Beer’s Law and the Optimum Transmittance in Absorption Measurements

Harold K. Hughes

doi:10.1364/AO.2.000937

Applied Optics
Vol. 2,
Issue 9,
pp. 937-945
(1963)
•https://doi.org/10.1364/AO.2.000937

Beer’s Law and the Optimum Transmittance in Absorption Measurements

Harold K. Hughes

Not Accessible

Your library or personal account may give you access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Harold K. Hughes, "Beer’s Law and the Optimum Transmittance in Absorption Measurements," Appl. Opt. 2, 937-945 (1963)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Abstract

Beer’s law is used widely in the determination of cross sections and of unknown solution concentrations in the form τ = 10⁻^A, where τ is the transmittance and A is the absorbance. By the criterion of minimum experimental error, the best value of the transmittance depends upon numerous parameters and varies between 2% and 90%. This paper considers the influence of the source, background, noise, counting time, and other parameters on the optimum transmittance.

Full Article | PDF Article

More Like This

Correction to the Beer-Lambert-Bouguer law for optical absorption

Haim Abitan, Henrik Bohr, and Preben Buchhave
Appl. Opt. 47(29) 5354-5357 (2008)

Ocean analysis by means of Beer’s law

John E. Tyler
Appl. Opt. 15(10) 2565-2567 (1976)

In Defense of Beer’s Law

Arthur C. Hardy and F. Mansfield Young
J. Opt. Soc. Am. 38(10) 854-858 (1948)

Previous Article Next Article

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Figures (2)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Tables (2)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Equations (74)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

a₁	Absorptivity of the solvent (m⁻¹)
a₂	Absorptivity of the solid sample or of the component of interest in the solution (m⁻¹)
a′	Mass absorptivity = a/ρ (m²/kg)
1/ac	Free path (m)
A	Absorbance; dimensionless
A_w	Absorbance of the windows
b	Breadth (thickness) of sample or cell interior (m)
c₂	Mass fractional concentration of the component of interest in the solution; dimensionless
e	Naperian base = 2.71828 …
m	Relative background; see Sec. II-D-1
n	Number of absorbing or scattering centers per unit volume in m⁻³; counts/sec with sample in the beam
n₀	Ccunts/sec incident on sample
N	Total count with sample in the beam
N₀	Total count incident on the sample
P	Radiant power transmitted by the sample or cell; W (or quanta/sec or particles/sec) per m²
P₀	Radiant power incident on the sample or cell
P_g	Radiant power of background
t	Time of counting transmitted beam (sec)
t₀	Time of counting incident beam
T	Total time of counting
μ	Absorption coefficient (m⁻¹)
μ′	Mass absorption coefficient = μ/ρ (m²/kg)
ρ	Density (kg/m³)
σ	Cross section for absorption or scattering in m²; standard deviation
τ	Transmittance; dimensionless

Conditions	Opt A	Opt τ	Error
Sec. II. Linear Detector
A. Constant source
1. 2-Beam, solvent comparison	0.43	36.8%	2.72 Δτ
3. 2-Beam, standard solution	0.43	36.8	Δτ
4. 2-Beam, independent error
a. Pure solvent	0.48	33.0	2.87 σ_P₁/P₀
b. Absorber	0.87	13.5	0.707 σ_P₁/P₁
5. 1-Beam, independent errors, 2 zeros
a. Pure solvent	0.54	28.7	3.11 σ_P₀/P₀
b. Absorber	0.88	13.1	σ_P₀/P₀
6. 1-Beam, independent error, 1 zero
a. Pure solvent	0.52	30.5	3.01 σ_P₀/P₀
b. Absorber	0.88	13.3	0.70 σ_P₀/P₁
7. Penetrating background	See Eq. (46)
B. Noise
1. Johnson noise	0.43	36.8
2. Shot noise	0.87	13.5
C. Poisson source
1. Fixed time, t = t₀	0.96	10.9	0.45 (1/N₀ + 1/N)¹/²
2. Fixed time, t₀ = ∞	0.87	13.5	0.50 N¹/²
3. Absorber in comparison beam	Do not use
4. Fixed total time	1.11	7.8	1.79 (n₀T)⁻¹/²
5. Fixed count	1.74	1.8	0.35 N⁻¹/²
D. Poisson source with background
1. Fixed area	τ = 7.8% to 31% depend on bg
2. Variable area	τ = 22.8% to >90% depend on bg
Sec. III. Limited Detector
With respect solut.	0.43	36.8
Absolute	Much less, even as low as 2%. See discussion.

a₁	Absorptivity of the solvent (m⁻¹)
a₂	Absorptivity of the solid sample or of the component of interest in the solution (m⁻¹)
a′	Mass absorptivity = a/ρ (m²/kg)
1/ac	Free path (m)
A	Absorbance; dimensionless
A_w	Absorbance of the windows
b	Breadth (thickness) of sample or cell interior (m)
c₂	Mass fractional concentration of the component of interest in the solution; dimensionless
e	Naperian base = 2.71828 …
m	Relative background; see Sec. II-D-1
n	Number of absorbing or scattering centers per unit volume in m⁻³; counts/sec with sample in the beam
n₀	Ccunts/sec incident on sample
N	Total count with sample in the beam
N₀	Total count incident on the sample
P	Radiant power transmitted by the sample or cell; W (or quanta/sec or particles/sec) per m²
P₀	Radiant power incident on the sample or cell
P_g	Radiant power of background
t	Time of counting transmitted beam (sec)
t₀	Time of counting incident beam
T	Total time of counting
μ	Absorption coefficient (m⁻¹)
μ′	Mass absorption coefficient = μ/ρ (m²/kg)
ρ	Density (kg/m³)
σ	Cross section for absorption or scattering in m²; standard deviation
τ	Transmittance; dimensionless

Conditions	Opt A	Opt τ	Error
Sec. II. Linear Detector
A. Constant source
1. 2-Beam, solvent comparison	0.43	36.8%	2.72 Δτ
3. 2-Beam, standard solution	0.43	36.8	Δτ
4. 2-Beam, independent error
a. Pure solvent	0.48	33.0	2.87 σ_P₁/P₀
b. Absorber	0.87	13.5	0.707 σ_P₁/P₁
5. 1-Beam, independent errors, 2 zeros
a. Pure solvent	0.54	28.7	3.11 σ_P₀/P₀
b. Absorber	0.88	13.1	σ_P₀/P₀
6. 1-Beam, independent error, 1 zero
a. Pure solvent	0.52	30.5	3.01 σ_P₀/P₀
b. Absorber	0.88	13.3	0.70 σ_P₀/P₁
7. Penetrating background	See Eq. (46)
B. Noise
1. Johnson noise	0.43	36.8
2. Shot noise	0.87	13.5
C. Poisson source
1. Fixed time, t = t₀	0.96	10.9	0.45 (1/N₀ + 1/N)¹/²
2. Fixed time, t₀ = ∞	0.87	13.5	0.50 N¹/²
3. Absorber in comparison beam	Do not use
4. Fixed total time	1.11	7.8	1.79 (n₀T)⁻¹/²
5. Fixed count	1.74	1.8	0.35 N⁻¹/²
D. Poisson source with background
1. Fixed area	τ = 7.8% to 31% depend on bg
2. Variable area	τ = 22.8% to >90% depend on bg
Sec. III. Limited Detector
With respect solut.	0.43	36.8
Absolute	Much less, even as low as 2%. See discussion.

Abstract

Cited By

Figures (2)

Tables (2)

Equations (74)

Applied Optics