| Author | Message | ||
     Bruce H. Campbell (Campclan) |
We all use Beer's Law as if it is perfectly linear, which it isn't because the definition of Beer's Law requires the use of only monochromatic radiation (and I don't know of any commercial spectrophotometer that is exactly monochromatic - there is a finite bandwidth), the absence of specular reflection from particles, the absence of fluoresence, and the re-radiation of photons from the excited species that reach the detector. My question then is; Does anyone know the exact theoretical expression(is there one) or where can I find it? | ||
| DJDahm |
Bruce: I�m not trying to be argumentative, but �What do you mean by Beer�s law�? As I see our situation, the problem is more fundamental than your question suggests. There is no �Beer�s law� for diffuse reflectance from particulate samples. Beer�s law, in some form, has been around since the 1850s. The definition of �Beer�s law� and related terms from the new Handbook of Vibrational Spectroscopy follows. Beer-Lambert law: The absorbance of a sample is the product of the Beer-Lambert absorption coefficient, the concentration and the path-length of the absorber Beer-Lambert coefficient: The absorbance divided by the path-length and concentration of absorber. Due to the possibility of decadic or napierian absorbance and the possibility of amount concentration or mass concentration, several different quantities are all Beer Lambert absorption coefficients. Absorbance: Widely used term for decadic absorbance, napierian absorbance and other different quantities. Chemists usually mean the decadic absorbance, frequently uncorrected for reflection and other non-absorption effects that must be allowed for if the absorbance is to have absolute numerical significance. Confusingly, the term absorbance is also widely used for the negative log of the ratio of the final to the incident intensities of processes other than transmission, such as attenuated total reflection and diffuse reflection. Notice the implication here. Beer�s law is defined only for transmission measurements on homogeneous samples using data that has been corrected for non-absorption effects. You list some other �definitional� limitations for the law. My own take is that the law is not defined on the basis the things that you list. Rather, they are reasons why the law may not hold. For example, the law is applicable to non-monochromatic radiation if there is only one absorber, but problems come in if there is overlap of absorption peaks from different absorbers within the band width. But however we choose to label the factors, non-absorption effects such as fluorescence and reflection cause a departure from Beer�s law. In my work, I have chosen only to worry about the effects of scatter (a term that applies to any radiation that deviates from a straight path through the sample.) What I call Beer�s law basically says that the contribution made to the absorbing power of a material by a single species is proportional to the concentration of that species. It also implies that the contributions of the various absorbers to the total absorbing power of a mixture are additive. I think we can state a starting point for particulate samples, which has been embodied in our Representative Layer Theory. The starting point is for a sample that has no scatter (and therefore the reference to transmission measurements below is a bit redundant). The absorbing power of a material is defined as the linear absorption coefficient in the absence of scatter (and other non-absorption effects). For a sample of particles, each of homogeneous composition, suspended in a non-absorbing medium that has the same refractive index as the sample, for measurements made in transmission, the Absorbance contribution from a material = absorbing power of the pure material (k) x volume fraction of sample represented by said material (v) x the path length through the sample (d). and the Total Absorbance of the sample = the sum of the "kvd" product for the componennts. I find this a more useful (compared to the Handbook definition) form for thinking about the problem of particulate samples, but it is a long way from the theoretical expression you asked for. I have derived expressions for the scattering case, some of which have been published and am continuing to work on the problem in my �spare time�. I�ll be interested in what others might have to say. | ||
| Gabi Levin |
For practical view of life, while working with individual corn seeds in forward diffuse "reflectance" or to put in practical words, transmission through the seed, some may be as skinny as 3mm and some are chubby as much as 8mm, yet, they all fit a nice regression for oil, protein, etc. Where does Beer's law fit here. Thanks, Gabi Levin | ||
| Bruce H. Campbell (Campclan) |
I can see that my including diffusivity from possible particles has confused the question to a degree. I was thinking of a liquid rather than a solid but also considering very few liquids are perfectly clean of particles. There are also what might be called eddy currents, Schlieren ones being an example, that can cause beam misalignment. With respect to the monochromatic radiation, I first heard of this in graduate school and was proven by the following (a picture would be nicer). Imagine a beam with finite bandwidth overlaid on an absorption band. The beam thus contacts a region of measureable size. The detector in essence integrates over that region. When you remember your calculus, you will realize that such an integration will not give an answer exactly equal to a rectangle with height the same as the midpoint of the region's height. Thus some non-linearity is introduced in the absorbance measurement. The deviations from the law due to very high absorbances are, in part, due to re-emission of the radiation from the relatively large concentration of absorbers that are excited. Some of this re-emited radiation will reach the detector. Even more, there is a statistical basis for the law and that is that only a very small fraction of the species of interest is excited by the beam. I think that the reason Beer's Law, OK, actually Beer-Lambert, works is the variations in it from ideality are small with respect to the other variations we experience. But we are coming to better and better instruments and sometime I project we will have to face the small variations in the Beer-Lambert Law. Plus I'm curious. If there are developments that show I am wrong with the above comments, please let me know. | ||
| DJDahm |
Gabi, Ah, the curse of the theorist: �Get practical�. I�d get upset except I have not discounted the possibility that Bruce faked a discussion starter as a plea for help (It is his list, after all). Actually, I�m paranoid enough to think Tony Davies put him up to it, so I will be motivated to finish the book draft I owe him on this subject. Seriously, it is necessary to describe the situation to know how Beer�s law fits. Remembering that Beer�s law says that that concentration will be a linear function of absorbance: 1st) The directly transmitted light will follow Beer�s law. Let us assume that any scattered radiation will depart in path just a bit from straight ahead. If we have a small area detector placed far enough away from the sample, there are two independent but additive sources of loss of intensity: absorption and scatter. This is what the physicists call �extinction�, and in the case of astrophysics (where distances traveled are very long compared to detector size or for that matter the earth) scatter tends to just be a background absorbance level on which the absorption pattern rides. Your situation of transmittance measurements (I assume you are not using an integrating sphere) is a gross approximation to this, so Beer�s law will be a much better approximation than in remittance measurements. To the extent that this is a good approximation, the size of the seeds doesn�t matter here. 2nd) All linear regressions assume that something akin to Beer�s law is in operation. In the case of PLS on a clean mixture, it takes one factor to describe each component and the factors after that are mostly accounting for the departure from Beer�s law. In the case of a dirty mixture like natural products the situation is not as obvious, but again, a successful calibration hides the departure from Beer�s law. The benefits of narrowing the wavelength range used in the regression is largely due to the fact that we are operating over a narrower range of absorbance values and the narrower the range being used, the less the departure from Beer�s law. So from a practical point of view for quantitative purposes: You are wise to choose transmission when it is practical. You would do well to keep the sample thickness as thin as you practically can. (For example, a thin sample in transflectance is more linear than an infinitely thick sample.) Cut the wavelength range down to the regions where the signal of interest exists. Don�t be shy about including additional factors as long as you can see shapes of parts of the spectra in them. And practically speaking, you didn�t need this practical advice, since you have already had success. | ||
| DJDahm |
Bruce, As I said, for one who worries about the theory of absorption spectroscopy of scattering samples, there is a more fundamantal problem: We don't have a Beer's law. I am not expert in the effect of the factors other than scatter, and I will happily sit back and listen while others discuss that aspect. I am also curious. I agree that the effects you mention tend to be small compared to the effects of scattering when it is present. And I now understand that you are talking about departures from Beer's law for the case for which it has been defined. Incidently, I'm not worried about how many names we attach to the law, the "Beer-Lambert" designation was a quote from the Handbook. Don Dahm | ||
| Bruce H. Campbell (Campclan) |
Don, Gabi, and others, Howard Mark sent me a reference (Spectrosopy 13(11), p.18-21 (1998)) that thoroughly explains many if not all the departures from the Beer-Lambert Law. Basically, the two main effects I see from the article (I could be wrong that they are the main ones) are: 1) The law is based on having an exactly monochromatic beam. In order to achieve this, there would have to be zero energy in the beam rendering it useless for practical spectrophotometry. 2) The beam path through the sample is not perfectly collimated, so some of the beam travels a longer path than others. This affects the pathlength. Still, the article didn't have an equation for all the perturbing effects, so one part of my original question remains: Is there an equation that relates all the non-instrument pertubations? Note I added the non-instrument restriction. | ||
| Bruce H. Campbell (Campclan) |
As is usual with a speedy mind like mine, I sent the latest email too soon. I should have copied Howard's comments. Howard was having trouble connecting. That is why he sent the comments to me directly instead of posting it here. If anyone else has trouble connecting, please let me know by sending a direct email to me at {[email protected]}. Howard's comments: Bruce, you're quite right in your comments. Don is also correct, but is answering a different question than the one you asked. Gabi is also asking a different question, the answer to which has to do with how closely non-linearities at different wavelengths track each other, and the way in which non-linearity at one wavelength can compensate for non-linearity at another. But none of those address your question. As I said, your comments are correct: Beer's law is a mathematical idealization of reality, and therefore the "real world" does not meet the conditions for which it was derived. Some of this is discussed in our column: Spectrosopy 13(11), p.18-21 (1998) If you don't have or can't get hold of a copy of this article, let me know and I can send one to you. Basically, there are at least two fundamental, unavoidable causes of departure from Beer's law (and the article lists several others, some of which are avoidable). The two are: 1) the finite bandwidth of a measuring instrument 2) the finite solid angle of an optical beam Both of these are inevitable consequences of the Law of radiative Transfer and the Law of Conservation of Energy: if either of them was zero, the beam would contain zero energy. Sticking with the bandwidth question for now, your second message addresses the issue: the measurement you make is the convolution of the underlying "real" spectrum, convolved with the instrument bandshape due to the instrument function. Convolution is the integral fo the cross-product between the two. But you can't do that to the absorbance spectrum, (if you want to simulate the effect, for example) you have to go two steps further back and calculate the convolution for the actual measured data, which is the reference energy spectrum and sample energy spectrum, each convolved with the instrument passband function. Then you can divide one by the other to compute T (or R, as the case may be) and further to the absorbance. Since convolution is essentially a process of weighted averaging, if either of the functions is constant, then the cross-product is the same as the non-constant function. Hence, if the bandwidth of an instrument is "sufficiently small" - "small" compared to the changes in spectral energy with wavelength of the signal, then the signal will be (or at least can be considered to be) essentially constant across the bandwidth of the instrument and the measured output will track the actual underlying "real" spectrum closely enough to make no difference. This is often the case in NIR, especially with natural products, and other samples with broad absorbances. The question then becomes: how small is "small enough" and you have to run numbers to see if it matters in a case of interest to you, since theory says that it will always be non-zero. But if the sample has narrow, sharp absorbance bands, or the source energy or detector sensitivity has sharp changes in response, then the energy will change across the bandwidth of the measurement, and the effect of the convolution will matter. Here again, though, you'd have to do the calculations to figure out exactly what the effect is. There are two flys in the ointment: knowing the instrument function and knowing the "real" spectrum. As you are probably aware, there are several methods available to try to "deconvolve" a measured spectrum and recover the "true" spectrum. Even if you know the instrument function, this process is fraught with problems, since the actual physical convolution of the instrument function with the underlying spectrum, that gives the measured spectrum, destroys some information. When you try to deconvolve them, this loss of information is usually manifested as "divide by zero" errors during the calculations, unless you are satisfied with an approximate deconvolution. So that, in a nutshell is the answer to your question. There are longer treatises, but I can't give you any references, I'm afraid. Howard | ||
| hlmark |
Bruce - for some reason, my connection to the discussion group "suddenly started working" again, so I can comment directly (at least for now). The "equation" you're seeking exists, but is not a simple one: it's the ratio of the two convolutions: of ([instrument function] * [source energy] * [detector response] * [sample transmission]) / ([instrument function] * [source energy] * [detector response]), where the "*" represents the convolution function. And that's just for the finite bandwidth effect. Other sources of non-linearity will each have their own formula, depending on where and how the effect enters into the equations. For example, stray light converts: Equation 1: A = log (T) = log (Io / I) into Equation 2: A = log (T) = log (Io + S) / (I + S) where S represents the fraction of the energy due to stray. For equation 1, both the actual and the calculated T are constant as Io changes, and A is linear with changes in concentration. In the presence of stray, as in equation 2, the actual T is constant, of course, but the calcualated T is not constant as Io changes, and the absorbance is not linear with concentration. Each type of effect has it's own nehavior and you have to take each into account at the point where it enters the (optical) system. Howard \o/ /_\ | ||
| Tony Davies (Td) |
Dear, Bruce, Don, Gabi and Howard: Thanks for a very interesting discussion (no Don I did not initiate it!); I'm sorry not to have joined in but I had a deadline and a mound of data. I'm not sure that I actually have much to offer. I do not think the lack of a "Beer's" law for diffuse reflection is a current limitation. However, if we had one, it might persuade instrument makers to design an instrument which measured transmission and reflection at the same time (as in Krivoshiev's potato peeling instrument). But until Don and friends are successful nobody wants to listen. We need new instruments to make new advances One thought: when I attempt to explain diffuse reflection to students I talk about photons. Is anyone aware of any theory based on photons rather than rays? This discussion has reminded me about two other questions which I will post as new questions. So again, thanks! Best wishes, Tony | ||
| hlmark |
Tony - you brought up several topics in one message. I certainly agree that having a better theoretical foundation for everything we do will help promote our understanding. You may be putting the cart before the horse, though. Notwithstanding Karl's original development of NIR based on empirical studies, most advances in instrumentation, as well as science in general have occured when new theory was developed and then practitioners implemented the implications of the theory. So while new instruments do help understand what theory tells us, the developments are usually the other way 'round. Photons are not often considered in NIR because they represent the "particle" aspect of the duality. "Particle" properties tend to become more important as wavelength decreases, and "wave" properties tend to become more important as wavelength increases. The visible portion of the spectrum is where the two aspects are of rougly equal import. NIR, being on the long-wavelength side of that, is more affected by the "wave" properties. The only place I can think of where the quantum effects are important is in the operation of the semiconductor detectors we use. Those operate (mostly) by having electrons (or "holes") promoted from the valence band to the conduction band by absorbing a photon. Otherwise I can't think of any phenomena that are operative in the NIR region. Howard \o/ /_\ | ||
| Bruce H. Campbell (Campclan) |
To all, Don and Mark exchanged some comments with me as one to stir the water(?) here and there. I have collected the exchanges and, I hope, in the correct order. At least they make sense to me. If you are puzzled, you may want to query one of us. Anyway, the collection is here. I have not edited them. Also, the headers are not there, but I think you can figure out who sent each part. If you can't, scroll down to the sign off and look for the name. Howard, Thanks for your response. I have not really thought much about these issues except in regard to scattering, so I was looking for correction on many of these things rather than trying to defend a position.I'll get the Spectroscopy issue the next time I get to Villanova. Our library at Rowan is not yet what it should be, and doesn't have it. My picture from what you've said is that it covers the instrumental part very thoroughly. I think Bruce has raised an important topic, and I think I listed some sources of deviation beyond the instrumental, but I'm still hoping that he will be able to hit the literature and write a short review article. I could help a bit with it, but I don't really have time to do the literature search necessary. Don Howard and Bruce: I thought I would try to summarize where we are in our recent discussions on departures from "Beer's" Law. If you guys want to help improve them before they get posted for the others that would be great. If you don't think it's worth posting them, let me know. Of course, it would be better still if someone (maybe a retired person, Bruce!) would do a piece on this for NIR News. What I refer to as the Bouguer-Lambert law says that light, when moving a distance d through an homogeneous medium, will be attenuated by absorption according to the formula: I/Io = exp(-kd), where k is the linear absorption coefficient of the material. There are deviations from this ideal case though the law itself is thought of as being "exact". However, there is an implicit (frequently unstated) assumption that in the range for which the law holds, the absorption capacity of the material is infinite. 1) Instrumental deviations: The ideal case requires that the incident beam be non-divergent and the measured beam be monochromatic. Departure from these gives results discussed in Spectroscopy xxx . 2) Surface reflection: In a typical transmission experiment, there are several interfaces thorough which the light passes (for example, air/cell wall; cell wall/sample; sample/cell wall; cell wall/air). At each there is the potential for reflection and refraction. For the ideal case (above) and a flat, smooth cell wall oriented perpendicularly to the incident beam; reflection causes a loss of light which is a constant offset in the absorbance function. [Loss at these surfaces is frequently corrected for approximately by placing an empty cell in the reference beam.] For surfaces that are not flat and smooth, there is an additional effect of refraction that tends to broaden the beam and introduces effects such as encountered in 1). 3) Scatter: Inhomogeneity of the refractive index throughout the sample introduces departures from the ideal case. The effects of these range from being minor perturbations to the major departures from the law. When the departures are significant enough, it becomes practical to make measurements using geometry other than transmission. No completely satisfactory theory for transmission through or remission from a highly scattering sample has been worked out. The theories are summarized in the Handbook of Vibrational Spectroscopy, especially in two chapters in Vol. II, one by Griffiths and Olinger (p 1125) , one by my son and I (p 1140). Our contribution has been to develop the Representative layer Theory, which includes the Dahm equation that bridges results for continuous and discontinuous two flux cases. Burger et al have the latest publications that use the Equation of Radiation Transfer as their basis. David Burns and Britton Chance have been using time of flight measurements, a technique that gives added information. 4) Fluorescence: There is a treatment by Schuster (circa 1905) that deals with material that is both scattering and luminescent. It uses a continuous model, so can be expected to have the same limitations as the Kubelka-Munk theory. What did I miss???? So just what is Beer's law? What I refer to as Beer's law makes an addition to the Bouguer-Lambert law in that that we replace the absorption coefficient for the material by the absorption coefficient (K) for the absorber in pure from times its concentration (c). [If there is more than one absorber, it is replaced by the sum of such terms.] A new equation is obtained by taking the logarithm of both sides and calling it absorbance: Absorbance = ln(I/Io) = Kcd [The form that is most often seen by spectroscopists uses the base 10logs.] This law written in this form is frequently referred to as Beer's law in common speech, but in print is increasingly called the Beer-Lambert law. Notice that we now are not talking exclusively about transmission through a single sample, but relating a group of samples in which the concentration of the absorbers may vary. The Beer-Lambert law is not thought of as being exact, but as being a very good approximation at low to moderate concentrations of absorbers. We commonly use the expression "deviations from Beer's law" to mean any plot of sample absorbance versus concentration that is not linear. Such deviation from linearity may have additional causes to those mentioned above. By and large, for a single absorber, the deviations from Beer's law are minor. However, for two or more absorbers, deviations are routinely encountered by spectroscopists. Two important causes for these deviations are: 1) Units: The absorbance contribution of an absorber is proportional to the number of moles of absorber per unit volume. Units such as weight percent or even molarity are not proportional to "proper" units, especially at high concentrations. 2) Bandwidth: The total absorbance from a mixture follows Beer's law reasonably well for very broad bandwidths as long as the detector is equally responsive at all wavelengths. The case is similar with truly monochromatic radiation. However, the absorbance taken over a narrow wavelength range will not be linear with one component even in the presence of a constant amount of a second one if the band is centered on the side of an absorption maxima of the second, absorber. Don Don - it'll take me some time to read the whole thing, too - It's too much to read on-screen. But I got as far as your asking "What did I miss????" and here it is: as I said before, Beer's law is a theoretical construct: an idealization of reality. As such it is independent of the measurement system. That means it addresses the question: how does the light behave in the sample itself, after passing through the cell window on the way in and before passing through the window on the way out? All the items you bring up are functions of the measurement, not of Beer's law itself. To that extent, your responses don't address the fundamental answer to Bruce's question. To the question: "Does Beer's law EVER hold in the real world?" the answer is still NO. I've addressed this somewhat more extensively in the column for which I previously gave the reference, but I'll give it again: Spectrosopy 13(11), p.18-21 (1998) If you haven't had a chance to read it yet, I recommend you get hold of a copy and read it; if you need, I can send a copy to you. Basically, for a lossless optical system (i.e., no scatter, no absorbance - this also counts out fluorescence, since that is radiation emitted at a different wavelength, after the original light was absorbed) the Law of Radiative Transfer applies. The Law of Radiative transfer states that the energy transferred by an (lossless) optical system is expreesed as follows: dE = I * d<lambda * dA * d<omega * dt where: dE is the energy transferred in time dt I is the optical intensity lambda is the wavelength A is the area of the beam omega is the solid angle subtended by the beam Since the optical system is (by assumption) lossless, E is constant throughout. The Law of Radiative Transfer then arises out of the second law of thermodynamics. It can be shown that, since in any given beam I * d<lambda is fixed, the product dA * d<omega must be constant. If that were not true, than it would be (theoretically) possible to focus all the radiation coming from a body onto a part of itself, that part would then heat up, and you would have created a temperature difference without doing any work: a violation of the second law. Under these circumstances, neither dA, d<omega or d<lambda can be zero. Since the energy is the product of those three terms, if any of them were zero, then the beam energy would of necessity also be zero: i.e., the beam would be non-existent. Immediate practical consequences are that you can never have a perfectly collimated beam (even a laser has finite divergence), you can never increase the resolution of an optical system indefinitely (the usual rules given for the limitations of microscopes and telescopes are consequences of this fundamental underlying law) and a few other things. It also means you can't have a spectrometer with "infinite" resolution, or you would again wind up with zero energy. Therefore, all three terms must have finite values. Of interest to us is that due to the lack of collimation, Beer's law fails in any sample with absorbance, since rays at different angles will experience a different pathlength through the sample and therefore undergo different amounts of absorption - a REAL, fundamental difference in the behavior of the light through the sample, independent of how it's measured. These differences in pathlength result in non-linearity of the absorbance-versus-concentration characteristic. This is the case I discuss in the column. The requirement for finite resolution has similar consequences. Anywhere the sample has non-constant absorbtivity as a function of wavelength will give rise to different amounts of absorbance at different wavelengths, resulting in non-linearity of the absorbance-versus-concentration response. The saving grace is that we can make optical systems that are sufficiently collimated, and have sufficiently narrow bandwidth for these effects to be negligible, and the resulting measurements to be useful and sufficiently linear for practical use. But theory says that it's never perfect. All the other effects you mention (and some other instrumental effects, such as stray light) are over and above this fundamental limitation, and those each depend on how (and for what) the measurement is made. There are also several others, some of which are listed in the column, effects such as interactions of an absorber in a sample sample with the solvent, with other constituents in the sample or with itself that cause the absorbance to change non-linearly with concentration; those effects are over and above even the ones you list. Now I'll go print out your whole message and see if there's anything else I want to comment on Howard Don - Minor addendum I forgot to include in the previous message: the effects of non-constant pathlength or non-constant absorbance across the bandwidth have standard treatments in the literature on chemical analysis. I haven't checked it out but advanced texts such as Ingle & Crouch's "Spectrochemical Analysis", Prentice-Hall (1988) should cover those topics. Howard Don - Now that I've printed out your message and had a chance to read the hard copy, I see where I missed a couple of your points, reading it on the screen. I also now think that we're in substantially more agreement than I originally did. Basically, you're saying that WHEN there is non-collimation or non-uniform absorbance over then bandwidth of the measurement, there is non-linearity. I agree fully with that, but in addition I'm saying that there is ALWAYS non-collimation AND non-uniform absorbance over the bandwidth (except, perhaps, in those rare cases where the sample is perfectly grey). Therefore, I'm saying that even in principle there is never a linear relationship between absorbance and concentration, because of these effects. The only remaining difference is the quibble as to how much is "too much" to call linear. In practice there is a wide range of cases where the linearity is sufficiently good to be useful, in both single and multiple-constituent cases. In thoery, none of them are "really" linear. I think you would probably agree with this assessment. Some other comments: Even for a single absorber there can be non-linearity from (chemical) interactions. Water and alcohol, for example, are known to have such strong interactions that even such a gross characteristic as the volume is affected: it is well-known for example, that mixing 1 quart of water with 1 quart of alcohol gives about 1.95 quarts of booze. The spectrum is affected just as much. I have some data, but not at enough different mixture levels to be really useful. Maybe I can get some more data - I'll work on it. Regarding units: the chemist normally does not use weight percent as a unit, that's an artifact arising out of the history of NIR, where those units were used by the instrument companies (followoing Karl's lead) because that was what the users were used to. It was good in that it's what made the technology "take off", but now it leave is scientific types with this non-standard unit to work with. The chemist historically used Molarity (moles per liter) as the concentration unit (for liquids, at least), and the historical expressions for ansorbance were put in terms of tha unit. Even so, weight fraction could be used jsut as well, I think, simply by changing the value of the absorption coefficient; after all the difference is constant factor: the molecular weight. So all in all, I think all our arguing boils down to the difference between the "purist"'s view and the practitioner's view of what constitutes "linearity". And the prectitioner has to decide on a case-by-case basis whether a given situation is "sufficiently linear" for his current purposes. Howard The exponential expression that Don showed is theoretically correct (as long as you consider it in the limit of d<lambda - 0 and d<omega - 0, and ignore the fact that such a beam will have zero energy). As for the series expression, you can play the usual mathemtical game of expanding the exponential term as a power series and then just keep as many terms as you need for "sufficient" accuracy. I don't know how old that transformation is, but it probably goes back to the time of Gauss. Back then such "tricks" were necessary in order to evaluate all transcendental functions, since no computers were available to do the computations, and individual terms of the series are usually relatively simple to calculate. For that matter, that's mostly how computers evaluate them today, although there are some other techniques that have been developed to minimize the amount of computation needed. But that's off the topic: as far as I know, the bottom line is that the series Re Don's comment about the limitation on collimation due to the wave character of light: I don't know if that argument is self-sufficient to explain it or if it's a consequence of the thermodynamic argument. I can imagine a (water) wave tank with a narrow channel, with one-dimensional waves propagating down the channel. I also recall the "particle in a one-dimensional box" from quantum mechanics. But in both cases, those may be merely idealizations, which don't actually occur in the real world. Howard There are two parts that haven't been addressed. One is that I was told, and I believe it even more strongly now that I have read what you two have written, that the Beer-Lambert law is theoretically met at what I will call infinite monochromicity of the beam. I know this disregards the other aspects, such as beam angle, etc. The other part is I learned the law is actually a series equation and the higher terms are neglible at very low dilutions, relatively speaking. This comes from the probability of a molecule absorbing once in a measurement instead of, at very high concentrations, the molecule could absorb more than once during the measurement. You can see that as the concentration becomes higher, the probability gets larger that it will not only absorb twice but even more than that. Thus, the law only will hold for cases when the number of times a species absorbs is the same. Bruce | ||
| Bruce H. Campbell (Campclan) |
I found out, or was gently reminded, that I didn't include two references given by Fred McClure. These two are books by Wendlandt & Hecht and Kortum, excellent and classical treatments of nonlinear spectroscopy - including scattering. | ||
| leena |
what is the effect of complex formation on applicability of Beer-Lambert LAW | ||
| DJDahm |
What is the effect of complex formation on applicability of Beer-Lambert Law? Given the mess above in this chain (to which I contributed), it seems we are all confused. Part of the confusion is definitional and part is theoretical. The short answer is: �Beers law does not hold for complex formulations!�. First, there is the Bouguer-Lambert law. Chemists tend to ignore it, and just think of it as wrapped up in Beer�s law. This has been fueled by references to the Beer-Lambert law. But the Bouguer-Lambert law says that within a homogeneous material (notice we have no interfaces under consideration), the falloff in the intensity of a transmitted beam is given by exp(-kd), where k is a measure of the absorptivity of the sample and d is the distance traveled. From a theoretical point of view, the Bouguer-Lambert law is thought of as exact. From a practical point of view, there are limitations on when measurements will conform to the law. These practical matters have been discussed by Howard Mark (and I believe Jerry Workman) in a very nice series of articles. Beer�s law says that if you know k for a pure material (we�ll call it k(pure), and you know the concentation (conc) of the material in a sample, the contribution of the material to the absorbtivity of the sample will be given by conc*k(pure). From the Bouguer�Lambert law, the mathematical expression log(1/T), called absorbance, yields a parameter that is proportional to kd and therefore, by Beer�s law, proportional to the concentration of the absorbing species (if we hold d constant). Remember, the above relationship holds only within a homogenous material. For a mixture containing several components, the absorbtivity of the mixture may be approximately related to the absorbtivity of the individual components by an expanded form of Beer�s law. k(mixture) = SUMMATION k(i) conc(i) Rigorously, the measurement of concentration for which Beer�s law holds is number of absorption sites per unit volume (though it is possible to adjust k so that the summation is a good approximation for other definitions of conc(i).) The implied assumption is that each component of the mixture has discrete absorption sites, acting individually, and that mixing does not change the nature of absorption by an individual site in any way. Thus when one mixes two types of absorbers, there is no effect on the absorption contribution of an individual component except the decrease in its concentration through dilution. |