📄 chemistry

Resolving Ontological Circularity and Pedagogical Confusion Inherent in the 2019 Mole Definition by Recognizing the Elementary Amount and Adopting the Entity as the Appropriate Atomic-Scale Amount Unit

This paper proposes resolving the ontological circularity and pedagogical confusion of the 2019 mole definition by introducing the "elementary amount" ( $n_a$ ) as a fundamental constant and the "entity" (ent) as an atomic-scale unit, thereby establishing amount of substance as a distinct physical dimension independent of the mole.

Original authors: Brian Phillip Leonard

Published 2026-10-19

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Brian Phillip Leonard

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of chemistry, there is a fundamental question that has puzzled students and teachers for decades: what exactly is a "mole"? It is not a small animal, nor is it a unit of time. In science, it is a specific way of counting the invisible building blocks of matter, such as atoms and molecules. Because these particles are so incredibly tiny, scientists cannot count them one by one. Instead, they use a massive number to represent a specific amount of substance, much like a baker might use a "dozen" to represent twelve eggs. This concept is essential for mixing chemicals correctly, but the official definition of the mole, updated in 2019, has created a confusing loop. The definition relies on a number called the Avogadro constant, yet that constant is defined in terms of the mole itself. It is a circular explanation that leaves many wondering if the mole is a real physical quantity or just a fancy way of saying "a number." This confusion has made it difficult to teach the subject clearly and has sparked debates about whether the mole is a legitimate measurement or merely a dimensionless count.

A researcher named Brian Phillip Leonard has proposed a way to untangle this knot by changing how we think about the smallest possible amount of a substance. His paper suggests that the confusion stems from trying to define a large collection of particles without first defining the single particle itself. Leonard argues that just as we have a standard unit for the mass of a single atom, we need a standard unit for the amount of a single atom. He calls this the "elementary amount," which is the quantity of matter contained in exactly one entity, whether that entity is an atom, a molecule, or an ion. To make this idea practical, he proposes a new unit of measurement called the "entity." This unit represents the amount of a single particle, serving as the atomic-scale equivalent of the mole, just as the "dalton" serves as the atomic-scale equivalent of the gram.

By introducing the "entity" as a real unit with its own dimension, Leonard breaks the circular logic that has plagued the 2019 definition. Instead of saying a mole is defined by a mysterious constant, his approach defines the mole as a specific collection of these "entities." In this view, a mole is simply a very large pile of entities, specifically a pile containing 6.02214076 times 10 to the power of 23 of them. This makes the relationship between the microscopic world and the macroscopic world much clearer. The amount of a substance is no longer a vague "measure" of a count; it is a physical aggregate, just like the mass of a substance is an aggregate of the masses of its individual particles. If you have a bag of marbles, the total mass is the sum of the mass of each marble. Similarly, the total amount of a substance is the sum of the "entity" amount of each particle.

This shift in perspective resolves a long-standing philosophical debate about whether the mole is a dimensionless number or a physical quantity. Leonard demonstrates that without an underlying physical amount at the atomic scale, there is nothing to count. The "entity" unit ensures that the concept of "amount" remains a physical quantity with its own dimension, distinct from a simple number. This distinction is crucial for understanding how chemistry works across different scales. In the microscopic world, where we deal with individual particles, the "entity" is the natural unit to use. In the macroscopic world, where we weigh chemicals in a lab, the "mole" remains the natural unit. The new framework allows scientists to move seamlessly between these two worlds without losing the physical meaning of what they are measuring.

The paper also clarifies how other important chemical constants fit into this picture. For example, the "molar mass constant," which relates the mass of a substance to its amount, is shown to be exactly one dalton per entity. This is a much more intuitive relationship than the previous, more complex formulas that involved multiplying huge numbers by tiny ones. By using the "entity" unit, the math becomes a simple ratio of mass to amount, mirroring the way we understand mass in everyday life. This approach also simplifies the teaching of stoichiometry, the part of chemistry that deals with calculating how much of each substance is needed for a reaction. Instead of relying on confusing conversion factors that often lead students to treat the mole as a magic number, students can see the reaction as a direct exchange of entities. One entity of nitrogen reacts with three entities of hydrogen to produce two entities of ammonia, a relationship that holds true whether you are looking at a single reaction or a massive industrial batch.

Leonard's proposal does not change the actual numbers used in laboratories; the mole remains the same size, and the Avogadro constant remains the same value. What changes is the story we tell about what those numbers mean. The paper argues that the current definition creates unnecessary confusion by treating the mole as a primary concept rather than a derived one. By recognizing the "elementary amount" as the fundamental building block, the mole becomes a logical and understandable aggregate. This clarity helps dissolve the paradoxes that arise when trying to count discrete particles in a continuous world. It confirms that while we often treat large numbers of particles as smooth, continuous quantities for practical purposes, they are fundamentally made of individual, countable units.

The ultimate goal of this work is to bring clarity to a concept that is central to science education and practice. By adopting the "entity" as the appropriate unit for atomic-scale amounts, the paper suggests that we can finally stop treating the mole as a mysterious bridge between the seen and the unseen. Instead, it becomes a clear, physical connection, grounded in the reality that matter is made of individual parts. This approach respects the history of the mole while fixing the logical gaps in its modern definition. It offers a way forward where the confusion about whether the mole is a number or a quantity is replaced by a solid understanding that it is a specific amount of physical stuff, built up from the smallest possible unit of that stuff. In the end, the paper concludes with a simple, profound truth: without an actual amount of matter to begin with, there is nothing to count at all.

Technical Summary: Resolving Ontological Circularity and Pedagogical Confusion in the 2019 Mole Definition

Problem Statement
The paper identifies a persistent crisis in the understanding of the "mole concept" within STEM education, exacerbated by the 2019 redefinition of the SI base unit for amount of substance. The core issue is an apparent ontological circularity: the mole is defined in terms of the Avogadro constant ( $N_A$ ), yet $N_A$ is defined in terms of the mole. Furthermore, the SI Brochure's description of amount of substance ( $n$ ) as a "measure" of the number of entities ( $N$ ) has fueled a debate regarding whether "amount" is a legitimate physical dimension or merely a dimensionless count. This lack of clarity leads to pedagogical confusion, where students and educators struggle to distinguish between counting discrete entities and measuring a continuous macroscopic quantity, often conflating the mole with the Avogadro number or treating it as a dimensionless conversion factor.

Methodology
The author proposes a conceptual resolution by leveraging previously introduced concepts to break the circular reasoning:

Elementary Amount ( $n_a$ ): The paper adopts the concept of the elementary amount, previously introduced by the author in 2025, defined as the amount of a single atomic-scale entity of any substance. Analogous to the elementary charge ( $e$ ) or the atomic mass constant ( $\mu_u$ ), $n_a$ is posited as a physical constant of Nature with the dimension of amount ( $N$ ).
The Entity Unit (ent): The paper recommends the adoption of the atomic-scale amount unit, the "entity," which was previously introduced by the author in 2007. It is defined exactly as one elementary amount ( $1 \text{ ent} = n_a$ ). This parallels the dalton (Da), which is defined as one atomic mass constant ( $1 \text{ Da} = \mu_u$ ).

The methodology involves reformulating the relationships between number of entities ( $N$ ), amount of substance ( $n$ ), and mass ( $m$ ) using these established definitions. The author demonstrates that $N_A$ is fundamentally the reciprocal of the elementary amount ( $N_A = 1/n_a$ ), independent of any chosen macroscopic unit. The paper then reconstructs the definition of the mole and molar quantities (molar mass, gas constant, Faraday constant) based on this atomic-scale foundation, showing how macroscopic units emerge as aggregates of elementary amounts.

Key Contributions and Results

Resolution of Circularity: By defining $n_a$ as a fundamental constant, the Avogadro constant is uniquely defined as $N_A = 1/n_a$ . The mole is then defined as an aggregate of a specific number (the Avogadro number, $N_{Avo}$ ) of these elementary amounts: $1 \text{ mol} = N_{Avo} \cdot n_a$ . This breaks the circular logic where $N_A$ and the mole define each other.
Clarification of Dimensionality: The paper establishes that amount of substance is a distinct physical dimension ( $N$ ), not a dimensionless count. The relationship $n(X) = N(X) \cdot n_a$ (or $n(X) = N(X) \text{ ent}$ ) mirrors the mass relationship $m(X) = N(X) \cdot m_a(X)$ . This validates "amount" as an aggregate quantity, similar to mass, rather than a mere number.
The "Entity" Unit: The introduction of the entity (ent) provides a coherent atomic-scale unit for amount. This allows for the seamless transition between discrete and continuous regimes (resolving the Sorites paradox) by treating $n(X)$ as a continuous variable only when $N(X)$ is sufficiently large, while retaining the dimension $N$ even at the atomic scale.
Reformulation of Constants:
- Molar Mass Constant ( $M_u$ ): Defined exactly as $1 \text{ Da/ent}$ (dimension $M/N$ ). While practically equal to $1 \text{ g/mol}$ , the paper notes that due to the 2019 redefinition fixing $N_{Avo}$ but not the dalton exactly in terms of the kilogram, $M_u$ is not exactly $1 \text{ g/mol}$ theoretically, though the deviation is negligible for practical purposes.
- Other Constants: The molar gas constant ( $R$ ) is reinterpreted as $k/n_a$ (Boltzmann constant per elementary amount), and the Faraday constant ( $F$ ) as $e/n_a$ , providing more intuitive atomic-scale interpretations than the traditional products involving $N_A$ .
Stoichiometry: The paper demonstrates that stoichiometric relationships become more transparent when derived from the atomic-scale extent of reaction using the entity unit. The change in macroscopic amount is shown to be a direct scaling of the atomic-scale changes, avoiding the confusion often caused by "Factor-Label" conversion rules that treat the mole as a dimensionless number.

Significance and Claims
The paper claims that adopting the elementary amount and the entity unit resolves both the ontological debate regarding the dimension of "amount" and the pedagogical confusion surrounding the mole. By treating amount as an aggregate of elementary amounts (just as mass is an aggregate of entity masses), the distinction between "counting" and "measuring" is clarified: one cannot count without an underlying amount.

The author asserts that this framework:

Eliminates the circularity inherent in the current SI definition.
Provides a consistent dimensional analysis across the micro-to-macro spectrum, preventing the erroneous treatment of the mole as a dimensionless number.
Offers a more intuitive foundation for teaching stoichiometry and physical chemistry, allowing students to understand molar quantities as aggregates of entities rather than opaque conversion factors.
Suggests that the "Avogadro constant," as currently named and defined in the SI, is largely a derived concept that obscures the fundamental nature of the elementary amount; the true defining constant is the elementary amount ( $n_a$ ) itself.

The paper concludes that without an underlying physical amount, there is nothing to count, thereby defusing debates about the legitimacy of the mole as a unit of a physical quantity.

Technical Summary: Resolving Ontological Circularity and Pedagogical Confusion in the 2019 Mole Definition

More like this