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Quantities, symbols, and units

Description

In the context of quantities, units, and symbols, symbols are used as a code for physicochemical quantities. 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). When the context is expanded, the symbols have to be expanded too, including more detail to avoid confusion (Qth versus Qel; Nce versus NO2). Then symbols may appear confusingly complicated, loosing the function of sending their message quickly. There is no single best way to design the right symbol — all depends on context. 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 2020-06-03

Body mass - an example

The symbol and meaning of body mass of humans — in comparison with cell mass of platelets and 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? 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 X=B. Similarly, the platelet is a countable object X=ce, considering platelets as a special case of 'cells' ce without nucleus. Since mass m is an SI base quantity with SI base unit kilogram [kg], the consistent consequence is to use the symbol m for body mass mB [kg] and cell mass mce [kg]. Subscripts B and ce define the countable object type X. But are the terms 'body' and 'cell' with symbols B and ce actually signals for the same fundamental meaning? This question may not even pop up in our mind as being relevant. The relevance, however, to separate the code from the 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 in a particular example of handling chemical substances. 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 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 of matter 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 number of bodies hu. 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; in our explicit signal: 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 term 'body mass' is applied to the term 'PLT mass'. How do we find an acceptable solution?
The mass of bodies (=mass of countable objects, = mass of elementary entities mX [kg]), mhu and mPLT, must be distinguished from the mass per count of bodies, MNX [kg·x-1].


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. In consistencies in the use of symbols, however, have to be carefully and explicitly pointed out. Otherwise, the signal may be misunderstood, if the medium is the message, and the message is not translated into a 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.
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 nB [mol], electric charge Q [C], mass of sample s ms [kg], volume of B VB [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 NX with a value of NX = 1 in the SI format (but NX = 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 QB in the equation defining the charge number of B, zB,
zB = QB·e-1
In the SI, elementary charge e has the unit coulomb [C]. 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 QB 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 QB have the same SI units but different dimensions, both with equally negative consequences. In the explicit system, the meaning of QB [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 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 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
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
cell-count concentration Concentration Cce [x∙L­-1] Tab. 4; Cce = NceV-1; count concentration C versus amount concentration c; subscript indicates the entity X=ce, but does not signal 'per entity' ('per entity' can only mean 'per count of entity')
cell-mass concentration in chamber Concentration Cmce [kg∙L­-1] see Cms: Tab. 4; Cmce = mceV-1; upper case C alone signals 'count concentration' (CN would be more explicit), whereas the signal for 'mass concentration' is in the combination Cm


Per mass: mass-specific quantities

Count-, amount-, and charge-specific quantities

Per amount: molar quantities

Per count: elementary quantities

Term Symbol Unit Links and comments
elementary charge Elementary charge e [C·x-1 SI; e = QB·NB
electric charge per substance 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). 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, QO2 cannot be understood as charge per substance, 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
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


Per charge: .. quantities

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