D
J. Chem. Chem. Eng. 6 (2012) 585-590
DAVID
PUBLISHING
Spectrophotometric Error Due to Uncertainty in Reading
the Signal
Mavroudis A. Demertzis*, Anastasios I. Palios, Alexandru V. Călin, Cristina-Ioana M. Vijdeluc and George A.
Chrysafis
Department of Chemistry, School of Sciences, the University of Ioannina, Ioannina 45110, Greece
Received: July 07, 2012 / Accepted: July 20, 2012 / Published: July 25, 2012.
Abstract: Based upon the standard deviations for the intercept and the slope of a linear regression line, as well as by differentiating
both Beer’s law and a linear relationship between transmittance and concentration at high transmission, it is proved that the relative
spectrophotometric error of a measurement becomes greater as the sample concentration only decreases. Further, it is demonstrated that
the present knowledge with regard to the error in absorption spectrophotometry is necessary to be reexamined. The total scale of
transmittance can literally be used for measurements, unfolding workable dynamic ranges about two orders of magnitude lower than
usually and thus absorption spectrophotometry can efficiently compete with other methods of analysis with respect to detection limits.
Key words: Spectrophotometric error, transmittance uncertainty, sensitive spectrophotometry, transmittance noise.
1. Introduction
In most measurements of physical properties of
analytes, by every instrumental method of analysis, the
average strength of a noise is constant and independent
of the magnitude of the signal. From this particular
point of view the effect of noise on the relative error of
a measurement becomes greater and greater as the
quantity being measured decreases in magnitude [1].
But, regarding spectrophotometry, the view about
the error is different. The effect of noise on the relative
error of a transmittance measurement becomes greater
and greater as the quantity being measured not only
decreases but also increases in magnitude. Clearly
there is a concentration for which the error becomes
minimum. The transmittance at which the propagation
of error is smallest is equal to 0.368% or 36.8% [2].
In this work, the concept of error in absorption
spectrophotometry, elucidated by three different
processes, does not constitute any exception to the
*
Corresponding author: Mavroudis A. Demertzis, Assistant
Professor, research field: analytical chemistry. E-mail:
mdemert@cc.uoi.gr.
former view mentioned above. Yet, it is shown up that
the extant thought in re the error in absorption
spectrophotometry must be reevaluated. The subject
has already been duly communicated [3-5].
2. Experiments
2.1 Reagents and Chemicals
ECR (eriochrome cyanine R) purchased from Fluka
was used to prepare aqueous solutions that meet the
requirements of the experimental part of this work.
2.2 Apparatus
All absorbance and transmittance measurements were
made using a Hitachi, Model U-2001 spectrophotometer,
matched with 10.0 mm quartz cells.
3. Results and Discussion
3.1 Error Assigned to Standard Deviations for
Intercept and Slope
For a linear relationship between the analytical
signal (y) and the concentration (x), making use of the
586
Spectrophotometric Error Due to Uncertainty in Reading the Signal
method of least squares, the best-fitted straight line
through the calibration graph points is calculated. This
line, known as the line of regression of y on x, is used to
estimate the concentrations of the test samples [6]. A
linear regression line is represented by the first-order
equation:
y = a + bx
(1)
In the case of Beer’s law y = A, x = C and a = 0.
By including the standard deviation of the intercept,
sa, which is proportional to the instrumental noise or
the uncertainty in reading the signal [6], Eq. (1)
becomes
y = (a ± sa ) + bxs
(2)
Where, xs is related to the individual x-value and
depends upon sa.
By combining Eqs. (1) and (2), it is taken
xs − x = ±
sa
b
(3a)
The difference xs-x obviously represents the
absolute error in x, which is constant for individual
x-values. Therefore, the standard deviation of the
intercept involves a parallel displacement on both sides
of the regression line, as shown in Fig. 1a for a negative
value of sa. Moreover, the absolute error in x, relating
to the standard deviation of the intercept can also be
unmistakably found from Fig. 1a geometrically. Since
the standard deviation of the intercept for a linear
regression line involves an identical absolute error in
Fig. 1a
concentration for any solution, the relative error of a
measurement, Eq. (3b), clearly becomes greater as the
concentration of a standard or a sample solution
decreases.
xs − x
s
=± a
x
bx
(3b)
On the other hand, by including the standard
deviation of the slope, sb, which also is proportional to
the uncertainty in reading the signal, Eq. (1) becomes
y = a + (b ± sb ) xs
(4)
y = a + bxs ± sb xs
(5)
or
Eqs. (1) and (4) given
xs = ±
bx
b ± sb
(6)
while Eqs. (1) and (5) given
xs − x = ±
sbx s
b
(7)
sbx
b ± sb
(8)
Finally, Eqs. (6) and (7) given
xs − x = ±
Here the absolute error in x, xs-x, is directly
proportional to the individual x-values, while the
relative error, Eq. (9), is constant for any solution. It is
evident that the standard deviation of the slope brings
along an angular displacement on both sides of the
regression line, as shown in Fig. 1b for a positive
change of sb.
Effect of a negative standard deviation of intercept on a linear regression line of analytical signal vs. concentration.
Spectrophotometric Error Due to Uncertainty in Reading the Signal
Fig. 1b
587
Effect of a positive standard deviation of slope on a linear regression line of analytical signal vs. concentration.
The absolute error in x can also be geometrically
found from Fig. 1b.
xs − x
sbx
s
=±
=± b
x
(b ± s b ) x
b ± sb
(9)
ΔA
ΔC
=
A
C
(12)
Based upon the data of Fig. 2a and using Eq. (12), a
plot of ΔC/C vs. C is shown in Fig. 2b.
Thus, the final effect is that the relative error
increases as the measured x-value decreases in
magnitude. This state is inviolable for every straight
regression line, independently of the technique used
to take experimental data.
3.2 Error Inherent to Absorbance Measurement
A host of measurements performed in our
laboratory for several absorbing species and
covering the total scale of reading, furnished
working graphs of very good linearity between
absorbance and concentration, as shown in Fig. 2a.
As already mentioned, a linear relationship between
absorbance and concentration indicates that the relative
error must be greater for smaller concentrations. This
conclusion is confirmed by a direct differentiation of
Beer’s law,
(10)
A = εbC
that gives
(11)
dA = εbdC
By substituting A/C for εb and using finite values
instead of infinitesimals, Eq. (11) becomes
Fig. 2a Absorbance of ECR solutions for concentrations
0.001 to 1 mM at 377 nm.
Fig. 2b Relative error of ECR solutions for concentrations
0.001 to 1mM at 377 nm; ΔΑ = 0.0002.
588
Spectrophotometric Error Due to Uncertainty in Reading the Signal
Clearly, the relative error in concentration is inversely
proportional to the absorbance, viz. greater only for
smaller concentrations.
3.3 Error Inherent to Transmittance Measurement
Corresponding to the data of Fig. 2a, a plot of
transmittance vs. concentration is shown in Fig. 3a.
For about T > 0.8 a linear relationship between the
variables is apparent, representing an equation of the
first degree as follows:
(13)
T = a - kC
For C = 0, T = a. Because incident beam, Po, is equal to
emergent beam, P, T = P/Po = 1. Thus, a = 1 and Eq.
(13) is definitely written as
(14)
T = 1 - kC
By differentiating Eq. (14) and proper substituting
for k, dT and dC it is finally taken:
ΔΤ ΔC
−
=
(15)
1− Τ C
From a plot of ΔC/C vs. C, shown in Fig. 3b, it is
evident that the error anew is greater only for smaller
concentrations.
Further, substituting P/Po for T and readjusting, Eq.
(14) can be changed into the following forms:
(16)
P = Po - kPo C
Po - P
= kC
(17)
Po
and
Po - P = kPo C
(18)
Here, it is noticed that Eq. (18) is analogous to the
next one,
Po - P = 2.303εbCPo
(19)
which is a result of Beer’s law in the form shown in Eq.
(20), provided the fraction of the absorbed light,
(Po - P)/Po , is ≤ 0.05; namely Eq. (19) is applicable at
high transmission for T ≥ 95%.
P
= e -2.303εbC
Po
(20)
An efficient, well accurate and precise
spectrophotometric procedure, based on Eq. (19) and
providing remarkably law detection limit has been
Fig. 3a Transmittance of ECR solutions for concentrations
0.001 to 1 mM at 377nm.
Fig. 3b Relative error of ECR solutions for concentrations
0.001 to 0.006 mM at 377 nm; ΔΤ = 0.0004.
proposed; in this case transmission measurements were
performed using a fluorescence spectrometer not a
spectrophotometer [7].
3.4 Current Thought about the Error
Absorption spectrophotometry is one of the most
useful and widely used techniques available to the
chemists for quantitative analysis, numbering
numerous applications that touch upon every field in
which quantitative chemical information is required
[1].
In spite of that, highly sensitive spectrophotometric
methods have not been developed as a result of large
error in concentration, that was thought to be cohered
with readings not only more than 80% T but also less
Spectrophotometric Error Due to Uncertainty in Reading the Signal
than 10% T [2], according to the coming equation
dC 0.4343 dT
=
TlogT
C
(21)
A plot of Eq. (21), using the data of Fig. 2a, is shown
in Fig. 4.
However, Eq. (21) is a product of wrong
differentiation calculus. Beer’s law was faulty
transposed to the form
0.4343 lnT
C= −
(22)
εb
which then was differentiated to provide Eq. (21).
In mathematical language, when y is a function of x,
then this relation is expressed by the equation:
y = f(x)
(23)
It may be said that the dependent variable, y, changes,
when the independent variable, x, changes [8]. But C in
Eq. (22) as a dependent variable does not change, when
the independent variable, T, changes. Thus, from a
physical point of view differentiation of Eq. (22), as
well as Eq. (21), are both without any sense.
Conversely, differentiation of Beer’s law, shown in
Eq. (24), finally provides Eq. (25),
(24)
- 0.4343lnT = -logT = εbC
d(-logT) dC
=
(25)
C
- logT
which is correct and equivalent to Eq. (12). In addition,
d(-logT) is just a differential, while the derivative of
-logT with respect to C, Eq. (26), is equal to the slope,
that is right for a linear relationship [8], such as Beer’s
589
law. Oppositely, the derivative of C with respect to T is
not equal to the slope in the case of Eq. (22), referring
to a non-linear relationship between the variables.
d(-logT) - logT
=
= εb
dC
C
(26)
Eq. (21), published some decades ago, was adopted
and well established from the total of the educational
manuals, viewing instrumental chemical analysis. In
this way, Eq. (21) constituted the touchstone for the
level of growth of the absorption spectrophotometry
with respect to impossibility of attainment lower
detection limits. A misconception for concentration
error has prevented the technique’s promotion long
ago.
4. Conclusions
The present-day perception regarding the error in
absorption spectrophotometry must be reviewed.
Accurate and sensitive spectrophotometric methods
can be acquired by going to the edge of the
transmittance scale, for instance, the relative error in
concentration for a transmittance value equal to 95% is
hardly less than 1%, while for T% = 99 the relative
error is about 4%. Absorption spectrophotometry is an
authentic, opportune and valuable technique for trace
analysis and can efficiently compete with
contemporary methods with respect to detection limits.
Finally, the linear relationship of transmittance or its
secondary dependent variables vs. concentration is
particularly practical for quantitative analysis at high
transmission.
References
[1]
[2]
[3]
Fig. 4 Relative error of ECR solutions for concentrations
0.001 to 1 mM at 377 nm; ΔΤ = 0.0004.
Skoog, A. D.; Holler, F. J.; Nieman, T. A. Principles of
Instrumental Analysis; 5th Ed., Thomson Learning
Academic Resource Center, United States, 1998; pp
99-342.
Hiskey, C. F. Principles of Precision Colorimetry. Anal.
Chem. 1949, 21, 1440-1446.
Călin, A. V.; Vijdeluc, C. I.; Fiamegos, Y. C.; Stalikas, C.
D.; Kovala-Demertzi, D.; Demertzis, M. A. In Sensitive
Spectrophotometric Determination of Copper with
di-2-pyridyl Ketone, Proceedings of the 9th Eurasia
Conference on Chemical Sciences (EuAs C2S-9),
Antalya, Turkey, 2006, Ohtaki, H.; Şener, B. Eds.; 2006,
590
[4]
[5]
Spectrophotometric Error Due to Uncertainty in Reading the Signal
B-OP-4, p 26.
Demertzis, M. A.; Călin, A. V.; Vijdeluc, C. I.;
Fiamegos, Y. C.; Stalikas, C. D.; Kovala-Demertzi, D.
In Errors in Spectrophotometry, Proceedings of the 5th
International Conference on Instrumental Methods of
Analysis-Modern Trends and Applications, Rio-Patras,
Greece, Christopoulos T. K., Ioannou, P.C. Eds., 2007,
OR 274.
Demertzis, M. A.; Palios, A. I.; Călin, A. V.; Vijdeluc, C.
I.; Chrysafis, G. A. In Spectrophotometric Error Due to
Uncertainty in Reading the Signal, Proccedings of the
[6]
[7]
[8]
12th Eurasia Conference on Chemical Sciences (EuAs
C2S-12), Corfu, Greece, Kitagawa, S., Hadjiliadis, N.,
Eds., 2012, S10-OP9.
Miller, J. C.; Miller, J. N. Statistics for Analytical
Chemistry; John Wiley and Sons, New York, 1988; pp
109-112.
Demertzis,
M.
A.
Low
Detection
Limit
Spectrophotometry. Anal. Chim. Acta 2004, 505, 73-76.
Farrington, D. Mathematical Preparation for Physical
Chemistry; McGraw-Hill Book Company, Inc., New
York, 1956; pp 70-91.