Introduction to Spectroscopic Methods

 Introduction to Spectroscopic Methods

Introduction:

     An introduction into any field requires that one learn the terms and symbols associated with work in that field. Unfortunately, the terms used in spectroscopy and Spectrophotometry are somewhat confusing. The common terms and alternative names/symbols employed in spectroscopy are listed in the table below. The recommended terms and symbols are listed under the column labeled Term and Symbol.

Term and Symbol	Alternative Name and Symbol
Radiant Power, P, P0	Radiation Intensity, I, I0
Absorbance, A	Optical Density, D
Transmittance, T	Transmission, T
Path Length, b	l, d
Absorptivity, a	Extinction Coefficient, k
Molar Absorptivity, e	Molar Extinction Coefficient

The letters P and P_o of the above table represent the power of a beam of light before and after passage through an absorbing species respectively. Absorbance, like the table shows, can be defined as the base-ten logarithm of the reciprocal of the transmittance :

A = log 1/T = -log I/I_o = -log P/P_o

It is important to note that the absorbance of a solution increases as the attenuation of the beam becomes greater.

The transmittance and absorbance of a solution as defined in the table above cannot be measured in

the laboratory because the analyte solution must be held in some sort of transparent container and as the diagram below shows the attenuation of the incident is not totally due to absorption.

Reflection and scattering losses are significant and to compensate for these effects, the power of the beam transmitted by the analyte solution is ordinarily compared with the power of the beam transmitted by an identical cell containing only the solvent. An experimental absorbance that closely approximates the true absorbance is then obtained with the equation

A = log P_solvent/P_solution x log P_o/P

In order to make manual photometers and spectrophotometers, which are often equipped with a display that has a linear scale extending from 0 to a 100%, operate in such a way that their readings are in percent transmittance, two preliminary adjustments are required. These are the dark current or 0%T adjustment and the 100%T adjustment.

The 0%T adjustment is performed with the detector screened from the source by a mechanical shutter. In this adjustment, the instrument is thus counter signaled so as to read zero in the absence of any radiation from the source and the dark current that many detectors exhibit in the absence of radiation is eliminated.

The 100% adjustment is made with the shutter open and the solvent in its light path. Normally the solvent is contained in a cell that is as nearly as possible identical to the one containing the samples. The 100% adjustment may involve increasing or decreasing the radiation output of the source electrically ; alternatively, the power of the beam may be varied with an adjustable diaphragm or by appropriate positioning of a comb or optical wedge, which attenuates the beam to a varying degree depending upon its position with respect to the beam.

Beer’s Law:

Bouguer, and later Lambert, observed that the fraction of the energy, or the intensity, of radiation absorbed in a thin layer of material depends on the absorbing substance and on the frequency of the incident radiation, and is proportional to the thickness of the layer. At a given concentration of the absorbing species, summation over a series of thin layers, or integration over a finite thickness, leads to an exponential relationship between transmitted intensity and thickness. This is generally called Lambert’s law. Beer showed that, at a given thickness, the absorption coefficient introduced by Lambert’s law was directly proportional to the concentration of the absorbing substance in a solution. Combination of these two results gives the relationship now commonly known as Beer’s law. This law states that the amount of radiation absorbed or transmitted by a solution or medium is an exponential function of the concentration of the absorbing substance present and of the length of the path of the radiation through the sample.

Beer’s law can be derived as follows. Consider the block of absorbing matter shown below.

A parallel beam of monochromatic radiation with power P_o strikes the block perpendicular to a

surface after passing through a length b of the material , which contains n absorbing particles , the beam’s power is decreased to P as a result of absorption. Consider now a cross-section of the block having an area S and an infinitesimal thickness d_x. Within this section there are d_n absorbing particles ; associated with each particle we can imagine a surface at which photon capture will occur. That is, if a photon reaches one of these areas by chance it will be absorbed. The total projected area of these capture surfaces within the section is designated as dS ; the ratio of the capture area to the total area, then , is dS/S. On a statistical average this ratio represents the probability for the capture of photons within this section.

The power of the beam entering the section, P_x, is proportional to the number of photons per square centimeter per second, and dP_x represents the quantity removed per second within the section ; the fraction absorbed is then - dP_x/P_x, and this ratio also equals the average probability for capture. The term is given a minus sign to indicate that P undergoes a decrease. Thus,

- dP_x/P_x = dS/S

Since dS is the sum of the capture areas present in section, dS must be proportional to the number of particles or dS = d_n where d_n is the number of particles and is the proportionality constant. Substitution into the previous equation and integrating from 0 to n yields

- In P/P_O = n/S

Upon converting to base ten logarithms and inverting to change the sign, we obtain

log P_o/P = n/2.303S

Since S (ie area under consideration) can be written as V/b where V is the volume of the block then the last equation can be written as

log P_O/P = n_b/2.303V

Noting that n/V has the units of concentration i.e. number of particles per cubic centimeter, conversion to moles per liter yields the following equation

log P_o/P = N_A x b_c/2.303 x1000

where NA is Avogradro’s number. Finally , collecting the constants into a single term  gives

log P_o/P = abc = A

The derivation of this law assumes (a) that the incident radiation is monochromatic, (b) the absorption occurs in a volume of uniform cross-section, and ( c) the absorbing substances behave independently of each other in the absorbing process. Thus, when Beer’s law applies to a multi component system in which there is no interaction among the various species, the total absorbance may be expressed as

A_total = a₁bc₁ + a₂bc₂ + .........+ a_nbc_n

Deviations from Beer’s Law:

Beer’s law states that a plot of absorbance versus concentration should give a straight line passing through the origin with a slope equal to ab. However, deviations from direct proportionality between absorbance and concentration are sometimes encountered. These deviations are a result of one or more of the following three things ; real limitations, instrumental factors or chemical factors.

Real Limitations:   Beer’s law is successful in describing the absorption behavior of dilute solutions only ; in this sense it is a limiting law. At high concentrations            ( > 0.01M ),the average distance between the species responsible for absorption is diminished to the point where each affects the charge distribution of its neighbors. This interaction, in turn, can alter the species’ ability to absorb at a given wavelength of radiation thus leading to a deviation from Beer’s law.

Deviations also arise because  is dependent upon the refractive index of the solution. Thus, if concentration changes cause significant alterations in the refractive index  of a solution, departures from Beer’s law are observed. It is not  which is constant and independent of concentration, but the expression

                                                              a = a_true x a/( a² + 2)²

where a is the refractive index of the solution. At concentrations of 0.01 or less, the refractive index is essentially constant, but at high concentrations the refractive index may vary considerably and so will a. This does not rule out quantitative analyses at high concentrations, since bracketing standard solutions and a calibration curve can provide sufficient accuracy.

Chemical Deviations:  Chemical deviations from Beer’s law are caused by shifts in the position of a chemical or physical equilibrium involving the absorbing species. A common example of this behavior is found with acid/base indicators. Deviations arising from chemical factors can only be observed when concentrations are changed. 

Instrumental Factors: Unsatisfactory performance of an instrument may be caused by fluctuations in the power-supply voltage, an unstable light source, or a non-linear response of the detector-amplifier system. In addition the following instrumental sources of possible deviations should be understood :

Polychromatic radiation:   Strict adherence to Beer’s law is observed only with truly monochromatic radiation. This sort of radiation is only approached in specialized line emission sources. All monochromators, regardless of quality and size, have a finite resolving power and therefore minimum instrumental bandwidth. A good picture of the effect of polychromatic radiation can be presented as  follows. When radiation consists of two wavelengths,  and ₁, and assuming that Beer’s law applies at each of these individually the absorbance at  is given by

log ( P_o/P ) = A = abc   

a P_o/P = 10^bc

Similarly, at ₁,

                                                                   P_1o/P₁ = 10^1bc

The radiant power of two wavelengths passing through the solvent is given by P_o + P_1o, and that passing through the solution containing absorbing species by         P + P_1. The combined absorbance is                                                  A_c = log ( P_o + P_1o)/P + P₁

Substituting for P and P₁ , we obtain

                                             A_c = log ( P_o + P_1o)/(P_o10^-bc +  P_1o10^-1bc)

In the very special case where ₁ = , the above equation becomes Beer’s law. As shown in the figure below , the relationship between A_c and concentration is no longer linear when the molar absorptivities differ ; moreover, greater departures from linearity can be expected with increasing differences between ₁ and .

 It is also found that deviations from Beer’s law resulting from the use of a polychromatic beam are not appreciable, provided the radiation used does not encompass a spectral region in which the absorber does not exhibit large changes in absorption as a function of wavelength. This observation is illustrated in the figure below.

Stray Radiation:  Stray light affects absorption measurements because stray radiation often differs in wavelength from that of the principal radiation and, in addition, may not have passed through the sample. When measurements are made in the presence of stray radiation, the observed absorbance is given by

                                                         A = log( P_o + P_s)/(P + P_s)

where P_s is the power of nonabsorbed stray radiation. The figure below shows a plot of A versus concentration for various ratios between P_s and P_o. It has been deduced that positive deviations from Beer’s law occurs when stray radiation is absorbed , and negative deviation if it is not.

Useful Websites Dealing With Spectroscopy

Chemical Abstracts Service:           

http://www.cas.org

Chemical Center Home Page:         

http://www.chemcenter/org

Journal of Chemistry and Spectroscopy:

http://www.kerouac.pharm.uky.edu/asrg/wave/wavehp.html

The Analytical Chemistry Springboard at Umea U. http://www.anachem.umu.se/jumpstation.htm

Spectrum Chromatography:                                               

http://www.lplc.com/