7. Beer’s Law and It’s Implications for Instrument Construction

dS dn

1. Derive Beer’s Law

ASSUMPTIONS1. No light is emitted

2. dx infinitesimal

3. Monochromatic light uniform on the surface, S

4. dn molecules in a section volume

5. Capture cross sectional area is

M h M kT *

V dx S

dP

P

dn

Sx

xP

P n

0 0

dP

P dn

S0

Photons cap tured

Pho tons im ping ing cap ture area

to ta l area

P dS

S

( ) 0

dS dn

Po P1P2

Consider a large # of boxes

This is an integration

Set up Derivation

ln P

n n

Sx P

P

0

0

dP

P

dn

Sx

xP

P n

0 0

ln lnP Pn

S0

ln

P

P

n

S0

log

.

P

P

n

S0 2 303

AP

PT

n

S

log log

.0 2 303

Substitutions

SV

b n MN

L

cmVa

1

10 3 3

A

P

P

MNL

cmV

V

b

a

log.0

3 3

1

10

2 303

A

NL

cmM b bM

a

1

102 303

3 3

.

A T bM log

What is the absorbance when the light transmitted is 50% of the initialbeam in a 2 cm path length cell for a concentration of 10-3 M?

Deviations

1. Assumed each molecule was independent of the other

When will the assumptions fail?

Molecules not independent when:

Neighbors experience each other

1. High concentrations

2. High electrolyte

3. Large local fields due to large absorption probability (alpha)

Apparent Instrumental Deviations

**polychromatic radiation***

What is the source of polychromatic radiation?

12

AP

PbC

1

1

101

log

Rearrange

P P bC

1 1

10 10

Similarly

AP

PbC

2

2

0 22

log

P P bC

2 2

20 10

Total absorbanceA

P P

P P

P P

P Pmeasured

log log

1

1

2

0 2 0 2

0 2 0 2

2

AP P

P Pmeasured bC bC

log 0 2 0 2

0 0 21

1 210 10

Consider several cases using this equation

AP P

P Pmeasured bC bC

log 0 2 0 2

0 0 21

1 210 10

1. Monochromatic light

1 2

AP P

P Pmeasured bC

log 0 2 0 2

0 0 2110

Ameasured bC

log

1

10

A bCmeasuredbC log 10 Reality check ok

AP P

P Pmeasured bC bC

log 0 2 0 2

0 0 21

1 210 10

2. Case 2

P P Po , , 1 0 2 0

AP P

P Pmeasured bC bC

log 0 0

0 010 101 2

AP

Pmeasured bC bC

log2

10 100

01 2

Ameasured bC bC

log2

10 101 2

Example Calculation

B=1M=0.001Molar absorptivity at 1=2000 at 2 = 200

Ameasured

log2

10 102000 1 10 200 1 103 3

Ameasured

log .

.

2

10 100 494

2 0 2

.494

M

When would this situation apply?

AP P

P Pmeasured bC bC

log 0 2 0 2

0 0 21

1 210 10

3. Stray Light

P P ando , , 1 0 2 2 0

AP P

P Pmeasured bC bC

log 0 1 0 2

0 1 0 2010 101

AP P

P Pmeasured bC

log 0 1 0 2

0 1 0 210 1

What happens when light at 1 is strongly absorbed?

P PbC0 1 0 210 1

AP P

Pmeasured

log 0 1 0 2

0 2

Example Calculation

Stray light is 0.5% of total light

P P0 2 0 10 005 .

AP P

Pmeasured

log

.

.0 1 0 1

0 1

0 005

0 005

Ameasured

log.

..

1 005

0 0052 303

The maximum absorbance the Instrument is capable of measuring is2.303

Comparison of InstrumentsInstrument %stray light maxASpect 20 0.5 2.3McPherson 0.1 3McPherson +filter 0.01 4Double monochromator 0.001 5

Physical Dimensions: 89.1 mm x 63.3 mm x 34.4 mm

Weight: 190 grams

Detector: Sony ILX511 linear silicon CCD array

Detector range: 200-1100 nm

Pixels: 2048 pixels

Pixel size: 14 μm x 200 μm

Pixel well depth: ~62,500 electrons

Sensitivity: 75 photons/count at 400 nm; 41 photons/count at 600 nm

Design: f/4, Symmetrical crossed Czerny-Turner

Focal length: 42 mm input; 68 mm output

Entrance aperture: 5, 10, 25, 50, 100 or 200 µm wide slits or fiber (no slit)

Grating options: 14 different gratings, UV through Shortwave NIR

Detector collection lens option: Yes, L2

OFLV filter options: OFLV-200-850; OFLV-350-1000

Other bench filter options: Longpass OF-1 filters

Collimating and focusing mirrors: Standard or SAG+

UV enhanced window: Yes, UV2

Fiber optic connector: SMA 905 to 0.22 numerical aperture single-strand optical fiber

Spectroscopic Wavelength range: Grating dependent

Optical resolution: ~0.3-10.0 nm FWHM

Signal-to-noise ratio: 250:1 (at full signal)

A/D resolution: 12 bit

Dark noise: 3.2 RMS counts

Dynamic range: 2 x 10^8 (system); 1300:1 for a single acquisition

Integration time: 3 ms to 65 seconds

Stray light: <0.05% at 600 nm; <0.10% at 435 nm

Corrected linearity: >99.8%

Electronics Power consumption: 90 mA @ 5 VDC

Data transfer speed: Full scans to memory every 13 ms with USB 2.0 or 1.1 port, 300 ms with serial port

Czerny-Turnerconstruction

What would be The maximumA this could measure?

What is the maximum amount of absorbance you can measure if the stray light in an instrument is 8%?

If it is 0.05% at 600 nm as for the Ocean Optics?

1. Where does stray light come from?

2. Is stray light likely to be more important for 200 or for 900 nm light?

3. Is stray light likely to be more or less important near a region where solvent interferes?

Double Dispersion Reduces the Stray Light

Comparison of Instruments

Name $ ∆ range Ps/Po%

Spect 20 2-4k 2-8 190-1000 0.5Double Beam 4-15k 195-850 0.1PE-57 >5k 0.2 190-750 <0.1Double dispersive 0.07 185-3125 0.0008Multichannel Array 7-9k 200-920

Beer’s Law and Standard Additions

QUANTITATION

1. Wide chromophore range (universality)-extended by color forming reactionsfor example complexation

2. Good sensitivity

3. Selectivity

4. Accuracy

5. Ease

1. Standard CurvesChoose a wavelength where the molar absorptivity does not change

where would this be?why choose this wavelength region?

Need clear cells and no greasy fingers. Why?

Need to control: temperature; pH; electrolyte/solvent. Why?

2. Standard addition method is useful when matrix (the solution containing the sample analyte) effects complicate matters

Overcoming Matrix Effects in Calibration Curves

Ppm Metal

SignalSolvent

MatrixExample: FlameAtomic Absorption forPb in SeaWater, PbCl2 Is lost lowering the signal

Matrix EffectIf we don’t have a Clear idea whatThe matrix effect isThen we drasticallyMisjudge the concOf the sample fromThe measured signal

Our standardsSuggest this Sample conc.

Standards madeUp in the matrix ofThe sample wouldSuggest this sampleConc.

Matrix

SampleSignal

A bto ta l m oles

to ta l vo lum emeasured

A bn n

V Vnew m easuredunknown added

sam ple o f unknown added,

A bV M V M

V Vmeasuredunkown unknown stam dard s dard

unkown s dard

tan

tan

A bV M V M

Vmeasuredunkown unknown stam dard s dard

to ta l

tan

A bV M

Vb

M

VVmeasured

unkown unknown

to ta l

s dard

to ta lstam dard

tan

Overcoming Matrix Effects in Calibration Curves

50

40

30

20

10

0

A bV M

Vb

M

VVmeasured

unkown unknown

to ta l

s dard

to ta lstam dard

tan

xslopeintercept

y

slope bM

Vs dard

to ta l

tan

in t ercep t bV M

Vunkown unknown

to ta l

in t

tan tan

ercep t

slope

bV M

V

bM

V

V M

M

unkown unknown

to ta l

s dard

to ta l

unkown unknown

s dard

in t tanercep t

slope

M

VMs dard

unkownunknown

M=slope=0.03912

B=intercept=0.2422

Vunknown= 10 ml

Mstandard=11.1ppm

in t tanercep t

slope

M

VMs dard

unkownunknown

0 2422

0 03812

11 1

107 01

.

.

..

ppmppm

A bto ta l m oles

to ta l vo lum emeasured

You did standard addition for the flame lead analysis. You found:

Your unknown volume is 10 mL and the standard you added is 20 ppb.

What is the unknown concentration?

A ppb 00 0521 0 433. . ( )

Two Component Spectra

A bC bCM M N N1 1 1

A bC bCM M N N2 2 2

Must be known

measure

Result is two equations in two unknowns – can be solved