3.3 Absorption vs absorbance
It is of interest to review the Beer-Lambert law to understand the difference between absrobance and absorption.
\[\begin{alignat}{2} \label{eq:abs1} I_n &= I_0 \times e^{(\epsilon \times c \times L)}\\ \end{alignat}\]where:
- \(I_0\) the intensity of emitted light (before it passes through the sample).
- \(I_n\) the intensity of received light (after it passes through the sample).
- \(\epsilon\): the molar absorption coefficient (\(m^2 \times \text{mole}^{-1}\)).
- \(c\): the concentration of the colored material in the solution (\(\text{mole} \times m^{-3}\)).
- \(L\): the pathlenght traveled by the light throught the solution. It is of a common usage to express it per meter (\(m^{-1}\)).
where \(a = \epsilon \times c\) is the absorption coefficient (\(m^{-1}\)).
\[\begin{alignat}{2} \label{eq:abs3} \frac{I_n}{I_0} &= e^{-(a \times L)}\\[10pt] \frac{I_0}{I_n} &= e^{(a \times L)}\\[10pt] log(\frac{I_0}{I_n}) &= a \times L\\[10pt] a &= \frac{log(\frac{I_0}{I_n})}{L}\\ \end{alignat}\]We notice here that the Beer-Lambert law is written in log base \(e\). However, for historical reasons (basically because it was esier to calculate in the old days), spectrophotometer use log base 10 than rather than \(e\). To convert from base 10 logarithm to a natural logarithm (i.e. base \(e\)), one need to use the conversion factor calculated as \(log(10)\) which is equato to 2.3025851. Then, the equation becomes:
\[\begin{alignat}{2} \label{eq:abs4} a &= \frac{log(\frac{I_0}{I_n}) \times 2.303}{L}\\ \end{alignat}\]Note that absorbance measured by spectrophotometers is noted \(A\). Finally,
\[\begin{equation} a = \frac{A \times 2.303}{L} \label{eq:absorption1} \end{equation}\]Because we are specifically measuring absorption of CDOM, often at a specific wavelength \(\lambda\), the recommended writing is as follow:
\[\begin{equation} a_{\text{CDOM}}(\lambda) = \frac{A(\lambda) \times 2.303}{L} \label{eq:absorption2} \end{equation}\]Where \(A(\lambda)\) is the measured absorbance and \(L\) the measurement cuvette thickness (or pathlenght) expressed per meter (\(m^{-1}\)).