13.1 Measuring Absorption
• derivation of Beer's Law using a chemical kinetic
approach
• definition of transmission and absorption
• absorption cross-section and molar absorptivity
• transmission is multiplicative and absorption is
additive
• expected performance when measuring
concentration
• expected performance when measuring spectra
• using solution color to guess at absorbed
wavelengths
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Kinetic Derivation of Beer's Law (1)
The process of absorption can be written as a
"reaction" involving light and two energy
states.
ΔE = hν
S2
B
1,2
S1 + hν ⎯⎯⎯
→ S2
A second-order rate equation can be written
for the reaction,
B1,2
S1
where ρ(ν) is the density of light in cm-3, N1 is the number density
of S1 in cm-3, and B1,2 is the second order rate constant (called the
___________________ for absorption). B1,2 has units of cm3 s-1.
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Kinetic Derivation of Beer's Law (2)
Of more interest is the change in
photon density with distance
through the sample. This is
obtained using, dx/dt = c. σ has
units of cm2 and is called the
absorption cross-section.
The total change in the photon
density over a distance, l, is
obtained by integration.
By converting photon density into
intensity, I = ρ(ν)×c, the final
equation can be written in terms
of an intensity ratio, I/I0.
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d ρ (ν )
B1,2
d ρ (ν ) dt
ρ (ν ) N1
=−
dt dx
c
=
dx
B1,2
σ≡
c
l
d ρ (ν )
∫ ρ (ν )
0
l
= −σ N1 ∫ dx
0
⎛ ρ (ν ) ⎞
l = −σ lN
ln ⎜
⎟
1
⎜ ρ (ν ) ⎟
0⎠
⎝
Transmission and Absorption
Transmission is defined as the intensity of light
leaving the sample, divided by the intensity of
light entering the sample. Transmission can be
related to number density using the last equation
on the previous transparency.
I
= 10
T≡
I0
−
σ lN
2.3
I0
I
= 10−ε lC
Although the previous derivation used number density and crosssection, the more common units of molar absorptivity, ε, and molar
concentration, C, give the same numeric result.
Absorption is defined as the negative logarithm (base 10) of the
transmission. This gives a parameter linear in concentration.
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Cross-Section and Absorptivity
Cross-sections and number densities are used in virtually every
type of spectroscopy ______ solution-phase, molecular absorption.
When comparing different spectroscopies, it is necessary to use
cross-sections.
The conversion between ε and σ involves nothing more than units.
)
σ ( cm ) =
ε ( L mol-1 cm-1 )
6.02 × 1023 ( mol-1 )
2
(
2.3 × 103 cm3 L−1
In the UV/visible the maximum molar absorptivity is ______,
which corresponds to a cross-section of ________________.
The cross-section has nothing to do with the size of the molecule.
Indeed, atoms have absorption cross-sections of ~10-13 cm2.
The maximum cross-section comes from antenna theory and is
(λ/2)2, which for 500 nm radiation is 6.25×10-10 cm2.
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Measuring Absorption
Optical Parameters with I0 = 5 μW
A
T
I(μW)
0.001
0.01
0.1
0.9977 0.9772 0.7943
1
2
3
0.1
0.01
0.001
4.9885 4.8862 3.9716 0.5000 0.0500 0.0050
_____ absorption problem: I and I0 are nearly the same value.
In order to measure an absorption of 0.001 the instrument has to
measure I to a precision of at least 0.23%.
_____ absorption problem: I is difficult to distinguish from 0. In
order to measure an absorption of 3, the instrument has to have
three orders of magnitude of linearity, i.e. I = 10-3 I0.
The detector and electronics must have a high precision and large
___________________. Top-of-the-line: 10-4 to 5; intermediate:
10-3 to 3; inexpensive: 0.01 to 2.
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Absorption Adds
Consider a solution in a 1-cm cell that has a transmission of 0.5. It
doesn't matter how much light enters the cell, only half will exit.
Now place two such cells back-to-back. The combination will
transmit 0.5×0.5 = 0.25 of the light, thus it is seen that
transmission is multiplicative.
Since absorption is the negative log of transmission, it can be seen
that absorption adds.
-log(Ttotal) = -log(T1 × T2) = -logT1 + -logT2
Atotal = A1 + A2
The most common use of additive absorption involves two or more
molecules in the same solution.
A(λ1) = ε1,1(λ1)C1 + ε2,1(λ1)C2
A(λ2) = ε1,2(λ2)C1 + ε2,2(λ2)C2
Solve by matrix algebra.
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Measuring Concentration
Performance can be estimated from Beer's Law. Assume that an
absorbance of 0.01 will provide a satisfactory signal-to-noise ratio.
An optimistic estimate: assume ε = 105 and l = 10 cm.
C=
A
0.01
=
= 10−8 M
ε l 105 × 10
A realistic estimate: assume ε = 104 and l = 1 cm.
C=
A
0.01
=
= 10−5 M
ε l 104 × 1
For a given determination, the _________________ can be
extended by using cells of different length. Commercially available
lengths are: 10, 5, 2, 1, 0.5, 0.2, 0.1 and 0.01 cm.
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Measuring Spectra
The SNR at the peak has to be sufficiently large to observe small
features.
SNR = ____ data cannot be
compared to a reference
spectrum.
0.04
SNR = 30
0.03
SNR = ____ data cannot be
identified with great
reliability.
SNR = ____ is required to
obtain a faithful
representation of the true
spectrum.
absorption
SNR = 3
0.02
0.01
0
400
450
500
-0.01
wavelength (nm)
Minimum concentrations for a spectrum are ~100× the minimum
concentration for quantitation.
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550
Color and Absorption Maximum
An artist's color wheel can be used to determine
the absorption maximum.
Absorption occurs at the _____________ of the
solution color. Thus, a yellow-colored solution
absorbs in the ________.
violet
yellow
Use the table below to convert color to
wavelength. Thus, a yellow-colored solution
absorbs near 400 nm.
A colorless solution absorbs in the __________.
Color and Approximate Wavelength
color
violet
blue
green
yellow
orange
red
λ (nm)
400
450
500
550
600
650
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