Measurement, units and the laboratory balance

Classical analysis rests on two measurements, mass and volume. A titration is only as good as the mass of primary standard weighed to make the titrant and the volumes delivered by the burette and pipette; a gravimetric result is only as good as the weighings of sample and precipitate. This lesson examines the tools that make these measurements, how to use them, how to correct their small systematic errors, and how a number measured in an ordinary teaching laboratory connects, through an unbroken chain of comparisons, to the international definitions of the kilogram and the metre.

SI units and traceability

The International System of Units (SI) defines seven base units. Since the 2019 redefinition, all seven are fixed by assigning exact values to constants of nature: the kilogram through the Planck constant, the mole through the Avogadro constant (exactly 6.022 140 76 × 10²³ per mole), the kelvin through the Boltzmann constant, and so on. The mole is no longer tied to the mass of carbon-12, although for practical chemistry the difference is far below anything you will ever measure.

Traceability means that a measurement result can be related to a reference, usually an SI unit, through a documented, unbroken chain of calibrations, each with a stated uncertainty. Your laboratory balance is calibrated with test weights; the weights are calibrated against heavier-duty standards at a calibration laboratory; those are compared with national standards held by the national metrology institute; and the national standards are realized from the SI definition. When a laboratory reports 0.1023 M NaOH, traceability is what lets a regulator in another country trust the number.

In Canada the national metrology institute is the National Research Council (NRC), which also produces certified reference materials, and Measurement Canada regulates trade measurement.

The analytical balance

An analytical balance weighs up to about 100 to 200 g with a readability of 0.1 mg (0.0001 g); semi-micro balances read to 0.01 mg and microbalances to 1 µg. Modern balances use electromagnetic force restoration: the pan is held in place by a coil in a magnetic field, and the current needed to balance the load is proportional to its weight. Because the balance actually measures force, it must be calibrated with known masses at its location, and many balances contain an internal calibration weight that is applied automatically.

Good weighing practice is simple but easy to neglect:

• Level the balance with its bubble indicator and keep it on a heavy, vibration-free bench, away from drafts, doors and heating vents. Close the draft-shield doors before reading.
• Let objects reach room temperature before weighing. A warm crucible creates convection currents that make it read light, and a cold one condenses moisture and reads heavy.
• Handle dried objects with tongs or tissue, not fingers, because fingerprints add oil and moisture of tens of micrograms.
• Never weigh chemicals directly on the pan; use a weighing boat, paper or bottle, and clean up spills at once.
• Weigh hygroscopic or volatile substances in a closed weighing bottle and use weighing by difference.

Weighing by difference

To transfer an accurately known mass of a solid, weigh a capped bottle containing the solid, tip some solid into the receiving flask, and weigh the bottle again. The mass transferred is the difference. This method avoids errors from material left on a weighing paper and from moisture picked up during the transfer, because the material weighed is exactly the material delivered. Taring, by contrast, sets the display to zero with a container on the pan; it is convenient for routine work but each reading still has its own uncertainty.

Buoyancy

A balance is calibrated with dense steel weights (about 8.0 g/cm³), but most samples are much less dense. Every object on the pan is buoyed up by the weight of air it displaces. A low-density object displaces more air than the steel weights that would balance it, so it reads light. The correction is

where m′ is the balance reading, d_a is the density of air (about 0.0012 g/cm³ near sea level and room temperature), d_w is the density of the calibration weights (8.0 g/cm³) and d is the density of the object. For water (1.00 g/cm³) the true mass is about 0.1 % larger than the reading; for a dense metal it is negligible. Buoyancy matters when calibrating glassware with water and when weighing primary standards for the most accurate work; for most titrations its effect cancels or is far below other errors. Winnipeg and Minneapolis lie at modest elevations, so the air density there is close to the sea-level value, but at high altitude it is noticeably lower.

def true_mass(reading, d_obj, d_air=0.0012, d_wt=8.0):
    """Correct a balance reading (g) for air buoyancy."""
    return reading * (1 - d_air / d_wt) / (1 - d_air / d_obj)

items = [('water', 1.00), ('KHP solid', 1.636), ('sodium carbonate', 2.54), ('NaCl', 2.16), ('gold', 19.3)]
for name, d in items:
    m = true_mass(10.0000, d)
    print(f"{name:17s} d = {d:5.2f} g/cm3   reading 10.0000 g -> true {m:.4f} g   ({(m - 10) / 10 * 100:+.3f} %)")

water             d =  1.00 g/cm3   reading 10.0000 g -> true 10.0105 g   (+0.105 %)
KHP solid         d =  1.64 g/cm3   reading 10.0000 g -> true 10.0058 g   (+0.058 %)
sodium carbonate  d =  2.54 g/cm3   reading 10.0000 g -> true 10.0032 g   (+0.032 %)
NaCl              d =  2.16 g/cm3   reading 10.0000 g -> true 10.0041 g   (+0.041 %)
gold              d = 19.30 g/cm3   reading 10.0000 g -> true 9.9991 g   (-0.009 %)

Volumetric glassware

Three kinds of glassware measure volume accurately. A volumetric flask contains a stated volume when filled to its etched mark at a stated temperature (usually 20 °C); it is used to prepare solutions of known concentration. A transfer (volumetric) pipette delivers a single fixed volume. A burette delivers variable, measured volumes and is the heart of a titration. Graduated cylinders, beakers and Erlenmeyer flasks are not measuring devices for analytical purposes; their marks are rough guides.

Glassware is marked TC (to contain) or TD (to deliver). A volumetric flask is TC: it holds the stated volume, but if you pour it out some liquid stays on the walls. A transfer pipette is TD: it is calibrated to deliver its volume when drained under gravity with the tip touched to the wall, leaving a small, reproducible drop in the tip that must not be blown out. Glassware is made in two tolerance classes; Class A is about twice as accurate as Class B and is used for analytical work.

Reading a burette

The surface of an aqueous solution in a glass tube curves downward to form a meniscus. Read the bottom of the meniscus with your eye level with it to avoid parallax. A burette is graduated in 0.1 mL divisions; estimate to 0.01 mL by interpolating between marks, and record every reading to two decimal places, including the initial reading (which should not be assumed to be 0.00). A card with a dark stripe held just behind and below the meniscus makes it much easier to see. Before use, rinse the burette with several small portions of the titrant, check that the stopcock does not leak, and remove the air bubble from the tip, because a bubble that escapes during the titration adds its volume to the reading.

A burette reading has a random uncertainty of about ±0.02 mL, and the delivered volume is the difference of two readings, so its uncertainty is about ±0.03 mL. A drop from a burette is about 0.05 mL. For a relative uncertainty of 0.1 % from reading alone, the titrant volume should be at least 30 mL, which is why titrations are designed to use most of a 50 mL burette.

Calibrating glassware

The tolerance is a manufacturer's guarantee, not the actual volume. For the most accurate work, glassware is calibrated by weighing the water it contains or delivers. Because water's density depends on temperature, and because the glass expands slightly when warm, tables give the volume of 1 g of water at each temperature, corrected for buoyancy and referred to 20 °C. At 20 °C, 1.0000 g of water read on a balance occupies 1.0029 mL; at 25 °C it occupies 1.0040 mL.

# Density of air-saturated water (g/mL) by a standard fit, valid about 5-40 C
def water_density(t):
    return 0.999974 - (t - 3.983) ** 2 * (t + 301.797) / (522528.9 * (t + 69.348))

def volume_per_gram(t, d_air=0.0012, d_wt=8.0, glass_alpha=1.0e-5):
    rho = water_density(t)
    mass_true = (1 - d_air / d_wt) / (1 - d_air / rho)    # buoyancy-corrected mass of a 1 g reading
    v_t = mass_true / rho                                   # volume at t
    return v_t * (1 - glass_alpha * (t - 20))               # refer the glass volume to 20 C

readings = [24.9563, 24.9581, 24.9549]
t = 22.0
f = volume_per_gram(t)
vols = [r * f for r in readings]
mean = sum(vols) / len(vols)
sd = (sum((x - mean) ** 2 for x in vols) / (len(vols) - 1)) ** 0.5
print(f"density at {t} C = {water_density(t):.5f} g/mL; 1 g reading -> {f:.4f} mL at 20 C")
print("volumes (mL):", ", ".join(f"{x:.4f}" for x in vols))
print(f"calibrated pipette volume = {mean:.3f} mL, s = {sd:.4f} mL")

density at 22.0 C = 0.99777 g/mL; 1 g reading -> 1.0033 mL at 20 C
volumes (mL): 25.0379, 25.0397, 25.0365
calibrated pipette volume = 25.038 mL, s = 0.0016 mL

Temperature and volumetric solutions

Liquids expand when warmed. Dilute aqueous solutions expand by about 0.02 % per degree near room temperature, so a solution prepared at 20 °C and used at 25 °C has a concentration about 0.1 % lower. In most teaching laboratories this is smaller than other errors, but in a laboratory that is cold in winter and warm in summer it can be significant, and organic solvents expand three to five times more than water. Glass expands much less (borosilicate glass about 0.001 % per degree) and is usually negligible.

Primary standards and standard solutions

A standard solution has a known concentration. It can be prepared directly by dissolving an accurately weighed amount of a primary standard and diluting to the mark in a volumetric flask, or indirectly by preparing an approximate solution and standardizing it by titration against a primary standard.

A primary standard is a compound pure enough to be weighed and used directly. It should be at least 99.9 % pure (with a certified purity), stable in air and on drying, not hygroscopic, readily soluble, of high molar mass so that weighing errors are relatively small, and react stoichiometrically and quickly with the analyte or titrant.

Sodium hydroxide is the classic example of a reagent that is not a primary standard. Its pellets absorb water and carbon dioxide from the air, so their mass does not correspond to a known amount of NaOH. A NaOH titrant is made approximately and then standardized against weighed KHP. Hydrochloric acid is similarly standardized against sodium carbonate or TRIS.

Drying and desiccators

Primary standards and samples for gravimetry are dried to constant mass, usually in an oven at 105 to 120 °C for a few hours, then cooled in a desiccator, a closed container holding a drying agent such as anhydrous calcium sulfate or silica gel. Constant mass means that successive heating and cooling cycles change the mass by no more than a small amount, typically 0.2 to 0.3 mg. Hot objects should be allowed to cool partly before going into the desiccator, and the lid should be slid rather than lifted.

Significant figures and rounding

Significant figures show the precision of a number: all certain digits plus the first uncertain one. A balance reading of 0.5132 g has four significant figures; a burette reading of 23.47 mL has four. In calculations, keep at least one or two extra digits until the final result and round only at the end; rounding intermediate results adds error. For a logarithm such as pH, only the digits after the decimal point are significant: [H⁺] = 2.0 × 10⁻³ M (two significant figures) gives pH 2.70. Lesson 3 replaces significant-figure rules with explicit uncertainties, which are more honest, but significant figures remain the everyday shorthand.