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Flow injection analysis methods for determination of diffusion coefficients
ANALWICA
CHIME4 ACTA
ELSEVIER
Analytica Chimica Acta 350 (1997) 359-363
Flow injection analysis methods for determination of diffusion coefficients Gongwei Zou*, Zhen Liu’,
Congxiang
Wang
Department ofChemistry, Nanjing University, Nanjing 210093, China
Received 1 November 1996; received in revised form 12 March 1997; accepted 13 March 1997
Abstract ‘lko flow injection analyses (FIA) methods for the determination of diffusion coefficients in a straight single tube FIA system were developed. Based on the analytical solution of the convection-diffusion equation, linear relationships of the logarithmic values of the dispersion coefficient (0) and the half-peak width (IV& with the diffusion coefficient (D,) were obtained. Experiments were designed to verify these methods. For example, for potassium hexacyanoferrate (III) a D,,, value of 0.72 x 10’ cm2 s-l was found versus a literature value of 0.76 x IO5 cm2 s-l (error, 5%). For potassium hexacyanoferrate (II) a D, value of 0.67x lo5 cm2 s-l was obtained versus a literature value of 0.63 x 10’ cm2 s-l (error, 6%). The diffusion coefficients of some important biomedical compounds, such as dopamine, epinephrine, norepinephrine and ascorbic acid, were then determined. The values of lo5 D&m’ s-’ are 0.604~0.03, 0.44f0.02, 0.6O~tO.01 and 0.6StO.06, respectively. Keywords:
Plow injection
analysis;
Diffusion
coefficients
1. Introduction As a method for automated analysis and sample pretreatment, flow injection analysis(FIA) is already in widespread use in many fields, such as agriculture, industry, environmental science, clinical analysis, etc. Its great success lies in its accuracy, reliability and reproducibility as well as its ease of automation and its high sample throughput. In FIA, the reproducibility is achieved through dispersion-control under non-equilibrium and non-homogeneous mixing conditions.
*Corresponding author. ‘Present address: National Dalian Institute of Chemical 116011, China.
Chromatographic R & A Center, Phvsics. Academia Sinica. Dalian. _
0003-2670/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PI2 SOOO3-2670(97)00253-5
FIA response curves reflect not only information about sample concentration, but also kinetic information about sample dispersion. The research on the kinetics in FIA may lead to the exploitation of physical or chemical information from the response curves. In a classic paper by Taylor [l], the idea of using the dispersion of a solution plug in a laminar flow for measurements of the diffusion coefficient of the solution molecules was first presented. At that time, the FIA method had not been developed. Based on Taylor’s convection-diffusion model, Reijn et al. [2], Kolev et al.[3,4] investigated successfully the dispersion kinetics of the sample slug in FIA systems without chemical reaction. Their results are attractive as they offer FIA methods for evaluation of diffusion coefficients, in which the statistical moments of the
G. Zou et al./Analytica
360
Chimica Acta 350 (1997) 359-363
FIA peak [2] or curve-fitting [3,4] are used. These methods seem to be reliable. However, they are lacking in ease of appliciability because of the complexity of the data processing. For this reason it seems to be necessary to develop the quantitative relationship between the molecular diffusion coefficients and simple peak parameters (such as peak height, half-peak width, baseline-to-baseline peak width, etc.). Vanderslice et al.[5] presented an empirical relationship between the baseline-to-baseline peak width and the diffusion coefficients. Gerhardt and Adams [6] confirmed this relationship and determined the diffusion coefficient of some biochemical compounds. However, since the peak in FIA is usually asymmetric, it is difficult to decide the exact point at which the signal disappears. The determination of peak width might be accompanied with a significant error. This report concentrates on the relationships of the dispersion coefficient and the half-peak width with the diffusion coefficients. Two FIA methods are suggested for the determination of diffusion coefficients. Subsequently, the diffusion coefficients of some important biomedical compounds are determined by these methods.
2. Theory
concentration gradient; 3. convective dispersion that deforms the sample zone. Taylor [l] was the first to put forward the convection-diffusion equation that includes three dispersion actions:
(1) where D, is the molecular diffusion coefficient, C is the concentration of the sample, r is the distance from the central axis, X is the distance along reaction tube, t is time, U is the mean linear flow rate and R is the inner radius of sample tube and reaction tube. For the case where the sample is injected as a very small slug of length L, and the reaction tube length L, is relative long, the analytical solution to the convection-diffusion equation has been given [7-91 in the following form: C,(L,
t)/Cc = i 1
erf k (L, + L, - Ut)K-‘/2t-‘/2 [
1
-L)K-‘/2t-‘/2 I> (2) [(Uf
+erf i where
The flow of fluids in a straight single tube FIA system is usually laminar. Therefore, the convectiondiffusion equation can be used to describe the sample dispersion in the FIA system without chemical reaction. Under laminar flow conditions, the velocity distribution across the tube is parabolic. There are three factors that control the dispersion of an injected sample (shown in Fig. 1): 1. radial diffusion due to the radial concentration gradient caused by the parabolic velocity profile; 2. axial diffusion due to the axial t>o
110
-L*
0
X
Fig. 1. Sample dispersion in a lam&u-flow carrier stream. R is the inner radius of sample tube and reaction tube; 17 is the mean linear rate of flow; L, is the length of sample tube; L, is the length of reaction tube, X is the distance along the tube.
R GdX,4
rC(X, r, t)dr
=$
(3)
s 0
and K = R2U2/48D,
(4)
Cm (X, t) is the mean concentration over the crosssection of the tube at position X and time t, COis the initial concentration, and K can be regarded as the effective diffusion coefficient of the sample under laminar flow conditions. Eq. (2) describes the Fcurve, a plot of sample concentration detected at the exit of reaction tube as C/Cc versus time, of the FIA system. Eq. (2) is valid only under the restriction [8] L,D,,,/UR2 > 0.8, but for some specially designed FL4 experiments this condition can be satisfied. When the center of the sample zone passes the detector, the F-curve reaches a maximum. The time
G. Zou et al. /Analytica
Chimica Acta 350 (1997) 359-363
at the maximum (t,) can be approximately calculated from the location of the center of the sample zone, which is expressed as follows: x=UtP-;Ls
(5)
So tp can be given as the expression: $I = (LT +;L,)lu Substituting
(6)
Eq. (6) into Eq. (2) gives
(2) max= erf [iLsKe1f2t;1f2] = en[~~-‘L~(L,+~L,)-“2D~2]
(7)
According to Ruzicka’s definition [lo], the dispersion coefficient (D) is expressed as D=C&‘,,, where C,,, is the maximum concentration detected from the peak of response curve. The error function can be calculated by means of the following equation: erf(x) =2
fi
3 1x5 ~--~~+~~-5?f~$-...
[
.
,
1x7
.
The
.
9
1 63)
value of erf(x) can be given approximately by 2x/fi for x c 0.2 (error is less than 0.3%). Therefore, when D 2 5, the dispersion coefficient can be expressed as follows: l/2
D=
D,‘f2
(9)
Eq. (9) shows that the 1ogD versus logD, curve is linear and the slope is -l/2. A computer program based on numerical method was designed to investigate relationship between halfpeak width and diffusion coefficient from Eq. (2). The following results were obtained: (1) When L,
+ A
(10)
where A and B are constants for given U, Lr, L, and R values, B is usually -0.5 to -0.4. A increases with increase of L,.lU, while B decreases with increase
361
Table 1 Theoretical
values of A and B in different FIA systems
Condition a
YU
(s)
A WJ
B
(s)
200
5
-1.66
-0.491
200
2
-1.71
-0.495
100
5
-1.91
-0.482
100
2
-2.01
-0.489
75
5
-1.98
-0.476
75
2
-2.13
-0.488
50
5
-2.07
-0.467
50
2
-2.26
-0.481
25
2
-2.45
-0.466
10
1
-2.67
-0.441
5
0.5
-2.71
-0.408
a R=0.025
cm; such a value is usually used.
of &NJ. Some theoretical values of A and B are listed in Table 1. (2) When .L,flOL, Eq. (10) is invalid, and Wi,2 is independent of 0,. Eq. (9) and Eq. (10) suggest two FIA methods for the determination of diffusion coefficient: (1) Dispersion coefficient method: at high dispersion (D>S), the diffusion coefficient can be measured directly from Eq. (9). For this purpose, Eq. (9) can be rewritten in the following way: 1ogD = - ;logD,
+f
(11)
where f is a combined factor that includes all the parameters of the FIA system employed. The f value can be obtained theoretically according to Eq. (9). However, it can also be obtained experimentally as a calibration factor from a standard whose D, value is well known. Once the f value is calculated, the diffusion coefficient can be evaluated from Eq. (11). (2) Half-peak width method: under the condition of L,<_LJlO, the diffusion coefficient of the sample can be calculated theoretically from Eq. (10) from the measured value of WI,. In practice, there are two approaches for calculating the values of A and B. One is a single-point calibration method in which B is considered as a constant that can be obtained theoretically and A is obtained as a calibration factor from a standard. The other is a double-point calibration
362
G. Zou et al. /Analytica
Chimica Acta 350 (1997) 359-363
method in which both A and B are considered as calibration factors and calculated from two standards.
3. Experimental 3.1. Instrumentation The numerical calculation program was written in QUICK BASIC language and run on an IBM personal computer. A Micro feeder (Azumadenkikogyo Co. Ltd., Japan, Model MF-2) was employed to deliver the carrier solution at a steady flow rate. The flow rates varied by adjusting the magnitude of syringe and feeding speed and ranged from 0.1 to 10.0 cm/s. A sampling valve (Rheodyne, Cotati, California, USA, Model 7125-075), fitted with a 25 cm or 10 cm x 0.5 mm i.d. sampling loop was used to inject the sample solutions. The system flow path was a 250 cm x 0.5 mm i.d. straight PTFE tube. The detection system was composed of a voltammeter (New Medical Science Institute, Chengdu, China, Model VAD-D) and a self-made thin-layer amperometric detector (its configuration was described in detail in Ref. [ 111) with a glassy carbon disk working electrode and a flow cell of about 1 uL. A chart recorder (Dahua Instrument and Meter Plant, Shanghai, China, Model XWT-204) was utilized to record the current response. The experiments were performed at ambient temperature of 2551°C controlled by an air-conditioner. 3.2. Reagents Potassium hexacyanoferrate (III) and (II) were chosen as internal standards whose diffusion coefficients are well known [ 12-151, whose concentration were l-5 mmol L-’ in 1 mol L-’ aqueous KC1 solution. All solutions were prepared with double-distilled water and A.R. grade chemicals. The standard and sample solutions were prepared by dissolving required quantities of commercial chemicals in the flow carriers. The concentration of the sample solutions (ranging from 1 to 10 mg mL-‘) were chosen to match those of the calibration of standards. All the solutions were deoxygenated with nitrogen to decrease air oxidation of the solutions.
3.3. Procedure In the dispersion coefficient method, after obtaining the peak currents of samples or standards, the carrier solutions were replaced by the sample or standard solutions to obtain plateau currents. Then the D values were calculated by dividing the plateau currents by the peak currents of FIA. In the half-peak width method, all the sample and standard solutions were injected through the sampling valve.
4. Results and discussion The experimental data for hexacyanoferrate(I1) and (III) ions were explored by cross-checking. Both potassium hexacyanoferrate(I1) and (III) solutions were used to obtain calibration factors. Then the calibration factor of the hexacyanoferrate(I1) ion was used to calibrate an experimental D, value for hexacyanoferrate(II1) ion and vice versa. The results obtained are shown in Table 2. It is clear that the results are in excellent agreement with each other. The error in the D, values relative to literature values is about 6%, which is in general acceptable for the determination of diffusion coefficients. Thus, the two FIA methods for determination of diffusion coefficient were verified. Using the single-point calibration half-peak width method, the diffusion coefficients of some important biomedical compounds, such as dopamine, epinephrine, norepinephrine and ascorbic acid were determined. The corresponding data reported by Gerhardt and Adams [6] using the peak width method Table 2 Comparison
of literature and experimental
D, values a
Compound
Detecting Dm,erp. (1O-5 cm%) potential b (V vs. SCE) method 1’ method 2 d
D,,la. ’ ( 10e5 cm%)
KsFe(CN)s We(CN)s
-0.20 +0.80
0.76 0.63
0.72f0.01 0.67f0.01
0.72f0.01 0.67f0.02
a Temperature was 25fl”C, concentration was about 1.0x 10e4 to 1.0x 10e5 mol L-’ in 1 mol L-’ aqueous KC1 solution, flow rate was 3.5 cm/s, number of determinations was 8. b Ref. [16]. ’ Dispersion coefficient method. d Single-point calibration half peak width method. e Refs. [12-151.
G. Zou et al. /Analytica Table 3 Diffusion coefficients
of four biomedically
important
compounds
Compound
Detecting potential b (V vs. SCE)
D msxpt. ( 10m5 cm%)
Dm,lit. ’ (1 0d5 cm’/s)
dopamine epinephrine norepinephrine ascorbic acid
+oso +0.54 +0x +0.80
0.60f0.03 0.44*0.02 0.60f0.01 0.68f0.06
0.60 0.51 0.55 0.53
Chimica Acta 350 (1997) 359-363
a
’ Temperature was 25fl”C, concentration was 1 to 10 mg mL_’ in 0.1 mol L-’ phosphate buffer (pH 7.4), flow rate was 3.5 cm/s, number of determinations was 8. b Refs. [17,18]. ’ Ref. [6].
are listed in Table 3. Except for the D, value of dopamine, the D, results from this work appear to differ from the previously reported values. However, since the determination of peak-widths might be accompanied by a relatively large error, the half-peak width method is preferable to the peak width method. Theoretically, knowing the exact value of all the parameters in Eq. (9) and Eq. (lo), diffusion coefficients can be calculated directly. However, calibration factor methods are often employed because the experimental system usually deviates from the ideal conditions. For example, the sample tube is not straight, and the inner radius of the sample tube and the reaction tube are not matched exactly. Furthermore, the injection process and the disturbances in the tube joints contribute to the total dispersion. All these effects result in an extra dispersion. which can be proved by the fact that the D, value calculated directly from Eq. (9) is about half of that obtained from the singlepoint calibration method. For the single-point calibration of the half-peak width method, the theoretical value of B can be taken. However, the double-point calibration method can be employed without the theoretical values of A or B. In the double-point calibration, the results show the same relationship between WI,2 and D, as Eq. (10). In the case that L, is 250 cm, L, is 10 cm, R is 0.025 cm and U is 3.4 cm/s, the theoretical values of A and B are -2.11 and -0.486, respectively, while the corresponding experimental values are -2.00 and -0.504. It is clear that the experimental results are in good agreement with the theoretical values. And thus, the D, value can
363
be measured by the double-point calibration method even though the theoretical values of A and B were unknown. The present methods allow for a quick and precise estimation of the molecular diffusion coefficient from some simple FIA parameters such as dispersion coefficient and half-peak width, which can be easily measured. An additional advantage of these methods is that the experimental conditions can slightly deviate from the ideal conditions of the theoretical model. The disadvantage of the methods which should be also stressed is that the use of a relative long straight tube leads to the difficulty in thermostating. However, this drawback can be overcome by the use of a rather narrow bore tube (or capillary), where only a short reaction tube is needed.
References VI G. Taylor, Proc. R. Sot. London, Ser. A, 219 (1953) 186. 121J.M. Reijn, W.E. van der Linden and H. Poppe, Anal. Chim. Acta, 126 (1981) 1. [31 SD. Kolev and E. Pungor, Anal. Chem., 60 (1988) 1700. [41 S.D. Kolev and W.E. van der Linden, Anal. Chim. Acta, 268 (1992) 7. r51 J.F. Vanderslice, K.K. Stewart, A.G. Rosenfeld and D.J. Higgs, Talanta, 28 (1981) 11. @I G. Gerhardt and R.N. Adams, Anal. Chem., 54 (1982) 2618. r71 A. Boumia, J. Coull and G. Houghton, Proc. R. Sot. London, Ser. A, 261 (1961) 227. AIChE J., 13 (1967) 801. PI W.N. Gill and V Ananthakrishnan, [91 G. Zou and Z. Liu, Chem. J. Chinese Univ., 16 (1995) 1185. r101 J.Ruzicka and E.H.Hansen, Flow Injection Analysis, 2nd ed., Wiley, New York, 1988, Chinese Version, Translated by S. Xu, Peking University Press, Peking, 1991, p. 23. Ull G. Zou, R. Shi, X. Xie, H. Cheng and R. Neeb, Anal.Chim.Acta, 208 (1988) 337. M. Pilgram and V.Z. Toome, WI M. Von Stackelberg, Elektrocbem, 57 (1953) 342. at Solid Electrodes, Marcel u31 R. N. Adams, Electrochemistry Dekker, New York, 1969, p. 214.. [I41 M. Eisenberg, C.W. Tobias and C.R. Wilke, J. Electrochem. Sot., 101 (1954) 306. Hydrodynamics, Prentice-Hall, r151 V.G. Levich, Physicochemical Englewood Cliffs, NJ, 1962, p. 326. 1161 J. Yang, Z. Yao and Z. Gao, Anal. Chim. Acta, 246 (1991) 341. Cl71 M. Goto, G. Zou and D. Ishii, J. Chromatogr., 275 (1983) 271. D81 W. Ye, Chinese J. Chromatogr., 7 (1990) 159.
CHIME4 ACTA
ELSEVIER
Analytica Chimica Acta 350 (1997) 359-363
Flow injection analysis methods for determination of diffusion coefficients Gongwei Zou*, Zhen Liu’,
Congxiang
Wang
Department ofChemistry, Nanjing University, Nanjing 210093, China
Received 1 November 1996; received in revised form 12 March 1997; accepted 13 March 1997
Abstract ‘lko flow injection analyses (FIA) methods for the determination of diffusion coefficients in a straight single tube FIA system were developed. Based on the analytical solution of the convection-diffusion equation, linear relationships of the logarithmic values of the dispersion coefficient (0) and the half-peak width (IV& with the diffusion coefficient (D,) were obtained. Experiments were designed to verify these methods. For example, for potassium hexacyanoferrate (III) a D,,, value of 0.72 x 10’ cm2 s-l was found versus a literature value of 0.76 x IO5 cm2 s-l (error, 5%). For potassium hexacyanoferrate (II) a D, value of 0.67x lo5 cm2 s-l was obtained versus a literature value of 0.63 x 10’ cm2 s-l (error, 6%). The diffusion coefficients of some important biomedical compounds, such as dopamine, epinephrine, norepinephrine and ascorbic acid, were then determined. The values of lo5 D&m’ s-’ are 0.604~0.03, 0.44f0.02, 0.6O~tO.01 and 0.6StO.06, respectively. Keywords:
Plow injection
analysis;
Diffusion
coefficients
1. Introduction As a method for automated analysis and sample pretreatment, flow injection analysis(FIA) is already in widespread use in many fields, such as agriculture, industry, environmental science, clinical analysis, etc. Its great success lies in its accuracy, reliability and reproducibility as well as its ease of automation and its high sample throughput. In FIA, the reproducibility is achieved through dispersion-control under non-equilibrium and non-homogeneous mixing conditions.
*Corresponding author. ‘Present address: National Dalian Institute of Chemical 116011, China.
Chromatographic R & A Center, Phvsics. Academia Sinica. Dalian. _
0003-2670/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PI2 SOOO3-2670(97)00253-5
FIA response curves reflect not only information about sample concentration, but also kinetic information about sample dispersion. The research on the kinetics in FIA may lead to the exploitation of physical or chemical information from the response curves. In a classic paper by Taylor [l], the idea of using the dispersion of a solution plug in a laminar flow for measurements of the diffusion coefficient of the solution molecules was first presented. At that time, the FIA method had not been developed. Based on Taylor’s convection-diffusion model, Reijn et al. [2], Kolev et al.[3,4] investigated successfully the dispersion kinetics of the sample slug in FIA systems without chemical reaction. Their results are attractive as they offer FIA methods for evaluation of diffusion coefficients, in which the statistical moments of the
G. Zou et al./Analytica
360
Chimica Acta 350 (1997) 359-363
FIA peak [2] or curve-fitting [3,4] are used. These methods seem to be reliable. However, they are lacking in ease of appliciability because of the complexity of the data processing. For this reason it seems to be necessary to develop the quantitative relationship between the molecular diffusion coefficients and simple peak parameters (such as peak height, half-peak width, baseline-to-baseline peak width, etc.). Vanderslice et al.[5] presented an empirical relationship between the baseline-to-baseline peak width and the diffusion coefficients. Gerhardt and Adams [6] confirmed this relationship and determined the diffusion coefficient of some biochemical compounds. However, since the peak in FIA is usually asymmetric, it is difficult to decide the exact point at which the signal disappears. The determination of peak width might be accompanied with a significant error. This report concentrates on the relationships of the dispersion coefficient and the half-peak width with the diffusion coefficients. Two FIA methods are suggested for the determination of diffusion coefficients. Subsequently, the diffusion coefficients of some important biomedical compounds are determined by these methods.
2. Theory
concentration gradient; 3. convective dispersion that deforms the sample zone. Taylor [l] was the first to put forward the convection-diffusion equation that includes three dispersion actions:
(1) where D, is the molecular diffusion coefficient, C is the concentration of the sample, r is the distance from the central axis, X is the distance along reaction tube, t is time, U is the mean linear flow rate and R is the inner radius of sample tube and reaction tube. For the case where the sample is injected as a very small slug of length L, and the reaction tube length L, is relative long, the analytical solution to the convection-diffusion equation has been given [7-91 in the following form: C,(L,
t)/Cc = i 1
erf k (L, + L, - Ut)K-‘/2t-‘/2 [
1
-L)K-‘/2t-‘/2 I> (2) [(Uf
+erf i where
The flow of fluids in a straight single tube FIA system is usually laminar. Therefore, the convectiondiffusion equation can be used to describe the sample dispersion in the FIA system without chemical reaction. Under laminar flow conditions, the velocity distribution across the tube is parabolic. There are three factors that control the dispersion of an injected sample (shown in Fig. 1): 1. radial diffusion due to the radial concentration gradient caused by the parabolic velocity profile; 2. axial diffusion due to the axial t>o
110
-L*
0
X
Fig. 1. Sample dispersion in a lam&u-flow carrier stream. R is the inner radius of sample tube and reaction tube; 17 is the mean linear rate of flow; L, is the length of sample tube; L, is the length of reaction tube, X is the distance along the tube.
R GdX,4
rC(X, r, t)dr
=$
(3)
s 0
and K = R2U2/48D,
(4)
Cm (X, t) is the mean concentration over the crosssection of the tube at position X and time t, COis the initial concentration, and K can be regarded as the effective diffusion coefficient of the sample under laminar flow conditions. Eq. (2) describes the Fcurve, a plot of sample concentration detected at the exit of reaction tube as C/Cc versus time, of the FIA system. Eq. (2) is valid only under the restriction [8] L,D,,,/UR2 > 0.8, but for some specially designed FL4 experiments this condition can be satisfied. When the center of the sample zone passes the detector, the F-curve reaches a maximum. The time
G. Zou et al. /Analytica
Chimica Acta 350 (1997) 359-363
at the maximum (t,) can be approximately calculated from the location of the center of the sample zone, which is expressed as follows: x=UtP-;Ls
(5)
So tp can be given as the expression: $I = (LT +;L,)lu Substituting
(6)
Eq. (6) into Eq. (2) gives
(2) max= erf [iLsKe1f2t;1f2] = en[~~-‘L~(L,+~L,)-“2D~2]
(7)
According to Ruzicka’s definition [lo], the dispersion coefficient (D) is expressed as D=C&‘,,, where C,,, is the maximum concentration detected from the peak of response curve. The error function can be calculated by means of the following equation: erf(x) =2
fi
3 1x5 ~--~~+~~-5?f~$-...
[
.
,
1x7
.
The
.
9
1 63)
value of erf(x) can be given approximately by 2x/fi for x c 0.2 (error is less than 0.3%). Therefore, when D 2 5, the dispersion coefficient can be expressed as follows: l/2
D=
D,‘f2
(9)
Eq. (9) shows that the 1ogD versus logD, curve is linear and the slope is -l/2. A computer program based on numerical method was designed to investigate relationship between halfpeak width and diffusion coefficient from Eq. (2). The following results were obtained: (1) When L,
+ A
(10)
where A and B are constants for given U, Lr, L, and R values, B is usually -0.5 to -0.4. A increases with increase of L,.lU, while B decreases with increase
361
Table 1 Theoretical
values of A and B in different FIA systems
Condition a
YU
(s)
A WJ
B
(s)
200
5
-1.66
-0.491
200
2
-1.71
-0.495
100
5
-1.91
-0.482
100
2
-2.01
-0.489
75
5
-1.98
-0.476
75
2
-2.13
-0.488
50
5
-2.07
-0.467
50
2
-2.26
-0.481
25
2
-2.45
-0.466
10
1
-2.67
-0.441
5
0.5
-2.71
-0.408
a R=0.025
cm; such a value is usually used.
of &NJ. Some theoretical values of A and B are listed in Table 1. (2) When .L,flO
+f
(11)
where f is a combined factor that includes all the parameters of the FIA system employed. The f value can be obtained theoretically according to Eq. (9). However, it can also be obtained experimentally as a calibration factor from a standard whose D, value is well known. Once the f value is calculated, the diffusion coefficient can be evaluated from Eq. (11). (2) Half-peak width method: under the condition of L,<_LJlO, the diffusion coefficient of the sample can be calculated theoretically from Eq. (10) from the measured value of WI,. In practice, there are two approaches for calculating the values of A and B. One is a single-point calibration method in which B is considered as a constant that can be obtained theoretically and A is obtained as a calibration factor from a standard. The other is a double-point calibration
362
G. Zou et al. /Analytica
Chimica Acta 350 (1997) 359-363
method in which both A and B are considered as calibration factors and calculated from two standards.
3. Experimental 3.1. Instrumentation The numerical calculation program was written in QUICK BASIC language and run on an IBM personal computer. A Micro feeder (Azumadenkikogyo Co. Ltd., Japan, Model MF-2) was employed to deliver the carrier solution at a steady flow rate. The flow rates varied by adjusting the magnitude of syringe and feeding speed and ranged from 0.1 to 10.0 cm/s. A sampling valve (Rheodyne, Cotati, California, USA, Model 7125-075), fitted with a 25 cm or 10 cm x 0.5 mm i.d. sampling loop was used to inject the sample solutions. The system flow path was a 250 cm x 0.5 mm i.d. straight PTFE tube. The detection system was composed of a voltammeter (New Medical Science Institute, Chengdu, China, Model VAD-D) and a self-made thin-layer amperometric detector (its configuration was described in detail in Ref. [ 111) with a glassy carbon disk working electrode and a flow cell of about 1 uL. A chart recorder (Dahua Instrument and Meter Plant, Shanghai, China, Model XWT-204) was utilized to record the current response. The experiments were performed at ambient temperature of 2551°C controlled by an air-conditioner. 3.2. Reagents Potassium hexacyanoferrate (III) and (II) were chosen as internal standards whose diffusion coefficients are well known [ 12-151, whose concentration were l-5 mmol L-’ in 1 mol L-’ aqueous KC1 solution. All solutions were prepared with double-distilled water and A.R. grade chemicals. The standard and sample solutions were prepared by dissolving required quantities of commercial chemicals in the flow carriers. The concentration of the sample solutions (ranging from 1 to 10 mg mL-‘) were chosen to match those of the calibration of standards. All the solutions were deoxygenated with nitrogen to decrease air oxidation of the solutions.
3.3. Procedure In the dispersion coefficient method, after obtaining the peak currents of samples or standards, the carrier solutions were replaced by the sample or standard solutions to obtain plateau currents. Then the D values were calculated by dividing the plateau currents by the peak currents of FIA. In the half-peak width method, all the sample and standard solutions were injected through the sampling valve.
4. Results and discussion The experimental data for hexacyanoferrate(I1) and (III) ions were explored by cross-checking. Both potassium hexacyanoferrate(I1) and (III) solutions were used to obtain calibration factors. Then the calibration factor of the hexacyanoferrate(I1) ion was used to calibrate an experimental D, value for hexacyanoferrate(II1) ion and vice versa. The results obtained are shown in Table 2. It is clear that the results are in excellent agreement with each other. The error in the D, values relative to literature values is about 6%, which is in general acceptable for the determination of diffusion coefficients. Thus, the two FIA methods for determination of diffusion coefficient were verified. Using the single-point calibration half-peak width method, the diffusion coefficients of some important biomedical compounds, such as dopamine, epinephrine, norepinephrine and ascorbic acid were determined. The corresponding data reported by Gerhardt and Adams [6] using the peak width method Table 2 Comparison
of literature and experimental
D, values a
Compound
Detecting Dm,erp. (1O-5 cm%) potential b (V vs. SCE) method 1’ method 2 d
D,,la. ’ ( 10e5 cm%)
KsFe(CN)s We(CN)s
-0.20 +0.80
0.76 0.63
0.72f0.01 0.67f0.01
0.72f0.01 0.67f0.02
a Temperature was 25fl”C, concentration was about 1.0x 10e4 to 1.0x 10e5 mol L-’ in 1 mol L-’ aqueous KC1 solution, flow rate was 3.5 cm/s, number of determinations was 8. b Ref. [16]. ’ Dispersion coefficient method. d Single-point calibration half peak width method. e Refs. [12-151.
G. Zou et al. /Analytica Table 3 Diffusion coefficients
of four biomedically
important
compounds
Compound
Detecting potential b (V vs. SCE)
D msxpt. ( 10m5 cm%)
Dm,lit. ’ (1 0d5 cm’/s)
dopamine epinephrine norepinephrine ascorbic acid
+oso +0.54 +0x +0.80
0.60f0.03 0.44*0.02 0.60f0.01 0.68f0.06
0.60 0.51 0.55 0.53
Chimica Acta 350 (1997) 359-363
a
’ Temperature was 25fl”C, concentration was 1 to 10 mg mL_’ in 0.1 mol L-’ phosphate buffer (pH 7.4), flow rate was 3.5 cm/s, number of determinations was 8. b Refs. [17,18]. ’ Ref. [6].
are listed in Table 3. Except for the D, value of dopamine, the D, results from this work appear to differ from the previously reported values. However, since the determination of peak-widths might be accompanied by a relatively large error, the half-peak width method is preferable to the peak width method. Theoretically, knowing the exact value of all the parameters in Eq. (9) and Eq. (lo), diffusion coefficients can be calculated directly. However, calibration factor methods are often employed because the experimental system usually deviates from the ideal conditions. For example, the sample tube is not straight, and the inner radius of the sample tube and the reaction tube are not matched exactly. Furthermore, the injection process and the disturbances in the tube joints contribute to the total dispersion. All these effects result in an extra dispersion. which can be proved by the fact that the D, value calculated directly from Eq. (9) is about half of that obtained from the singlepoint calibration method. For the single-point calibration of the half-peak width method, the theoretical value of B can be taken. However, the double-point calibration method can be employed without the theoretical values of A or B. In the double-point calibration, the results show the same relationship between WI,2 and D, as Eq. (10). In the case that L, is 250 cm, L, is 10 cm, R is 0.025 cm and U is 3.4 cm/s, the theoretical values of A and B are -2.11 and -0.486, respectively, while the corresponding experimental values are -2.00 and -0.504. It is clear that the experimental results are in good agreement with the theoretical values. And thus, the D, value can
363
be measured by the double-point calibration method even though the theoretical values of A and B were unknown. The present methods allow for a quick and precise estimation of the molecular diffusion coefficient from some simple FIA parameters such as dispersion coefficient and half-peak width, which can be easily measured. An additional advantage of these methods is that the experimental conditions can slightly deviate from the ideal conditions of the theoretical model. The disadvantage of the methods which should be also stressed is that the use of a relative long straight tube leads to the difficulty in thermostating. However, this drawback can be overcome by the use of a rather narrow bore tube (or capillary), where only a short reaction tube is needed.
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