How Does Light Absorption Intensity Depend on Molecular Size? Linda B. ~ i ~ h tJay , ' S. Huebner, and Robert A. Vergenz Department of Natural Sciences, University of North Florida, Jacksonville, FL 32224-2645
The molecular photon-absorption cross section a is a useful conceot in teachiue about soectroscoov because it makes possible direct comparisons between absorption intensity and molecular size. This cross section is the effective area presented by a molecule to an approaching photon. The concept nicely supplements the usual introductory treatment of absorption, which includes the Lambert-Beer law and Bohr atomic theory. The relationships of molar extinction coefficient e to absorption cross section and transition moment are discussed. Comparisons of transition moments, molecular len&hs. and results of particle in a box calcul&ions illustrate~pedagogicallyusehi connections between spectroscopy and quantum mechanical theory appropriate for more advanced undergraduate courses. Spectroscopic data from the literature illustrate that for rr + r' transitions, order of magnitude estimates of absorption intensities are possible from Lewis structures.
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General Considerations The color of a compound is an indication of the frequency ranges in which it absorbs light and the intensity of the absorption. This paper is concerned with the relation between absorotion intensity and molecular size. (The relationship between absorptibn frequency and molecular size is not included here because it is adequately discussed elsewhere (131.)
Scope For com~oundsin dilute solution. absorotion results from tran2tions between molecular states. Absorption intensity is typically characterized by the molar extinction coefficient ( 4 , 5 ) .I t is determined in theory by the quantum mechanical transition moment and by the initial popula. . tion of the states. The extinction coefficient permits us to compare intensities of absorption over a wide range of frequencies. For example, the intensity of the visible color of iodine or bromine can be compared to that of the invisible ultraviolet absorption of many organic compounds. The relationship of this coefficient to the absorption cross section and transition moment is derived in the next section. This derivation applies to all individual molecular electronic transitions in dilute solutions. Limitations One limitation of the derivation is that it does not take into account overlappiw transitions caused by near dewneracies of molecul&~ta&. An example of this is the cutoff frequenw in the far W. Most common solvents and many solib materials that are transparent in the visible range have limited usefulness in f a r - W spectroscopy due to their cutoff frequencies. The W cutoff is not due to any 'Current address: College of Veterinary Medicine, University of Tennessee. Knoxville. TN 37901-1071. 'In this paper, units necessary for the validity of an equation will followthe appropriate variable in parentheses.
single particularly intense transition (6).Rather it results from a large density of chromophores in the condensed phase and the many nearly degenerate occupied valence shells that participate in a + a" or n + a* transitions with possibly overlapping absorption bands. Also ignored in the derivation are internal reflection, scattering, and other surface and geometric effects (71, such as preferential orientation of molecules. It is usual to ignore these effects, although they can be significant in a varietv of imoortant bioloeical. chemical. and industrial probleks deaiing with coniensed phases, k c h as suspenand photographic emulsions (10). sions, membranes (8,9), Empirical Relationships The Molar Extinction Coefficient
For dilute solutions, the extinction coefficient is defined for chromophore concentration C (molar),2 and light path length b (in cm) by the Lambert-Beer law. A =E C ~ (1) The absorbance A, t h e intensity I of light passing through a solution, and the incident intensity loare related by
These d e f ~ t i o n describe s light absorption in a thin section of solution of thickness dx as a first-order orocess in the amounts of chromophore and light, that is, -dr=KCzdz
(3)
Rearranging, integrating over appropriate limits, and applying eq 2, we show that eq 1is equivalent to r=(lnlO)& (4) From eq 1,the units of E are M-'cm-'. These units obscure any direct connection to other molecular properties. As shown below, the units of the absorption cross section are area, allowing direct comparison to molecular size. The Photon-Absorption Cross Section The photon-absorption cross section can be defined as the effective cross-sectional area presented by a chromophore to a single approaching photon. For a beam of photons spread uniformly over an area sl the absorption cross sectionis defined from the results of a large number of photons (11). ( I - _ .(no.of interacting beam particles) A - trotol no. of henrn pnniclrs, ,rotnl no. ofrnrgcr panicles) ( 3 , ,
If N is the number density of chromophore molecules, then Nsl dx is the number of target molecules in a thin section of solution of thickness dx. For n incident photons, -dn is the number of both molecules and ohotons-that interact. Substituting these values into eq 5 and solving for dn, we get the following for the proportion of photons absorbed. Volume 71 Number 2 February 1994
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longest length of the molecule. For this reason r k l is also the fundamental link between the absorption intensity and molecular size. The last equality results because n is proportional to I. This, with eqs 3 and 4, shows that o and e differ only in units, as
where NAis Avogadro's number. Students fmd this equation useful because it allows direct conversion between o and E. Quantum Relationships The extinction coefficient can be related to molecular size using quantum theory (4, 12-16). For electronic charge e and coordinate r , the potential energy of interaction between molecular dipole moment, y = er, and the time-dependent electric field of a photon, h ( t ) ,is the scalar product,
.+
V(t) = -p t. (t)
Derivation Connecting Theory and Experiment The proportionality constant, K from eq 3, can also he expressed in terms of the probability of absorption per chromophore, the transition energy hv a t frequency v, and the number density of chromophores CNA.Because intensity is the energy transmitted per unit time through a cross-sectional area, the decrease in intensity in a section of thickness dz will be -dI=PklhvCNA& (10) Each rotational-vibrational (rovibrational) transition + 1' that contributes to the electronic absorption band . a definite intensity exfor k + 1 has a discrete v ~ iand pressed by either P a or EK. Comparison of eq 10 with eqs 3 and 4 for each k' and I' relates the microswpic property PI? to the macroscopic property EM..
k'
(8)
This is treated as a perturbation to the electronic Hamiltonian, and the corresponding wave function Y is expanded as a linear combination of unperturbed stationary state wave functions, YL.
The last equality uses the relation,
of intensity to the total radiative energy density (energylvolume),4defined by
The Transition Moment For a transition from an initial state Yk to a final state Yl the coefficient values initially are ck = 1and cl = 0, but after the transition they are ck = 0 and cl= 1.The probability P k l of transition for each chromophore molecule per unit time is I ci 1 '.For a distribution p (energy x timelvolume)of radiation density over frequency, Einstein (12,131showed that and
\
J
where c is the speed of light in a vacuum, and q is the refractive index. The transition probability Pal for the electionic transition is found by summing the discrete rovibrational probabilities Pw. Each rovibrational transition is stimulated by the portion Uw of the total radiation density at the correct frequency and Uk?.= p(vWi.Av)from eq 12 for width Av of the rovibrational oeak. We have assumed constant distrihution of radiation density over the narrow rovibrational band. The discrete sum is then approximated using integration, requiring E to be treated as a wntinuous function of v. Summing eq 11,we get
where h is Plank's constant, and e+ is the vacuum permitivit^.^ Transition moment
is the fundamental wnnection between absorption intensity and the wave function. The ground state wave function Yk rapidly goes to zero beyond the nuclear skeleton of the molecule. Because q l i s a n integral over all space containing Yk as a factor, the latter acts as an envelope function to limit the spatial extent of the integrand. Thus, rkl= I r k l j cannot be larger than the 3We use the remmmended SI units of meter-kilogram-sewndamp (mksa).The centimeter-gram-second (cgs) system has no base unit of electricity, so in electromagnetic applications it is dimensionally different from mksa, though the cgs system is still used. For information on necessary transformations of electromagnetic equations, see ref 17. 4Clearly distinguishing between the integrated energy density U and the energy density distribution p eliminates much confusion in the treatment of spectroscopic intensities. Failure to do so has resulted in dimensional inconsistencies in both the secondary and primary literature (e.g..ref 14, eq 11; ref 4, eqs 4-14, 17, and 23b).
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Journal of Chemical Education
Integrating is accomplished with the aid of two simplifyinp assumptions: The variat~onin v over the hand is neali&le compared to the magnitude of v, and p is approkimately constant in this range. The latter assumption is valid for narrow absorption bands. Using constants for v and p corresponding to the maximum of the absorption hand, we get
The value of em, can he related to the integrated absorption intensity E(V)
dv
by assuming a Gaussian disthbution for ~ ( v ) ,
