40
K. D. Dahm and D. J. Dahm
that real solutions can exhibit “departures” from Beer’s Law at high concentrations.
We understand these departures as symptoms of the fact that in using Beer’s Law κ
is generally assumed constant, when in reality, there is reason to expect that κ would
become dependent upon composition as c increases and solute–solute interactions
become more significant.
Now imagine we are still using the experimental arrangement in Fig. 3.1, but
the sample produces both scattering and absorption. Like the sunlight in Bouguer’s
experiments, any light that is either absorbed or scattered will fail to reach the
detector. Thus, both scattering and absorption contribute to the extinction modeled
by κ. Here again, the distinction between dilute vs. concentrated solutions could
complicate the use of Beer’s Law.
By contrast, if we use an experimental arrangement like the one as shown in
Fig. 3.2, the light that is scattered at least once but still exits the sample can be
detected. A hemispherical detector can be used to measure as “transmission” (I in
Eq. 3.3) all of the light that penetrates the sample, regardless of the specific path. The
presence of scatter also gives rise to the phenomenon of “remission,” which is light
that emerges from the sample’s front surface, and which can be quantified by another
hemispherical detector. (In practice, the experiment pictured in Fig. 3.2 is most
straightforwardly carried out using an integrating sphere.) The classical definition
of “absorbance” in Eq. 3.2 does not acknowledge the phenomenon of remission. In
most of the models we discuss in Sects. 3.6–3.10, we will instead consider A, R, and
T, which we define as the fractions of incident light that were absorbed by, remitted
from, and transmitted through the sample, respectively. Absorbance can then be more
broadly defined as:
Absorbance = −log 10 (1 − A)
(3.4)
Note that when R = 0, 1-A is equal to T, which is in turn equal to I/I 0 . Thus, this
definition is equivalent to Eq. 3.2 for the special case of a non-scattering sample.
In sum, Beer’s Law is an equation that has great practical appeal, because it
represents a simple linear relationship between a readily measured quantity (I/I 0 )
and the concentration c of the absorber, which is usually what we are trying to
deduce in spectroscopy. However, Beer’s Law can only be expected to be a good
model in specific circumstances (clear solution, dilute solution, etc.). Furthermore,
even when Beer’s Law proves to be a good model, one must recognize the limitations
of the value of κ. A chemical compound does not have a single “absorptivity” that
is uniformly applicable. The value of κ depends upon factors (e.g., solvent, sample
thickness, and sample geometry) that are specific to the context of an experiment and
should only be considered valid in that specific context.
K. D. Dahm and D. J. Dahm
that real solutions can exhibit “departures” from Beer’s Law at high concentrations.
We understand these departures as symptoms of the fact that in using Beer’s Law κ
is generally assumed constant, when in reality, there is reason to expect that κ would
become dependent upon composition as c increases and solute–solute interactions
become more significant.
Now imagine we are still using the experimental arrangement in Fig. 3.1, but
the sample produces both scattering and absorption. Like the sunlight in Bouguer’s
experiments, any light that is either absorbed or scattered will fail to reach the
detector. Thus, both scattering and absorption contribute to the extinction modeled
by κ. Here again, the distinction between dilute vs. concentrated solutions could
complicate the use of Beer’s Law.
By contrast, if we use an experimental arrangement like the one as shown in
Fig. 3.2, the light that is scattered at least once but still exits the sample can be
detected. A hemispherical detector can be used to measure as “transmission” (I in
Eq. 3.3) all of the light that penetrates the sample, regardless of the specific path. The
presence of scatter also gives rise to the phenomenon of “remission,” which is light
that emerges from the sample’s front surface, and which can be quantified by another
hemispherical detector. (In practice, the experiment pictured in Fig. 3.2 is most
straightforwardly carried out using an integrating sphere.) The classical definition
of “absorbance” in Eq. 3.2 does not acknowledge the phenomenon of remission. In
most of the models we discuss in Sects. 3.6–3.10, we will instead consider A, R, and
T, which we define as the fractions of incident light that were absorbed by, remitted
from, and transmitted through the sample, respectively. Absorbance can then be more
broadly defined as:
Absorbance = −log 10 (1 − A)
(3.4)
Note that when R = 0, 1-A is equal to T, which is in turn equal to I/I 0 . Thus, this
definition is equivalent to Eq. 3.2 for the special case of a non-scattering sample.
In sum, Beer’s Law is an equation that has great practical appeal, because it
represents a simple linear relationship between a readily measured quantity (I/I 0 )
and the concentration c of the absorber, which is usually what we are trying to
deduce in spectroscopy. However, Beer’s Law can only be expected to be a good
model in specific circumstances (clear solution, dilute solution, etc.). Furthermore,
even when Beer’s Law proves to be a good model, one must recognize the limitations
of the value of κ. A chemical compound does not have a single “absorptivity” that
is uniformly applicable. The value of κ depends upon factors (e.g., solvent, sample
thickness, and sample geometry) that are specific to the context of an experiment and
should only be considered valid in that specific context.