Analytical Chemistry & Spectroscopy: How Anyone Knows What Is in the Glass

A city tests its tap water and reports lead at 3 micrograms per litre. A stadium lab says an athlete's sample is clean. A court is told that a white powder contained a particular drug. In every case someone looked at a small sample and put a number, or a yes or no, on what was in it. Behind each sentence is a chain: a way of getting a signal out of a molecule, a way of turning the signal into an amount, and an honest estimate of how wrong that amount could be.

That chain is analytical chemistry. Its graduates carry one habit above all: they never read a number without asking how was it measured, against what, and how sure are we? This piece builds that habit with seven small tools you can play with. It is chemistry applied, so it needs the chemistry of acids and equilibria, and the statistics of error bars.

Light as a ruler, and the calibration line

Many solutions are coloured, and the more solute, the darker. The scientific version came in steps: Pierre Bouguer in 1729 and Johann Lambert in 1760 showed that light is absorbed in proportion to the thickness it crosses, and August Beer in 1852 showed that the same is true of concentration. The result is the Beer–Lambert law. Each equal slice of the solution removes the same fraction of the light that reaches it, so the light left falls off exponentially, and the logarithm of that fall is a straight line.

Absorbance is the useful number because it is proportional to concentration. A = 1 means 10% of the light gets through, A = 2 means 1%, A = 3 means 0.1%. The molecule NADH, central to the metabolism in Biochemistry (35/43), absorbs at 340 nm with ε = 6,220 L mol⁻¹ cm⁻¹; in a 1 cm cuvette a 100 micromolar solution reads A = 0.622, so 24% of the light passes. Try the slider. Real instruments also leak a little stray light, which makes high absorbances read too low, and that is why chemists keep readings under about A = 1 or 2.

Now the key step. You do not trust ε from a table. You calibrate: make standards of known concentration, measure each, draw the best straight line, and read the unknown off the line. Because every reading has noise, the line has uncertainty, and so does the answer. The fit is ordinary least squares, the same one in Probability & Statistics (34/05). The uncertainty in the unknown is smallest at the middle of the standards and grows toward the ends, and shrinks with more standards.

Two habits follow. Never extrapolate beyond your top standard; the line may bend there. And a result is written value ± uncertainty with its confidence level, never as a bare number.

Detection limits: how little is "nothing"?

A blank, a sample with none of the substance, still gives a wobbling signal. So "I see a signal" is not enough to say "it is there". The question is a decision under noise, and it can go wrong in two ways: a false positive (calling a blank "present") and a false negative (missing a real trace). Lloyd Currie set this out in 1968, and it is the basis of most modern definitions.

A common rule of thumb puts the limit of detection at 3.3 σ divided by the slope of the calibration, where σ is the noise of the blank or the residual scatter of a low calibration (the ICH guideline for pharmaceuticals uses it). The picture is two bell curves, one for blanks and one for samples near the limit. Put a decision threshold between them. Lower it and you catch more traces but cry wolf more often; raise it and the reverse.

At a threshold of 3 σ, a blank is falsely flagged only 0.13% of the time, but a sample whose true signal is exactly 3 σ is missed half the time. Hence a "detection limit" is not a wall; it is a stated pair of error rates. Below it, honest labs write "not detected at the limit of X", which is not the same as "zero".

Titration: counting molecules by neutralising them

The oldest method is still the cleanest. Add a reagent of known concentration drop by drop until the reaction is exactly complete (the equivalence point), and the volume you used tells you the amount. Titration is a counting method: one mole of NaOH neutralises one mole of a monoprotic acid, whatever the acid. The curve of pH against volume is the story of the reaction, and every point on it comes from one balance: charges must sum to zero.

Take 25 mL of 0.1 M acetic acid (the acid in vinegar, pK_a 4.76) and add 0.1 M sodium hydroxide. The pH starts at 2.9, climbs slowly, equals the pK_a (4.76) exactly halfway at 12.5 mL, where half the acid has become acetate, then jumps near 25 mL and ends past 12. At equivalence the pH is 8.73, not 7, because acetate is itself a weak base. A strong acid like hydrochloric acid gives an equivalence point at exactly pH 7 and a much taller jump. The size of the jump decides which indicator works: phenolphthalein changes colour between about pH 8.2 and 10, so it suits acetic acid; methyl orange, 3.1 to 4.4, does not.

Chromatography: separate first, then measure

Real samples are mixtures: coffee, blood, river water. Most measurements need the components pulled apart first. In about 1900 the Russian-Italian botanist Mikhail Tsvet poured a plant extract through a column of chalk (calcium carbonate) and watched the pigments separate into coloured bands; in 1906 he named the method chromatography, "colour writing". Archer Martin and Richard Synge shared the 1952 Nobel Prize in Chemistry for the partition chromatography that grew from it.

The principle is a race. A liquid or gas carries the mixture past a stationary material. Molecules that stick to the material more (a larger retention factor k) spend less time moving and arrive later. Two things decide whether two peaks come out separately: how different their stickiness is (the selectivity α, the ratio of their k values) and how sharp the peaks stay (the column's plate number N, a measure of how little each band spreads). Together they give the resolution.

Worked: N = 10,000, α = 1.10, first peak k = 2 so the second is k = 2.2. Then R_s = 25 × 0.0909 × 0.6875 = 1.56, just separated. The lesson is the square root: quadrupling N only doubles the resolution, whereas nudging α from 1.10 to 1.20 can do more than all the plates in the world.

Mass spectrometry: weighing molecules

A mass spectrometer ionises molecules, sends the ions through electric and magnetic fields, and sorts them by mass-to-charge ratio. J. J. Thomson built the first in the early 1910s; Francis Aston's improved instruments revealed isotopes in many elements and won him the 1922 Nobel Prize in Chemistry. For most of the century it only handled small, volatile molecules. In the late 1980s John Fenn (electrospray) and Koichi Tanaka (a laser method) found ways to launch large molecules such as proteins into the gas phase intact, and shared the 2002 Chemistry Nobel (with Kurt Wüthrich, honoured for NMR of large molecules), which is why Biochemistry (35/43) now reads proteins by their masses.

The first thing a graduate reads is the isotope pattern. Chlorine is 75.76% ³⁵Cl and 24.24% ³⁷Cl; bromine is 50.69% ⁷⁹Br and 49.31% ⁸¹Br; carbon is 98.93% ¹²C and 1.07% ¹³C (IUPAC values). So one chlorine gives a pair of peaks two mass units apart in roughly a 3:1 ratio, one bromine gives a near 1:1 pair, and each carbon adds about 1.1% to a peak one unit up. A chemist glancing at two peaks of nearly equal height two units apart says "bromine" before reading anything else.

Reading IR and NMR spectra

Two more techniques ask a molecule about its structure rather than its mass. Infrared (IR) spectroscopy measures which frequencies of infrared light make bonds vibrate. Every bond type has a typical band, so the spectrum is a bar-code of functional groups. A strong sharp band near 1715 cm⁻¹ says a carbonyl (C=O) is present, and a broad hump around 3300 says O–H.

Bond	Typical band (cm−1)	Look of the band
O–H (alcohol)	3200 to 3550	broad, rounded
C–H (alkyl)	2850 to 3000	medium, several
C≡N (nitrile)	2210 to 2260	sharp, medium
C=O (ketone, ester)	about 1700 to 1750	strong, sharp

Nuclear magnetic resonance (NMR) puts the sample in a strong magnetic field and listens to hydrogen nuclei. Bloch and Purcell shared the 1952 Physics Nobel for the discovery; Richard Ernst's Fourier-transform method earned the 1991 Chemistry Nobel. A proton spectrum gives three clues per group of equivalent hydrogens: its position (the chemical shift, in ppm), which reveals the neighbouring atoms; its area, proportional to how many hydrogens there are; and its splitting. The splitting rule is one of the most useful in chemistry: a hydrogen with n equivalent hydrogen neighbours on adjacent carbons shows up as n + 1 lines, with heights from Pascal's triangle (1:2:1 for a triplet, 1:3:3:1 for a quartet).

Look at ethyl acetate, the fruity solvent in nail-polish remover. The CH₃ of the ethyl group has two neighbours, so it is a triplet; the CH₂ has three, so it is a quartet; the CH₃ on the carbonyl side has no hydrogen neighbours, so it is a lone singlet. Areas 3 : 2 : 3. Those three patterns are enough to deduce the structure, and chemists do exactly that, daily, to prove a product is what they meant to make (see Organic Chemistry, 35/38).

How labs know what they claim

A good lab does not trust any one number. It runs a blank to catch contamination, a spiked sample (a known amount added) to see how much of the substance the method recovers, a certified reference material from a standards body to check the whole chain, and duplicates to measure scatter. Many labs follow ISO/IEC 17025, the international standard for testing and calibration labs, and join proficiency tests, where the same unknown goes to many labs and their answers are compared. A positive in forensic or anti-doping work is usually confirmed by a second method that works on different principles, such as chromatography joined to mass spectrometry.

Limits come from policy as well as chemistry. The World Health Organization's long-standing guideline for lead in drinking water is 10 µg/L, and the European Union's 2020 drinking water directive lowers its limit to 5 µg/L, with a transition to January 2036. A method must therefore detect well below those numbers, with a stated error, or the guideline cannot be enforced. For scale: a micrograms-per-litre is a part per billion, about 2.5 grams of substance spread through an Olympic swimming pool. Methods this sensitive can see contamination that older ones could not. That is progress, and it also means that "detectable" and "harmful" are separate questions, which only toxicology can answer.

What a graduate just sees

• A measurement is a number, an uncertainty and a method. A bare number is an anecdote.
• Absorbance is proportional to concentration, and 1 unit means 90% of the light is gone.
• A calibration line is most trustworthy in the middle, and the answer's error bar grows toward the ends.
• A detection limit is a pair of error rates, not a wall. "Not detected" is not "zero".
• A titration curve is a charge balance; the equivalence point is only pH 7 for a strong acid and strong base.
• Resolution grows with the square root of the plates; changing selectivity is the strong lever.
• Two peaks of equal height two mass units apart mean bromine; a 3:1 pair means chlorine.
• In NMR, n neighbours make n + 1 lines, and the areas count the hydrogens.

Where it shows up next

• Physical Chemistry (35/40): why bonds absorb the frequencies they do, derived from quantum mechanics (34/83).
• Biochemistry (35/43): enzyme assays are absorbance measurements over time; proteins are identified by mass.
• Environmental Engineering (34/68): water standards are written as concentrations that a lab can actually measure.
• Medicine: blood tests are calibrated analytical chemistry, run thousands of times a day with quality control.
• In the world: food safety, anti-doping, crime labs, pharmaceutical quality control, and the sensors in phones and air monitors.

If you want to go deeper

• 📘 Textbook. Daniel Harris and Charles Lucy, Quantitative Chemical Analysis (W. H. Freeman / Macmillan, 10th edition). The standard for the course; strong on error and calibration.
• 📘 Textbook. Douglas Skoog, F. James Holler and Stanley Crouch, Principles of Instrumental Analysis (Cengage, 7th edition). The usual year-3 text for spectroscopy and chromatography.
• 📘 Gentler. James Miller and Jane Miller, Statistics and Chemometrics for Analytical Chemistry (Pearson). A readable path from error bars to calibration and detection limits.
• 🌐 Free course. MIT OpenCourseWare 5.310 Laboratory Chemistry, with analytical methods in practice. A free online text: the analytical sections of LibreTexts Chemistry.