Quantities, symbols, and units

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Quantities, symbols, and units

Description

In the context of quantities, symbols, and units, a code is required to convert terms defining physicochemical quantities into symbols (encoding) and to decode symbols as used in equations, text, and figures. Then symbols and abbreviations gain meaning. Simple symbols — such as Q or N — are used with different meanings depending on context (think of Q for heat and Q for electric charge; or N for number of cells and N for number of O₂ molecules). The context provides the code. When the context is extended, the symbols have to be expanded too, including more detail to avoid confusion (Q_th versus Q_el; N_ce versus N_O2). Then symbols may appear too complicated, loosing the function of sending their message quickly. There is no single best way to design the right symbol or to replace meaningful symbols (Q_el) by ambiguous abbreviations (Q) — all depends on context. We need to use the adequate medium (words, symbols, and abbreviations; equations, text, and figures; videos and slide presentations) and provide the code to achieve communication. The medium is the message, the message is the meaning — from Marshall McLuhan to Hofstadter.

When a code is familiar enough, it ceases appearing like a code; one forgets that there is a decoding mechanism. The message is identical with its meaning (Hofstadter 1979 Harvester Press).

Communicated by Gnaiger Erich 2020-06-04

What comes to mind from body mass to amount of substance

Using the symbol and meaning of body mass of humans in comparison with cell mass of platelets — or molecular mass of O₂ — takes an expression used in common language (body mass) into a different experimental context (cell mass, O₂ mass), with devastating consequences for the decoding of the seemingly similar messages. Diverting briefly from the quantity of mass m to the quantity of volume V: what is the meaning of V_O2? Is it the volume of O₂ in a bag filled up with pure O₂? Or — moving from 'body mass' to 'body volume' and taking O₂ as the 'body' — is it the volume of a single O₂ molecule? This question does not pop up when I take a measurement of my body mass and note that it is 64 kg. As explained below, a more generally consistent statement should be: the mass of a sample of humans was determined at 64 kg. The sample contained a number N_B of human bodies. The mass per human body was 64 kg divided by N_B. Since in this particular experiment the number of bodies was one, the mass was 64 kg per body. Thus body mass has the meaning of mass per body.

From the term 'body' connected with mass it is clear that neither the 'body and mind' debate nor 'body intimissimy' are in our mind. The physical body B is a countable object or elementary entity X. The SI explains entity X: "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). And IUPAC comments on amount of substance n_B: "The definition applies to entities B which should always be indicated by a subscript or in parentheses, e.g. n_B or n(B)." .. "The symbol [B] is often used for amount concentration of entities B."

These are messages elaborated by eminent international consortia. The messages are deceivingly simple — the message is the meaning. The term and symbol 'entity X used in the SI document may be connected to the IUPAC phrases using 'entities B', rather suggesting that X = B in the context of pure and applied chemistry. The argument for rejecting this decoding system is not intuitively clear, but provides a simple solution in the end. Just as a human body can be defined as an elementary entity, so can a platelet or molecule be the object of a study and defined as elementary entity X. It is virtually a paradigm shift to step back from defining the entity in different contexts as X = B, X=ce, X = O₂. Whereas B, ce, O₂. Consider entity X a s a fundamental qunatity with the meanig 'countable object', appart from the specification of entity type, then the symbols make a different sense: X_B, X_ce, X_O2. No more mixing up the concept of 'single elementary entity X and entity type.

Considering platelets as a special case of 'cells' ce without nucleus, we have the entity X_PLT. Since mass m is an SI base quantity with the SI base unit kilogram [kg], the consistent consequence is to use the symbol m for body mass m_B [kg], cell mass m_ce, and oxygen mass m_O2 [kg]. Subscripts B, ce, and O₂ define the countable object type X. But are the terms 'body', 'cell', and 'oxygen' with symbols B, ce, and O₂ actually signals for the same fundamental meaning? This question may not even pop up in our mind as being relevant. The relevance, however, to spot ambiguity in the code for gaining meaning is explained in two steps: (1) Change the term 'body' and symbol 'B' to the term 'human' and symbol 'hu'. In the same way, we may be more specific in terms of cell type, replacing ce by PLT (for platelets). And then comes 'substance': give B the message of O₂ as a chemical substance. Then we get m_hu [kg], m_PLT [kg], and m_O2 [kg]. Little has been gained, except for the realization that the term body with symbol B represents any type of countable object (human organism B, cell type B, substance B), whereas hu, PLT, and O₂ define more specifically the elementary entity X. And here comes perhaps a surprise. (2) In the term 'body mass' there is a hidden message: the concept 'mass consisting of type B' is interpreted implicitly in a more restricted sense, including information on the number of bodies concerned. Whereas mass m_B of type B has the restricted meaning of mass m [kg] of type B, body mass gives another fundamental message: meaning the mass of a single body hu, with the number of bodies equal to one. The number of bodies is a count N_B. N_hu is the number of human persons (in physics: number of human objects; in medicine: number of human subjects or patients; in our explicit symbol: N_hu). The meaning of the signal m_PLT is much more open for debate: is it the mass of a sample of platelets, involving an experimental number N_PLT of platelets? It cannot be assumed that in every context the concept 'body mass' is applied to decode the term 'PLT mass'. How do we find an acceptable solution?

Body mass — the mass of bodies (= mass of countable objects, = mass of elementary entities m_X [kg]), m_hu and m_PLT and m_O2 — must be distinguished from body mass with the meaning of mass per count of bodies,

Eq. 1: MNX = mX·NX-1 [kg·x-1]

Body volume V_O2 [L] is a concept entirely different from volume per body,

Eq. 2: VNO2 = VO2·NO2-1 [L·x-1]

M_Nhu is a quite complicated symbol for body mass as understood in the context of the body mass index (or body mass excess). With this context narrowly defined, however, it is practical to use the simple abbreviation M for body mass in the sense of 'mass per count of humans' [kg·x^-1], which is easily distinguished from the symbol m in the sense of 'mass of bodies' [kg].

Count and unit [x]

The following tables explain in detail the rationale of symbols used for extensive quantities, based on the International System of Units (SI), and specific quantities as used in 'Mitochondrial physiology' (BEC 2020.1). A system of units (SI) has to be consistent, whereas a system of symbols cannot be fully consistent without ignoring conventional definitions in various field of application. Inconsistencies in the use of symbols, however, have to be carefully and explicitly pointed out. Otherwise, the signal may be misunderstood, if the message is taken as an unintended meaning. There are some cases where the signal 'symbol' is more clear than the name of the corresponding quantity, such that the combined use of name and symbol adds to clarity. In all cases, adding the units to the names and symbols helps for clarification of the meaning and is frequently the shortest approach to consistency and provision of an adequate code.

Since units are of such fundamental importance for consistency of meaning, it should not be surprising that one of the biggest areas of confusion is the application of the quantity 'count', as a consequence of the lack of an explicit unit in the International System of Units (SI). In the SI the quantity 'count' is given the unit 1, which is not written. Then the units of extensive quantities amount of B n_B [mol], electric charge Q [C], mass of sample s m_s [kg], volume of B V_B [m³] or [L] are not different from the units of these quantities expressed per count. Quantity per count means quantity of X per single X. The term 'single X refers to a count N_X with a value of N_X = 1 in the SI format (but N_X = 1 x in the explicit format). Since the SI gives identical units to the extensive quantities and the 'per count quantities', a check for consistency is impossible on the basis of units. Compare the extensive quantity electric charge Q [C] with Q_B in the equation defining the charge number of B, z_B,

Eq. 3: zB = QB·e-1

In the SI, elementary charge e has the unit coulomb [C]. However, e does not have the dimension of electric charge, but electric charge per count (unit [C·x^-1] in the explicit system). The dimension of Q_B cannot be deduced from the units in the SI: Quantities relating to counting .. are just numbers (Bureau International des Poids et Mesures 2019 The International System of Units (SI) p. 151). As a consequence of the quantity 'count' given the meaning of 'just numbers' in the SI, count has neither a unit nor a dimension in the SI. Q and Q_B have the same SI units but different dimensions, both with equally negative consequences. In the explicit system, the meaning of Q_B [C·x^-1] is signalled in the units: it is a count-specific ('per count') quantity in contrast the extensive quantity Q [C]. Neither the name 'elementary charge' nor the SI unit [C] reveal the important meaning of this quantity, which is a universal constant declared by the SI on 2019-05-20. Add to 'elementary charge' the term 'per proton' and the counting unit [x] to the message, then the meaning is immediately clear — 'elementary' means 'per count:

Term	Symbol and definition	Unit
electric charge	Qel = Iel·t	[C] = [A·s]
elementary charge (per proton)	e = Qel·NH+-1	[C·x-1]
elementary charge per substance B	QB = Qel·NB-1	[C·x-1]
count of protons	NH+ = Qel·e-1	[x]
count of substance B	NB = Qel·QB-1	[x]
charge number per count of protons, elementary charge number of H+	zH+ = QH+·e-1 = 1	nondimensional
charge number per count B, elementary charge number of B	zB = QB·e-1 = NH+·NB-1	nondimensional

Extensive quantities

Term	Symbol	Unit	Links and comments
count of X	Count	NX	[x]	SI; see number of entities X
cell count	Count	Nce	[x]	number of cells. The symbol N contains the message 'count = number of' with the counting unit [x]. The subscript ce indicates the type of countable objects, X=ce. Importantly, the subscript ce does not contain the message number, it indicates only the type of countable entity. In other contexts, the symbol N may be used for 'pure' (nondimensional) numbers. To distinguish between these meanings, the symbol N should be used only for a dimensionless number, and the symbol NX for a count, i.e. for 'number of X'.
amount of substance X	Amount	nX or n(X)	[mol]	SI; amount n of X versus count N of X
electric charge	Electric charge	Qel	[C]	SI; Qel = Iel [A] · t [s]; Qel versus Qth
cell mass	Body mass	mce	[kg]	Tab. 5; Fig. 5; mass of cells m versus mass per cell (per cell count) MNce

Size-specific quantities

Per volume: density and concentration

Term	Symbol	Unit	Links and comments
concentration of B, amount	Concentration	cB = nB∙V­-1	[mol∙L­-1]	SI: amount of substance concentration [[[Cohen 2008 IUPAC Green Book |24]]]; the molar and count formats are distinguished as nB and NB, respectively.
concentration of O2, amount	Concentration	cO2 = nO2∙V­-1	[mol∙L­-1]	[O2]
concentration of X, count	Concentration	CX = NX∙V­-1	[x∙L-1]	(number concentration, IUPAC); the signal for count concentration is given by the upper case C in contrast to c for amount concentration. In both cases, the subscript X indicates the entity type, not to be confused with a number of entities.
cell-count concentration	Concentration	Cce	[x∙L­-1]	Cce = Nce∙V-1; count concentration C versus amount concentration c; subscript ce indicates the entity X=ce: concentration of ce. But it does not signal 'per entity' ('per cell' can only mean 'per count of cells').
cell-mass concentration in chamber	Concentration	Cmce	[kg∙L­-1]	see Cms; Cmce = mce∙V-1; upper case C alone would signal 'count concentration' (CN is more explicit), whereas the signal for 'mass concentration' is in the combination Cm.

Per mass: mass-specific quantities

Term	Symbol	Unit	Links and comments

Count-, amount-, and charge-specific quantities

Count, amount and charge are a group of 'numerical' quantities, expressing the count in the most elementary format N with counting units [x], whereas amount and charge are formats which can be converted to the count format by universal constants: (1) the Avogadro number N_A = N_X·n_X^-1 [x·mol^-1] for the amount format n,

Eq. 4 n to N : nB · NB·nB-1 = nB · NA = NB

(2) For any substance B, the charge number z_B = Q_B·e^-1 is a constant. Therefore, the electrical format e is simply converted into the count format N, using Q_B = z_B·e,

Eq. 5 e to N : Qel · NB·Qel-1 = nB · (zB·e)-1 = NB

The terms and symbols used in the above equations are explained in the following tables.

Per count: elementary quantities

Term	Symbol	Unit	Links and comments
elementary charge (per proton)	Elementary charge	e	[C·x-1	SI; e = QH+ = Qel·NH+-1
electric charge per count B	Electric charge	QB	[C·x-1]	QB is the electric charge per entity B; QO2 = 4 C·x-1. IUPAC does not define the symbol QB separately, but uses it in Section 2.13 in the definition of charge number, zB = QB·e-1; therefore, QB = zB·e. The symbol Q signals the extensive quantity Qel [C], whereas the subscript B in this case signals 'per count of B' (per NB). For the special case of the proton, B = H+, we get QH+ = zH+·e. By definition, zH+ = 1. Therefore, QH+ = e. A code is required to reveal the identical meaning of the two symbols QH+ and e. This causes confusion: Compare VO2 [L] which is the volume of O2 in a sample, where the subscript O2 contains the message of entity type X=O2. In contrast, QH+ and QO2 are charge per count of substance, which cannot be understood if the subscript O2 contains only the message of entity type X=O2 (as in VO2), but subscript O2 has the meaning of 'divided by NO2', confusing the symbol for an entity type X=O2 with the number of a single elementary entity, NX = NO2 = 1 x. X ≠ NX.
charge number per entity B	Charge number	zB	1	zB = QB·e-1 (IUPAC); zO2 = = QO2·e-1 = 4; IUPAC uses the term 'charge number of an ion' which should be changed to 'charge number per ion', or more clearly to 'charge number per ion number'. The symbol z carries the message 'number of elementary charges per number', and the subscript carries the message on the type of entity X.
cell mass, mass per cell	Body mass	MNce	[kg∙x­-1]	Tab. 5; Fig. 5; mass per cell count MNce; upper case M and subscript N signal 'per count', subscript ce signals the entity X=ce; in a context restricted to cells or molecules or a particular organism such as humans, the abbreviated symbol M [kg∙x­-1] provides a sufficiently informative signal, particularly in combination with the explicit unit.

Per charge: .. quantities

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