Quantitative Analytical Techniques

While the previous lessons focused on qualitative analysis (identifying what a compound is), this lesson covers quantitative analysis — determining how much of a substance is present. We cover calibration curves, the Beer–Lambert law, standard addition, percentage purity by titration, and gravimetric analysis.

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Calibration Curves

A calibration curve (or calibration graph) is a graph of instrument response against known concentration of standard solutions. It is used to determine the concentration of an unknown sample.

How to Construct a Calibration Curve

1. Prepare a series of standard solutions of known concentration of the analyte.
2. Measure the instrument response (e.g., absorbance in UV-Vis spectroscopy, peak area in GC, peak height in HPLC) for each standard.
3. Plot instrument response (y-axis) against concentration (x-axis).
4. Draw a line of best fit (often linear through the origin for spectroscopic methods).
5. Measure the instrument response for the unknown sample.
6. Read off the concentration of the unknown from the calibration curve.

Exam Tip: In exam questions, you may be asked to read off a value from a calibration graph. Draw clear construction lines on the graph from the measured response to the x-axis, and state the concentration to an appropriate number of significant figures.

Sources of Error

• Standards must be prepared accurately using volumetric flasks and pipettes.
• The unknown must be measured under exactly the same conditions as the standards.
• The calibration curve may not be linear at high concentrations (deviations from Beer–Lambert law).

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The Beer–Lambert Law

The Beer–Lambert law relates the absorbance of a solution to its concentration and the path length of the light through the solution:

A = εcl

where:

• A = absorbance (no units, also called optical density)
• ε (epsilon) = molar absorption coefficient (dm³ mol⁻¹ cm⁻¹), a constant for a given substance at a given wavelength
• c = concentration (mol dm⁻³)
• l = path length (cm), the distance the light travels through the solution (typically 1 cm in standard cuvettes)

Key Points

• Absorbance is directly proportional to concentration (at low to moderate concentrations).
• A graph of A against c gives a straight line through the origin with gradient = εl.
• The law holds for dilute solutions only. At high concentrations, deviations occur due to molecular interactions.

Worked Example 1: Using Beer–Lambert Law

A solution of KMnO₄ has a molar absorption coefficient ε = 2455 dm³ mol⁻¹ cm⁻¹ at 525 nm. A sample in a 1.00 cm cuvette gives an absorbance of 0.735. Calculate the concentration.

A = εcl

0.735 = 2455 × c × 1.00

c = 0.735 / 2455 = 2.99 × 10⁻⁴ mol dm⁻³

Worked Example 2: Calibration Curve Data

Standard solutions of a dye are prepared and their absorbances measured at 480 nm:

Concentration (mol dm⁻³)	Absorbance
0.00	0.000
1.00 × 10⁻⁴	0.152
2.00 × 10⁻⁴	0.305
3.00 × 10⁻⁴	0.460
4.00 × 10⁻⁴	0.612
5.00 × 10⁻⁴	0.763

An unknown solution of the dye gives an absorbance of 0.400.

From the graph (or by calculation), the gradient = 0.763 / (5.00 × 10⁻⁴) = 1526 dm³ mol⁻¹ (this equals εl where l = 1 cm, so ε = 1526 dm³ mol⁻¹ cm⁻¹).

Unknown concentration: c = A / (εl) = 0.400 / 1526 = 2.62 × 10⁻⁴ mol dm⁻³.

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Standard Addition

Standard addition is a technique used when the sample matrix (other substances present) affects the instrument response, making a simple calibration curve unreliable.

Method

1. Divide the unknown sample into several equal portions.
2. Add known, increasing amounts of standard (known concentration of analyte) to each portion. One portion receives no added standard (the “blank” addition).
3. Measure the instrument response for each portion.
4. Plot response (y-axis) against the amount of standard added (x-axis).
5. Extrapolate the line back to the x-axis (where response = 0). The magnitude of the x-intercept gives the concentration of the analyte in the original sample.

This method compensates for matrix effects because the standards are measured in the same matrix as the unknown.

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Percentage Purity by Titration

Titration can be used to determine the percentage purity of a substance.

General Method