Quantities, symbols, and units

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Quantities, symbols, and units


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 (Qth versus Qel; Nce versus NO2). 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 (Qel) 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 VO2? 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 NB of human bodies. The mass per human body was 64 kg divided by NB. 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 nB: "The definition applies to entities B which should always be indicated by a subscript or in parentheses, e.g. nB or n(B)." .. "The symbol [B] is often used for amount concentration of entities B." (Cohen 2008 IUPAC Green Book).
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 phrase "entities B", suggesting that X = B. Wheat remains unclear is the answer to a question which may appear strange at first: Does entity X define a single countable object, or does entity X define the types of countable objects, such as molecules of type B, cells of type fibroblast, organisms of type C. elegans, humans of type 'healthy control'? Just as human bodies can be defined as elementary entities, so can platelets or molecules be the object of a study and defined as elementary entities X. It is virtually a paradigm shift to step back from defining the entity in different contexts as X = B, X=ce, X = O2 and to separate the concept of the quantity "unit entity U" [x] from the specification of "elementary type X": the unit entity U of elementary type X is expressed by the symbol UX, just like the number of entities of type X is expressed by the symbol NX. Consider the unit entity U as a fundamental quantity with the meanig 'single countable object', appart from the specification of entity type X. Then 'cells' ce as a sample type of countable objects X=ce gives meaning to the unit entities Uce as the building blocks of the sample with a cell count Nce of units Uce in the sample. No more mixing up the concepts of 'elementary entity' in the sense of unit entity UX and elementary entity in the sense of elementary type X.
Elementary entity type Symbol Unit entity Count Mass or volume of X Mass or volume per entity
general elementary type X UX [x] NX [x] mX [kg] or VX [L] MUX [kg·x-1] or VUX [L·x-1]
(general) body B UB [x] NB [x] mB [kg] or VB [L] MUB [kg·x-1] or VUB [L·x-1]
human (body) hu Uhu [x] Nhu [x] mhu [kg] MUhu [kg·x-1]
cell ce Uce [x] Nce [x] Vce [L] VUce [L·x-1]
oxygen molecule O2 UO2 [x] NO2 [x] VO2 [L] VUO2 [L·x-1]
Considering platelets as a special case of 'cells' ce without nucleus, we have the unit entity Uce. 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 mB [kg], cell mass mce, and oxygen mass mO2 [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, if the previous section has not delivered the message: (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 mhu [kg], mPLT [kg], and mO2 [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' as used in the context of BMI, 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 mB 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 NB. Nhu is the number of human persons (in physics: number of human objects; in medicine: number of human subjects or patients; in our explicit symbol: Nhu). The meaning of the signal mPLT is much more open for debate: is it the mass of a sample of platelets, involving an experimental number NPLT 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 mX [kg]), mhu and mPLT and mO2 — must be distinguished from body mass with the meaning of mass per count of bodies,
Eq. 1: MUX = mX·NX-1 [kg·x-1]
Body volume VO2 [L] is a concept entirely different from volume per body,
Eq. 2: VUO2 = VO2·NO2-1 [L·x-1]
MUhu 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 unit body 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.

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) MUce

Size-specific quantities

Per volume: density and concentration

Term Symbol Unit Links and comments
concentration of B, amount Concentration cB = nB-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 = nO2-1 [mol∙L­-1] [O2]
concentration of X, count Concentration CX = NX-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 = NceV-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 = mceV-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 NA = NX·nX-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 zB = QB·e-1 is a constant. Therefore, the electrical format e is simply converted into the count format N, using QB = zB·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 amount: molar quantities

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. XNX.
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 MUce [kg∙x­-1] Tab. 5; Fig. 5; mass per cell count MUce; 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

MitoPedia concepts: MiP concept, MitoFit Quality Control System 

MitoPedia topics: BEC