confidence limits, and the NBS certified value are given in Table I. Recovery Experiments. One of the advantages of the isotopic dilution technique of analysis is that only enough spiked sample required for an accurate isotope ratio determination need be recovered from the original sample. A knowledge of the recovery efficiencies is not required for computation of results of analysis. The efficiency of recovery of a preconcentration technique however, is of interest because the amount of recovery will effect the detection limits for the method. Experiments were performed in order to determine the recovery for the procedures up to and including the electrodeposition step. A set of samples was run using the previously described experimental procedure but omitting the spike additions. For comparison, electrodeposition periods of one hour were also used in addition to the 3-hour periods used in the procedure. The material recovered on the high-purity gold cathodes was
redissolved in 0.2 ml of warm HNOI, spiked with the stable isotopes, and neutralized with aqueous ammonia. The recovered trace elements were then electrodeposited onto the same gold electrodes. The electrodes were sparked in the SSMS and the amounts of material recovered on the electrodes were computed from Equation 1. This procedure gave results for Ag, Cu, and Ni; however, Mo could not be detected in this second plating. Increasing the plating time from one to three hours increased the total recovery for Ag, Cu, and Ni from 20 to 4 0 x . The high sensitivity of the spark source mass spectrograph method is illustrated by realizing that only 0.2 pg of silver on the gold electrodes was more than adequate for the determination of silver.
RECEIVED for review December 2, 1969. Accepted February 24, 1970.
Guggenheim Exponential Method for Unequal Intervals Robert L. Cleland Department of Chemistry, Dartmouth College, Hanover N. H . 03755
A PROPERTY y(x) which varies exponentially with x may be expressed where y(0) is the value of y(x) at x = 0, and p is a parameter independent of x. Many physical processes are described by Equation 1, including all first-order processes following differential equations of the form: dy(x)/dx = py(x). Experimental determination of the constant 0 requires explicit values of y(x), which are not always directly available from observations. First-order kinetic processes for which x is the time t and p is a negative constant -a, require that at infinite time y( 03 ) = 0. The observations will often consist of readings z(t), which reach a limiting asymptotic value z( m ), so that y ( t ) = z(t)
- z(m)
=
(2)
where B = z(0) - z( m ). A problem arises concerning the value of z( m ) when the latter cannot be observed experimentally. Guggenheim (1,2) showed long ago that measurement of z(t) at equal time intervals Tz led to a solution to this problem. A treatment is presented below for the more general case of unequal time intervals, which is applicable to a variety of experimental situations. Data are presented to demonstrate the precision which this solution can be expected to give in the determination of a. THEORETICAL TREATMENT
By subtraction of zl = z(tl) from zz = z(tz), we may eliminate z( a ) from Equation 2: Az
E
z2 - z1 = B(e-Qf’ - e-Qfl) = Be-Q‘f’(e-QTz- 1) (3)
(1) E. A. Guggenheim, Phil. Mug., 2 (7), 538 (1926). (2) D. P. Shoemaker and C. W. Garland, “Experiments in Physical Chemistry,” McGraw-Hill, New York, N. Y., 1962, p 222.
where TZ tz - tl. For a fixed interval Tz a series of pairs of measurements of z(t) may be plotted as In Az for each pair against t ( = t l for each pair) to yield -a as the slope of a straight line (Guggenheim method). Equation 3 may be transformed by use of the variable Tl = tl tz to give
+
-Az
=
2Be-QT1/2sinh(aTz/2)
Expansion of the hyperbolic sine in a power series in aTz and division by TZyields -Az/Te
=
Bae-QT1/2X
[I
+ (ffTz)’/24 f (ffTz)‘/1920 +
. ] (4)
The natural logarithm of Equation 4, correct to order ( C ~ T ~ ) ~ , is In (- R )
In( -Az/Tz) = ln(Ba) - aT1/2
+ ( a T ~ ) ~ / 2 4( 5 )
where R is the average rate of change of z during the time interval tl, tz. When aTz is sufficiently small, the final term in Equation 5 may be neglected. For successive pairs of measurements a plot of In ( - R ) against T1/2, the average time for a given pair, then has a slope -a and an intercept ln(Ba). The constant B and hence the value of z( m ) may be estimated from the intercept at Tl = 0. The quantity Az/Tz in Equation 4 is a finite difference approximation over the range tl, tzto dy(t)/dt = -Bae-Qf = - a y ( t )
(6)
which results from differentiation of Equation 2. Taking the slope of a plot of Equation 5 amounts, for small intervals Tz, to a second differentiation of the experimental data. Thus, differentiation of Equation 6 gives (dzy/dt2)/(dy/dt) = -a, which is just the derivative of Equation 5 with respect t o T1/2 as Tz -+ 0, TI + 2t. The Guggenheim methods are therefore suitable only for experimental data with good precision. ANALYTICAL CHEMISTRY, VOL. 42, NO. 6, MAY 1970
675
Table I. Comparison of Methods of Data Treatment Relative standard deviation Equation Minimum Maximum Average
(z)
2 5, modified Guggenheim 5, standard Guggenheim
0.06 0.7 2.0
0.7 7.0 3.9
0.27 2.7 3.0
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).
