general chemistry - Beer's law and molar absorptivity

[Monkeymaniac]

I'm a bit confused regarding the molar absorptivity, or solute specific constant e in Beer's law Absorbance = eCl.

In TBR, it is stated that

Beer's law is expressed in Equation 1.3, where e is a constant for the solute at lamdamax (the wavelength of greatest absorbance), C is the solute concentration and l is the width of the cuvette.

Absorbance = eCl (1.3)

So the paragraph is definiltey alluding that the constant e is a constant specific for the wavelength that yields the greatest absorbance.
But later while solving a TBR practice passage, I got a question whose answer contradicts with what the book says above.

The question is,

Which of the following relationships may be TRUE?
I. As the molarity increases, the absorbance increases.
II. Some straight-forward wrong answer.
III. Absorbance = eCl at all lamda where absorbance of light can occur.

A. I only
B. II only
C. I and III only
D. II and III only

Because what I read earlier, I thought that III was a wrong statement and chose A. as the answer, but the book says C. is the right answer.

And in the solution the book justifies why III is a correct statement by saying that

Beer's law applies at all wavelengths (lamdamax is chosen because it is the greatest value, and thus is the easiest wavelength at which to obtain an accurate measure of absorbance). Because statement III is valid, choice C is the best answer.

WTF is it saying? Does e even vary with different light wavelength? Then during the exam, should I assume that e may correspond to any wavelength and not necessarily lamdamax?

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[sleepy425]

ok, so Beer's law holds for all wavelengths. Beer's law simply states that absorbance increases linearly with increasing concentration of the analyte. That's all it is.

Now, in the equation, A=ebC (I use this notation, it's more common, e is the molar extinction coefficient/molar absorptivity, b is the path length, C is the concentration of the analyte), the molar extinction coefficient varies depending on wavelength. At lambda max, the molar extinction coefficient is highest, and on either side of lambda max, the extinction coefficient falls off. So, for any given wavelength, you can determine the extinction coefficient.

It might be helpful to explain how spectrophotometric measurements are usually taken (beer's law applies to other things besides spectrophotometry, but UV/Vis is the most common application):

One common way to take a spectrophotometric measurement is to set the spectrophotometer to a particular wavelength, and then measure the absorbance of the sample at that wavelength. The extinction coefficient is specific to that wavelength, and if you know it, and you know the path length, you can determine the concentration of the sample from the equation. If you don't know the extinction coefficient at that wavelength, you can make a standard curve (so take the absorbance of different concentrations of the analyte and plot conc. versus absorbance). The slope of the standard curve is the extinction coefficient times the path length (A=ebC, and you plot C vs A, so the slope is eb). Then you just divide the slope by the path length and you get the extinction coefficient.

The other common way to take a spectrophotometric measurement is to scan a spectrum. What does this mean? Well, you take a sample with your analyte in it, and you tell the spectrophotometer to slowly change the wavelength and determine the absorbance at each wavelength. So you might scan all wavelengths from 250 nm to 500 nm. You'll get a graph of wavelength versus absorbance. Absorbance will be highest at lambda max. Since the concentration of the sample (and the path length) is not changing while you're scanning the spectrum, the graph will show you the relative extinction coefficients at each wavelength.

Sooo, example time. http://omlc.ogi.edu/spectra/PhotochemCAD/abs_gif/tryptophan

That graph shows a spectral scan of the amino acid tryptophan. If you'll notice, the y axis is labeled "molar extinction coefficient." Basically what they did to make that graph was they got the graph of wavelength versus absorbance, and then divided the absorbances by the concentration and the path length. Please ignore the left part of the curve (the part that goes really really high, since all compounds absorb at wavelengths below 220ish nm, so you'll see that in all spectra). Just look at the bell-shaped part. The lambda max is at 278 nm, where the molar extinction coefficient is highest. On either side, the extinction coefficient falls off. However, beer's law still holds on either side, it's just that the coefficient is different.

 

[kentavr]

I would agree with Monkeymaniac.
The wording for this question is awkward. If they mean that the Beer law is liner dependency for concentration and length then that will be OK.
But since they use the same character "e" - without realizing that "e" is different for different wavelengths, they are wrong. "e" is a function of lambda. and correct question should not use simple "e" notation for all lambda, without subscript or direct specification of function like e(lambda).