aoa
Anal. Chem. 1984, 56,808-810
bolic reactions will allow fast and simple optimization of biological systems. ACKNOWLEDGMENT We wish to thank Jay Stern from Joy Manufacturing Co. for initiating and continuously encouraging this research. We also wish to express our appreciation to John Findley for his helpful remarks. Registry No. Sulfate, 14808-79-8;lactic acid, 50-21-5; acetic acid, 64-19-7; carbonate, 3812-32-6. LITERATURE C I T E D
0
5
io
15
20
25
30
minutes
(1) Postgate, J. R. “The Sulphate-Reducing Bacteria”; Cambridge University Press: Cambridge, 1979. (2) Rich, W.; Johnson, E.; Lois, L.; Kabra, P.; Stanford, B.; Marton, L. Clin. Chem. (W/flSfOn-S8/em,N.C.) 1980, 26, 1492. (3) Koppel, I. R., Paper Presented at the Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, 1981; Paper 241. (4) Rockiin, R.; Johnson, E. Anal. Chem. 1983, 55, 4 (5) Ai-Hiili, I. K.; Moody, G. R.; Thomas, J. D. R. Analyst (London) 1983, 108, 43. (6) Ai-HiIIi, I. K.; Moody, G. R.; Thomas, J. D. R. Analyst (London) 1983, 708, 1209.
Figure 1. Ion chromatograph readings: (a) initial (b) final (Dionex 20001 with ICE column connected to HP3390A integrator).
The accuracy of the Dionex ion chromatography system was up to i 5 % for the detection of sulfate, lactate, and acetate species and &lo% for the carbonate. This is satisfactory for an engineering design. In conclusion, ion chromatography can successfully contribute to the understanding and the exploration of bacterial metabolism patterns. Its ability to simultaneously quantify the major ions which are involved in various bacterial meta-
Alon Lebel T e h Fu Yen* Department of Civil Engineering Environmental Engineering Program University of Southern California Los Angeles, California 90089
RECEIVED for review October 24, 1983. Accepted January 4, 1984. We thank Joy Manufacturing Co. for providing the funding for this research.
Definition of the Response Time of Ion-Selective Electrodes and Potentiometric Cells Sir: One of the critical limiting factors in the use of ionselective membrane electrodes, especially in routine analysis, is their so-called response time. Uemasu and Umezawa ( I ) pointed out the logical paradox involved in the internationally accepted definitions (2,3)oft, and t*, namely, that one cannot determine t , and t* values without knowing E , (equilibrium potential) or t,, because t , or t* is defined as time required for the ion-selective electrode to reach a% of its equilibrium potential (2)or to become equal to its steady-state value within 1 mV (3),respectively, after a concentration step change in the sample. Since these definitions (2, 3) provide no aid to practical analytical work, Uemasu and Umezawa (1) defined a value, called differential quotient, t(At,aE),as a measure of practical response time and compared it with other conventional definitions. The differential quotient in fact is a limiting value of the slope of the potential-time curves and in this sense it was applied first by Lindner, Tdth, and Pungor (4) for characterizing and comparing the transient functions of ionselective electrodes. Pungor and Umezawa suggested a way to draw an unambiguous distinction between the response time of the potentiometric cell (containing an ion-selective electrode) and that of the ion-selective electrode itself ( 5 ) . These two response time values may become equal only under special experimental conditions (high mass transport
rate and low concentrations) (6) or in the case of ion-selective electrodes with relatively long response times (7-9). In Uemasu and Umezawa’s paper some characteristic features of the slope method suggested for determining response times have been summarized as follows: it has the advantage that “it is concentration independent when At is chosen properly short and AE is small” and “it has no direct relation to mathematical formulation of response curves like the time constant in an exponential” equation. The aim of the present paper is to discuss the above statements in more detail based on the simple diffusion model (7, IO), generally valid for describing the dynamic response of potentiometric cells under practical analytical conditions (low mass transport rate and high sample activities, i.e., relatively slow measuring setup and electrodes with short response times) ( 6 , I I ) . The diffusion model was selected to illustrate the fact that the differential quotient, Le., slope value has direct relation to the mathematical equation describing the response time curves as Uemasu and Umezawa worked under conditions fulfilling the diffusion model assumptions. In light of this, the activity dependence of the slope values will become obvious. Naturally the same treatment can be carried out for other types of mathematical equations also if different model assumptions are valid (e.g., for different types of electrodes (8) or other experimental conditions (13, 14)).
0003-2700/84/0356-0808$01.50/00 1984 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 4, APRIL 1984
-Table I. Theoretical Response Time Values Calculated with Equation 1 (Values, Seconds, Calculated with
activity increase 10 ai/flio = 100 ai/aio =
activlty decrease ai/aio = 0.1 ai/qo = 0.01
0.27 0.09
1.47 1.47
4.36 3.77
3.16 3.25
3.84 3.93
5.44 5.54
1.43 2.39
3.41 5.13
6.66 8.36
5.42 1.82
6.12 8.52
10.14
The mathematical back up of the activity dependence of the differential quotient under different limiting cases is discussed and classified into groups (depending upon users convenience), in order to justify the use of a differential quotient on a general base as the definition of response time of ISE's. The advantages, disadvantages, and other characteristic features involved in this definition of response time will also be summarized. By use of the activity step method for studying the dynamic characteristics of ISE's, the various experimental ways of investigations can be understood on the basis of the following equation:
aim= K a t
+ 2(ai" - a:)
a{ = q"
=
7.76
6.16
8.88
+ (aim- a:)(l
- e-t/7')
.( !gt .( 2%) =
10.37 10.47
10.37 11.06 12.67 12.77 13.46 15.07 Time constant value used for practical
7'.
cases: ba, activity increase, K >> 1;bp, activity decrease, K > 1
where B = (l/7')e-t/r', From eq 5a and 6a it follows that as long as the second term in the denominator can be neglected compared to unity, dE/dt should be proportional to E,but if this second term is much larger than one, dE/dt becomes independent of K. The first case is faced when the initial slope of response time curves is determined as lim erfc 6/2(D't)1/2 = erfc
erfc (6/2(D't)1/2)
a / dt
t-c
(2)
(3)
where 6 is the thickness of the diffusion layer, 7' = fi2/2D'and D'is the diffusion coefficient of ions within the aqueous layer. By inserting the surface activity values into the Nernst equation, we obtain the slope of the response time curves, given as
(
8.07
1s b ,
(1)
where a: and a," are the activity of the primary ion in the bulk of the sample solution prior to and following the activity step applied, respectively, and K is a proportionality factor. Thus, (a) The effect of the activity level (a: or aim)on the potential vs. time function can be studied at constant activity ratios ( K ) . For instance, when K = 10, we may apply an activity step from a? = to M or from a: = to M. (b) The effect of activity ratios ( K ) on the response-time curves can be studied a t a constant value of a: or aim.For instance when a: = lo4 M, we may apply an activity step to M ( K = 10) or to (K = 100). Naturally both (a and b type) types of the study can be carried out for activity increase ( K > 1)and decrease ( K < 1) in the same way. According to the diffusion model (7) the electrode potential is defined by the solution activity of the primary ion (a:) a t the electrode surface and the shape of potential-time functions is related to transport processes between the bulk of sample solution and the electrode surface. The primary-ion activity a t the electrode surface can be described by eq 2 and 3 for relatively short (12) and long (7) times, respectively
= a:
7' =
7.74
a In the case of t, and t* the response time data increase proportional with analytical conditions ( 7, 10).
a(
809
--
(4)
where t outside the brackets refers to a definite time value. In an earlier paper we showed that the slope of response time curves is activity level independent (case a, K = constant) as long as the diffusion of the primary ion is rate controlling (Figure 7 in ref 6). But when investigating the effect of activity ratios (case b, a? or a; is constant) one can distinguish between two limiting
m
=o
(7)
and lim (1 t-c
=0
(8)
while the second case holds when determining the slope a t the asymptotic part of the response time curves. The latter was proved experimentally by Uemasu and Umezawa (2) in determining the differential quotient of response time curves of copper(I1) ion selective electrodes (Table I1 in ref 1). At the same time Lindner et al. (6) showed experimentally that the initial slope values ( m ) increased proportionally with log K instead of K (eq 5a). Both experimental findings (ba case) can be interpreted on the basis of the same principle if one considers the concentration gradient within the adhering layer (Figure 5a in ref 6) a t different activity levels. Thus it becomes clear that in a series of experiments where ai" = constant and one applies activity steps a t different activities (e.g., lo4 10-l; lom1; 10-1 M; ...) after a short initial period (t') the
-
-
-
810
Anal. Chem. 1984, 56,810-813
surface activity (ai') and the concentration gradient within the adhering layer become the same, independently from the initial a? values. Thus in the introductory period (0 < t < t? the slope of the transient signals should be proportional to AE = E(t? - E(a:) = log aim/a:, i.e., to log K , as after t > t'the transient signals run on the same path. At bp for an activity decrease, when the activity steps directed to more diluted solutions K
