Lesson 1: The “General Beer-Lambert law”:

1. The Beer-Lambert law ^[1] is much more actual as ever before, but it is mostly not obeyed correct by almost every spectroscopist!!!!     The law is all time true! — except in its absolutely extremes. IF YOU or ANY SPECIALIST thinks, that a solution is not longer following the law of Beer - Lambert, don't believe you or him !!!!!!
   Think first, what you probably was not obeying enough, which spectroscopic circumstance you was not putting into your calculation! But to understand what I mean, you must first learn to think in spectra, and not in single wavelength!  —

   I have never seen a spectroscopist with a single wavelength in his solution, he always had some substance 'solved' in his cuvette, and that means spectra !!!!!       It's high time to learn it now !!!

   (We were never able, to find a real case, when a solution was not following strictly the law of Beer - Lambert! In every case, we finally found, that we was not taken enough care to a “spectroscopic effect”, — we was not counting it in a correct way! But with “spectroscopic effects”, it is not meant all this “pseudo correlations” of a single wavelength measurement against a “physically effect”, reported so many times in the common literature!)
   
2. Be aware that the following four steps has to be taken with very much diplomacy and care !!
3. Ask your spectroscopist how many litres/gallons of Ethanol(Alcohol) he normally needs for preparing the solutions for measuring, during a year.
4. Now ask him, specially if he told you, that most measurements are made in water, to estimate how many litres/gallons of water he would normally need for preparing the solutions for measuring, during a year.
5. Now, if you dare the impossible, ask him, how many litres/gallons of buffer (For the first time it doesn't matter, what for a kind of buffer. If he is using always a strong acid or a strong base, count it as a buffer.) he would normally need for preparing the solutions for measuring, during a year.
6. Finally count the amount of Ethanol, water, and buffers together, and calculate how high the percentage part of buffer will be. At last it's time to think about, if the amount of buffers will be high enough, when you are knowing, that almost every chromophor, that has an absorption in the visible range of the spectrum, is also more or less an indicator! But there are also a lot of pharmacy products, they will show you a certain reaction on pH changes!
7. What have we learned from doing this four steps? Use, — when ever possible, and it is almost every time possible, a solvent mixed with a buffer, for preparing your measuring solutions. If you have troubles to find a buffer, that does not “precipitate“, when you mix it together, please mail me. I will give you our experience in this field, and I can tell you, we are using several solvent mixtures for a lot of problems.
8. What have we learned, too?   Yes, — that all atlases, at least the UV/Vis-atlases, the NNIR-atlases, and the NIR-atlases, are of absolutely disputable value, because if you take a look a the solvents they have used for measurements, you find mostly Alcohol (Methanol, Ethanol), Hexane, Water, Acetone, and similar solvents, but never “pH adjusted”! You can be lucky, if you find a spectrum taken in presence of a strong acid or a strong base!
   If you need an additional example,^[2] I have to go very far back into our history(1983), because this mistake we never did again.
9. A spectroscopic influence you should already know, is of course, that the spectrum of a compound is very strong depending on the solvent you are using to prepare your stock-solution^[3] and the necessary dilutions.
   That there are other effects under there misleading name: 'nonlinear optics' what is not meaning, that it has to be an effect, which is not obeying the Beer-Lambert law. I will only tell it you for the moment, and discuss it later.
10. Now I think, it is coming the most significant learning-step for every spectroscopist, and specially for all of them, they are stamped on their memory with single wavelength measurements during a whole life !
    We have to change from the “Special Beer-Lambert law”, you have learned in step 1, to the "General Beer-Lambert law", in German: "Allgemeines Beer-Lambert Gesetz".
    You ask me, what I will more generalise at the 'old' law ?   NOT I will, WE SPECTROSCOPISTS, have to do it! I only think, it is high time to do it !
    You remember the “Special Beer-Lambert law”:
    

    (Sorry, but I like it a little bit more correct!)

    Don't forget: It isn't defined, neither on your lamp, nor at your detector! It is defined at both surfaces of your sample! That means, scattering light is not absorption of light, it is loosing light, — never mingle it !!!
    Question: For what is the product  (c * d)  standing (representing what?)?  —
    NO, Its much easier!:
    

    Number of Species[4] in your Measuring Beam:  nsmb   (c * d). [4]

    Read:  -> direct proportional !!!

    So it is much more than ONLY wise, to never forget it any more !!!
    
    Now let's have a look at the "General Beer-Lambert law":
    
    
    Can you understand, what it is telling you ? What you have to do from now on, till the end of your “life as a spectroscopist” ? I don't believe, but let us discuss it in Lesson 2 !

Epilog to Lesson 1:

Lesson 2: Consequences from the “General Beer-Lambert law”:

Let's start our discussion with a look at the methodic's of the different spectroscopist's:

	1.)	What “almost every” spectroscopist does: cS = cR * ( AS / AR )           (1)   Where:   cS    is the concentration of the sample solution to be determined.   cR   is the known concentration of the reference solution.   AS  is the recorded absorbance of the sample solution in the cuvette of the size d.   AR  is the recorded absorbance of the reference solution in the cuvette of the size d. For the first view it doesn't seem to have very much to do with the law of Beer-Lambert ! You may be surprised, but I would never try to argue against this, your conclusion !!!
	2.)	What a “putative good” spectroscopist additionally does: A putative good spectroscopist prepares a few additional dilutions different ratio, from the prepared reference-stocksolution, or in the best case, prepares a few reference-solutions with different amounts, and try to show you a kind of linearity of equation (1). And if they try to be very good, they start to 'forecast' you the allowed range of linearity of equation (1) but only for the reference-solution !
	3.)	What a really Good Modern Spectroscopist (GMS) must do: A really good spectroscopist would apply the general law of Beer-Lambert to the equation (1) and collect, — and apply it to his further work, all consequences that follow from interpreting the new equation! Let's try to do it together:                         (2) STOP !   STOP !  : That would be typically "spectroscopist like", do only the half of the work, and tell everybody to be not responsible for all that follow ! I — told you, to apply the   “General” Beer-Lambert law:               (3) SO !   That's much better. With this I can live for the moment !: Of course, this is mathematical absolutely not correct !   But this equation is already enough precise, to define ALL responsibilities we have as real good, “modern” spectroscopists (GMSs), and let's be told, that from the exact equation no additional responsibilities will follow, specially not, as we will only describe the situation inside the cuvette, and not on the active element of your detector: (You will realise it, when we will talk about the Real Multi Component Analysis (MCA)[5] together !)   Where:   cS    is the concentration of the sample solution to be determined.   cR   is the known concentration of the reference solution.     is the concentration of the component i in the sample solution.     is the extinctioncoefficient of the component i in the sample solution.     is the concentration of the component j in the reference solution.     is the extinctioncoefficient of the component j in the reference solution.      is the path length of the cuvette. It is according to the “old” methodic mostly identic for referenz und sample.     is the complete number of components of the reference solution.      is the complete number of components of the sample solution.
For the gentle reader, who has already understood the "General Beer-Lambert" law, including this three steps we did, it should, at least now, be clear, that the (general) Beer-Lambert law will NEVER set a limit for the linearity !!!![5]     Rather, it is the physics that sets us a limit for linearity, as it gives us the limit for the highest possible “concentration” we are able to prepare and bring into the measuring light beam. !!! Peter Forster It is also clear, that there are a few exceptions, but really a few, and most of it are of such kind, that a spectroscopic effect is tried to be measured and interpreted as an absorbance what in reality never will be true. A typical example are the so called “pseudo solutions”. Their apparent Absorption is coming from the fact, that this type of “solution” is primary scattering the light and not absorbing it. That a lot of spectroscopist handle this fact of loosing light equal, as if it would be a real absorbance, can only be thought in contempt of Mr. Beer und Mr. Lambert. Moreover, only for completeness, I will list the class of compounds, which show you an optical effect, for normal spectroscopist's misleading called “non linear optics”, as they are not following the law of Beer-Lambert, in the way, that at least a part of the light I0 will be reemitted at, or transformed to, an other wavelength! Under this class some other effects are collected, too. To this 'nonlinear optics', the same can be told as to the 'pseudo solutions', what not means, that they are less interesting effects, but in an absolutely different kind of research!   Peter Forster Moreover, it should now be clear, that almost any correlation, which tries to find a equation for a part of the “nonlinear” range of the Beer-Lambert law to any physical property, must be a “pseudo-correlation”, as in reality, in the best case, the correlation should be determined against a concentration of a specific compound ("species") for the observed range !!! Peter Forster
	4.)	Consequences for a “really good, modern” spectroscopist (GMS) from equation (3): The spectroscopist must be evidence of, that m = 'complete' and n = 'complete' holds ! That means, too, that the solvent needs to be included, if it shows some significant absorbance at the wavelength of measuring. It will have fatal consequences in such a situation, if a spectroscopist feels himself on the secure side, in case he uses a double-beam, double-monochromator instrument, and has to determine higher sample concentrations (keyword: linearity till 4.5 AU's). The spectroscopist must be evidence of, that m = n holds ! Moreover he must be evidence of, that  =   for every index i = j holds ! With other words: He must be evidence of, that the identical compounds (species) will be contained in the sample as in the reference. That must also hold, to be not misunderstood, specially when in the view of the spectroscopist, only one compound should be contained in the sample as in the reference. But that meas, that the spectroscopist has to verify each sample ! (sample-method-validation sends you Kind Regards.) In the same way, the spectroscopist must be evidence of, mainly if spectra would be stored, that dsample = dreference holds ! Otherwise he must calibrate the used cuvettes and put their path length into calculations ! By the way: The consequence that must be fulfilled to make the equation (3) correct, is: For every index i = j        = r = const.  must hold ! That's telling us, that equation (3) is only a special, but limited version of the correct, general equation, that should be used! How about: - Are you able to write the mathematical correct equation (3) ? Try it once !    It's not so easy as you may think ! (Hint: If you are using matrixes/vectors it becomes much easier !)
	5.)	How a “really good, modern” spectroscopist (GMS) conforms the requirements from equation (3): Already by the step of developing a spectroscopic method, I would like to distinguish, for better understanding, between two cases:   A.)  When an “alleged” pure compound is presented: The “really good, modern” spectroscopist prepares two till three stock solutions slightly different concentrations, but he already accounts for the later sample concentrations. Moreover, he prepares an additional stock solution with approximately twice, or three times the concentration of the solutions unter 1. Now he prepares from every stock solution approximately eight till ten, the more - the merrier, dilutions over a real expanded concentration range ! Subsequently he records the spectrum from every dilution, and that in a way, if the spectrophotometer allows it, over a expanded wavelength range, that contains also a curtain part (usually one third) of the “base line” !     If he tries to do this step with a recording speed of more than approximately 100 nm/min., he will fail latest at step 6 ! As first step after recording the spectra, he will proof the linearity of each dilution raw at the wavelength he selects to be the best for “measuring” the concentration of samples. As next, he will compare each linearity raw against the others, too. As second step, he plots all spectra of each dilution raw for a normalized concentration on a separate sheet and shows you the evidence of, that all spectra are absolutely congruent, expect a small increasing noise for the very weak solutions That's real linearity and how you have to test it ! Now as you have all results, sign it all and stitch it all together. This will be the minimal report about your development of a spectroscopic method !.     Stitch all together and personaly sign it!  — That's all!! This whole procedure of developing a method will be done by a “really good, modern” spectroscopist in about a little more than one hour ! ! !         ( see: Linearity Report![6])   (Are you interested to see how such Linearity looks?)[6]   B.)  When we are knowing, that we will have a mixture of different compounds: Never use mixtures for the following step (2.) !!! If you don't have access to “pure” compounds, there is an alternative procedure, how you developed a spectroscopic method, but about that we will talk later! For each compound that may be present in the sample mixture you have to repeat the procedure described in steps 5.)A.1 till 5.)A.6 ! Now you prepare several artificial sample mixtures, by “pipetting” known different volumes from the already prepared stock solutions to simulate a certain variability of mixtures. From this artificial samples the “really good, modern” spectroscopist records the spectra and determines the recovery rate for each individual compound of the mixture !!! At the same time he recalculates from the found recovery rates the “sample spectrum” and shows you, that the real measured and the recalculated spectrum are also absolutely congruent !!! With doing so, he gives you the evidence of, that there will be not any unknown “reaction” between the compounds of the mixtures, not a chemical, nor a physical one !!!  You are asking me, what we will understand as a physical reaction? Under this term we understand, for example, a loose complex in the particular solvent (existent only in this combination). It must be clear, that at a such reaction all time the chromophor system of two or more molecules are participating, and that means, that the spectrum of the mixture will significantly differ! But you have never seen it, because you were all time measuring only one wavelength and telling, that the observed system were not following the law of Beer-Lambert. In case, such a chemical or physical effect is detected, the model for the mixture needs to be corrected !!!! (You need at least an additional compound!) Now as your model is correct and you have all results, sign it all and stitch it all together. This will form the minimal report about your development of a spectroscopic method for mixtures !. That this, a little more complex method development costs also a “really good, modern” spectroscopist some more time, as in the case of an “alleged” pure compound, can easily be understood! This procedure for mixtures can be shorted and made more reliable at the same time with less stress. We will see it, when we are talking about “real Chemometrics”.
From all what is explained on this page, now two facts are absolutely clear formed: For the future for us “really good, modern” spectroscopists (GMSs), there will be NOT any excuse any more, to be responsible for a absolutely correct developed spectroscopic analytical method and, of course, take very much care for a necessary (required!) and adequate Sample-Method-Validation (SMV)!!! That this, our responsibility, can not be fulfilled, even with the newest generation of the “scanning” spectrophotometers !!! To solve our problem, we need very reliable and very well designed Diode Array Spectrophotometers !!! We need additional a device called "Dynamic Dilutor" to realize the whole lot of required dilutions in a time as short as possible, and with as less solvent as ever possible, too !!! You can be lucky, as both devices are easily available today. Peter Forster
I will soon continue.