Interactive Case Study: BeerLambert law  Linear calibration
Produced by Graham Currell, University of the West of England, Bristol, in association with:
● Royal Society of Chemistry, 'Discover Maths for Chemists' website, and
● Essential Mathematics and Statistics for Science, 2nd Edition
Graham Currell and Antony Dowman, WileyBlackwell, 2009
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Introduction
The absorbance, A, (sometimes known as optical density, OD), of light passing through a solution is given by the BeerLambert law as:
A = ε×b×C
where
ε (Greek letter, epsilon) is the molar absorptivity of the solute with units of M^{1 }cm^{1} (or (mol L^{1})^{1 }cm^{1 } or mol^{1 }dm^{3 }cm^{1})
b is the path length of the light through the solution in units of cm.
C is the concentration of the solution in mol^{ }L^{1} (or mol^{ }dm^{3 }or M)
This straight line relationship works well for dilute solutions (low values of C), but it is important to remember that, at higher concentrations, the equation can cease to be valid and the relationship between A and C becomes nonlinear.
Spectrophotometric measurement of concentration
(download the following question as a Word document for educational use)
The following exercise gives a worked example with video answers to illustrate the mathematics of this problem.
You can then follow study links to other resources to help you with the mathematics.
In a simple spectrophotometric experiment to measure the concentration of a solution of potassium permanganate:

the spectrophotometer is zeroed to give A = 0 when C = 0, by using a 'blank' solution (this can occur automatically in a doublebeam instrument),

the absorbances, A, are measured for a series of standard concentrations, C, of potassium permanganate at a wavelength of 522 nm,

a 'bestfit' line of regression in a graph of the measured values of A against C, gives a calibration line for the particular measurement,

the absorbance, A_{o}, of the solution being tested is measured and the equivalent value of concentration,C_{o}, is calculated from the calibration line.
The absorbance, A_{o}, of the solution being tested is measured three times, giving replicate values,
A_{o} = 0.763, 0.741, 0.749.
The calibration values of absorbances, A, of the six standard concentrations, C, are given in the table below:
C (mg^{ } L^{1}) 
0 
20 
40 
60 
80 
100 
120 
A 
0 
0.267 
0.583 
0.824 
1.120 
1.313 
1.499 
Molar mass of potassium permanganate, KMnO_{4}, M_{m} = 158 g mol^{1}.
Answer the following questions:
Question 1
Use the experimental data to calculate a 'bestestimate' for the concentration, C_{o}, of the test solution.
Calculate a 'bestestimate' for the molar absorptivity, ε, of potassium permanganate at this wavelength, assuming that the path length of the cell used to hold the solution is 10 mm.
You will need to consider the applicability of Beer's law for this particular set of experimental data.
Question 2 (advanced)
Calculate the 95% confidence interval for your value of, C_{o}.
Numerical Answers:
Question 1 
The bestestimate answer for the concentration is: C_{o}. = 53.7 mg L^{1} . 

The slope of the 'bestfit' line of regression, m = 0.0140 (mg L^{1})^{1}. 

Using molar mass M_{m} = 158 g mol^{1}, gives slope, m = 0.0140×1000×158 = 2212 (mol L^{1})^{1}. 

Using m = ε×b, with b = 1.0 cm, gives a 'bestestimate' molar absorptivity, ε = 2212 (mol L^{1})^{1 }cm^{1} 
Question 2 
Standard deviation uncertainty, u(C_{o})= 0.97 mg L^{1} . 

The 95% confidence interval for the concentration is: C_{o} = 53.7 ± 3.1 mg L^{1} . 
Video Answers:  including specific links to revision/help in the relevant mathematical skills:
Download the Excel file used to perform the above calculations
Please send any comments and suggestions to:
graham.currell@uwe.ac.uk
17/09/09