The Beer-Lambert Law

Introduction 


This post is part of a primer on infrared spectroscopy and global warming. The previous post  looks at the features of the spectra of molecules of interest molecules and radiation and discusses how molecules give rise to infrared spectra.  This post looks at the question of how much radiation is absorbed by gas phase molecules in a laboratory setting and examines some of the differences between the laboratory gas cell and the earth's atmosphere.

Absorption in a Gas Cell


Imagine a sample of gas in an enclosed cell.  The cell has IR-transparent windows on either side.  There is a source of IR.  Suppose that the source is very intense relative to ambient IR emission, and  that the cell is at constant pressure and temperature. Additionally, assume an ideal detector.  The absorbing gas, represented here by dots is homogeneously spread throughout the cell.  The length of the cell is L and is measured in meters.  The concentration C is in units of milligrams per meter cubed (mg/m³).  I could also have chosen units of parts-per-million by volume (ppm or ppmv).


The incident radiation has an initial intensity, I0. After passing through the cell, some of that radiation is absorbed.  The radiation reaching the detector, Is is  therefore less than I0.  The amount of radiation absorbed is proportional to the length of the cell, L, and the concentration C.

Sometimes these two quantities are paired together as CL, called the concentration-pathlength.  CL has units of (mg/m³) x m or (mg/m²).  It can also be expressed as parts-per-million by volume times meters or ppm-m.  CL is sometimes called the column density, or the burden. There is a third quantity that also determines how much radiation is absorbed.  The absorptivity, α, is an intrinsic measure of the ability of the the sample to absorb radiation.  It is wavelength dependent and specific to the molecule.

The absorption of the cell or the absorbance, A = αCL.  Absorbance is a unitless number so the absorptivity, α, has units of  m²/mg.  Absorbance is proportional to the strength of the absorber, i.e., α, and how many absorbers are in the way, CL.

Now consider a thin slice of the left hand side of the original cell.  Suppose the gas in this thin slice absorbs one millionth of the incident radiation.  Let the amount of radiation that gets through this first layer be I1. How much radiation gets through?  The answer is


          I1 =(999, 999/1,000,000) x I0


Now imagine adding a second thin layer of the cell:

How much radiation gets through?  The situation is identical to the previous situation except that the intensity of radiation entering the second slice has been reduced by the first slice.  So


          I 2 = (999, 999/1,000,000) x I1


Or


           I2 = (999, 999/1,000,000) x (999, 999/1,000,000)  x I0 


Imagine now calculating  the amount of radiation absorbed in each slice to arrive at the amount of radiation that gets through the sample.  This reasoning forms the basis of Beer's Law.  The derivation requires a bit of calculus.


Derivation of Beer's law


Imagine now that the slices are infinitesimal and that the length of a slice is dx.  Each slice absorbs an infinitesimal amount of radiation dI.  The amount of radiation absorbed by an infinitesimal slice depends on the the concentration of absorbers in that slice, the absorptivity of the absorbers, and the radiation incident on that slice:


            dI = -αCIdx


Or

           dI/I = -αCdx


Integrating yields:


          ln (I) = -αCX + K


The integration constant, K, vanishes when evaluating the boundary conditions.  The cell length is L, so the expression needs to be evaluated from 0 to L.  The amount of radiation that enters the cell was previously defined as I0 and that exiting as Is.  


         ln(Is) -  ln(I0 )  = -αCL + K + 0 - K


Or


          ln( Is/I0) = -αCL


Taking the exponential function of both sides and multiplying by  I0, yields:


         Is =I0 exp(-αCL)

This expression is the Beer-Lambert Law, or Beer's Law. Note that the validity of this expression, as written, depends on several assumptions discussed below.


Getting Familiar With  Beer's Law


Beers Law, as written here, indicates that the amount of radiation that gets through the sample is proportional to the incident radiation.  The constant of proportionality is called the transmittance,

           t = Is/I0


Again note that the validity of this expression, as written, depends on several assumptions discussed below. It is necessary to modify this expression for IR in the atmosphere as shown in subsequent posts in this series. The transmittance is can be defined as:

           t  = exp(-αCL)


The transmittance can be measured by measuring the intensity of radiation hitting the detector with the sample present, measuring the the intensity of radiation with the sample removed, and then dividing the two values.  The astute reader will notice an implicit assumption in such a measurement. That assumption is examined in subsequent posts in this series.


It is important to note that there are different conventions.  Sometimes the exponential is expressed in base e and sometimes it is expressed in base 10.  One can account for the difference by changing the value of α.  Conceptually, there is no difference, but when one wants to actually use literature values of the absorptivity, one must make sure whether the citation uses  the e- or the 10- based exponential.  It would be nice if sources were consistent, but they are not.


Transmittance


Plotted below is the percent transmittance as a function of concentration for a fictional gas.

Notice that for low concentrations that adding more concentration has a very large effect on how much radiation is transmitted. At larger concentrations, increasing the concentration has a smaller effect. The effect of increasing the pathlength is the same.  The implication in a laboratory setting is that if the concentration-pathlength product is high enough, doubling it will not substantially increase the amount of radiation that is absorbed because virtually all of the radiation has already been absorbed.  This effect is sometimes termed saturation.  The units in this graph are arbitrary: I just invented them to illustrate the point, but the physic is valid.


Beer's Law is not enough to understand quantitatively how IR absorbers such as carbon dioxide absorb in the atmosphere.


To get the results shown here, two assumptions were made:

1. The source is very intense relative to ambient IR emission.    
   (We are neglecting emission from the gas in the cell!)
2. The cell is at constant pressure and temperature.

In the atmosphere, neither of these assumptions is valid; so before concluding that there is nothing to worry about, it is necessary to look a little deeper.  The next post looks a little deeper into the issue of saturation of the 15-micron band of carbon dioxide under these assumptions.


Sources:

• Atkins, P. W. Physical Chemistry, New York: W. H. Freeman and Company, New York, 3rd edition, 1986
• Steinfeld, Jeffrey I, Molecules and Radiation, The MIT Press, Cambridge, MA, 2nd edition, 1985
• Struve, Walter S., Fundamentals of Molecular Spectroscopy, John Wiley & Sons, New York, 1989 
• http://en.wikipedia.org/wiki/Beer%E2%80%93Lambert_law

Postscript (8/31/2010):


The formatting of this post was not correct. I fixed most of the problems and in the course of doing so, I improved some of the descriptions in this post. More work needs to be done to fix the format.


(9/01/2010):


The html editor does not appear to handle br tags properly.