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Linearity In LC Detectors
A detector operates by registering an output in response to
sample detection. Chromatographers expect a linear relationship
between response of the detector and concentration of the sample,
and calibration techniques are designed to promote this
relationship. However, not all detectors are linear and effects
from other components of the HPLC system may cause the detector
to deviate from a linear response between response and
concentration. The following section discusses how a detector's
linearity can be measured and evaluated, and it provides
illustrations of these techniques. Much of this section follows
guidelines and terminology adopted by the American Society for
Testing and Materials (ASTM).Two important terms when describing
detector performance are linear range and dynamic range.
Linear range is based upon the following equation:
For a linear detector the equation
beomces R = SC since Ro should equal zero. A plot of
response versus concentration would result in a line with slope =
S, where the value of S = R/C. Thus, a linear detector is one
where the sensitivity is constant at all sample concentrations.
Subsequently, linear range refers to concentrations where S is
constant within a certain tolerable range, usually +/- 5% (Dorschel,
1989). ASTM refers to dynamic range as "that range of
concentrations of the test substance, over which a change in
concentration produces a change in detector signal." The
lower limit of the dynamic range is the concentration where the
detector signal is equal to twice the detector's noise level
(random variations in detector signal). The upper limit is the
concentration where the slope of the detector response curve
approaches zero. If the curve does not have a plateau, then the
highest measured concentration is understood to be the upper
limit.When measuring and evaluating linearity in detectors, off-line
(static) test or on-line (dynamic) test is used. The
static test is where the detector is disconnected from the HPLC
system and the sample concentrations are injected directly into
the detector cell via a syringe, pump, etc., where no constant
flow through the detector is seen. It is the procedure to use for
nondestructive detectors and it allows the user to know the exact
concentration of sample reaching the detector.On the other hand,
the dynamic test is performed using the entire HPLC system. This
is more suitable for destructive detectors, such as
electrochemical, and it is easier to use. It also allows testing
the detector to the lowest of its output response levels since
any nonsample components or gases in the flow solvent can be
resolved or separated from the desired sample.The process for
measuring linear or dynamic ranges for nondestructive detectors
involves using solutions with known dilutions of the sample
compound. The sample compound must also have known detection
properties, such as absorbance or refractive index. The solutions
are injected into the detector, the detector response is
recorded, and the detector is washed with pure solvent in between
each test dilution. It should be noted that certain factors may
change results of this test, such as choice of sample or detector
operating conditions. Factors such as these are beyond the scope
of this section and will not be discussed.Using ASTM guidelines,
Dorshel et al. evaluated the linearity of an anonymous UV
detector using acenaphthene as the test sample and employing a
static system (Dorschel,
1989). The following response vs. concentration plot was
obtained:
(Dorschel, 1989)Graph 1 would seem
to imply a linear plot since all but the final data point falls
on the line beginning at the origin and extrapolated out.
However, subsequent results seemed to prove that the
chromatographer needs to view the experimental results more
closely before a conclusion of linearity for this detector can be
decided.Graph 1 shows that the dynamic range upper limit is 0.25
g/L (the highest measured concentration since no plateau) and
Graph 2 (logarithmic axes) shows the lower limit to be 1.75 X 10-6
g/L. Thus, the dynamic range, expressed as a ratio of the upper
and lower limits, is 1.43 X 105 .The linear range was
found using ASTM guidelines since all but one data point was
included on the extrapolated line. First, the data points are
connected using a smooth curve. Next, a best-fit line is drawn
from the origin through the data points. Finally, another line
with 95% slope of the best-fit line is drawn from the origin and
extrapolated out. The concentration for the upper limit of the
linear range is the point where the smooth curve intersects the
95% slope line: 
(Dorschel,
1989)The lower limit is the same as with the dynamic range.
Thus, the linear range for the ASTM procedure is 1.36 X 105.Dorshel
et al. note that the ASTM method for linear range assumes that
the detector is linear and the method may tend to smooth data
which comes from a non-linear detector. Therefore, they
calculated the linear range using the definition stated earlier,
where the sensitivity values must fall within a certain tolerable
region (+/- 5%). For this UV detector, the response should adhere
to the Beer-Lambert Law: 
Since absorbance is equal to detector
response, and the product of the extinction coefficient and
optical pathlength, b, is equal to sensitivity, then R/C = A/C =
Extin.Coeff. x b = S. So a R/C vs. Log C plot can be made, where
the +/- 5% tolerance region (dotted lines) is chosen arbitrarily
(in this example, 0.031g/L was chosen): 
(Dorschel,
1989)The linear range in this technique (the concentrations
which fall within the dotted lines) is 9.52. The linearity plot
indicates that the detector is only linear over a certain, narrow
range of concentrations. The significance of this result is not
to omit use of this detector at certain concentrations, but
instead to allow the chromatographer to recognize nonlinearity.
Dorshel et al. insist that "it is not the existence of
nonlinearity but rather the analyst's failure to anticipate and
adjust for nonlinear behavior that can lead to poor
quantification" (Dorschel,
1989)
.Dorschel,
C.A.; Ekmanis, J.L.; Oberholtzer, J.E.; Warren, F.V. Jr.;
Bidlingmeyer, B.A. Analytical Chemistry, 1989, Vol. 61,
pp. 952.
Dorschel,
p. 951A-968A.
Dorschel,
p. 953A.
Dorschel,
p. 954A.
Dorschel,
p. 954A.
Dorschel,
p. 956A.
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linearity, LC detectors,
detectors, detection, linear range, dynamic range, range,
dynamic, linear, American Society for Testing and Materials,
ASTM, performance, HPLC, high performance liquid chromatography,
concentration, off-line test, static, static test, on-line test,
dynamic test, non-linear detector, UV, UV detector, output,
slope, line, equation, sample, chromatography, chromatographers,
relationship, response, chemistry, analytical chemistry,
analytical, university of kentucky, college of pharmacy
For a linear detector the equation
beomces R = SC since Ro should equal zero. A plot of
response versus concentration would result in a line with slope =
S, where the value of S = R/C. Thus, a linear detector is one
where the sensitivity is constant at all sample concentrations.
Subsequently, linear range refers to concentrations where S is
constant within a certain tolerable range, usually +/- 5% (Dorschel,
1989). ASTM refers to dynamic range as "that range of
concentrations of the test substance, over which a change in
concentration produces a change in detector signal." The
lower limit of the dynamic range is the concentration where the
detector signal is equal to twice the detector's noise level
(random variations in detector signal). The upper limit is the
concentration where the slope of the detector response curve
approaches zero. If the curve does not have a plateau, then the
highest measured concentration is understood to be the upper
limit.When measuring and evaluating linearity in detectors, off-line
(static) test or on-line (dynamic) test is used. The
static test is where the detector is disconnected from the HPLC
system and the sample concentrations are injected directly into
the detector cell via a syringe, pump, etc., where no constant
flow through the detector is seen. It is the procedure to use for
nondestructive detectors and it allows the user to know the exact
concentration of sample reaching the detector.On the other hand,
the dynamic test is performed using the entire HPLC system. This
is more suitable for destructive detectors, such as
electrochemical, and it is easier to use. It also allows testing
the detector to the lowest of its output response levels since
any nonsample components or gases in the flow solvent can be
resolved or separated from the desired sample.The process for
measuring linear or dynamic ranges for nondestructive detectors
involves using solutions with known dilutions of the sample
compound. The sample compound must also have known detection
properties, such as absorbance or refractive index. The solutions
are injected into the detector, the detector response is
recorded, and the detector is washed with pure solvent in between
each test dilution. It should be noted that certain factors may
change results of this test, such as choice of sample or detector
operating conditions. Factors such as these are beyond the scope
of this section and will not be discussed.Using ASTM guidelines,
Dorshel et al. evaluated the linearity of an anonymous UV
detector using acenaphthene as the test sample and employing a
static system (Dorschel,
1989). The following response vs. concentration plot was
obtained:
(Dorschel, 1989)Graph 1 would seem
to imply a linear plot since all but the final data point falls
on the line beginning at the origin and extrapolated out.
However, subsequent results seemed to prove that the
chromatographer needs to view the experimental results more
closely before a conclusion of linearity for this detector can be
decided.Graph 1 shows that the dynamic range upper limit is 0.25
g/L (the highest measured concentration since no plateau) and
Graph 2 (logarithmic axes) shows the lower limit to be 1.75 X 10-6
g/L. Thus, the dynamic range, expressed as a ratio of the upper
and lower limits, is 1.43 X 105 .The linear range was
found using ASTM guidelines since all but one data point was
included on the extrapolated line. First, the data points are
connected using a smooth curve. Next, a best-fit line is drawn
from the origin through the data points. Finally, another line
with 95% slope of the best-fit line is drawn from the origin and
extrapolated out. The concentration for the upper limit of the
linear range is the point where the smooth curve intersects the
95% slope line: 
(Dorschel,
1989)The lower limit is the same as with the dynamic range.
Thus, the linear range for the ASTM procedure is 1.36 X 105.Dorshel
et al. note that the ASTM method for linear range assumes that
the detector is linear and the method may tend to smooth data
which comes from a non-linear detector. Therefore, they
calculated the linear range using the definition stated earlier,
where the sensitivity values must fall within a certain tolerable
region (+/- 5%). For this UV detector, the response should adhere
to the Beer-Lambert Law: 
Since absorbance is equal to detector
response, and the product of the extinction coefficient and
optical pathlength, b, is equal to sensitivity, then R/C = A/C =
Extin.Coeff. x b = S. So a R/C vs. Log C plot can be made, where
the +/- 5% tolerance region (dotted lines) is chosen arbitrarily
(in this example, 0.031g/L was chosen): 
(Dorschel,
1989)The linear range in this technique (the concentrations
which fall within the dotted lines) is 9.52. The linearity plot
indicates that the detector is only linear over a certain, narrow
range of concentrations. The significance of this result is not
to omit use of this detector at certain concentrations, but
instead to allow the chromatographer to recognize nonlinearity.
Dorshel et al. insist that "it is not the existence of
nonlinearity but rather the analyst's failure to anticipate and
adjust for nonlinear behavior that can lead to poor
quantification" (Dorschel,
1989)