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Difference between revisions of "Quantities, symbols, and units"

From Bioblast
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:::: 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''<sub>B</sub> [mol], electric charge ''Q'' [C], mass of sample s ''m''<sub>s</sub> [kg], volume of B ''V''<sub>B</sub> [m<sup>3</sup>] 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''<sub>''X''</sub> with a value of ''N''<sub>''X''</sub> = 1 in the SI format (but ''N''<sub>''X''</sub> = 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''<sub>B</sub> in the equation defining the charge number of B, ''z''<sub>B</sub>,
:::: 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''<sub>B</sub> [mol], electric charge ''Q'' [C], mass of sample s ''m''<sub>s</sub> [kg], volume of B ''V''<sub>B</sub> [m<sup>3</sup>] 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''<sub>''X''</sub> with a value of ''N''<sub>''X''</sub> = 1 in the SI format (but ''N''<sub>''X''</sub> = 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''<sub>B</sub> in the equation defining the charge number of B, ''z''<sub>B</sub>,
  ''z''<sub>B</sub> = ''Q''<sub>B</sub>·''e''<sup>-1</sup>
  ''z''<sub>B</sub> = ''Q''<sub>B</sub>·''e''<sup>-1</sup>
:::: 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<sup>-1</sup>] in the explicit system). The dimension of ''Q''<sub>B</sub> 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''<sub>B</sub> have the same SI units but different dimensions, both with equally negative consequences. In the explicit system, the meaning of ''Q''<sub>B</sub> [C·x<sup>-1</sup>] 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 elementary charge ''e'' = ''Q''<sub>B</sub>·''N''<sub>B</sub> [C·x<sup>-1</sup>].
:::: 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<sup>-1</sup>] in the explicit system). The dimension of ''Q''<sub>B</sub> 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''<sub>B</sub> have the same SI units but different dimensions, both with equally negative consequences. In the explicit system, the meaning of ''Q''<sub>B</sub> [C·x<sup>-1</sup>] 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:
:::: {| class="wikitable"
|-
! Term !! Symbol and definition !! Unit
|-
| electric charge || ''Q''<sub>el</sub> = ''I''<sub>el</sub>·''t'' || [C] = [A·s]
|-
| elementary charge || ''e'' = ''Q''<sub>el</sub>·''N''<sub>H<sup>+</sup></sub><sup>-1</sup> || [C·x<sup>-1</sup>]
|-
| elementary charge per substance B || ''Q''<sub>B</sub> = ''Q''<sub>el</sub>·''N''<sub>B</sub><sup>-1</sup> || [C·x<sup>-1</sup>]
|-
| count of protons || ''N''<sub>H<sup>+</sup></sub> = ''Q''<sub>el</sub>·''e''<sup>-1</sup> || [x]
|-
| count of substance B || ''N''<sub>B</sub> = ''Q''<sub>el</sub>·''Q''<sub>B</sub><sup>-1</sup> || [x]
|-
| charge number per count of protons, elementary charge number of H<sup>+</sup> || ''z''<sub>H<sup>+</sup></sub> = ''Q''<sub>H<sup>+</sup></sub>·''e''<sup>-1</sup> = 1 || nondimensional
|-
| charge number per count B, elementary charge number of B || ''z''<sub>B</sub> = ''Q''<sub>B</sub>·''e''<sup>-1</sup> = ''N''<sub>H<sup>+</sup></sub>·''N''<sub>B</sub><sup>-1</sup> || nondimensional
|-
|}
</div>
</div>
<br />




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:::: {| class="wikitable"
:::: {| class="wikitable"
|-
! Term !! Symbol !! Unit !! Links and comments
|-
|-
| cell count || [[Count]] || ''N''<sub>ce</sub> || [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 ''N''<sub>''X''</sub> for a count, ''i.e.'' for 'number of ''X'''.
| cell count || [[Count]] || ''N''<sub>ce</sub> || [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 ''N''<sub>''X''</sub> for a count, ''i.e.'' for 'number of ''X'''.
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:::: {| class="wikitable"
:::: {| class="wikitable"
|-
! Term !! Symbol !! Unit !! Links and comments
|-
|-
| elementary charge || [[Elementary charge]] || ''e'' || [C·x<sup>-1</sup>|| SI; ''e'' = ''Q''<sub>B</sub>·''N''<sub>B</sub>
| elementary charge || [[Elementary charge]] || ''e'' || [C·x<sup>-1</sup>|| SI; ''e'' = ''Q''<sub>B</sub>·''N''<sub>B</sub>
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:::: {| class="wikitable"
:::: {| class="wikitable"
|-
! Term !! Symbol !! Unit !! Links and comments
|-
|-
| cell-count concentration || [[Concentration]] || ''C''<sub>ce</sub> || [x∙L­<sup>-1</sup>] || Tab. 4; ''C''<sub>ce</sub> = ''N''<sub>ce</sub>∙''V''<sup>-1</sup>; 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-count concentration || [[Concentration]] || ''C''<sub>ce</sub> || [x∙L­<sup>-1</sup>] || Tab. 4; ''C''<sub>ce</sub> = ''N''<sub>ce</sub>∙''V''<sup>-1</sup>; 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')

Revision as of 16:32, 3 June 2020


high-resolution terminology - matching measurements at high-resolution


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 or N for number of O2 molecules). When the context is expanded, the symbols have to be expanded, 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
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



Elementary quantities: per count

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 voume: 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



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