American Journal of Analytical Chemistry, 2010, 2, 73-82
doi:10.4236/ajac.2010.12010 Published Online August 2010 (
Copyright © 2010 SciRes. AJAC
Application of H-Point Standard Addition Method and
Multivariate Calibration Methods to the Simultaneous
Kinetic-Potentiometric Determination of
Cerium(IV) and Dichoromate
Mohammad Ali Karimi1,2*, Mohammad Hossein Mashhadizadeh3,
Mohammad Mazloum-Ardakani4, Fatemeh Rahavian2
1Department of Chemistry & Nanoscience and Nanotechnology Research Laboratory (NNRL),
Faculty of Sciences, Payame Noor University (PNU), Sirjan, Iran
2Department of Chemistry, Faculty of Sciences, Payame Noor University (PNU), Ardakan, Iran
3Department of Chemistry, Tarbiat Moallem University of Tehran, Tehran, Iran
4Department of Chemistry, Faculty of Sciences, Yazd University, Yazd, Iran
E-mail: &
Received June 27, 2010; revised July 20, 2010; accepted July 26, 2010
A kinetic-potentiometric method for simultaneous determination of Cerium(IV) and dichoromate (Cr2O72-)
by H-point standard addition method (HPSAM), partial least squares (PLS) and principal component regres-
sion (PCR) using fluoride ion-selective electrode (FISE) is described. In this work, the difference between
the rate of the oxidation reaction of Fe(II) to Fe(III) in the presence of Ce4+ and Cr2O72- is based of the
method. The rate of consume fluoride ion for making complex is detected with a FISE. The results show that
simultaneous determination of Ce4+ and Cr2O72- can be done in their concentration ranges of 1.0-30.0 and
0.1-20.0 µg/mL, respectively. The total relative standard error for applying the PLS and PCR methods on 8
synthetic samples was 2.97 and 3.19, respectively in the concentration ranges of 1.0-30.0 µg/mL of Ce4+ and
0.1-20.0 µg/mL of Cr2O72-. In order for the selectivity of the method to be assessed, we evaluated the effects
of certain foreign ions upon the reaction rate and assessed the selectivity of the method. The proposed meth-
ods (HPSAM, PLS and PCR) were evaluated using a set of synthetic sample mixtures and then applied for
simultaneous determination of Ce4+ and Cr2O72- in different water samples.
Keywords: Multivariate Calibration, HPSAM, Simultaneous Kintic-Potentiometry, Dichromate, Cerium(IV)
1. Introduction
Dichromate (Cr2O72-) and cerium(IV) are strong oxidants
in chemistry that are used widely as oxidizing agent in
diverse chemical reactions in the laboratory and industry
for the synthesis of many different kinds of chemical
compounds. Dichromate is used as a common inorganic
chemical reagent, most commonly used as an oxidising
agent in various laboratory and industrial applications
[1,2]. It also used to oxidize alcohols, determination
ethanol, cleaning laboratory glassware of organic con-
taminants [3]. Cerium(IV) is also a common inorganic
chemical reagent and industrially important and is used
in nuclear reactors, alloys with nickel and chromium,
microwave devices, lasers, agriculture and miscellaneous
[4-6]. In analytical chemistry, standardized aqueous
solutions of Cr2O72- and Ce4+ are sometimes used as
oxidizing titrants for redox titrations. Therefore, deter-
mination of these oxidants is very important. Several
methods have been reported for the determination of
them such as spectrophotometry [7,8], spectrofluorome-
try [9] and electroanalytical techniques [10,11]. In rela-
tion to the simultaneous analysis of Cr2O72- and Ce4+,
Masin et al. have been reported a titrimetric method for
the simultaneous determination of Ce4+, MnO4- and
Cr2O72- using phosphate ion as precipitant of Ce4+ and
Fe2+ as titrant [12]. To the best our knowledge, there is
not any report for simultaneous determination of Cr2O72-
and Ce4+ using HPSAM and chemometrics methods.
In recent years the usage of chemometrics methods in
electroanalytical chemistry, as in other areas of analytical
chemistry, has received considerable attention as these
methods can help us with extraction of more information
from experimental data. Applications of HPSAM and
chemometrics methods have been frequently reported for
the calibration of overlapped voltammetric signals [13-
16]. In the field of potentiometry, several methods have
been reported based on flow injection system and titra-
tion using PLS, ANN and Kalman filter as modeling
methods [17-21]. We are reported the first application of
PLS and PCR multivariate calibration methods and
HPSAM to the simultaneous kinetic-potentiometric de-
termination of binary mixtures of hydrazine and its de-
rivatives [22,23] and binary mixture of levodopa and
carbidopa drugs [24]. The methods were based on the dif-
ferences observed in the production rate of chloride ions in
reaction of these species with N-chlorosuccinimide. The
reaction rate of production of chloride ion was monitored
by a chloride ion-selective electrode. Recently, we also
reported the application of HPSAM, PLS and PCR
methods for the simultaneous determination of binary
mixtures of Fe(III) and Al(III) and ternary mixtures of
Fe(III), Al(III) and Zr(IV) [25,26]. This method was
based on the complex forming reaction of these metallic
ions with fluoride ion that has a differential rate at cer-
tain reaction conditions. Therefore, the rate of fluo-
ride-ion reaction with Fe(III), Al(III) and Zr(IV) was
monitored by an fluoride ion-selective electrode (FISE).
This paper reports the first application of HPSAM,
PCR and PLS to the simultaneous determination of oxi-
dants binary mixtures such as Cr2O72- and Ce4+ using po-
tentiometric technique. The methods are based on the
difference observed in the oxidation reaction rate of Fe(II)
to Fe(III) in the presence of Ce4+ and Cr2O72- as oxidants
and complexing reaction between Fe(III) with fluoride
ion at certain reaction conditions. The very fast response
of the fluoride ion selective electrode (FISE) and its
Nernstian behavior with respect to fluoride ions in acidic
solutions indicated that this electrode might be employed
effectively in kinetic studies of reactions involving
changes in the fluoride ion concentration [25,26]. There-
fore, rate of the complexing reaction of fluoride ion with
Fe(III) was monitored by a FISE.
2. Experimental
2.1. Apparatus and Software
A solid-state Fluoride-selective electrode (Metrohm
Model 6.0502.150) was used in conjunction with a dou-
ble junction Ag/AgCl reference electrode (Metrohm
Model 6.0726.100), whose outer compartment was filled
with a saturated KCl solution. The Metrohm Model 780
potentiometer, attached to a Pentium (IV) computer, was
used for recording the kinetic potentiometric data. All
measurements were carried out in a thermostated (25.0 ±
0.2˚C), double-walled reaction cell with continuous
magnetic stirring. The electrode was stored in 1 × 10-3
mol/L potassium fluoride solution when not in use. For
pH measurements, a Metrohm Model 780 pH meter with
a combination glass electrode was used. PLS and PCR
analyses was performed using a MATLAB 7.0 software.
2.2. Materials and Reagents
All chemicals were of analytical reagent grade and dou-
bly distilled water was used throughout. A stock solution
of iron (1000 µg mL-1) was prepared by dissolving 0.524
g of iron (II) sulfate (FeSO4·7H2O) in water and diluted
to 100 mL. Stock solutions Ce4+and Cr2O72- were pre-
pared in 100-mL flasks by dissolving 0.2885 g of cerium
sulfate tetrahydrate and 0.1369 g of potassium dichro-
mate in water and diluting with water to the mark. Ce-
rium(IV) sulfate and potassium dichromate and salts of
Fe(II) and fluoride were purchased from Merck (Ger-
many). Acetate buffer solution (0.05 mol/L, pH 3.0) was
prepared using acetic acid and NaOH solutions and ad-
justing its pH with a pH meter.
2.3. Procedure
Twenty five milliliters of double distilled water, 2.0 mL
of buffer solution, 1.0 mL of standard fluoride solution
(0.1 mol/L) and 1.0 mL of 5×10-3 mol/L of iron (II) solu-
tion were added to the thermostated (25.0 ± 0.2˚C) reac-
tion cell. Five milliliter of the standard or sample solu-
tion of Cr2O72-, Ce4+ or a mixture of them were injected
into the cell quickly, and after the stabilization of the
potential (about 30 s), all data were recorded. The poten-
tial changes versus time were recorded at the time inter-
vals of 1.0 s. Synthetic samples containing different
concentration ratios of Cr2O72- and Ce4+ were prepared
and standard additions of Cr2O7 2- were made. Simulta-
neous determination of Cr2O72- and Ce4+ was conducted
by recording the potential changes for each solution from
10 to 500 s. After each run the cell was emptied and
washed twice with doubly distilled water.
Using the standard analyte solutions, we can construct
a calibration graph of (10E/S-1) versus concentration
(fixed-time method) [22-27], where E is the potential
variation in a selected time interval t and S is the slope
of the fluoride electrode response, which is determined
periodically by successive additions of micro-amounts of
100 µL of 1.0 × 10-5 -1.0 mol/L of NaF standard solu-
tions in 25 mL of water mixed with 2 mL of buffer solu-
The simultaneous determination of Cr2O72- and Ce4+
standard solutions with HPSAM was performed by
measuring the potential changes (E) at 60 and 80 s after
Copyright © 2010 SciRes. AJAC
initiation of the reaction for each sample solution. Then
plots of HPSAM of (10E/S-1) versus added concentration
of Ce4+ were constructed for mixtures of Cr2O72- and Ce4+.
Simultaneous determination of Cr2O72- and Ce4+ with
PLS and PCR methods was performed by recording the
potential for each solution from 10 to 500 s.
3. Results and Discussion
It is require finding the system that shows different ki-
netic behavior for the reaction with Cr2O72- and Ce4+. It is
well-known that the rate of oxidation of Fe2+ with
Cr2O72- is much higher than that with Ce4+ [12]. There-
fore, we could use from this different kinetic behavior
for simultaneous determination of Cr2O72- and Ce4+. The
concept for the simultaneous analysis of Cr2O72- and
Ce4+- in this work is based on the difference in their oxi-
dizing power. Preliminary studies showed that iron(II)
ion as reagent at presence of fluoride ion using FISE is
suitable for our purpose. Upon the addition of the oxi-
dant (Ce4+ and Cr2O72-) into solution of Fe2+ in the pres-
ence of F¯, the oxidation reaction of Fe2+ by Ce4+ and
Cr2O72-take place as follows:
Ce4+ + Fe2+ + 2 F¯ [FeF2]+ + Ce4+ (1)
Cr2O72- + 2 Fe2+ + 4 F¯ + 14 H+
2 [FeF2]3+ + 2 Cr3+ + 7 H2O (2)
When using this ions and FISE for monitoring differ-
ence in the reaction rate, the linear range and differences
of reaction rate for both two species (Cr2O72- and Ce4+)
were suitable. In order to simultaneous kinetic potenti-
ometric determination of Cr2O72- and Ce4+ by HPSAM,
PCR and PLS a series of experiments were conducted to
establish the optimum system to achieve maximum sen-
sitivity. Therefore, all experimental parameters affecting
the reaction rate of Fe2+ with Ce4+ and Cr2O72- (response
time, concentration of F¯ and Fe2+, pH, etc) were care-
fully optimized.
3.1. Study of the Electrode Characteristics
The very fast response of FISE and its Nernstian behav-
ior toward fluoride ions in acidic solutions indicates that
this electrode might be employed effectively in the ki-
netic studies of reactions involving changes in the fluo-
ride ion concentration [25,26]. The characteristics of the
fluoride-selective electrode in the HNO3-H3BO3-NaOH
buffer were studied. In order to evaluate the operating
characteristics of the FISE at pH < 5, calibration graphs
were constructed for sodium fluoride in the concentration
range of 1.0 × 10-5-1.0 × 10-1 mol/L at pHs 4.0, 3.0, 2.5
and 2.0. The slope was found to be 56.9 mV/decade and
remained almost constant to 0.2 mV over 6 months of
usage in this system at pH 3.0.
3.2. Effect of Fluoride Concentration
The effect of F concentration over the ranges of 1.0 ×
10-5-1.0 × 10-1 mol/L fluoride ion on the linear range of
calibration graph and reaction rate with Fe(II) was inves-
tigated (Figure 1). The results also indicate that the con-
centration of F has a great effect on the linear range and
the change potential value. When the concentration of F
is low, a gradual slope in the calibration graph is realized
while a high concentration of F produces is high a steep
slope in the calibration graph. So the fluoride concentra-
tion must be in excess, but by increasing the fluoride
concentration, the potential change is decreased and the
sensitivity is lower. Since, maximum differences in ki-
netic behaviour of Fe(III) (resulted from oxidation reac-
tion of Fe2+ to Fe3+ by Ce4+ and Cr2O72-) and was ob-
served in concentration of 1.0 × 10-3 mol L-1 fluoride and
both species also had larger values of potential change
(E) in this concentration. Therefore, a concentration of
1.0 × 10-3 mol/L fluoride was selected as the optimum
concentration for further studies.
3.3. Effect of Fe2+ Concentration
The effect of Fe2+ concentration over the ranges of 1.0 ×
10-5-1.0 × 10-1 mol/L Fe2+ ion on the reaction rate of Fe2+
with Ce4+ and Cr2O72- and linear range of calibration
graph was investigated (Figure 2). The results have
shown that the increase of Fe2+concentration, up to 5.0 ×
10-3 mol/L, causes an increase in the reaction rate of Fe2+
with both Ce4+ and Cr2O72- and the potential change, but
decreased at higher concentrations. There- fore, a con-
centration of 5.0 × 10-3 mol/L Fe2+ ion was selected as
the optimum concentration for further studies.
3.4. Effect of pH
Acidity of the solution influences potential response of
Figure 1. Effect of F¯ concentration on the reaction of F¯
and Fe2+ with 5.0 μg/mL of Ce4+ () and 10.0 μg/mL of
Cr2O72- (). Conditions: 5 × 10-3 M Fe2+, pH 3.0, 25 °C.
Copyright © 2010 SciRes. AJAC
-Log [Fe
Figure 2. Effect of Fe2+ concentration on the reaction of F¯
and Fe2+ with 5.0 μg/mL of Ce4+ () and 10.0 μg/mL of
Cr2O72- (). Conditions: 1 × 10-3 M F¯, pH 3.0, 25 °C.
FISE, the oxidation reaction rate of Fe2+ with Ce4+ and
Cr2O72- and complextion reaction rate of F with Fe3+.
The results show that maximum difference in kinetic
behavior of Fe3+ was observed at pH 3.0 (Figure 3). In
addition, Fe3+ had larger values of potential change (E)
in this pH. Above pH 3.0, the potential change decreased
evidently due to the occurrence of the hydrolysis reaction
competing with the complexation between fluoride and
Fe3+, and under pH 3.0, the potential change decreased
too, probably owing to the formation of hydrogen fluo-
ride, to which the fluoride electrode is insensitive. Thus,
pH of 3.0 was selected as the optimum pH for further
3.5. Composition Effect of Ground Buffer
The change of potential value (
E) for reaction of Fe2+
with Ce4+ and Cr2O72- in the presence of certain amount
fluoride ion in different acidic solutions was investigated
(Table 1). The results have shown that in the solution of
HNO3–H3BO3 (pH 3.0),
EFe3+ has larger values. Ac-
cording to obtained results, the 0.1 mol L-1 boric acid-
0.1 mol/L nitric acid- mixed solution (pH 3.0) containing
1.0 × 10-3 mol/L fluoride was chosen as the ground
buffer solutions.
3.6. Temperature Effect of Reaction Rate
The temperature of solution evidently affects the reaction
rate of the kinetic reaction. But higher temperatures do
not have a positive effect on the reaction of Fe2+ with
Ce4+ and Cr2O72- and complexing reaction of Fe3+ with
fluoride. Therefore, the temperature of solution was kept
at 25 ± 0.2˚C by thermostatic water bath in all of the
3.7. Potential-Time Behavior
The potential-time behavior of reactions of Fe2+ with
Ce4+ and Cr2O72- in the presence of Funder the opti-
mized conditions is shown in Figure 1. Figure 2 shows
typical reaction curves for the reaction of Fe2+ with Ce4+
and Cr2O72- at different concentrations. As can be seen in
Figures 4 and 5, the reaction of Cr2O72- is faster Ce4+ and
was almost completed in 60 s after initial reaction but the
reaction of Ce4+ was completed almost in 500 s. This
difference in the reaction rates allowed us to design the
HPSAM, PCR and PLS methods for simultaneous de-
termination of Ce4+ and Cr2O72-. Characteristics of cali-
bration graphs for the determination of Ce4+ and Cr2O72-,
under the optimum conditions, are also given in Table 2.
Therefore, above difference mentioned in the reaction
rates allowed us to design the HPSAM, PCR and PLS a
method for simultaneous determination of Ce4+ and
Cr2O72-, in the concentration ranges of 1.0-30.0 and 0.1-
20.0 µg/mL, respectively.
3.8. Requirements for Applying HPSAM
The basis of using HPSAM for treating kinetic data un-
der the conditions that the reaction of one component is
Figure 3. Effect of pH on the reaction of F¯ and Fe2+
with 5.0 μg/mL of Ce4+ () and 10.0 μg/mL of Cr2O72- ().
Conditions: 1 × 10-3 M F¯, 5 × 10-3 M Fe2+, 25 °C.
0100200 300400 500
Time / Sec
E / m
5 mV
Figure 4. Potential-time curves for the reaction of F and
Fe2+ with 10.0 μg/mL of Cr2O72-(a), 5.0 μg/mL of Ce4+ (b)
and mixture of them (c).
Copyright © 2010 SciRes. AJAC
Table 1. The values of
E for reaction of 10 µg/mL of Fe2+
with Ce(IV) and Cr2O72- in the presence of 1.0 × 10-3 M F
ion in different acid solutions (pH = 3.0).
Acid solution H3PO4-H3BO3 KCl-HCl HNO3-
EFe3+ (for
Cr2O72-)/mv 2.3 7.5 8.5 13.0
EFe3+ (for
Ce4+/mv 1.8 5.5 6.7 10.4
Table 2. Characteristics of calibration graphs for the de-
termination of Ce4+ and Cr2O72-.
Species Linear
range Slope Intercept
(µg/mL) (mL/µg) (µg/mL)
Ce4+ 1.0-30.0 0.0681 0.0177 0.9995 (n =
10) 0.450
Cr2O72- 0.1-20.0 0.1119 0.2975 0.9994 (n =
10) 0.086
Figure 5. Typical potential-time curves for the reaction of
F- and Fe+2 with Ce4+(a) and Cr2O72- (b) at different con-
centrations (μg/mL).
completed, while that of other component is not com-
pleted yet, is described below. In this case, the vari- ables
to be fixed were the time variables t1 and t2, at which the
product of the reaction of Ce4+ had the same amount of
(10E/S-1) over the range between these two times, and
also there is an appropriate difference between the slopes
of the calibration lines.
Considering a binary mixture of Ce4+ and Cr2O72-, for
example, assume that the amount of R (10E/S-1) of the
oxidation in the reaction of Fe2+ with Ce4+ and then com-
plexation in the reaction of Fe3+ with F at time variables
t1 and t2 are Pi and Qi, respectively, while those for the
Cr2O72--Fe2+-F reaction under the same conditions are P
and Q, respectively (Figure 6). They are equal in this
The following equations show the relation between
For Ce4+: Q
i = Pi + mitj (t1 tj t2; i = 0,1,, n) (1)
For Cr2O72-: Q
= P + mtj (m = 0) (2)
where subscripts i and j denote different solutions for n
additions of Ce4+ concentration prepared to apply to
HPSAM and the time comprising the t1-t2 range, respec-
Thus, the overall amounts of (10E/S-1) (or R) of the
Ce4+-Cr2O72- mixture are:
At t1 Rt1 = P + Pi (3)
At t2 Rt2 = Q + Qi (4)
Simultaneous kinetic determination of concentration
of Ce4+ and Cr2O72- by HPSAM requires the selection of
two times t1 and t2. To select the appropriate times, the
following principles were observed. At the two selected
times t1 and t2, the amount of R of the Ce4+ must be linear
with the concentrations, and the amount of R for Fe3+
must remain constant even if the Cr2O72- concentrations
are changed. The amount of R for the mixture of Ce4+
and Cr2O72- should be equal to the sum of individual Rs
of the two compounds. In addition, the slope difference
of the two straight lines obtained at both t1 and t2 must be
-0. 5
050100 150200 250
Time / Sec
E / S
P +P
Figure 6. Plot of potential changes (10E/S-1) for the reaction
of F- and Fe2+ with 10.0 μg/mL of Ce4+ (a), 5.0 μg/mL of
Cr2O72- (b) and mixture of them (c).
Copyright © 2010 SciRes. AJAC
as large as possible to achieve good accuracy. Then
known amounts of Ce4+ are successively added to the
mixture and resulting potential changes are measured at
the two times and expressed:
Rt1 = (10E(t1)/S-1)t1 = P0 + P + Mt1Ci (5)
Rt2 = (10E(t2)/S-1)t2 = Q0 + Q + Mt2Ci (6)
where E(t1) and E(t2) are the potential changes meas-
ured at t1 and t2, respectively. P0 and Q0 are the amounts
of R for Ce4+ at a sample at t1 and t2, respectively. P and
Q are the amounts of R for Cr2O72- at t1 and t2, respec-
tively (Figure 7).
Mt1 and Mt2 are the slopes of the standard addition
calibration lines at t1 and t2, respectively. Ci is the added
Ce4+ concentration. The two obtained straight lines in-
tersect at the so-called H-point (-CH, RH), which at point
H (Figure 7), since Rt1 = Rt2, H(-CH, RH) (-CCeric, RDi-
chromate) from Equations (1) and (2) we have:
P0 + P + Mt1(-CH) = Q0 + Q + Mt2(-CH) (7)
-CH = [(Q – P) + (Q0 – P0)]/(Mt1 –Mt2) (8)
as species Cr2O72- is assumed not to evolve over the con-
sidered range of time,
Q = P
CH = (Q0 – P0)/(Mt1 – Mt2) (9)
which is equivalent to the existing CCeric (=P0 /Mt1 = Q0/
Mt2). Combining this with Equation (5) yields RH = P.
The overall equation for the potential at the H-point is
simply represented as:
Q = P = RH = RFe (10)
The intersection of the straight lines (Equations (5)
and (6) directly yields the unknown Ce4+ concentration
(CCeric) and the R for Cr2O72- species (RDichromate) corre-
sponding to t1 and t2 in the original samples, as the two
times were chosen in such a way that the later species
had the same R at both times. This analytical signal en-
ables us to calculate the concentration of Cr2O72- from a
calibration curve.
Since Ce4+ is selected as the analyte, it is possible to
select several pairs of time ranges which present the
same R for Cr2O72-. Some of the selected time pairs were
60-80, 80-110, 150-200, 250-300 and 350-420 s. Greater
time increments caused higher sensitivity and steeper
slopes of the two time axes, as shown previously by
Campins-Falco et al. [29]. Also, the accuracy of deter-
minations was affected by the slope increments of H-
point plots. However, the time pair that gives the greatest
slope increment, lower error, and shortest analysis time
was selected. For this reason, the time pair of 60-80 s as
the most suitable times was employed.
A summary of the results obtained for various analyte
concentrations is given in Tables 3 and 4. The concen-
tration was calculated directly by solving a system of
equations of two straight lines. Cr2O72- concentrations
were calculated in each test solution by the calibration
method with a single standard and ordinate value of R.
3.9. Multivariate Calibration
Multivariate calibration consists of establishment of a
relationship between matrices of chemical data. These
methods are based on a first calibration step in which a
mathematical model is built using a chemical data set
-20 -15 -10-50510
added (Dichromate)
/ µg mL
ΔE / S
Figure 7. Plot of HPSAM for simultaneous determination of
a mixture of Cr2O72- (15 μg/mL) and Ce4+ (25 μg/mL).
Table 3. Results of several experiments for analysis of Ce4+
and Cr2O72- mixtures in different concentration ratios using
HPSAM (T = 25˚C).
(μg/mL) Found (μg/mL)
R-C equation r
Ce4+ Cr2O72- Ce4+ Cr2O72-
R80 = 0.0322Ci +
0.4665 0.99945.0 2.0 4.90 ± 0.102.12 ± 0.08
R60 = 0.0509Ci +
0.5069 0.9990
R80 = 0.0209Ci +
0.3056 0.99903.0 3.0 3.14 ± 0.113.10 ± 0.10
R60 = 0.0286Ci +
0.3334 0.9999
R80 = 0.0164Ci +
0.5153 0.99915.0 10.0 5.16 ± 0.139.85 ± 0.18
R60 = 0.0239Ci +
0.5880 0.9990
R80 = 0.0057Ci +
0.9900 0.993013.0 15.0 13.20 ± 0.215.32 ±
R60 = 0.0153Ci +
1.1312 0.9970
R80 = 0.0052Ci +
1.5450 0.999920.0 10.0 19.76 ±
10.12 ±
R60 = 0.0155Ci +
1.6895 0.9930
R80 = 0.0101Ci +
1.9061 0.995025.0 15.0 24.70 ±
15.38 ±
R60 = 0.0058Ci +
1.8365 0.9995
R80 = 0.0014Ci +
0.9954 0.992015.0 5.0 15.27 ±
0.23 5.34 ± 0.15
R60 = 0.0107Ci +
1.0391 0.9992
Copyright © 2010 SciRes. AJAC
Table 4. Results of six replicate experiments for analysis of
Ce4+ and Cr2O72- mixture using HPSAM (T = 25ºC).
Spiked (μg/mL) Found (μg/mL)
R-C equation r
Ce4+ Cr2O72- Ce4+ Cr2O72-
R80 = 0.1090Ci
+ 1.0720 0.9902 13 15 12.98(99.8) 15.51(103.4)
R60 = 0.0046Ci
+ 0.9736 0.9900
R80 = 0.0094Ci
+ 1.0276 0.9980 13 15 12.79(98.3) 14.75(98.3)
R60 = 0.0040Ci
+ 0.9479 0.9999
R80 = 0.0153Ci
+ 1.1312 0.9980 13 15 13.10(100.
7) 14.80(98.6)
R60 = 0.0057Ci
+ 0.9900 0.9933
R80 = 0.0104Ci
+ 1.0300 0.9990 13 15 12.70(97.7) 14.40(96.0)
R60 = 0.0056Ci
+ 0.9611 0.9930
R80 = 0.0154Ci
+ 1.1488 0.9930 13 15 13.30(102.
3) 14.80(98.6)
R60 = 0.0056Ci
+ 1.0034 0.9940
R80 = 0.0156Ci
+ 1.0520 0.9990 13 15 12.60(96.9) 15.0(100.0)
R60 = 0.0055Ci
+ 0.9567 0.9940
Mean 13.01 14.87
SD 0.28 0.39
RSD (%) 2.16 2.60
(e.g., potential values) and a concentration matrix data
set [30-34]. The calibration is followed by a prediction
set in which this model is used to estimate unknown
concentrations of a mixture from kinetic profile. Multi-
variate calibration methods are being successfully ap-
plied to the multi-component kinetic determination to
overcome some of the drawbacks of classical methods.
The theories and applications of chemometrics methods
such as PCR and PLS, to the analysis of multi-component
mixtures, have been discussed by several workers [31-
35]. PCR and PLS modeling are powerful multivariate
statistical tools, which are successfully applied to the
quantitative analysis of spectrochemical and electro-
chemical data [33-36]. The first step in the simultaneous
determination of the species by PCR and PLS method-
ologies involves the construction of calibration matrix
for the binary mixture of Ce4+ and Cr2O72-. For con-
structing the calibration set, factorial design was applied
to five levels in order to extract a great deal of quantita-
tive information, using only a few experimental trials. In
this research, a synthetic set of 34 solutions, including
different concentrations of Ce4+ and Cr2O72-, was pre-
pared. A collection of 26 solutions was selected as the
calibration set (Table 5) and the other 8 solutions were
used as the prediction set (Table 6). Their composition
was randomly designed to obtain more information from
the calibration procedure. Changes in the solution poten-
tial were recorded during a time period of 500 seconds.
To select the number of factors in the PCR and PLS
algorithm, as a cross-validation method, leaving out one
sample method was employed [37]. The prediction error
was calculated for each species of the prediction set. This
error was expressed as the prediction residual error sum
of squares (PRESS):
Where m is the total number of calibration sample, Ĉi
represents the estimated concentration while Ci is the
reference concentration for the ith sample left out of the
Table 5. Calibration set for constructing PCR and PLS
methods in determination of Ce4+ and Cr2O72- (μg/mL).
Sample Ce4+ Cr2O72- Sample Ce4+ Cr2O72-
1 5.0 2.0 14 22.0 15.0
2 5.0 6.0 15 22.0 18.0
3 5.0 12.0 16 1.0 0.5
4 5.0 20.0 17 1.0 6.0
5 10.0 10.0 18 1.0 4.0
6 10.0 15.0 19 25.0 17.0
7 10.0 19.0 20 25.0 19.0
8 13.0 17.0 21 25.0 20.0
9 13.0 20.0 22 18.0 7.0
10 3.0 3.0 23 18.0 15.0
11 3.0 9.0 24 15.0 5.0
12 3.0 17.0 25 15.0 9.0
13 22.0 14.0
Table 6. Prediction set for constructing PLS and PCR
methods in determination of Ce4+ and Cr2O72-.
Predicted (μg/mL)
Solution Synthetic
(μg/mL) PLSa PCRa
Ce4+ Cr2O72- Ce4+ Cr2O72- Ce4+ Cr2O72-
1 15.80(105.3) 13.70(105.3) 15.60(104.0)
2 2.78(92.6)13.00(100.0)
3 4.80(96.0)9.90(99.0)
4 18.50(102.7) 13.50(103.8)17.80(98.8)
5 12.60(96.9)15.30(95.6)
6 15.70(98.1) 22.60(102.7)15.40(96.3)
7 9.80(98.0)13.10(100.7)
8 3.05.0 3.20(106.6)5.30(106.0) 3.10(103.3)4.90(98.0)
recovery 100.2 101.3 99.8 99.1
RMSEP 2.97 3.40 3.03 2.90
RSEP(%) 3.19 2.97
a Predicted mean (recovery percent)
Copyright © 2010 SciRes. AJAC
Copyright © 2010 SciRes. AJAC
Al3+, Fe3+, Zr4+, Ti4+ have a interference effect because
complex forming reaction of these metallic ions with
fluoride ion. The interference of iodide ion was also be-
cause oxidation reaction with Ce4+ and Cr2O72- as oxi-
calibration during the cross validation. Figure 8 shows a
plot of PRESS against the number of factors for a mix-
ture of components. To find out minimum factors, we
also used the F-statistics to carry out the significant de-
termination [33]. The optimal number of factors, for the
two components, was obtained as 2 for both PCR and
3.11. Application
For evaluating the predictive ability of a multivariate
calibration model, the root mean square error of predic-
tion (RMSEP), relative standard error of prediction (RSEP)
and squares of correlation coefficient (R2), which is an
indication of the quality fit of all the date to a straight
line, can be used as follows [33,37]:
/ 100
ii i
 
 (13)
To evaluate the analytical applicability of the proposed
methods (PCR, PLS and HPSAM), we spiked known
amounts of both Ce4+ and Cr2O72- into some water sam-
ples. The proposed methods were applied to determine
analytes simultaneously and satisfactory results were
obtained. The results are given in Table 8. The results
show that the proposed methods could accurately deter-
mine the concentration of these oxidants mixture under
investigation in real water samples, and there is no sig-
nificant difference between the results of PCR, PLS and
HPSAM for their simultaneous determination.
Table 7. Statistical parameters calculated for the prediction
set using PLS and PCR models.
 
Ce4+ 2.97 3.03 0.35 0.36 0.9980 0.9990
Cr2O72- 3.40 2.90 0.64 0.40 0.9993 0.9988
where Ĉi represents the estimated concentration, Ci and n
are the actual analyte concentration and the number of
samples, respectively.
Table 7 shows the values of RSEP, RMSEP and R2
for each component using PLS and PCR. It is shown that
the obtained values, for the statistical parameters, are
almost the same for both PLS and PCR methods.
Numbers of factors
3.10. Interference Study
The study of interference ions was performed by a stan-
dard mixture solution containing 10.0 μg/mL of each Ce4+
and Cr2O72- and a certain amount of foreign ions. The
following excesses of ions do not interfere (i.e., caused a
relative error of less than 5%): more than a 1000-fold
(largest amount tested) amount of K+, Zn2+, Cu2+, Bi3+,
As3+, Cd2+, Mg2+, Be2+, Cl¯, NO3¯, BO33-, C2O42- a
100-fold amount of Mn2+, Ni2+, Co2+, pb2+, Cr3+, Ca2+; a
10-fold amount of SO42-, PO43-, Hg2+ and a 1-fold amount
of Al3+, Fe3+, Zr4+, Ti4+, I¯. As was to be expected,
Figure 8. Plot of PRESS against the numbers of factors
PLS()and PCR().
Table 8. Simultaneous determination of Ce4+ and Cr2O72- in different water samples using HPSAM, PCR and PLS methods.
Found (μg/mL)
Sample Spiked (μg/mL) HPSAM PLS PCR
Ce4+ Cr2O72- Ce4+ Cr2O72- Ce4+ Cr2O72- Ce4+ Cr2O72-
1 4.0 0.5 4.20 ± 0.12 0.51 ± 0.10 4.10 ± 0.11 0.46 ± 0.05 3.97 ± 0.20 0.48 ± 0.02
10.0 10.0 10.20 ± 0.24 10.40 ± 0.30 10.10 ± 0.22 9.88 ± 0.20 9.79 ± 0.14 9.90 ± 0.10
2 3.0 15.0 2.90 ± 0.06 15.40 ± 0.25 2.97 ± 0.05 14.80 ± 0.28 3.10 ± 0.10 15.20 ± 0.20
7.0 15.0 7.20 ± 0.21 15.30 ± 0.33 7.14 ± 0.18 14.88 ± 0.15 6.85 ± 0.13 14.90 ± 0.11
3 25.0 4.0 24.60 ± 0.36 4.10 ± 0.10 24.8 ± 0.33 3.88 ±0.09 25.4 ± 0.28 4.16 ± 0.15
18.0 20.0 17.80 ± 0.28 19.30 ± 0.20 17.65 ± 0.37 19.10 ± 0.12 18.20 ± 0.13 19.8 ± 0.21
4. Conclusions
This work as the first application of PCR, PLS and
HPSAM in the simultaneous determination of the binary
mixture of Ce4+ and Cr2O72- shows the ability and excel-
lent performance of ISEs as detectors not only for indi-
vidually determination of produced or consumed ions,
but also in the simultaneous kinetic-potentiometric analy-
sis. In addition, this paper has also demonstrated that the
ability and advantages of the HPSAM and chemometrics
methods such as PCR and PLS, ISEs and kinetic meth-
ods produce a very attractive and excellent technique for
the analysis of multi-component oxidant mixtures. Other
chemometrics approaches like ANN, ISEs (flouride,
bromide, iodide, etc) and other kinetic reactions in which
the rate of production or consumption of the correspond-
ing ion is different can also be used. Our team has ob-
tained good results for the simultaneous determination of
other species using HPSAM, chemometrics methods,
different ISEs and various reaction systems and our
complete results will be presented for publication in the
5. Acknowledgements
The authors would like to express their appreciations to
Professor Afsaneh Safavi and Professor Hamid Abdol-
lahi for their valuable discussion and useful suggestions.
This research was supported by the Payame Noor Uni-
versities of Ardakan and Sirjan.
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