The Mathematics
We had just about reached the point of realizing that the data indicated that absorbances of mixtures —when measured in situations
where Beer's law applies — are proportional to volume fractions and were preparing a manuscript for publication (13). In preparation
for writing up the work, we did a literature search for work dealing with CLS. As reported in that paper, we did not find
any reference to the use of volume fractions in conjunction with CLS analysis. However, upon perusing the book about effects
of scattering samples written by our good friend Don Dahm (14), down near the bottom of page 5 in his description of a "representative
layer," we came across this remarkable statement: ". . . we will require that the volume fraction of each kind of particle
in the layer is the same as the volume fraction in the sample."
I (HM) immediately ran to my computer and sent Don an email. After explaining what we had just learned, I asked, "Don, what
made you take it into your head to use volume fractions for the description of the concentration in your book?" His response
was something equivalent to, "Well, every physicist knows that!"
After a few more backandforth exchanges, wherein Don said that it follows from Beer's law, we eventually got to the point
where I simply asked him, "Can you prove that? Starting with Beer's law can you derive the fact that components in a mixture
absorb in proportion to their volume fractions?" Don then developed a proof. Here is Don's proof exactly as he sent it, except
edited only so that the equation numbers conform to our style and numbering:
<Start of derivation>
This mathematical explanation is presented here. We start with Beer's law, which may be stated as
Because absorbance is a dimensionless quantity, the units on the righthand side must also be dimensionless. For example,
if "concentration" is expressed as the mass concentration (g/cm^{3}) and pathlength is measured in centimeters, then the absorptivity must be expressed in cm^{2}/g. "Volume fraction" would be a perfectly reasonable measure to use for "concentration" when applying Beer's law to a mixture
of liquids, and because volume fraction is dimensionless, absorptivity would be expressed in cm^{1} in such a case.
The intent of this derivation is to demonstrate that volume fraction is a fundamental property in determining the absorbance
of mixtures, regardless of the specific measurement used for concentration. We will do this by examining application of Beer's
law to a mixture, expressing "concentration" as mass concentration.
The following symbols and definitions will be used in the derivation:
V = the volume of a solution which is a mixture of pure compounds
A = the absorbance of a solution in a cell of unit pathlength
A
_{
i pure} = the absorbance of a pure sample of compound i
A
_{
i
}
= the contribution of compound i to the absorbance of the solution (A)
α
_{
i
} = the absorptivity of the ith component
ν
_{
i
}
= the volume of the ith pure component
m
_{
i
} = the mass of the ith pure component
ρ_{i} = density of the ith pure component (m
_{
i
}/ν
_{
i
})
{γ_{i}} = mass concentration of component i in solution (m
_{
i
}/V)
l = the pathlength.
Consider a sample that is formed by mixing pure materials having known volumes (ν
_{1}, ν
_{2}, and so on). The final volume of the mixture is given by V = ν
_{0} + Σν
_{
i
}, with the term ν
_{0} representing the volume change upon mixing. Here, we will consider the "ideal mixture" in which ν
_{0} = 0.
Beer's law is, again
Applying Beer's law to a pure component (i), and using the nomenclature established above, gives
While applying Beer's law to a single component (i) within the mixture gives
By definition m
_{
i
} = ρ_{i} × ν
_{
i
} and γ_{
i
} = m
_{
i
}/V. Combining these definitions gives
Or equivalently:
Note that ν
_{
i
}/V is the volume fraction of component i in the mixture. Substituting the definition of mass concentration given by equation 5 into equation 3 gives this expression
for Beer's law for a component in a mixture:
Thus, at constant pathlength, the contribution of a component to absorbance is directly proportional to its volume fraction.
Furthermore, note that we can group (α_{
i
}× ρ_{
i
} × l) together, and apply equation 2, giving
For transmission through a homogeneous solution, from Beer's law we know that the total absorbance of a mixture is the sum
of the contribution of each of the absorbers, and we express that mathematically with the following expression:
Combining equations 7 and 8 gives
Thus, according to Beer's law, the absorbance of the mixture is dependent upon the absorbances of the individual pure components,
the volume fraction of each, and nothing else.
<End of derivation>
