Interactive Video Case Study: Beer-Lambert 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, Wiley-Blackwell, 2009
Return to Study
Guide Index
Data Analysis:
Spectrophotometric measurement of concentration
(download the following
question as a Word document for educational use)
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
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 non-linear.
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, 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 'best-estimate' for the concentration, C_{o}, 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, C_{o}.
Study guide
You can click here for the answers and for study help with the relevant calculations.
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.
Question 1 | The best-estimate answer for the concentration is: C_{o}. = 53.7 mg L^{-1} . |
The slope of the 'best-fit' 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 'best-estimate' 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