mvil ItU
TERMS EMPLOYED IN ABSORPTION SPECTROSCOPY
Table 6-1 lists the common terms and sym-bols employed in absorption spectroscopy. In recent years, a consi~erable effort has been made by the American Society for Testing Materials to develop a standard nomencla-ture; the terms and symbols listed in the first two columns of Table 6-1 are based on ASTM recommendations. Column 3 contains alternative symbols that will be found in the older literature. A standard nomenclature seems most worthwhile in order to avoid am-biguities; the reader, is, therefore, urged to learn and use the ri:commended terms and symbols.
Transmittance
Figure 6-1 depicts a beam of parallel radia-tion before and after it has passed through a layer of solution with a thickness of b em and a concentration of c of an absorbing species.
As a consequence of interactions between the photons and absorbing particles, the power of the beam is attenuated from Po to P. The transmittance T of the solution is the fraction
1For more detailed trelitment or absorption spectros-copy. see: R. P. Bauman,Absorption Sp"ctroscopy. New York: Wiley, 1962; J. Ri Edisbury,Practical Hints on Absorption Sp"ctrometryi New York: Plenum Press, 1968;F. Grum, inPhysical Methods o/Chemistry,eds. A.
Weissberger and B. W. Roositer. New York: Wiley-Interscience, 1972, Vol. I, Part III B, Chapter 3; G. F.
Lothian,Absorption Sp"ctrophotometry, 3d ed. London:
Adam Hilger Ltd., 1%9; and J. E. Crooks,The Sp"ctrum inChemLstry.London: Academic Press, 1978.
Po,--I
-I'---~
P TA -log=!...Po PoIi"Absorbing solution of concentration c
FIGURE 6-' Attenuation of a beam of radiation by an absorbing solution.
of incident radiation transmitted by the solu-tion. That is,
T=PIPo (6-1)
Transmittance is often expressed as a percentage.
The absorbance of a solution is defined by the equation
I Po
A = - oglo Toc log-P
Note that, in contrast to transmittance, the absorbance of a solution increases as the attenuation of the beam becomes larger.
Absorptivity and Molar Absorptivity As will be shown presently, absorbance is directly proportional to the path length through the solution and the concentration of the absorbing species. That is,
A=abc (6-3)
where a is a proportionality constant called the absorptivity. The magnitude of a will clearly depend upon the units used for b and c. When the concentration is expressed in moles per liter and the cell length is in cen-timeters, the absorptivity is called the molar absorptivity and given the special symbol e.
Thus, when b is in centimeters and c is in moles per liter,
Experimental
M••••• rement of PandPo
The relationship given by Equation 6-1 or 6-2 is not directly applicable to chemical analysis.
Neither P nor Po, as defined, can be conven-iently measured in the laboratory because the solution to be studied must be held in some sort of container. Interaction between the ra-diation and the walls is inevitable leading to a loss by reflection at each interface; moreover, significant absorption may occur within the walls themselves. Finally, the beam may suffer a diminution in power during its passage through the solution as a result of scattering
by large molecules or inhomogeneities. Reflec-tion losses can be appreciable; for example, about 4% of a beam is reflected upon vertical passage of visible radiation across an air-to-glass or air-to-glass-to-air interface (see example, p. 103).
In order to compensate for these effects, the power of the beam transmitted through the absorbing solution is generally compared with that which passes through an identical cell containing the solvent for the sample. An experimental absorbance that closely approx-imates the true absorbance of the solution can then be obtained; that is,
A I p •••_. Po
= og--
=Iog-p•••8liea P
radiation after it has passed through a cell containing the solvent or the solution of the anaIyte.
M••••• rement of
Trllll8lltlttance and Absorbance Transmittance (or absorbance) measurements are made with instruments having· the com-ponents arranged as shown in Figure S-Ib (p. 115). The electrical output G of the detec-tor of such an instrument is given by (see p. 138).
G=KP
+
K' (6-5)where K' is the dark current that often exists when no radiation strikes the transducer and K is a proportionality constant.
In many instruments, the readout device will consist of a linear scale calibrated in units of 0 to 100% T. A transmittance measure-ment with such an instrumeasure-ment requires three steps. First, with the light beam blocked from the transducer by means of a shutter, an elec-trical adjustment is made until the pointer of the readout device is at exactly zero; this step is called the dark current or 0% T adjustment.
A 100% T adjustment is then carried out with the shutter open and the solvent in the light path. This adjustment may involve increasing or decreasing the power output or 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. After this step, we may write Equation 6-5 in the form
Go=100=KPo
+
0.00In the final step of the measurement, the solvent cell is replaced by the sample. The meter reading is then given by
G=KP +0.00
Dividing this equation by the previous one yields
The terms Po and P, when used hence-forth, refer to the power of a beam or
TABLE &-1 IMPORTANT TERMS AND SYMBOLS EMPLOYED IN
ABSORPTION MEASUREMENT
Term ••• Symbol" Definition Akerutin Name and Symbol
Radiant power, P, Po Energy ofradiation (in ergs) Radiation intensity, I, 10
impinging on a l-an2 area of a detector per second
Absorbance, A P
log~ Optical density, D;extinction, E
P
Transmittance, T P
Po Transmission, T
Path lent!th of I,d
radiation, in em b
AbsorptivitY,b a beA Extinction ~fficient, k
Molar absorptivity,' £ A
be
Molar extinction coefficient• TerminololY recommended by the American Chemical Society. Reprinted with permission from AMI. CItmt.,24, 1349 (1952); •••• 2298 (1976~ Copyrisht by the American Chemiall Society.
• e may be expressed in g/liter or other specified concentration units; b may be expressed in em or in other units or length.
, e is expressed in units or mol/liter.
G
= -
P x 100=
T x 100 PoThe meter is thus made direct reading in per-cent transmittance. Obviously, an absorbance scale can also be scribed on the meter face.
QUANTITATIVE ASPECTS OF
ABSORPTION MEASUREMENTS
The remainder of this chapter is devoted to an examination of Equation 6-4 (A = ebc~
with particular attention to causes of devia-tions from this reladevia-tionship. In addition, con-sideration is given to the effects that uncertainties in the measurement of P and Po have on absorbance (and thus concentration~
Bee". Law
Equations 6-3 and 6-4 are statements of Beer's law. These relationships can be ration-alized as follows.2 Consider the block of absorbing matter (solid, liquid, or gas) shown in Figure 6-2. A beam of parallel monochro-matic radiation with power Po strikes the block perpendicular to a surface; after pass-ing through a length b of the material, which
FIGURE 6-2 Attenuation of radiation with initial power Po by a solution con-taining c moles per liter of absorbing solute with a path length of b ern. P < Po·
2The discussion that follows is based on a paper by F. C. Strong. AMI.Ch"",.24. 338 (t952). For a rigorous derivation of tbe law, see: D.J.Swinehart. J. Chem.
Edue.• 39.333 (1972~
contains n absorbing particles (atoms, ions, or molecules ~ its power is decreased tp P as a result of absorption. Consider now a cross section of the block having an area Sand an infinitesimal thickness dx.Within this section there are dn 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, absorption will follow immediately.
The total projected area of these capture sur-faces 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 cap-ture of photons within the section.
The power of the beam entering the section, P", is proportional to the number of photons per square centimeter per second, and dP" represents the quantity removed per second within the section; the fraction ab-sorbed is then -dP"/P,,, 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" dS
-p;=S
Recall, now, that dS is the sum of the cap-ture areas for particles within the section; it must therefore be proportional to the number of particles, or
where tinis the number of particles and ais a proportionality constant, which can be called the capture cross section. Combining Equa-tions 6-6 and 6-7 and summing over the inter-val between zero and n, we obtain
_ (dP"
=(adn
·"0
P" ·0 S which, upon integration, giv~P
an
-In-=-Po S
Upon converting to base lO logarithms and inverting the fraction to change the sign, we obtain
among the various species, the total absorb-ance for a multicomponent system is given
by
Po an
log
Ii
=2.303S (6-8) A._I=Al+
A2+ ... +
A. . (6-11)=£Ibel
+
£2be2+ ... +
£.bc.where the subscripts refer to absorbing com-ponents I, 2, ... , n.
where n is the total number of particles within the block shown in Figure 6-2. The cross-sectional area S can be expressed in terms of the volume of the block V and its length b.
Thus, Limitation. to the
Applicability of B•••.•• Law
The linear relationship between absorbance and path length at a fixed concentration of absorbing substances is a generalization for which no exceptions are known. On the other hand, deviations from the direct proportional-ity between the measured absorbance and concentration when b is constant are fre-quently encountered. Some of these devia-tions are fundamental and represent reallimi-tations of the law. Others occur as a consequence of the manner in which the absorbance measurements are made or as a result of chemical changes associated with concentration changes; the latter two are sometimes known, respectively, as instrumen-tal deviations and chemical deviations.
Real Limitations to Beer's La". Beer's law is successful in describing the absorption be-havior of dilute solution only; in this sense, it is a limiting law. At high concentrations (usually >O.OlM~ the average distance be-tween 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 their ability to absorb a given wavelength of radiation. Be-cause the extent of interaction depends upon concentration, the occurrence of this phe-nomenon causes deviations from the linear re-lationship between absorbance and concen-tration. A similar effect is sometimes encountered in solutions containing low absorber concentrations and high concentra-tions of other species, particularly elec-trolytes. The close proximity of ions to the S=-cm
V
2b
Substitution of this quantity into Equation 6-8 yields
Po' anb
log
Ii
= 2.303V (6-9) Note that n/V has the units of concentration (that is, number of ~icles per cubic centi-meter); we can readily convert n/V to moles per liter. Thus, in particles . c=
---6.02 )( lO23 particles/mol
1000 cm3/liter lOOOn .
x Vem3 = 6.02 )( lO23V mol/llter Combining this relationship with Equation 6-9 yields
I Po 6.02 x lO23abe og
Ii
= 2.303 x 1000Finally, the constants in this equation can be collected into a single term £to give
Po .
log
Ii
=;ebc =A (6-10)Application of
B•••.•• Law to Mixtu ••••
Beer's law also applies to a solution contain-ing more than one kind of absorbing substance. Provided there is no interaction
absorber alters the molar absorptivity of the latter by electrostatic interactions; the effect is lessened by dilution.
While the effect of molecular interactions is ordinarily not significant at concentrations below O.OlM, some exceptions are en-countered among certain large organic ions or molecules. For example, the molar absorp-tivity at 436 nm for the cation of methylene blue is reported to increase by 88% as the dye concentration is increased from 10-5 to lO-2M; even below 10-6M, strict adherence to Beer's law is not observed.
Deviations from Beer's law also arise be-cause£is dependent upon the refractive index of the solution.3 Thus, if concentration changes cause significant alterations in the re-fractive index n of a solution, departures from Beer's law are observed. A correction for this effect can be made by substitution of the quantity en/(n2
+
2)2 for £ in Equation 6-10.In general, this correction is never very large and is rarely significant at concentrations less than O.OlM.
Chemical Deviatiol& Apparent deviations from Beer's law are frequently encountered as a consequence of association, dissociation, or reaction of the absorbing species with the solvent. A classic example of a chemical de-viation occurs in unbuffered potassium dichro-mate solutions, in which the following equilibria exist:
Cr20~-
+
H20~2HCr04 ~2H+
+
2CrO~-At most wavelengths, the molar absorptivities of the dichromate ion and the two chromate species are quite different. Thus, the total absorbance of any solution depends upon the concentration ratio between the dimeric and the monomeric forms. This ratio, however, changes markedly with dilution, and causes a• G. Kortum and M. Seiler.Allgew.
C"""'"
51, 687(1939~
pronounced deviation from linearity between the absorbance and the total concentration of chromium(VI). Nevertheless, the absorbance due to the dichromate ion remains directly proportional to its molar concentration; the same is true for the chromate ions. This fact is easily demonstrated by measuring the absorb-ance of chromium(VI) solutions in strongly acidic solution, where dichromate is the prin-cipal species, and in strongly alkaline solu-tion, where chromate predominat:s. Thus, deviations in the absorbance of this system from Beer's law are more apparent than real because they result from shifts in chemical equilibria. These deviations can, in fact, be readily predicted from the equilibrium con-stants for the reactions and the molar absorp-tivities of the dichromate and chromate ions.
EXAMPLE
Two solutions of the acid-base indicator HIn were prepared by diluting 1.02 x 1O-4F solutions of HIn (K. = 1.42 x 10-5) with equal volumes of: (a) O.2F NaOH; and (b) O.2F HCI. When measured in a 1.0Q.an cel~
the following absorbance' data were obtained at 430 and 570 nm:
SohIrioB Dilutell With 0.2FNaOH
O.2F HCI
1.051 0.032
0.049 0.363 Derive absorbance data at the two wavelengths for unbuffered solutions having indicator con-centrations in the range of 2 x 10-5 to 16X 10-5 F. Plot the data.
The absorbance of the various solutions depends upon the equilibrium
HIN~H+
+
In-K = 1.42X 10-5
=
[H+][In-]• [HI~
Substitution of the hydrogen ion concen-tration of the strong acid or base solutions
into the expression for K. makes it apparent that, in the presence of strong base, essentially all of the indicator is dissociated to In ~ ; there-fore,
[In-] ~ (1.02 x 10-4)/2 = 5.10 x 10-5 Similarly, in the strong acid [HIn] ~ 5.10 x 10-5. The molar absorptivities for the two species can thus be determined. At 430 nm,
,A 1.051
£in= be = 1.00 x 5.10 X 10-5
= 2.06 X 104
, _ 0.032 _ 2
tHin- 5.10X 10-5 - 6.3 x 10 and at 570 nm,
M 0.049 ,,2
£k,= 5.10X 10-5 = 9.6 x hr
M _ 0.363 _ 3
£Hln- 5.10X 10-5 - 7.12 x 10 Let us calculate the concentration of HIn and In- in an unbuffered 2.00 x 1O-5F solution of HIn. From the equation for the dissociation reaction, we see that
We are now able to calculate the absorbance at the two wavelengths. Thus,
A430 = e1nb[In-]
+
t:Hlnb[HIn]= 2.06 x 104x 1.00 x 1.12X 10-5
.+
6.3 x 102 X 1.00 x 0.88X 10-5= 0.236
A570 .: 9.6 x 102x 1.12X 10-5
+
7.12X 103x 0.88X 10-5=0.073
The accompanying data were obtained simi-larly.
ually, we may write for radiation A'
, I P'o 'be
A=og-=e P'
P'o =10"'"
P'
FH1n [HIn]
[lIt-]
A430 A5702.00X 10-5 0.88X 10-5 1.12X 10-5 0.236 0.073 4.00 x 10-5 2.22 X 10-5 1.78X 10-5 0.381 0.175 8.00 x 10-5 5.27X 10-5 2.73X 10-5 0.596 0.401 12.00
x
10-5 8.52X 10-5 3.48X 10-5 0.771 0.640 16.00 x 10-5 11.9X 10-5 4.11*
10-5 0.922 0.887Figure 6-3 is a plot of the data derived in the foregoing example and illustrates the kind of departures from Beer's law that arises when the absorbing system is capable of undergoing dissociation or association. Note that the direction of curvature is opposite for the two wavelengths.
Instrumental DeYiations with Polychromatic Radiation. Strict adherence to Beer's law is observed only when the radiation employed is truly monochromatic; this observation is yet another manifestation of the limiting character of the law. Unfortunately, the use of radiation that is restricted to a single wave-length is seldom practical; devices that isolate portions of the output from a continuous source produce a more or less symmetric band of wavelengths around the desired one (see Figures 5-14 and 5-17, for example).
The following derivation shows the effect of polychromatic radiation on Beer's law.
Consider a beam comprising of just two wavelengths A' and AM. Assuming that Beer's law applies strictly for each of these individ-Furthermore,
[In-]
+
[HIn] = 2.00 x 10-5 Substitution of these relationships into the expression for K. gives[In-]2 _ -5
2.00X 10-5 _ [In-] - 1.42 x 10 Rearrangement yields the quadratic expression
[In -]2
+
1.42 x 10- 5[In-]- 2.84X 10-10 = 0 which gives
[In-] = 1.12 x 10-5
[HIn] = 2.00 x 10-5 - 1.12X 10-5
= 0.88X 10-5
~ 0.600
1:?!
j
<l: 0.400
i
O.QOO
0.00 4.00 8.00 12.00
Indicator conCentration. F X 105
FIGURE 6-3 Chemical deviations from Beer's law for unbuffered solutions of the indicator HIn. For data, see example on this page.
When an absorbance measurement is made with radiation composed of both wave-lengths, the power of the beam emerging from the solution is given by (P'
+
P") and that of the beam from the solvent by (PO+
PO).Therefore, the measured absorbance is A _ I (P'o
+
PO)101 - og (P'
+
P") which· can be rewritten asAM =10g(P
o +
PO)- 10g(P'olO-""
+
POIO-'-") Now, whent
= t', this equation simplifies toAM =e'bc
and ~r's law is followed. If the two molar absofP;livities differ, however, the relationship betw~n AM and concentration will no longer be linear; moreover, greater departures from linearity can be expected with increasing dif-ferences between e' and t'. This derivation can be expanded to include additional wave-lengths; the relationships remain the ~e.
It is an txperimental fact that deviations from Beer's law resulting from the use of a polychromatic beam are not appreciable, provided the radiation used does not en-compass a spectral region in which the absorb-er exhibits large changes in absorbance as a function of wavelength. This observation is illustrated in Figure 6-4.
Instrumental Deviations in the Presence of Stray Radiation. We have noted earlier (Chapter 5)that the radiation exiting from a monochromator is ordinarily contaminated with small amounts of scattered or stray ra-diation which reaches the exit slit owing to reflections from various internal surfaces.
Stray radiation often differs greatly in wave-length 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
tration for various levels of p. as compared to Po·
Note that the instrumental deviations il-lustrated in Figures 6-4 and 6-5 result in absorbanees that are smaller than theoretical.
It can be shown that instrumental deviations always lead to negative absorbance errors.4
The Effect of
Instrumental Noise on the Precision of Spectrophotometric Analy••• S
The accuracy and precision of spectropho-tometric analyses are often limited by the un-certainties or noise associated with the instrument. A general discussion of in-strumental noise and signal-to-noise optimi-zation is found in Chapter 3; the reader may find it helpful to review this material before undertaking a detailed study of the effect of instrumental noise on the precision of spec-trophotometric measurements. In addition, a review of the contents of Appendix 1 on the use of standard deviation as a measure of precision may also prove worthwhile.
As was pointed out earlier, a spectro-photometric measurement entails three steps:
a 0% T adjustment, a 100% T adjustment, and a measurement of % T with the sample in the radiation path. The noise associated with each of these steps combines to give a net uncertainty for the final value obtained for T. The relationship between the noise en-countered in the measurement of T and the A' =Iog---Po
+
p.P+P.
where p. is the power of the stray radiation.
Figure 6-5 shows a plot of A' versus
concen-• I. M. Kolthoff and P. J. Elving. eds, Treatise on Anal.l·t-ical Chemistry. New York: Interscience Publishe ••• 1964.
Part I. vol.S,pp. 2767-2773.
• See: L. D. Rothman, S. R. Crouch, and J. D. Ingle, Jr.
Anal. Chem. 47. 1226 (197S); J. D. Ingle, Jr. and S. R.
Crouch, Anal. Chem., 44, 137S (1972); J.O. Erickson and T. Surles, American Laboratory, g (6~ 41 (1976); OplilflJJm Parameters for Spectrophotometry. Varian Instrument Division, Palo Alto, CA.
FIGURE &-4 The effect of polychromatic radiation upon the Beer's law relationship, Band A shows little deviation sinceedoes not change greatly throughout the band. Band B shows marked deviations sinceeundergoes significant changes
FIGURE &-4 The effect of polychromatic radiation upon the Beer's law relationship, Band A shows little deviation sinceedoes not change greatly throughout the band. Band B shows marked deviations sinceeundergoes significant changes