Key Concepts
• Lambert’s Law of Absorption
• Beer’s Law
• Beer-Lambert Law
• Absorption Cross-Sections
• Photometric quantities
• Spectrophotometer
• The Cary 50 Spectrophotometer
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Lambert described how intensity changes with distance in an absorbing medium.
• The intensity
I
0 if a beam of light decreases exponentially as it passes though a uniform absorbing medium with the linear decay constant
α
.
Restatement: In a uniform absorbing medium, the intensity of a beam of light decreases by the same proportion for equal path lengths traveled.
• The linear decay constant α is a characteristic of the medium. It has units of reciprocal length.
α is the path length over which the intensity is attenuated to 1/ e.
I ( x )
I
0
I ( x )

I
0 e
  x x l
α
I
I
Johann Heinrich Lambert
1728-1777

I
0 e
  l
I d I

The distance traveled through the medium is called the path length .

I

0 e

 
I x d x d d
I
I
0
I x

   e
  x
I
Photo: http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Lambert.html
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Typically base 10 is used in photometry.
k
  ln 10 I

I
0 e
  x 
I
0
10
 k x
I
I
0
 e
  x 
10
 k x k is the path length over which the intensity is attenuated to 1/10 .
I
I
0

10
 k x
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If one slab of absorbing material of thickness l reduces the intensity of a beam of light to half.
l
α
I
0
I
I
I
0

10
 k l 
1
2
Then two slabs of the same absorbing material will then reduce the intensity of a beam of light to one quarter . l
α l
α
I
0
I
I
I
0

10
 k 2 l 
1
2
2

1
4
And three slabs will reduce the intensity of a beam of light to one eight . l
α l
α l
α
I
0
I
I
I
0

10
 k 3 l 
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1
2
3

1
8

Beer found that Lambert’s linear decay constant k for a solution of an absorbing substance is linearly related to its concentration c by a constant, the absorptivity ε, a characteristic of the absorbing substance.
Restatement: The linear decay constant k is linear in concentration c with a constant of proportionality
ε.
(August Beer, 1825-1863) k
  c
Typical units are: k cm
−1
; c M (moles/liter);
ε
M
−1 cm
−1
A colored absorber has an absorptivity that is dependent on wavelength of the light ε ( λ ).
The absorptivity is the fundamental property of a substance. This is the property that contains the observable spectroscopic information that can be linked to quantum mechanics (also see absorption cross section.)
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In photometry we measure the intensity of light and characterize its change by and object or substance. This change is typically expresses as percent transmittance or absorbance .
Transmittance (T)
Frequently when your primary interest is the light beam
T

I
I
0 usually given in percent
Absorbance (A)
(AKA optical density, O.D.)
Used almost exclusively when your interest concerns the properties of the material
A
  log


I
I
0


  log T by convention, base 10 logs are used
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Lambert’s and Beer’s Laws are combined to describe the attenuation of light by a solution. It is easy to see how the two standard photometric quantities can be written in terms of this law.
I

I 10
0
  c x
Transmittance
T
T

I
I
0

10
  c x
Absorbance
A
  log


I
I
0
A
  c x


  log T
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The absorption of light by particles (and single molecules) is characterized by an absorption cross section C . In this model the particle is replaced by a perfectly absorbing sphere with a cross sectional area C . This cross section is a property of the particle and is not related to its geometric cross sectional area. The concentration of particles per unit volume is N .
 k

NC

NC ln 10 typical units are: C cm 2 ; N cm
−3
 
N
A
C ln 10


10

3 liter cm
3
The cross section can be directly related to the molar absorptivity. N
A is Avagadro’s number. units are: C cm 2 ; N cm
−3
; N
A mole
−1
; ε M
−1 cm
−1
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The absorption efficiency Q of a particle is the ratio of its absorption cross section C to its geometric cross section C geo
.
Absorption efficiency is dimensionless.
Q

C
C geo
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Attenuation of light by absorption and scattering both obey Lambert’s
Law. Thus we can extend our treatment of absorption to scattering and extinction. (Recall that extinction is the effect of absorption + scattering.)
C ext

C abs

C sca
Q ext
 ext
A



Q abs
 abs
 ext cx

Q sca



  sca abs
The scattering efficiency can be much larger than unity.
Extinction paradox: Q ext
= 2 ( Q abs
= 1; Q sca
= 1) for an perfectly absorbing particle very large compared to the wavelength of light.
  sca
 cx
Note:
•All of these quantities are in general wavelength dependent.
•Our discussion has not included the mechanism (cause) of absorption and scattering.
•There are many different mechanisms that cause of absorption and scattering.
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•
Spectrometer: measures I vs
λ
.
Simply measures the spectrum of the light (e.g. emission spectroscopy).
•
Spectrophotometer: measures I / I
0 vs
λ
.
Measures how the sample changes the spectrum of the light (e.g. transmission, reflection, scattering, fluorescence).
All spectrophotometers contain a spectrometer.
• -meter: the detector is electronic
• -graph: light intensity recorded on film
• photometer: measures I / I
0 without λ selection.
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Measures absorbance as a function of wavelength
Components: light source, monochromator, sample cell, detector, optical system.
monochromator sample cell detector light source
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Computer controlled acquisition of absorption spectra monochromator balance the forces: detector sample
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Can you find the diffraction grating and the slit?
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• First, measure the baseline using a blank sample. This is raw I
0
. The blank sample is the cuvette with deionized water (everything but your nanoparticles). This corrects for any absorption due to the cuvette, water, and variations of the light intensity of the light source, monochromator, etc.
• Second, measure the zero by inserting the beam block . This corrects the instrument for the detector background.
• Third, measure your sample. This is the raw
I . The Cary 50 automatically calculates the corrected intensities ( I and I
0 subtracting the zero from each of the raw intensities.
) by
• Subsequent measurements do not require re-measuring the blank and zero, simply repeat step 3.
T

I
I
0
 raw raw I
I
0

 zero zero
A
  log T
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• Spectroscopy
• Chemical Analysis: trace analysis, pH, forensic, in situ monitoring, remote monitoring, geology, astronomy, ....
•
Particle size
•
Thin film characterization
•
Color matching
•
Optics
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