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The core objective of a chemical composition measurement is to determine its true value. However, when measuring the composition of a macroscopic sample with a large number of atoms or molecules, realizing the true value of the measurand at both the macroscopic and microscopic levels remains an unsolved theoretical problem. We find that the true value of a sample composition exists in any subsample of a homogeneous molecular population of the sample. Here, we propose the Central Law of Measurement of the Amount of Substance: “The homogeneity of a sample molecular population represents the measurement accuracy of the sample composition in an analytical procedure”. The Central Law is based on a homogeneous molecular population axiom in which the molecular composition of a sample is identical for any homogeneous subsample. Furthermore, we point out that, at the microscopic scale, Avogadro’s law does not hold true.

The determination of a sample composition means the measurement of the amount of chemical elements. The International System of Quantities defines the quantity, the “amount of substance”, to be proportional to the number of specified elementary entities in a sample. The measurement of the amount of substance implies not only the determination of the size (number) of moles at the macroscopic scale but also the determination of the number of elementary entities at the microscope scale. In chemical composition measurement one’s concern focused on the uncertainty of the measured value for the reliability of measurement results [

Measurements of the amount of substance play a critical role in the understanding of a range of topics from particle physics to practical applications, such as biomedical testing, environmental monitoring, and quality control in industrial production. Each day, millions of such measurements are carried out worldwide, forming the bases for many important medical, environmental, economic, and legal decisions which rely on their accuracy [

In general, the true value of the measurand is elusive and unknown [

A central question for the measurement of the amount of substance is to search for the true value of a sample composition in the chemical measurement process at both the microscopic and macroscopic scales. In the context of the accurate determination of the Avogadro constant, we have formulated the amount of substance measurement homogeneity principle [^{−8}. Due to the limit of chemical measurement method and measurement instrument, the assessment limitation of homogeneity is not better than the sensitivity and resolution of a measurement method, and quantitative analytical chemistry to this level of accuracy was not readily available. Theoretically when a sample is chosen, if an analytical procedure does not disturb the sample composition and no measurement error occurs, then the true value can be determined.

By definition, for a homogeneous molecular population in a sample, we know that the molecular composition is identical for any homogeneous subsample. Based on the homogeneous molecular population axiom, if we mix all the molecules in a sample to afford a homogeneous molecular population, then any homogeneous subsample can represent the composition of the whole sample. Therefore, in order to realize the true value of the sample composition, we must maintain the molecular composition of the sample as unchanged and homogeneous in every analytical procedure. The homogeneous molecular population axiom is the theoretical basis of the Central Law. Accordingly, to measure the amount of substance, the measurement process must be performed such that the sample’s molecular population in each operational step is directly related to the true value of the measurand.

Starting from the homogeneous molecular population axiom for a sample―“the molecular composition of the sample is identical for any of its homogeneous subsamples”―we can conceive a homogenization thought experiment, as shown in

It is well known that matter consists of atoms and molecules. The measurement of the sample composition strives to determine the atomic and molecular constituents of a sample, i.e. to measure the atomic and molecular population(s). Therefore, the results of measuring the sample composition reflect the molecular population of the sample constituents. From the relationship between a sample molecular population and the measurement result of the sample composition, we can prove that the homogeneity of a sample’s molecular population represents the measurement accuracy of the sample composition in an analytical procedure. This is an important point for understanding accurate measurement of the amount of chemical elements on both the microscopic and macroscopic scales.

For clarity, some of the vocabulary and terminology used in this paper is explained here. According to the International Vocabulary of Metrology, “accuracy” is defined as the “closeness of agreement between a measured quantity value and a true quantity value of a measurand” [

the measurand) of the sample composition.

In order to prove the Central Law mathematically, the amount-of-substance fraction of a sample composition should be expressed and measured in terms of the number of microscopic elementary entities. Suppose a sample S contains all the possible, though not necessarily all, stable elemental substances in the periodic table of the elements [

the isotopic elements are identical, and the atom number of this element

where _{A} is the Avogadro constant.

According to the entry 2.11 of the VIM [

Hence,

As shown in equation (1), the true value of the amount-of-substance fraction of an analyte element is the ratio of the number of atoms of the analyte element to the total number of atoms of all elements.

When a chemical measurement procedure consists of a homogenization process of sample S, the molecular population of sample S will become homogeneous at the molecular level. For any subsample H of the homogeneous molecular population, the number of atoms of the element

where the subscript hmp stands for any homogeneous molecular population of the sample.

Due to the homogeneous molecular population axiom, in which the molecular composition of a sample is identical for any homogeneous subsample, we have

Hence,

Consequently, in a sample, the true value of the amount-of-substance fraction of an element exists in the entire molecular population of the sample as well as in any homogeneous molecular population of the sample. The true value of a sample composition is invariant for the homogeneous molecular population, as shown in equation (3). Then, the amount-of-substance fraction of the subsample H is equal to the amount-of-substance fraction of the original sample S, i.e. the true value of the sample composition.

Conversely, consider the chemical measurement process consisting of n steps: in the first step the amount-of-

substance fraction for the element

If we assume that

i.e. the true value of the amount-of-substance fraction

When a chemical measurement procedure leads to a change in the true value of the sample composition, it simultaneously changes the corresponding sample molecular population. In such an operation, if a measured value

of a measured sample M, where the atom number of the measured element

Combining equation (3) with equation (5), we arrive at

Equation (6) is a mathematical expression of the Central Law of Measurement of the Amount of Substance. The left hand side of the Equation (6) represents the measurement accuracy of the sample composition, and the right hand side of the Equation (6) represents the homogeneity of a sample molecular population. Therefore, the homogeneity of a sample molecular population represents the measurement accuracy of the sample composition. We thus answer Youden’s scientific question!

The Central Law of Measurement of the Amount of Substance tells us whether or not a sample is a homogeneous molecular population after the sample is chosen. To obtain the true value of a sample composition, we must measure all the molecules of the sample, or homogenize the sample at the molecular level and maintain this homogeneity in every measurement procedure. In this way, we can measure a sample composition with the highest accuracy at both the microscopic and macroscopic scales. Accordingly, an accurate chemical measurement process should also be a sample homogenization procedure. Successful homogenization depends on the related molecular properties and the pattern of the chemical conversion as well as the experimental capability and method.

By Equation (3) we know that for a chemical measurement process any change to the numerator or the denominator or the ratio between the numerator and the denominator will have an effect on the true value of a sample’s composition. Here, we notice that all influence factors and sources of uncertainty in a chemical composition measurement will lead the changes of the true value of a sample’s composition. That is why all the sources of uncertainty should be taken into account in the evaluation of overall uncertainty of the measurement results. Therefore, it is essential for obtaining reliable analytical results that the general demand on sampling or chemical preparation or instrument measurement is to maintain the true value of a sample composition, and Equation (3) is a theoretical basis of principle of sampling [

For a multistep analytical procedure, the degree of inhomogeneity of every step of the measurement process will become a sampling error in the next measurement step, and the result reflects the propagation of error over many analytical procedures. Furthermore, in chemical analysis the composition of a sample will be changed by the additional analyte from a preparative reaction process or by the analyte losses in a purifying extraction process. Therefore, the measured value of a sample composition will deviate from the true value of the sample composition. Thus, if we do not ensure a representative sampling, or if we do not obtain a homogeneous sample, then to obtain the true value of a sample composition we must measure the entire sample, i.e. all the molecules of the sample. However, the entire molecular population is rarely measured because of the cost or technical limitations. Accordingly, it is crucial to maintain the homogeneity of a sample’s molecular population. Additionally, the chemical measurement process may itself change the sample’s molecular population and thereby alter the true value of the composition of the original sample.

The impact of a chemical measurement process on the molecular population of the sample and the sample composition is examined as follows. The molecular population of a sample can be influenced in two ways for a chemical measurement process. The first artificially introduces additional analyte and matrix molecules, changing the original composition. Alternatively, some analyte and matrix components may be lost, which may also change the sample’s molecular population. In general, a chemical measurement procedure may include preparative reaction and purifying extraction processes, as well as detection and calibration steps, that can alter the composition of the original sample. For example, in a preparative reaction process, it is not only possible to change the sample’s molecular distribution, but the process may also be a source of additional analyte from the reagents or the environment. Consequently, these factors can change the population of the analyte molecules. However, the changes that result from contamination can be corrected for by accurately measuring the composition of the contaminant. Likewise, the changes from the recovery of the analyte can also be corrected for by accurately measuring the amount-of-substance fraction of the analyte loss in a purifying extraction process.

Let us assume that the number of atoms of the element

In the purifying extraction process, some loss of analyte may occur and alter the population of the analyte molecules. If the number of lost atoms of the element

The detection step involves a kinetic detection process and detector response determination. During the kinetic process, the population of the analyte molecules may further change: contaminants from other reagents and the environment may be present and there may be some loss of the analyte. To calibrate the detection results, we need a calibration sample that is homogeneous at the molecular level, and the detection process and conditions for the calibration sample should be the same as those used during sample assessment [

The Central Law of Measurement of the Amount of Substance provides the theoretical basis for seeking the true value of a sample composition in the development of new measurement methods and techniques.

The accurate measurement of the amount of substance ultimately involves the accurate measurement of the number of entities. A mole is defined as “the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of the isotope carbon 12” [^{23} [_{A} relates any quantity on the atomic scale to its corresponding macroscopic scale: a macroscopic quantity is N_{A} times its corresponding microscopic quantity [

To count the entities accurately, first one must correctly identify the entities to be counted. Even for larger en-

tities such as cells, it was reported that 20% of the published work using human cell lines was misidentified due to cell line cross-contamination [

In chemical analysis it is more difficult to correctly detect the sample molecular population for a complex sample. Matrices, for example, may affect the analyte molecular distribution of the sample during the kinetic process of detection, and may also interfere with the detector response [

The most widely used method for the measurement of the mole is to weigh pure material according to the equation

where n is the amount of substance, m is the mass of the pure material, and ^{−8} [^{17}. This is a significant number on the microscopic scale. If there was a change of less than 1 × 10^{16} atoms in the 1 kg silicon sphere, it would be impossible to identify the differences by weighing. Therefore, it is impossible to exactly count a microscopic number of atoms (with single-atom resolution) by a macroscopic weighing method.

Early in 1811, the Italian chemist Amedeo Avogadro stated that “equal volumes of gases at the same temperature and pressure contain the same number of molecules regardless of their chemical nature and physical properties”. This principle is known as Avogadro’s law or Avogadro’s hypothesis. Avogadro’s great contribution was to introduce the concept of molecules two hundred years ago. Approximately one hundred years later, the French physicist Jean Baptiste Perrin experimentally verified the existence of molecules. The concept of the mole originates in the “gram-molecule”, which refers to an amount of a compound equal to its molecular weight expressed in grams. Perrin explained the term “gram-molecule” as follows [

In fact, temperature and pressure are macroscopic descriptions of properties in thermal equilibrium. These thermodynamic state quantities are defined (and measurable) only at equilibrium [

where N_{a} is the number of particles of the system,

If N_{a} = 10^{23}, then the relative root-square fluctuation in the number of particles of the given system is on the order of 10^{−12} and, hence, negligible (small at macroscopic level). But the fluctuation in the number of particles of the given system is of the order of 10^{11} (big at microscopic level). This is to say for an ideal gas thermodynamic equilibrium system of a certain macroscopic temperature, pressure and volume with particle numbers 10^{23}, because the fluctuation in the number of particles is of the order of 10^{11}, the number of particles is not unique, it has a set of values. However, Avogadro’s law stated that equal volumes of gases at the same temperature and pressure contain the same number of molecules (i.e. the number of particles is unique). That’s not true. In a gas thermodynamic equilibrium system the macroscopic quantities such as temperature, pressure and volume cannot define the unique number of microscopic molecules. Actually, the number of particles has a set of values with equal volumes of gases at the same temperature and pressure. Due to fluctuations, the measurement of a change smaller than the fluctuations is impractical. Theoretically, based on a thermal equilibrium system, the uncertainties of the measurement results for the thermodynamic quantities are not better than their fluctuations.

Since Avogadro’s law is an experimental law, when

the microscopic molecules are in fact indistinguishable, according to the macroscopically measured values such as pressure, temperature, and volume, which exhibit uncertainties that are much higher than 1 × 10^{−12}. The measured number of molecules is meaningful only within the uncertainties of the measurement results of these macroscopic thermodynamic state quantities. It is impossible to define a unique number of molecules in a thermodynamic equilibrium system, in the case in which there are the thermodynamic fluctuations of the number of molecules. In summary, at the microscopic scale, Avogadro’s law does not hold.

Therefore, the Avogadro constant based on this law is not actually a physical constant, and the theoretical uncertainty of the number of molecules is no less than its fluctuation. Based on the 1971 definition of the mole and the 1980 supplement [

where n is the amount of substance, N is the number of specified elementary entities, and N_{A} is the Avogadro constant.

To ensure the accurate measurement of the amount of chemical elements, the definition and the method for realizing the true value of a sample composition should be consistent. The key issue is to measure the number of molecules of a sample composition at the microscopic scale. Unfortunately, in the measurement of the amount of substance, the results of a sample composition are usually achieved through a macroscopic measured quantity without ensuring the homogeneity of the sample molecular population. Moreover, the uncertainty of the macroscopic measured quantity is far from the requirement for an exact determination of the number of molecules.

Recognizing that matter is composed of atoms and molecules, a measurement result as displayed on a chemical detector reflects the molecular population reaching the detector at a given time. For an accurate measurement of the amount of substance at both the macroscopic and microscopic levels, the homogeneity of the sample at the molecular level is essential. The true value of a sample composition is preserved in any homogeneous molecular population of the sample as well as in the total molecular population of the sample. By its very nature, measurement should always aim for the true value of the measurand. To realize this objective, one must maintain the true value unchanged in every analytical procedure. In essence, the measurement of the amount of substance is not a simple problem of atomic or molecular counting, but whether or not one can sufficiently manipulate and control the large number of atoms and molecules to be measured. The true value of a sample composition can only be realized if we have the capability to correctly identify atoms individually. Without this capability, we can only obtain a best value of a sample composition with some statistical uncertainty. The homogeneous molecular population of a macroscopic sample inherently underlies the exact relationship between a certain number of molecules and a physical quantity of the molecular constituents, regardless of the number of molecules of a homogeneous subsample. Assuring the homogeneity of a sample’s molecular population is the microscopic mechanism by which to maintain the measurement accuracy of the sample composition. Manipulating and controlling the molecules of a sample are prerequisites to guarantee the realization of the true value of the sample composition. The Central Law of Measurement of the Amount of Substance is a fundamental law for the accurate measurement of the amount of substance at the macroscopic and microscopic levels.

The authors would like to thank Dr. Robert Wielgosz for inviting Hong Yi to present this work at CCQM/BIPM workshop on “The redefinition of the mole: A new era of chemical metrology” in April 2012. We are grateful to Prof. Dr. Paul De Bievre for his valuable comments on an earlier draft of this paper. All errors remain the authors’ responsibility.