Measuring spectra: The Beer-Lambert law

The figure below shows the experimental setup for taking a spectrum:

Figure: Schematic of the experimental setup needed to record a spectrum.

Radiation from a source is passed through a monochromator, which filters out all frequency components except one, generating monochromatic radiaion of a single frequency $\nu$. The beam is then split into to two by a beam-splitter. One of the two rays is allowed to pass through the a cell containing the sample whose spectrum is sought. The other is allowed to pass through a reference cell, which is just the cell without the sample. We'll come back to the role of the reference cell later. Let $I_0$ be the intensity of radiation entering the sample, and $I_S$ be the intensity of radiation emerging from the sample. The output beam with intensity $I_S$ is then sent to a photodetector that measures the actual intensity, the photosignal is then amplified and sent to a recording device, where a final readout of the intensity at frequency $\nu$ is noted.

Let us first analyze what happens as the beam is sent through the sample. Let the direction of propagation of the radiation through the sample be the $x$ direction and let the sample extend from $x=0$ to $x=l$ on the $x$ axis. Thus, $I_0$ is the intensity at $x=0$. We wish to determine the intensity $I(x)$ as the beam passes through the sample. As it does, there will be an attenuation of intensity due to absorption events. Remember that the beam contains many many photons, and the sample contains many molecules, so we are interested in a measure of the number of photons absorbed vs. the number of photons that pass through without being absorbed. This will give us a measure of the attenuation in the intensity, since intensity is directly proportional to the number of photons passing through a given area of the sample at any instant in time.

We expect the following. If $I(x)$ is the intensity at point $x$ in the sample, then the fractional loss of intensity $-dI/I(x)$ when the beam passes through a length $dx$ of the sample will be proportional to the concentration of the sample $C$ as well as $dx$:

$$-{dI \over I(x)} \propto Cdx$$

The constant of proportionality is denoted $\varepsilon'$ and is called the molar extinction coefficient:

$$-{dI \over I(x)} = C\varepsilon' dx$$

(The reason for the prime will be clarified below.) The units of the molar extinction coefficient are L$\cdot$mol$^{-1}\cdot$m$^{-1}$. Rearranging this as a simple first-order differential equation gives

$${dI \over dx} = -C\varepsilon' I(x)$$

The solution gives us the intensity $I(x)$ as a function of $x$ through the sample:

$$I(x) = I(x=0)e^{-C\varepsilon' x} = I_0e^{-C\varepsilon' x}$$

Or taking the log of both sides

$$\ln\left({I(x) \over I_0}\right) = -C\varepsilon' x$$

When $x=l$, $I(x) = I_S$, so we can write this as

$$\ln\left({I_S \over I_0}\right) = -C\varepsilon' l$$

The attenuation of the beam through the sample will be due to only in part to the actual material whose spectrum is sought. There is another contribution to the cell, itself, and whatever material it is composed of. This is where the reference beam comes in. By observing the attenuation of the beam through the reference cell, we can determine how much of the attenuation is due to the cell alone. Thus, we are not interested in the ration $I_S/I_0$ as much as the ratio $I_S/I_R$, where $I_R$ is the beam that emerges from the reference cell, partly attenuated. For the ratio $I_S/I_R$, we assign the molar extinction $\varepsilon$ and write

$$\ln\left({I_S \over I_R}\right) = -C\varepsilon l$$

or

$$\ln\left({I_R \over I_S}\right) = C\varepsilon l$$

The quantity $\ln(I_R/I_S)$ is called the absorbance $A$ (the quantity $\ln(I_S/I_R)$ is called the transmittance $T$). Thus, we have finally

$$A = C\varepsilon l$$

This result is known as the Beer-Lambert law, and it is one of the fundamental principles of molecular spectroscopy. The extinction coefficient $\varepsilon$ measures the extent to which the sample is able to absorb radiation at a given frequency $\nu$ or wavelength $\lambda$ (remember we can use either since $\nu = c/\lambda$). Hence, it is an intrinsic property of the material.