Elementary entity

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Elementary entity

Description

An elementary entity U_X [x] of entity type X is a single unit of countable objects (X = molecules, cells, organisms, particles, parties) or events (X = beats, collisions, emissions, decays, celestial cycles, instances, occurences, parties). "An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles" (Bureau International des Poids et Mesures 2019).

If an object is defined as an assembly of particles (a party of two, a molecule as the assembly of a stoichiometric number of atoms), then the entity is the assembly but not the assembled particle. A number of defined elementary units U_X is a count, N_X = N·U_X [x], where N is only a number and as such N is dimensionless. The elementary unit U_X has the dimension U of the count N_X. The elementary unit U_X has the same unit [x] as the count N_X, or more accurately it gives the count the defining 'counting-unit' [x]. From the definition of count as the number (N) of elementary units (U) of entity type X, it follows that count divided by elementary unit is simply a number, N = N_X·U_X^-1. The unit u_N of a count can neither be the entity X nor a number. The elementary entity type X defines the identity X of the elementary unit U_X with the unit 'counting-unit' with symbol [x]. Since a count N_X is the number of elementary entities, the elementary entity U_X is not a count (U_X; it is not identical with N·U_X).

Abbreviation: U_X [x]

Reference: BEC2020.1 doi10.26124bec2020-0001.v1

Communicated by Gnaiger Erich 2020-07-10
in: Anastrophe XX Entity X and the elementary unit x of A X-mass Carol

Entity-type, elementary entity, and elementary unit

In Euclid's Elements (Book VII) a "unit" is defined as 'a single individual thing'. Take 'individual thing' as equal to elementary entity. The Euclidean unit is tightly conjugated with the individual thing. In diametrical contrast, the SI unit is entirely abstracted from any individual type of thing. The SI unit is entirely disjugated from the individual thing. Put 1 SI unit into an experimental system to work with a volume of 1 unit [m³] or run an experiment with a mass of 1 unit [kg], it may be a unit of anything. 1 kg is always 1 kg of any entity X. The entity-type X does not even have to be known to measure volume in units of meters cubed or mass in units of kg.

We see Euclid's unit in reflections with the SI's unit not merely in a looking glass, but in a kaleidoscopic interplay of Number, Unit, Count, Entity — NUCE (in nuce), which remains a CASE to this day (CASE represents the Counting-Assembling-Sorting Experience).

Entity-type X is distinguished from the elementary entity U_{N_X}. Dual-message code may lead to misunderstanding depending on context. For example, take O₂ as entity X. V_O₂ and m_O₂ are the volume [m³] and mass [kg], respectively, of a sample of pure O₂. For the volume, it is important to specify the gaseous state at a given temperature and pressure, whereas these specifications are irrelvant for the quantity mass. For volume and mass it needs to be ensured that there is a pure sample of oxygen (X = oxygen), but the specification of the elementary entity is irrelevant. The values of oxygen volume and oxygen mass remain independent of having in mind molecules of O₂, atoms of O, or charges of oxygen. Observe now the subtle transition from extensive quantities Q_u (V_O₂ and m_O₂) to elementary quantities Q_U (N_O₂ and n_O₂).

N_O₂ is the count of O₂ molecules. N_O₂ is equal to the number of elementary entities U_O₂. The elementary entity U_O₂ for the count of oxygen molecules is different from the elementary entity U_O for the count of oxygen atoms. U_O₂ and N_O₂ are expressed in the unit 'elementary unit' [x]. n_O₂ is the amount of O₂ expressed in the unit mole [mol]. n_O₂ is equal to N_A^-1 times the number of elementary entities U_O₂.

X = ce in 'cell mass' is normally understood as meaning "the mass of all entities of entity-type ce in a sample", m_ce [kg]. In contrast, X = body in the context of body mass and body mass index (BMI) is used in dual-message code, m_body. Body mass with dual-message code means the mass of all entities of entity-type human body, m_body [kg], divided by the number of bodies, N_body, which leads to the explicit canonical expression M_{U_body} = m_body·N_body^-1.

Colour code: Colour red may be used in X, indicating that the term entity or symbol X are used in dual-message code, meaning both entity-type and unit-entity. When specifying "single entity X" it is sufficiently clear that X is used in dual-message mode, and the term "number of X" is equally clear to meaning "number of unit-entities X", N_X = N·U_{N_X}^-1. Similarly, in the term "O₂ flow per cell" I_O₂/ce" it is sufficiently clear that ce is used in in dual-message code.

Base quantities and count

Quantity	Symbol for quantity Q	Symbol for dimension	Name of abstract unit u_Q	Symbol for unit u_Q [*]
elementary entity ^*,$	U_X	U	elementary unit	x
count ^*,$	N_X = N·U_X	X	elementary unit	x
amount of substance ^*,§	n_X = N_X·N_A^-1	N	mole	mol
charge ^*,€	Q_el = z_X·e·N_X	I·T	coulomb	C = A·s

length	l	L	meter	m
mass	m	M	kilogram	kg
time	t	T	second	s
electric current	I	I	ampere	A
thermodynamic temperature	T	Θ	kelvin	K
luminous intensity	I_v	J	candela	cd

[*] SI units, except for the canonical 'elementary unit' [x]. The following footnotes are canonical comments, related to iconic symbols.

^* For the elementary quantities N_X, n_X, and Q_el, the entity-type X of the elementary entity U_X has to be specified in the text and indicated by a subscript: n_O₂; N_ce; Q_el.

^$ Count N_X equals the number of elementary entities U_X. In the SI, the quantity 'count' is explicitly considered as an exception: "Each of the seven base quantities used in the SI is regarded as having its own dimension. .. All other quantities, with the exception of counts, are derived quantities" (Bureau International des Poids et Mesures 2019 The International System of Units (SI)). An elementary entity U_X is a material unit, it is not a count (U_X is not a number of U_X). N_X has the dimension X of a count and U_X has the dimension U of an elementary entity; both quantities have the same abstract unit, the 'elementary unit' [x].

^§ Amount n_X is an elementary quantity, converting the elementary unit [x] into the SI base unit mole [mol] using the Avogadro constant N_A.

^€ Charge is a derived SI quantity. Charge is an elementary quantity, converting the elementary unit [x] into coulombs [C] using the elementary charge e, or converting moles [mol] into coulombs [C] using the Faraday constant F. z_X is the charge number per elementary entity U_X, which is a constant for any defined elementary entity U_X. Q_el = z_X·F·n_X

References

Bioblast link	Reference	Year
Gnaiger 2020 BEC MitoPathways	Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5^th ed. Bioenerg Commun 2020.2. https://doi.org/10.26124/bec:2020-0002	2020
Gnaiger 2020 MitoFit x	Gnaiger E (2021) The elementary unit — canonical reviewer's comments on: Bureau International des Poids et Mesures (2019) The International System of Units (SI) 9th ed. https://doi.org/10.26124/mitofit:200004.v2	2021
BEC 2020.1 doi10.26124bec2020-0001.v1	Gnaiger E et al ― MitoEAGLE Task Group (2020) Mitochondrial physiology. Bioenerg Commun 2020.1. https://doi.org/10.26124/bec:2020-0001.v1	2020

Bioblast links: SI base units - >>>>>>> - Click on [Expand] or [Collapse] - >>>>>>>

Entity, count, and number, and SI base quantities / SI base units

Quantity name	Symbol	Unit name	Symbol	Comment
elementary	U_X	elementary unit	[x]	U_X, U_B; [x] not in SI
count	N_X	elementary unit	[x]	N_X, N_B; [x] not in SI
number	N	-	dimensionless	= N_X·U_X^-1
amount of substance	n_B	mole	[mol]	n_X, n_B
electric current	I	ampere	[A]	A = C·s^-1
time	t	second	[s]
length	l	meter	[m]	SI: metre
mass	m	kilogram	[kg]
thermodynamic temperature	T	kelvin	[K]
luminous intensity	I_V	candela	[cd]