Supplementary Material for A Question of Balance? Part III: Weighing Isn’t an Uncertain Process, Is It?

In Part I of this series (1), the effect of buoyancy was discussed. In this supplement to Part III, we look at this in more detail and review a weighing by difference method that compensates for both buoyancy and drift. The result is confirmed using a Monte Carlo Simulation approach (2).

Buoyancy corrections can become important when the density of the object being weighed is very different from the density of the masses employed for the calibration function. For example, this is true when accurate masses of water are needed for the calibration of volumetric glassware.

When a sample is weighed on a balance, the observed weight is less than the true weight (in vacuo) owing to buoyance effects. In its simplest form, there are three factors involved: the density of air, ρ_Air, the density of the calibration masses, ρ_Mass, and the density of the sample, ρ_Sam, being weighed. Of these, the density of air ρ_Air is the most difficult to determine. However, it has a generally accepted value under normal laboratory conditions of 0.0012g mL. Equation 1.1 gives the relationship required to calculate the weight in vacuo.

EQUATION 1.1:[1.1]

Let us suppose that we need to calibrate a 100 mL volumetric flask. We weigh the flask empty and then fill it to the mark with water at 20 °C, which has a known density of 0.99705 g/mL and reweigh and find a value of w_air of 99.814 g. Dividing w_airby0.99705 yields a volume of 100.109 mL. Looking good you say, but is this the correct value? Well, the balance was calibrated using traceable austenitic stainless-steel masses, which have a density of 8.00g cm³.

Employing equation 1.2, we can find the weight in vacuo compensated for buoyancy effects:

EQUATION 1.2: [1.2]

which yields a volume of 100.215 mL. This is a small but significant systematic error of 0.11% in terms of calibrating the flask.

Monte Carlo Simulation of the Weighing by Difference Method

The Davison-Possolo method, as described in the main article [QA1: text missing?].

To simulate the sample weight and its uncertainty, it is necessary to have an estimate of the weight variability of the containers used. Sample containers like these are manufactured in large quantities, and the standard deviation for a batch will be well characterized. For this example, let us assume it is 0.005 g.

We can calculate the weighing uncertainty of the container by combining the uncertainty of the balance 0.000387 (see equation 2 in main article) with standard deviation of the container weight, 0.005, as per equation 1.3.

EQUATION 1.3:[1.3]

The uncertainty of weighing is the same for all the containers, irrespective of weight or content, as is illustrated in Figure S1.

The uncertainty of the difference in the reference weighings is given by:

EQUATION 1.4[1.4]

The factor of two is required as there are two weighings.

Let us assume that we need to estimate the weight of a sample of low-density powder of 50 g. The mass in the reference container is selected as 25 g. This is approximately half the weight of the sample needed. We perform the four-step Davison weighing process (3) and record the values for w1 to w4 of 28.428 g, 3.436 g, 53.768 g, and 28.476 g, respectively. The difference between w1 and w4 is the drift and is equal to -0.048 g.

Therefore, the weight of the powder is given by:

EQUATION 1.5[1.5]

We now calculate the uncertainty for w_pdrbased upon the u_cont and u_Diff

EQUATION 1.6[1.6]

The factor of two is again required as there are two weighings of the sample container.

However, the use of these error budget equations does not readily convey the dynamic nature of the weighing process. Running a Monte Carlo Simulation (5) on the three parameters, w3, w2, and (w1-w4) together with their standard deviations from equations 1.3 and 1.4 clearly does and visually demonstrates the spread of the weight data at k=2 (95.45% confidence) and k=3 (99.73% confidence) from n equal to 500,000 iterations.

This is shown in Figure S2. Note particularly that it is not necessary to calculate equation 1.6, but the u_wpdr value found is the same within rounding error.

References

(1) Burgess, C.; McDowall, R. D. A Question of Balance? Part 2: Putting Principles into Practice. LCGC Europe 2006, 3 (9).
(2) Burgess, C. The Basics of Measurement Uncertainty in Pharmaceutical Analysis. Pharm. Technol. 2013, [QA2: Volume and page nos?]
(3) Davidson, S.; Perkin, M.; Buckley, M. The Measurement of Mass and Weight; NPL Measurement Good Practice Guide No. 71; National Physical Laboratory, 2004. https://www.npl.co.uk/special-pages/guides/gpg71_mass.aspx (accessed 2026-01-14).
(4) Possolo, A. Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results: Example E1, Weighing; NIST Technical Note 1900; National Institute of Standards and Technology, 2015.
(5) Minitab Workspace 1.3.0.0; 2024