Gravimetric and Volumetric Preparation Error

A prepared solution is a chain of measurements, and its concentration is no better than the worst link. Most of that chain is invisible in the final record, which usually states a concentration and nothing about how it was arrived at. Formal treatments of measurement uncertainty divide the contributions into those evaluated statistically from repeated observation and those evaluated from any other information — manufacturer specifications, calibration certificates, judgement — and require both to be combined before a result is quoted 1. Bench preparation is a good example of why: the largest terms in an ordinary dilution are usually the ones nobody repeated.

Where error enters

Between a labelled container of solid and a working solution there are eight or nine opportunities to be wrong, and they are not of comparable size. Listing them in order is the only way to see which deserve attention.

Two entries in that table are not measurement problems and cannot be improved by better technique. The peptide fraction of the solid is a property of the material that must be supplied or determined. Adsorption is a physical loss: peptide leaves solution and attaches to glass or polypropylene, most severely in dilute, low-ionic-strength, unbuffered preparations, and the loss can reach tens of per cent before anything is even pipetted 3. It is mitigated by low-binding surfaces, by a carrier protein where the assay tolerates one, by avoiding unnecessarily dilute intermediate stocks, and by conditioning surfaces before use 4. It is not mitigated by calibrating the pipette.

Balance resolution against pipette tolerance

The two instruments fail differently, and the difference is the whole argument. A balance has an error that is essentially fixed in absolute terms: an analytical balance reading to 0.1 mg has a repeatability standard deviation of the same order regardless of whether the pan holds 5 mg or 5 g. Its relative error therefore falls as the quantity rises. A pipette has an error with a proportional component and a floor, so its relative error rises sharply as the delivered volume falls below its nominal capacity.

The pharmacopoeial minimum-weight rule follows directly from the first of those. Setting a required accuracy — a tenth of a per cent is the usual figure — and combining it with the balance repeatability standard deviation gives a smallest defensible sample of roughly two thousand times that standard deviation. Below that quantity the balance is still displaying digits, but they no longer meet the accuracy the work assumed.

Three rules fall out of the table and are worth applying without further thought. Choose the smallest pipette that delivers the required volume in a single stroke. Never operate a pipette below about twenty per cent of its nominal capacity. And prefer one stroke to several — repeated part-strokes into the same vessel accumulate the systematic error once per stroke rather than averaging it away.

Liquid properties change the picture further. Air-displacement pipettes work through a cushion of air, which volatile solvents saturate with vapour, so they under-deliver alcohols and organic solvents unless the tip is pre-wetted several times. Viscous liquids drain slowly and leave a film. Dense liquids load the air cushion differently. Positive-displacement pipettes, in which a piston contacts the liquid directly, remove all three effects and are the correct instrument for anything that is not dilute aqueous buffer.

Why weighing wins below a threshold

Gravimetric preparation replaces the volume measurement with a mass measurement and converts afterwards. The solvent is dispensed into a tared vessel on a balance and its mass recorded; the volume follows from the density. Consider water at room temperature: a millilitre weighs almost exactly a gram, so an analytical balance reading to 0.1 mg resolves a tenth of a microlitre, or one part in ten thousand. The same volume delivered by a well-maintained pipette carries a systematic uncertainty of six to ten parts in a thousand. The gravimetric route is one to two orders of magnitude better, using an instrument already on the bench.

The crossover sits at around a hundred microlitres for routine aqueous work, and moves upward — favouring the balance — for any solvent that is volatile, viscous or dense, and for any preparation where the same solvent must be delivered reproducibly across many vessels. It moves downward only when evaporation during the weighing becomes significant, which is a matter of seconds for volatile organics in an open vessel and is handled by a lid, a covered weighing boat or a solvent trap.

1. Place the receiving vessel on the balance, closed where possible, and allow the reading to settle before taring.
2. Tare, then dispense the solvent into the vessel without removing it from the pan.
3. Record the delivered mass to the full resolution of the balance. This number, not the intended volume, is the datum.
4. Record the solvent temperature at the time of delivery, measured rather than assumed from the room thermostat.
5. Convert mass to volume using the density of that solvent at that temperature.
6. Apply a buoyancy correction where the required accuracy is better than about a tenth of a per cent.
7. Record the balance identifier and the date of its last calibration alongside the result.
8. Carry the measured volume, not the nominal one, into every subsequent calculation.

Density, temperature and buoyancy

The conversion from mass to volume is where a gravimetric preparation can quietly reacquire the error it was meant to avoid. Density is a function of temperature, and the coefficient is not negligible at the accuracy a balance offers. Water falls from about 0.9982 g·mL⁻¹ at 20 °C to about 0.9970 at 25 °C — a shift of roughly 0.12 per cent, which is ten times the resolution of the weighing that preceded it. Volumetric glassware compounds the same problem from the other direction, since it is calibrated at a stated temperature, conventionally 20 °C, and is out of specification when used warm.

Solvent identity matters far more than temperature. Assuming the density of water for a non-aqueous solvent is not a small correction but a gross error: common laboratory organics range from below 0.8 to above 1.1 g·mL⁻¹, so the assumption can be wrong by twenty per cent or more in either direction. Mixed solvents are worse, because their densities are not the linear combination of the components. Look the value up for the actual solvent at the actual temperature, and record which value was used.

Buoyancy is the last term and the one most often skipped. A balance calibrated with stainless steel weights reports apparent mass in air, and a sample of much lower density displaces more air than the calibration weight did. For water-like materials the correction is around one part in a thousand — irrelevant at pipette accuracy, significant once the weighing is the accurate step. Apply it whenever the target uncertainty is tighter than about 0.1 per cent, and state that it was applied.

Propagation, and recording what was measured

Errors of different character combine by different rules, and treating them alike is the commonest mistake in a dilution series. Independent random errors add in quadrature: the combined relative uncertainty is the square root of the sum of the squares of the individual relative uncertainties, so five steps each carrying one per cent random error give about 2.2 per cent overall, not five. Systematic errors do not behave that way. A pipette biased one per cent low at every step biases the fifth point by about 5.1 per cent, in a known direction, and no amount of replication reveals it because every replicate is wrong identically 2.

That difference drives the choice of dilution architecture. A serial series is economical and gives even logarithmic spacing, but every point inherits every upstream error, so the points are correlated and the series has no internal check. Independent dilutions prepared separately from a single stock isolate the errors, cost more material and more pipetting, and produce points whose disagreement is informative. Where a concentration–response relationship is the output, independent preparation is worth its cost; where the requirement is only a wide dynamic range, serial preparation with an accurately characterised stock is defensible provided the correlation is acknowledged.

The record is what makes any of this recoverable. Write down the measured mass of solid and the balance used, the measured mass of solvent and the density and temperature applied to it, the pipette or balance behind every subsequent transfer, the number and architecture of dilution steps, and the derived concentration with its combined uncertainty. Quote uncertainty with its coverage factor so a reader knows whether the interval is a standard uncertainty or an expanded one 1. A concentration recorded without the measurements that produced it is a claim, not a result, and it cannot be audited once the vessel is empty.