel each other. For aT2 = 0.02, terms neglected in deriving Equation 5 are entirely negligible. We recall that a-l is the time required for y ( t ) to fall t o 1/ e of y(0). PRECISION AND ACCURACY OF GUGGENHEIM PLOTS
Analysis of data obtained in a study (3) of two-component diffusion in porous glass permits estimation of the precision to be expected from the standard and modified Guggenheim (3) R. L. Cleland, J. K. Brinck, and R. K. Shaw, J . Phys. Chem., 68,2779 (1964).
methods. Weights z(t) of porous disks suspended in liquid mixtures varied with time t o reach finally asymptotic limiting values z( rn ). The function y(t), after a n initial nonexponential period, varied exponentially with time, as required by Equation 2 . In these experiments a-l ranged from 55 t o 160 minutes; time intervals T2between readings were of the order 0.1 to 0.15 a-l in the parts of the experiments where rapid changes in z(t) occurred. Variations between readings ATz were of the order 0.02 to 0.05 a-l, so that the errors predicted by Equation 7 were less than OSZ. In a few experiments regular intervals Tz were used, as required by the standard Guggenheim technique. The data were fitted by the least-square method to Equation 2 in logarithmic form with use of the experimental value of z ( m ) and t o Equation 5 with omission of the final term. Values of a and the standard deviation of the fitted points were obtained in this way for 28 separate experimental runs. The relative standard deviations are given in Table 1. Based on the result of Equation 2 as the true result, the relative error in a from the Guggenheirn methods ranged from 10.3 to i 8 Z with an average difference of about These results indicate that the two Guggenheim methods are comparable in precision for this experimental technique, but that the calculation which uses z ( ~ directly ) is an order of magnitude better in precision. This difference is due to the use, mentioned above, of second derivatives of the experimental data in the Guggenheim techniques, and the consequent appearance of large fluctuations in In( -I?) introduced by random errors in the data. The Guggenheim method and its present modification may therefore be considered useful techniques when accuracies t o a few per cent in cy are acceptable. This will often be the case-e.g., in student experiments where time limitations on length of experiments exist. Sometimes instrumental arrangements also make direct evaluation of z( m ) impossible, and in these cases recourse to the Guggenheim techniques is essential. Where unequal intervals are involved, the present method is recommended.
+3z.
RECEIVED for review December 30, 1969. Accepted February 20, 1970. The author wishes to acknowledge the support of. this work in part by U S . Public Health Service Grant G M 081 13 from the National Institute of General Medical Sciences.
Activity Measurements Using a Potassium-Selective Liquid Ion-Exchange Electrode James N. Butler and Rima Huston Tyco Laboratories Inc., Waltham, Mass.02154 AVAILABILITY of a new liquid ion-exchange electrode system which has been reported t o be highly selective for potassium over sodium ( I ) has encouraged us t o study its thermodynamic behavior under conditions where quantitative measurements can be made rigorously (2, 3). The cell Ag/AgCl/K+, Na+Cl-/ion exchanger/K+, Cl-/AgCl/Ag (1) . , L. A. R. Pioda. V. Stankova. and W. Simon. Anal. Lett., 2, 665 (1969). (2) R. Huston and J. N. Butler, ANAL.CHEM.,41, 200 (1969). (3) Ibid., p 1695. 676
0
ANALYTICAL CHEMISTRY, VOL. 42, NO. 6, MAY 1970
was measured a t 25 O C over a wide range of compositions for the test (left-hand) solution. The reference electrolyte (right-hand compartment) was 0.01m KC1 saturated with AgCl and was held constant in composition throughout the measurements. The ion exchanger was obtained from Orion Research, Inc. (Type 92-19), and is believed t o consist of valinomycin in a n aromatic solvent (4). Potentials were measured using a Beckman Research p H meter with a digital voltmeter (Tyco DVM-404) as a readout device. Calibration of this system us. a n NBS-calibrated (4) M. S. Frant and J. W. Ross, Jr., Science, 167, 987 (1970).
Table 1. Measurements in KCl Solutions a t 25 “C AEobsr mV A E c a l e d r mV 0.579 166.6 f 1.6 180.8 0.581 119.6 i. 3.0 130.0 0.581 123.0 It 0 . l a 130.0 0.602 98.2 f 0 . 6 110.0 0.651 68.0 i 0 . 7 75.6 75.6 0.651 70.1 + O . l a 0 0.774 0 0.899 -95.5 f 0 . 9 -110.6 0.963 -201.0 i 0 . 8 -225.4 -342.4 0.989 -330.6 f 2.3 0.989 -332.8 i 0.3a -342.4
Error, mV -14.2 -10.4 -7.0 -11.8 -7.6 -5.5
Y=
mKCl
4.228 1.569 1.569 1.024 0.4854 0.4854 0.0936 0.00936 9.36 x 10-4 9.36 x 10-5 9.36 x 10-5 Fresh exchanger.
standard cell using a Leeds and Northrup Model K-3 potentiometer showed a maximum error of 0.2 mV in 200 mV. Routine calibration of the digital voltmeter was made using the slide wire of the Beckman pH meter, and this did not change more than 0.2 mV during the course of the day. The liquid ion-exchange electrode was transfmed back and forth between the test solution and a reference solution (usually O.lm KCl) and the potential recorded as a function of time. Equilibrium appeared to be reached within 5 min, except in the most dilute solutions. A stable potential was reached more quick11 if the electrode was shaken down (like a clinical thermometer) before it was wiped to remove solution. Matched Ag/AgCl electrodes prepared by the thermalelectrolytic process (5) were used in the two cells, and were equilibrated with the solution to be measured for several hours before the liquid ion exchange electrode was placed in the solution. Solutions were prepared by weight (except for concentrations below O.lm which were obtained by dilution) from ACS reagent grade salts (Fisher Certified). Concentrations were verified by potentiometric titration with standard AgN03. The NaCl used contained less than 0.005 % potassium and the KCl contained less than 0.005% sodium. Bromide content of both salts was less than 0.01 and iodide content was less than 0.002 %. The pH of the solutions was between 5.5 and 6, and thus no interference was expected from hydrogen or hydroxyl ion (4). All measurements were conducted in a water bath thermostated at 25.0 f 0.1 OC. The results of measurements in solutions containing only KCl are given in Table I. The potential differences (between the test solution and the 0.0936m reference solution) quoted are the average of four successive measurements (errors are standard deviations), except for those with a footnote, which were made with a fresh sample of the ion exchanger and are the average of two measurements. (For these the error is the range.) Mean activity coefficients for KC1 (y*) were interpolated from the tables of Robinson and Stokes (6) for concentrations O.lm and above; and calculated from the DebyeHuckel theory (with a = 4) for lower concentrations. The calculated values of potential were obtained using the Nernst equation:
x
(5) D. J. G. Ives and G. J. Janz, “Reference Electrodes,” Academic Press, New York, 1961. (6) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” Butterworths, London, 1959.
0
+15.1 $15.3 +11.8
+9.6
I 2o x
0.
0
E.*-
J-jLU-
+
o I
I
01
02
I 03
I 04
I
1
05
06
I 06
+ I 08
I 09
P
io
x2
Figure 1. Deviations of experimental measurements from calculated potential values based on the Nernst equation and isopiestic (7) data Total ionic strength I is indicated as follows: 0 0.095 m, 0.47 m, A 1.00 m, X 1.59 m, 0 4.15 m. Solid line calculated using a selectivity ratio K , = 2 X 31-‘
+
where m is the molal concentration of KCl, and E ois evaluated using the known concentration and activity coefficient of the reference solution. Systematic deviations from the Nernst equation are clearly apparent in the last column of Table I. These are in a direction opposite to the observed potential differences, and indicate that the liquid ion-exchanger is apparently somewhat permeable to chloride as well as to potassium ion. The deviations are smaller if the ion exchanger is fresh. At concentrations below 10-+m, other workers ( 4 ) showed that deviations are in the direction of still more positive potentials, indicating that the ion exchanger contributes a finite amount of potassium ion to the test solution. Empirically, a slope of approximately 90 of the Nernstian slope provides a relatively good fit (*2 mV) to the experimental data over the range from 0.001 to 4.3m. However, this slope depends on the age of the ion exchanger (e.g., 95 % for fresh and 35 % for aged material) and thus should be determined at the time of measurement if accurate analytical results are desired. The measurements made in NaCl-KC1 mixed electrolytes are summarized in Table 11. Each entry is the average of ANALYTICAL CHEMISTRY, VOL. 42, NO. 6, MAY 1970
677
Table 11. Mean Activity Coefficients of KCl in NaCl-KCI Electrolytes Total -log 7 2 1 ionic strength XZ - AE, mV -log 7 2 1 (corrected) K, 0,0936 1. 0000 0 0.1111 0.1115 ... 0.0940 0.6798 13.1 0.1403 0.1405 ... 0.0942 0.5061 17.6 0.1152 0.1153 ... 0.0946 0.2572 33.6 0.1045 0.1044 0.0948 0.1188 51.6 0.0897 0.0896 0. 0.0949 0.0434 76.0 0,0777 0.0776 0.0085 0.0949 0.0128 105.5 0.0624 0.0622 0.0035 0.0949 0.0012 141.0 ... 0.0029 0.4854 1.0000 0 0.1864 ... 0.4809 0.7309 7.0 0.1738 0.1732 0.4771 0.4984 16.4 0.1660 0.1658 ... 0.4735 0.2797 31.0 0.1610 0.1612 ... 0.4713 0.1409 48.4 0.1569 0.1573 0.0069 0.4702 0.0707 66.2 0.1568 0.1573 0.0032 0.4693 0.0177 96.6 0.1128 0.1133 0.0041 1.0241 1.0000 0 0.2204 0.2197 ... 1.0174 0.7056 7.7 0.2066 0.2062 1.0142 0.5608 13.3 0.2026 0,2023 ... 1.0073 0.2514 31.2 0.1772 0.1772 ... 1.0042 0.1094 50.2 0.1559 0.1561 ... 1.0030 0.0582 65.5 0.1479 0.1481 ... 1.0023 0.0225 86.7 0.1206 0.1209 ... 1.0018 0.0010 147.4 ... 0.0018 1,0018 0.0005 160.2 ... ... 0.0012 1.5687 1.0000 0 0.2360 0.2363 ... 1.5771 0.7265 5.8 0.2180 0.2181 ... 1.5843 0.4927 13.8 0.2034 0.2034 ... 1.5926 0.2271 29.7 0.1713 0.1713 ... 1.5947 0.1622 35.8 0.1505 0.1504 0.075 1.5978 0.0625 57.3 0.1266 0.1264 0.034 1.5991 0.0218 82.2 0.1082 0.1081 0.014 4.2279 1.0000 0 0.2375 0.2382 ... 4.2027 0.7780 4.6 0.2192 0.2196 ... 4.1756 0.5361 11.7 0.1957 0.1960 ... 4.1482 0.2887 23.3 0.1569 0.1567 ... 4.1299 0.1212 40.8 0.1139 0.1133 0.039 4.1229 0,0572 60.9 0.1201 0.1195 0.015 4.1182 0.0134 91.7 0.0644 0.0637 0.0065 Note: Component 1 is NaCI, component 2 is KCI. XZ = m / ( r n l m2). yzl values are corrected to ionic strengths of 0.0945, 0.4756, 1.0096, 1.5881, or 4.1605 (depending on the group) using isopiestic data to determine the correction factors (8). K, is calculated as described in text, A E is potential in test solution minus potential in reference solution (fist entry in each group). An error of 0.1 mV in E corresponds to an error of 0.0008 in log yzl, and therefore the last digit in columns 4 and 5 is not significant except for differences. .
.
t
.
.
I
+
of KC1 in the reference solution was obtained as described above. The deviations of the observed E M F values from the theoretical values are plotted in Figure 1. These were calculated assuming that the isopiestic data (7,8) for the mean activity coefficients of KC1 (y2J and NaCl ( n z ) in mixed electrolytes were correct. Individual measurements have been plotted to show the consistency of results in a given solution. At higher Na/K ratios, the deviations are always in the same direction (the observed potential is more positive than the calculated potential), consistent with the hypothesis of partial transport of Na+ along with K+. The selectivity ratio of this ion-exchange electrode has been reported ( 4 ) to be 2 X lo-' (for Na us. K). This value is obtained by comparing the potential of the electrode in a solution containing only KCl with that in a solution containing KCl at the same concentration together with 0.1M NaC1. A line calculated using this value is plotted in Figure 1. Our data can also be used to calculate a selectivity ratio, but at the high ionic strengths we are considering, we must take account of the differences in activity coefficient between NaCl and (720)
0005b // I
ob
OW5
0
$
0
+
8 I
I
0015
0010
I 0020
I
0025
XP
Figure 2. Extrapolation of selectivity ratio to zero potassium content four to eight separate measurements. For each ionic strength, the reference solution was the KCI stock solution (first entry in each group). The mean activity coefficient ( ~ 2 ~ of) KC1 in the mixed electrolyte was calculated assuming that the Nernst equation was obeyed with the theoretical slope and that the potassium-selective liquid ion exchanger did not respond at all to sodium ion (3). The mean activity coefficient 678
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ANALYTICAL CHEMISTRY, VOL. 42, NO. 6, MAY 1970
(7) R. A. Robinson, J . Phys. Chem., 65, 662 (1961). (8) R. M. Rush, AEC Report ORNL4402. Oak Ridge National Laboratory, Oak Ridge, Tenn., April 1969.
KCl in the same electrolyte mixture. We have defined the selectivity ratio (K,) by the equation (3) DT
where ml is the molal concentration of NaCl, m 2is the molal concentration of KCl, and the other symbols are as defined above. The values obtained for K, are listed in Table I1 for solutions where the Na/K ratio is greater than 5. For solutions consisting mostly of KCI, the uncertainties in the mean activity coefficient are much greater than the effects of selectivity . Note that the selectivity ratio is relatively independent of total ionic strength, but depends in a consistent way on the Na/K ratio. Extrapolation of K, to zero KCl Concentration is h w n in Figure 2. Values between 5 x and 4 x are consistent with the data. Although this is slightly higher than the published value (4), a slightly lower extrapolated value might be obtained if fresh exchanger were used for each measurement and correction were made for residual K+ in the NaCl.
Because the total ionic strength is held constant for each set of measurements, the gradient of chloride concentration across the ion exchange membrane is also constant, and thus deviations from Nernstian behavior due to chloride transport should contribute negligibly to these measurements. The fact that systematic deviations from the calculated potentials are observed (or alternatively, that the selectivity ratio is composition dependent) indicates that the transport mechanism is more complicated than might be supposed on the basis of a simple ion exchange model. ACKNOWLEDGMENT
The authors thank John C. Synnott for his assistance with the calculations and Martin Frant for providing unpublished data and for his helpful discussions of the results.
RECEIVED for review December 3, 1969. Accepted February 16, 1970. Work supported by the U.S . Department of the Interior, Office of Saline Water.
CORRESPONDENCE Calibration of Methanol Nuclear Magnetic Resonance Thermometer at Low Temperature SIR: In high resolution proton magnetic resonance, the temperature of a sample is commonly obtained by measuring the chemical shift of methanol. At low temperature, more hydrogen bonding occurs, and the O H peak shifts downfield. The temperature dependence has been determined with a thermocouple by Varian Associates ( I ) , and more accurately by Van Geet (2,3) using a static thermistor probe, in which the sample container can spin around the static probe, A cylindrical weight of polyperfluoroethylene (4) prevents lifting of the sample tube by the nitrogen flow which cools the sample. The static thermistor is more accurate than the spinning thermistor (5) which has a delicate coaxial electrical contact. CHEMICAL SHIFT OF METHANOL
We have measured the chemical shift Av (in Hz) between the CHI and OH groups down to the melting point of methanol, and have also repeated previous measurements (2). Part of the new results have been reported (3). A quadratic equation fits the data with an error (RMS) of 0.8 OK over the entire temperature range from 175 to 330 OK (convenient graphs are available from the author):
T
=
403.0
-
0.491 IAvI -66.2 (10-2Av)Z
(1)
(1) Varian Associates, Palo Alto, Calif. 94303, Publication Number 1481. (2) A. L. Van Geet, ANAL.CHEM., 40,2227 (1968). (3) A. L. Van Geet, Paper presented at the 10th Experimental N M R Conference, Mellon Institute, Pittsburgh, Pa., February 1969. (4) A. L. Van Geet, ANAL.CHEM.,40, 1914 (1968). (5) A. L. Van Geet, Reu. Sci.Instrum. 40, 177 (1969).
The coefficient of the quadratic term is small, and over a temperature range of 50 OK,the data can be fitted to a straight line : 175-225 OK: T = 537.4 - 2.3801AvI 220-270 OK: T = 498.4 - 2.083 Av
(2)
- 1.8101AvJ
(4)
I 1
265-313 OK:
T = 468.1
(3)
At the limits of these ranges, the straight line approximations still agree with Equation 1 within 0.7 OK. Shoup (6) used a thermocouple to measure the shift of methanol from 210 to 310 OK, and her results are in excellent agreement with Equation 1. Above 255 OK, Equation 1 is in excellent agreement with the previously reported result (2), but at 220 OK, the deviation has risen to 4 OK. The deviation results from the cooling of the sweep coil of the Varian A60 by the cold nitrogen gas, which causes the sweep width to drift (6). If one fits a straight line to the data of Equation 1 over’ the entire temperature range, one obtains very nearly Varian’s calibration chart ( I ) , but the fit is not good. Varian’s line intersects the curve of Equation 1 at 221 and at 312 OK. Near the extremes, at 190 and 325 OK, Varian’s chart gives a temperature higher by, respectively, 5.5 and 3.2 OK, while near the center of the range, at 270 OK, it gives 3.6 OK lower. At room temperature, the slope of the Varian line for methanol is 7 lower, leading to activation energies 7 lower. Above room temperature, ethylene glycol is frequently used to measure temperatures. We have also repeated (3) our (6) R. R. Shoup, National Institutes of Health, Bethesda, Md. 20014, private communication, 1968. ANALYTICAL CHEMISTRY, VOL. 42, NO. 6, MAY 1970
679
