Simple principles for metrology in chemistry: identifying and counting

We examine the problem of quantitative chemical measurement for well-identified substances, discuss the quantity called ‘amount of substance’, the means of expressing it, and its physical SI unit the mole. The everyday quantity which is a number of entities may be measured by the performance of two operations (identification and counting), the results of which may be communicated with two items of information (their name and the number of entities). We distinguish nominal, ordinal, interval and ratio scales of measurement and apply these to counting, referring to ordinal and cardinal numbers and Helmholtz’ analysis of measurement. Counting may be by direct serial numeration, direct parallel numeration, or comparative numeration. We discuss the limitations of serial numeration, the possibilities of parallel numeration, and the advantages of comparative numeration where a unit for counting in multiples (such as the analyst’s mole) may be used to define a scale on which equal numbers of objects correspond to equal values of some other physical quantity. We conclude that the numeration of very large numbers of objects is readily achieved but with unavoidable uncertainty, using operations which compare numbers of entities either to numbers of other entities or to some other quantity which accurately models numbers of entities.