Spectral intensity, Beer Lambert’s Law

Superposition principle
Onto each electronic state a vibrational level series is superimposed and onto each vibrational state a rotational level series is superimposed.

In Figure 1. a possible electronic transition is shown (grey vertical arrow) and a vibrational transition in the electronic ground state by (black vertical arrow).

An electronic transition may occur when a molecule absorbs photon with energy equals to one of the possible energy differences between excited and ground state, and transition dipole has a value different to zero.

Absorption of light, Beer-Lambert’s law

The light intensity functions
The flux of photons, ? is defined as the number of photons, np transmitted through a cross sectional area, q during time, t.
1.
If the source is monochromatic the equation can be transformed to energy flux multiplying the number of photons by h? which gives photon intensity:

. 1a.
unit: Wm-2.
The incoming radiation intensity I0 decreases when transferring an optically active sample. The outgoing radiation intensity, I equals the difference in

I = I0 – Ia 2.

between incoming and absorbed intensities, accordig to the law of energy conservation

Figure 2. The radiating power or intensity is reduced by an absorbing medium of b optical path.

In photometry, we use relative intensities, dividing Eq. 1. by I0 we get

3.
a quantity called transmittance.
4.
The percentage transmission, T%



Beer-Lambert law
Increasing the thickness l (optical pathlength) of an absorbing layer by dl the transmitted intensity decreases by dI. Therefore, this derivative



is negative, and directly proportional a material coefficient ?, the outgoing intensity, I and the concentration of light absorbing material c in the solution,.

5.

Integrating this equation between limits



the integrated function


Rearranging and introducing the wavelength dependence



The function form:

6.
This form assign variables (asorbance, molar absorbance) dependence of other variables. Be careful is not a product!
The absorbance at constant wavelength is linearly dependent on the concentration of absorbing material, optical path and specific absorbance (? (?)).

When concentration is given in mol dm-3 the coefficient ? is called molar absorbance. The usual non SI unit of thickness is cm, and the unit of ? is dm3mol-1cm-1.
? depends on the nature of material, the matrix and the wavelength of radiating source.
(Matrix refers to the components of a sample other than the analyte of interest.)

Intensity formulas
When the material is transparent: I = I0, T = 1 or T% = 100.
When the material absorbs all the incoming light: I = 0 or T% = 0

The logarithmic ratio of intensities called absorbance is frequently applied in quantitative determinations.
7.

The output data of spectrophotometers are transmittance, T transmittance percentage, T% or absorbance, A.

Absorbance can be given in terms of transmittance,
7a.

or in terms of transmittance percentage



From intensity functions only the absorbance is directly proportional to the concentration of analyte, therefore it is frequently used for concentration determination.

The light absorbing components determine the absorbance of a multicomponent system. E.g. dissolving iodine in CCl4 a violet coloured solution is obtained for which the absorbance at 520 nm (the maximum of iodine absorption) is:



At 520 nm the CCl4 is transparent thus




At constant c and l a spectral band can be observed. By increasing concentration a band series is obtained.

Figure 3. concentration series of an absorption band centered at 520 nm.

Figure 4. Translation of Fig. 3 at a single wavelength. The linear concentration dependence

We read absorbance data at ?max from band series and plot the concentration dependence of absorbance.

As BL law is valid in the concentration range studied, we have a straight line in Figure 4. As BL law contains no additive constant the ideal graph crosses the x axis at the origin i.e. at A=0, c=0 point. Experimental error causes the value of intercept to be different from zero.

Example 1.
The molar absorption coefficient of a substance dissolved in hexane is known to be 855 M-1cm-1 at ? = 270 nm. Calculate the percentage reduction in intensity when light of that wavelength passes through 2.5 mm of a solution of concentration 3.25 10-3 M.

Solution
Excess datum: ? = 270 nm
l = 0.25 cm
c = 0.00325 M
? = 855 M-1cm-1
Percentage reduction = T%

, .


Example 2.
Riboflavine in dilute acetate solution shows a maximum in absorption spectrum at 444 nm. Calculate the wavenumber in cm-1 and frequen