Interactive Case Study:    Beer-Lambert law - Linear calibration

book cover Produced by Graham Currell, University of the West of England, Bristol, in association with:
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Essential Mathematics and Statistics for Science, 2nd Edition
   Graham Currell and Antony Dowman, Wiley-Blackwell, 2009

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The absorbance, A, (sometimes known as optical density, OD), of light passing through a solution is given by the Beer-Lambert law as:  

        A = ε×b×C  

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 dm3 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 non-linear. 

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 double-beam instrument),

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

    • a 'best-fit' line of regression in a graph of the measured values of A against C, gives a calibration line for the particular measurement,

    • the absorbance, Ao, of the solution being tested is measured and the equivalent value of concentration,Co, is calculated from the calibration line.

The absorbance, Ao, of the solution being tested is measured three times, giving replicate values,
Ao = 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, KMnO4, Mm = 158 g mol-1.

Answer the following questions:

Question 1        

Use the experimental data to calculate a 'best-estimate' for the concentration, Co, of the test solution.

Calculate a 'best-estimate' 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, Co.

Study guide:

The following techniques are relevant to the calculations in this problem: 

Linear regression for the slope and intercept of a straight line
Excel for data analysis (web):
      X-Y graphs using Excel (video), Linear regression using Excel (video), Data Analysis Tools in Excel (video)

You can also download an Excel file that gives an interactive demonstration of the effect of Stray Light on absorbance values.

Numerical Answers:

Question 1 The best-estimate answer for the concentration is:  Co. = 53.7 mg L-1 .
  The slope of the 'best-fit' line of regression, m = 0.0140 (mg L-1)-1.
  Using molar mass Mm =  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 'best-estimate' molar absorptivity, ε = 2212 (mol L-1)-1 cm-1

Question 2 Standard deviation uncertainty, u(Co)= 0.97 mg L-1 .
  The 95% confidence interval for the concentration is:  Co = 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


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