An Introduction to Ultraviolet-Visible Molecular
Spectrometry (Chapter 13)
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Beer’s Law:
A = -log T = -logP0 / P = e x b x C
See Table 13-1 for terms.
In measuring absorbance or transmittance, one should
compensate for reflections and scatter occurring at the
interface of the cuvette.
Compensation is always done by running the blank.
128
Application of Beer’s Law to Mixtures
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Before applying these equations, you need to
know the following:
For compound 1:
e1 at l1 and l2
For compound 2:
e2 at l1 and l2
If their values are not know, you should obtain them
via calibration curves.
It is also common to obtain e values from single
standard solutions.
You should always remember: e is a constant
that depends on:
Wavelength
Temperature
Solvent
e1
e2
Absorbance
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l1
l2
129
Concentration, M
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Three types of limitations cause deviations
from linearity:
a) Real Limitations
b) Instrumental Limitations
c) Chemical Limitations
Real Limitations:
Result from analyte-analyte interactions at
high analyte concentrations (usually C >
0.01M).
The upper limit of the LDR depends on the
compound. For the same compound, it also
depends on the solvent and the
temperature.
You should always determine experimentally
the upper limit of the LDR.
Absorbance
Limitations of Beer’s Law
LDR
Concentration
High analyte concentration causes average
distances among analyte molecules to decrease
• Solute-solute interactions are mainly dipole-dipole interactions
• Electrostatic interactions can also occur among strong electrolytes
130
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Chemical deviations:
Arise when an analyte dissociates, associates, or reacts with a concomitant (including solvent) to
produce a product with different absorption spectrum than the analyte.
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Typical example: Acid-Base Indicators (HIn).
HIn
<=>
H+
(Color 1, l1)
Spectrum looks like:
=>
+
In(Color 2, l2)
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There is a dissociation constant associated to the equilibrium above:
Ka = [H+][In-] / [HIn]
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HIn and In- concentrations depend on the pH of solution:
[H+] = Ka x [HIn] / [In-] => log [H+] = logKa + log [HIn] / [In-] => - log [H+] = - logKa - log [HIn] / [In-]
=> pH = pKa + log [In-] / [HIn]
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Assuming an indicator with pKa = 5 (Ka = 10-5):
pH
log[In-] / [HIn]
[In-] / [HIn]
1
-4
10-4
2
-3
10-3
5
0
100
6
1
101
7
2
102
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[In-] = ? X [HIn]
0.0001
0.001
1
10
100
At any given wavelength, the total absorbance intensity of the solution is the sum of the individual
intensities of HIn and In-:
A Total, l = A HIn, l + A In-, l
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According to Beer’s law, the absorbance intensities of HIn and
In- are the following:
A HIn, l = e HIn, l . b . [HIn]
A In-, l = e In-, l . b . [In-]
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Substituting in the total absorbance equation:
A Total, l = e HIn, l . b . [HIn] + e In-, l . b . [In-]
=> At any given wavelength, the total absorbance intensity of the
solution (A Total) depends on the concentrations of HIn and In-.
=> Because [HIn] and [In-] depend on pH, A Total depends on pH.
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You should also remember that the indicator concentration is
given by:
C Indicator = C = [HIn] + [In-]
HIn
<=>
H+
+
InC–x
x
x
+
where x = [In ] = [H ]. From Ka:
Ka = x2 / C – x => x2 = Ka (C – x) => x2 + Ka.x – Ka.C = 0
=> x = -Ka ± {Ka2 + 4KaC}1/2 / 2
This equation demonstrates that the concentration of indicator
varies non-linearly with the ionized fraction of the acid. The same
is true for [HIn] ( = C – x).
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A graph of intensity of absorption as a function of concentration
of indicator of an un-buffered solution provides a non-linear plot.
At 430nm: the absorbance is
primarily due to the ionized In- form
of the indicator and is proportional
to the ionized fraction, which varies
non-linearly with the total indicator
concentration.
At 530nm: the absorbance is due
principally to the un-dissociated
acid HIn, which increases nonlinearly with the total concentration.
132
Instrumental deviations due to polychromatic light
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Beer’s law is only followed when
measurements
are
made
with
mocnochromatic radiation.
Wavelength selection of continuous
sources radiation made with filters or
monochromators provides a Gaussian
wavelength profile with a central
wavelength of maximum intensity.
Consider a beam of radiation consisting
of two wavelengths:
Only when the
two molar
absorptivities are
the same, this
equation
simplifies to A =
e.b.C and Beer’s
law is followed.
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This condition
is best met at
the maximum
absorption
wavelength of
the absorber.
133
Instrumental deviations in the presence of
stray radiation
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Similar considerations are true for stray
radiation.
In the presence of stray light radiation (Ps),
absorption is given by:
A’ = log [P0 + Ps] / [P + Ps]
Depending on its relative magnitude, stray
light radiation can cause significant
deviations from linearity.
At high stray levels and high concentrations,
i.e.
strong
absorbance
and
low
transmittance, the radiant power transmitted
through
the sample can become
comparable to or lower than stray-light level.
Mismatched Cells
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If the analyte and blank cells are optically different and/or have different path-lengths,
deviations from Beer’s law can occur.
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You should always use optically equivalent cells!
134
Effects of instrumental noise on spectrophotometric analyses
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The relative standard deviation (sc/c) of a concentration (c)
obtained via a transmittance measurement (T) is given by the
equation:
sc/c = 0.434sT / T. logT
where sT is the absolute standard deviation of the
transmittance measurement.
This equation shows that the uncertainty in a measurement
varies non-linearly with the magnitude of the transmittance.
Non-linearity as a function of relative standard deviation is
also true for absorbance measurements.
The sources of noise (uncertainties) in transmittance
(absorbance) measurements can be divided in three cases.
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K1, k2 and k3 are
proportionality constants.
Only for Case I sT is
independent of T.
If the limiting source for
uncertainty
in
a
measurement
is
instrumental noise, the
best standard deviation
within the calibration
curve will then be
obtained
at
the
absorbance value where
sc/c is minimum.
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Types of Instruments
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Single channel systems
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Multi-channel systems
Multi-channel systems present the advantage of
real-time spectra but the upper concentration limit
of their LDR is usually lower than single-beam
systems.
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