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Study Notes: Practical Application of Beer’s Law

We know that Beer’s Law states that:

A = epsilonlc

where A is absorbance, epsilon is the molar absorptivity constant, l is the optical path length and c is concentration.

In quantitative analysis, the analyst prepares a series of standard solutions and measures the absorbance of each (at the same wavelength) on the spectrometric instrument.

A calibration graph is prepared from the known concentrations and the measured absorbances. A straight line calibration graph of absorbance versus concentration prepared from standard solutions. For more information, contact your tutor.

The concentration of the sample is found by applying the measured absorbance to the calibration graph. The concentration of an unknown solution, containing the same species as the standards, can be read off the graph as soon as its absorbance has been measured.

Sample problem:

A Cu2+ solution gave an absorbance of 0.50 at 560 nm in a 1.0 cm sample cell. Under the same conditions, a standard series gave absorbances as follows:

[Cu2+] (mg/L)

What was the concentration of Cu2+ in the test sample if Cu2+ is the only species in the solution that absorbs at 560 nm?

Plot A vs c for the standard series to obtain a calibration graph.

This conversion graph is a straight line, going through the points (5, 0.2) and (10, 0.4) where c is the unit on the x axis and A on the y-axis.

Read the concentration of the unknown from the calibration graph.

c = 12.5 mg/L

Rather than use an instrument, visual colorimetry is a type of spectroscopic analysis that relies on the capacity of the eye to differentiate between, and match, the intensities of colours. This is the basis of methods A and B below.

A     Dilution Method
The analyst attempts to match the intensity of the colour of an unknown solution to the intensity of a standard solution in an identical tube (so the optical path length is the same). The darker of the two solutions is diluted by a measured amount until the intensity of the colour is the same as the other. The concentration of the unknown is then calculated from the concentration of the standard solution and the dilution factor.

Sample problem:
An unknown solution is compared to a 0.2 g/L standard of the same species. The standard solution is darker in colour so 5.0 mL is taken and diluted to 100 mL to get an exact match with the unknown. What is the concentration of the unknown?

Concentration of unknown = final concentration of standard =

Csubscript i times v subscript i, divided by v subscript f, which equals 0.2 times 5.0 divided by 1000, all divided by 100.0 divided by 1000, which equals 0.01 g/L.

B     Standard Series Method
The intensity of the colour of an unknown is compared to a series of standard solutions with a range of known concentrations. The intensity of colour of the unknown either matches a standard or falls between two of the standards. An intermediate intensity is estimated to one half the concentration increment between the standards either side. This method will be highly inaccurate unless the concentration increment for the standard series is small. A second standard series can be prepared to span the range between two of the original standards in a series of smaller steps.

Sample problem:

Five standard solutions are placed into identical test tubes:

Solution Number
Concentration (mg/L)

An unknown solution is compared with the standard series. The unknown is darker than solution 3, but lighter than solution 2. Estimate the concentration of the unknown.

Concentration of unknown = midpoint between solutions 2 and 3 =

this equals 1.50 plus 1.00, all divided by 2, which equals 1.25 mg/L.

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