So here we're going to say that Beer's Law represents a theoretical model that forms a correlation between a substance's absorbance and its concentration \( c \). Here it's given by the formula that \( a = \epsilon \cdot c \cdot l \). Here, again, \( a \) represents our absorbance, \( \epsilon \) represents our molar absorptivity, a physical characteristic of a compound that usually ranges from values of 0 to 15,000. \( c \) represents our concentration. Remember, concentration is synonymous with molarity, so units are usually moles per liter. \( l \) represents the path length or the width of the cuvette within the spectrophotometer, thus, this would be path length which is the width of the cuvette in centimeters. Then we're going to say here that absorbance equals \( \log \left(\frac{I_{0}}{I}\right) \), so \( I_{0} \) represents the intensity of the reference beam, and \( I \) here represents the intensity of the sample beam.
Alright. So here we're going to talk about the spectrophotometer. So, if we take a look here, we're going to say the application of Beer's law can be visualized with the use of a UV-VIS, or ultraviolet-visible light spectrophotometer, with a conjugated compound. When we talk about a conjugated compound, it just means it has alternating double and single bonds. A great example of that would be 1,3-butadiene. So, it would look like this: a conjugated system or conjugated compound because of its double bond, single bond, double bond sequence.
The way we have this spectrophotometer is basically it's going to irradiate a sample, with wavelengths of light usually ranging from 200 to 800 nanometers. Here the light source, which can come from two different types of lamps, you can have a tungsten lamp or some other type of metal lamp, and what happens here is it shoots a beam of light, which is reflected here on this monochromator, which then gets split. So it can get split into two beams and one beam is going to pass through a cuvette containing the organic reference beam, is going to pass through a cuvette containing only the solvent.
Alright. Here this is my reference, so this one has only solvent. And then here this is the sample, so that one's going to have the organic compound that is dissolved in the solvent. And then what's going to happen here the spectrophotometer basically it's going to compare the two intensities of the beams at a particular wavelength and basically, it's going to plot those results to show the absorbance as a function of wavelength. Here it's in terms of wavelength, in nanometers, and we're going to say here, it gives us this image here of a particular compound. We're gonna say this image here is your absorbance spectrum, and it just uses computer software to generate that absorbance spectrum.
Now here we're going to say when a conjugated system like butadiene is irradiated with UV light, a pi bond can be promoted to a higher energy level and produce a UV-VIS absorption spectrum shown below. Well actually, it is of isoprene, not butadiene. We used isoprene in this one actually, and all that's really happening if we look at butadiene, we'd say that butadiene has in it 4 carbons that are all sp² hybridized. Remember if you're sp² hybridized that means you're connected to three groups, three elements in this case, each carbon. So if we branch out the hydrogens that they're each connected to, just to show the real connections. Okay. So each carbon is connected to three things.
So we have four carbons that are sp² hybridized. This gives us a number of molecular orbitals, and then we're going to say that these molecular orbitals will be psi 1, psi 2, psi 3, and psi 4. We're going to say we have two pi bonds; this one's a pi bond, pi bond is just a double bond, and this one's a pi bond. In total, there are 4 electrons within those two pi bonds, so we plot those electrons here on the ground state molecular orbital diagram that we've constructed. One electron up, one electron down. One electron up, one electron down. This will represent the ground state of my butadiene compound.
When we irradiate it with UV light, we're able to promote one of the electrons up to a higher energy state. So all that happens here is one of these electrons is just going to jump up to a higher energy state. So there it goes right there. This shows us the electron excitation of an electron and then we compare it to the reference cuvette and seeing from these two what my absorbance spectrum could potentially look like with the right software used. Okay? So just remember, Beer's law is just a way of us trying to form a connection between absorbance and the concentration of our compound.
Go on to the next page and see some calculations that are pretty typical when it comes to Beer's law and this idea of the connection between absorbance and concentration.
