13.1 Measuring Absorption

13.1 Measuring Absorption

• derivation of Beer's Law using a chemical kinetic approach • definition of transmission and absorption • absorption cross-section and molar absorptivity • transmission is multiplicative and absorption is additive • expected performance when measuring concentration • expected performance when measuring spectra • using solution color to guess at absorbed wavelengths

13.1 : 1/10 Kinetic Derivation of Beer's Law (1)

The process of absorption can be written as a ΔE = hν "reaction" involving light and two energy S states. 2 B1,2 Sh12+⎯⎯⎯ν → S B A second-order rate equation can be written 1,2 for the reaction,

S1

-3 where ρ(ν) is the density of light in cm , N1 is the number density -3 of S1 in cm , and B1,2 is the second order rate constant (called the 3 -1 ______for absorption). B1,2 has units of cm s .

13.1 : 2/10 Kinetic Derivation of Beer's Law (2)

ddρ (νρν) ( ) dt B1,2 Of more interest is the change in ==−()N1 photon density with distance dx dt dx c through the sample. This is B ≡σ 1,2 obtained using, dx/dt = c. σ has c ρν units of cm2 and is called the absorption cross-section.

ll The total change in the photon dρν( ) density over a distance, l, is =− Ndx1 ∫ ()ρν ∫ obtained by integration. 00σ ⎛⎞() ln ⎜⎟ρνl =− lN ⎜⎟() 1 By converting photon density into ⎝⎠ρν0 σ intensity, I = ρ(ν)×c, the final equation can be written in terms

of an intensity ratio, I/I0.

13.1 : 3/10 Transmission and Absorption

Transmission is defined as the intensity of light

leaving the sample, divided by the intensity of I0 light entering the sample. Transmission can be related to number density using the last equation on the previous transparency. σlN I I − T ≡=102.3 = 10−εlC I0

Although the previous derivation used number density and cross- section, the more common units of molar absorptivity, ε, and molar concentration, C, give the same numeric result.

Absorption is defined as the negative logarithm (base 10) of the transmission. This gives a parameter linear in concentration.

13.1 : 4/10 Cross-Section and Absorptivity

Cross-sections and number densities are used in virtually every type of spectroscopy ______solution-phase, molecular absorption. When comparing different spectroscopies, it is necessary to use cross-sections.

The conversion between ε and σ involves nothing more than units. 2.3× 10331 cm L− 2-1-1( ) σε()cm= ()() L mol cm 6.02× 1023 mol -1

In the UV/visible the maximum molar absorptivity is ______, which corresponds to a cross-section of ______.

The cross-section has nothing to do with the size of the molecule. Indeed, atoms have absorption cross-sections of ~10-13 cm2.

The maximum cross-section comes from antenna theory and is (λ/2)2, which for 500 nm radiation is 6.25×10-10 cm2.

13.1 : 5/10 Measuring Absorption

Optical Parameters with I0 = 5 μW

A 0.001 0.01 0.1 1 2 3 T 0.9977 0.9772 0.7943 0.1 0.01 0.001

I(μW) 4.9885 4.8862 3.9716 0.5000 0.0500 0.0050

_____ absorption problem: I and I0 are nearly the same value. In order to measure an absorption of 0.001 the instrument has to measure I to a precision of at least 0.23%.

_____ absorption problem: I is difficult to distinguish from 0. In order to measure an absorption of 3, the instrument has to have -3 three orders of magnitude of linearity, i.e. I = 10 I0.

The detector and electronics must have a high precision and large ______. Top-of-the-line: 10-4 to 5; intermediate: 10-3 to 3; inexpensive: 0.01 to 2.

13.1 : 6/10 Absorption Adds

Consider a solution in a 1-cm cell that has a transmission of 0.5. It doesn't matter how much light enters the cell, only half will exit. Now place two such cells back-to-back. The combination will transmit 0.5×0.5 = 0.25 of the light, thus it is seen that transmission is multiplicative.

Since absorption is the negative log of transmission, it can be seen that absorption adds.

-log(Ttotal) = -log(T1 × T2) = -logT1 + -logT2 Atotal = A1 + A2

The most common use of additive absorption involves two or more molecules in the same solution.

A(λ1) = ε1,1(λ1)C1 + ε2,1(λ1)C2 A(λ2) = ε1,2(λ2)C1 + ε2,2(λ2)C2

Solve by matrix algebra.

13.1 : 7/10 Measuring Concentration

Performance can be estimated from Beer's Law. Assume that an absorbance of 0.01 will provide a satisfactory signal-to-noise ratio.

An optimistic estimate: assume ε = 105 and l = 10 cm. A 0.01 C == =10−8 M εl 105 × 10 A realistic estimate: assume ε = 104 and l = 1 cm. A 0.01 C == =10−5 M εl 104 × 1

For a given determination, the ______can be extended by using cells of different length. Commercially available lengths are: 10, 5, 2, 1, 0.5, 0.2, 0.1 and 0.01 cm.

13.1 : 8/10 Measuring Spectra

The SNR at the peak has to be sufficiently large to observe small features.

SNR = ____ data cannot be 0.04

compared to a reference SNR = 30 spectrum. 0.03

SNR = 3 SNR = ____ data cannot be 0.02 identified with great

reliability. absorption 0.01

SNR = ____ is required to 0 400 450 500 550 obtain a faithful representation of the true -0.01 spectrum. wavelength (nm)

Minimum concentrations for a spectrum are ~100× the minimum concentration for quantitation.

13.1 : 9/10 Color and Absorption Maximum

An artist's color wheel can be used to determine the absorption maximum.

Absorption occurs at the ______of the solution color. Thus, a yellow-colored solution absorbs in the ______. violet yellow

Use the table below to convert color to wavelength. Thus, a yellow-colored solution absorbs near 400 nm.

A colorless solution absorbs in the ______.

Color and Approximate Wavelength color violet blue green yellow orange red λ (nm) 400 450 500 550 600 650

13.1 : 10/10