# Review of the International Systems of Quantities and Units Usage

*Standards*)

## Abstract

**:**

## 1. Introduction

#### 1.1. The International System of Units

- The General Conference on Weights and Measures (Conférence Générale des Poids et Mesures, CGPM),
- The International Committee for Weights and Measures (Comité International des Poids et Mesures, CIPM) and,
- The International Bureau for Weights and Measures (Bureau International des Poids et Mesures, BIPM).

**S**ystème

**I**nternational d’Unités, SI [1]) with four additional “base units” to metre and kilogram: second for a time duration, ampere for electric current, kelvin for thermodynamic temperature, and candela for luminous intensity; 16 “derived units” with special names were accepted, too. The seventh base unit, mole for amount of substance was accepted by the CGPM in 1971. Six additional derived units with special names have been gradually adopted, bringing their number to 22. The SI base units have been redefined several times and the last time took place in 2018 following Feller’s proposal [2] of using “exact numerical values for seven defining constants expressed in terms of their SI units” (1, pp. 127–135); e.g., “the metre was redefined in terms of the speed of light, and the second was redefined based on the microwave frequency of a caesium atomic clock”. In the European Union (EU), the SI acceptance and adaptations are being led by the European Commission, and its directives are realized by member states.

#### 1.2. The International System of Quantities—ISQ

- General (2009)
- Mathematics
- Space and time
- Mechanics
- Thermodynamics
- Electromagnetism (2008)
- Light and radiation
- Acoustics (2020)
- Physical chemistry and molecular physics
- Atomic and nuclear physics
- Characteristic numbers
- Condensed matter physics
- Information science and technology (2008)

- Ensure that products and services are safe, reliable and of good quality;
- For business, they are strategic tools that reduce costs by minimizing waste, losses, and errors and increase productivity; they help companies to access new markets, level the playing field for developing countries, and facilitate free and fair global trade.

## 2. Quantities

#### 2.1. Quantity Names and Definitions

- The term “coefficient” should be used when two quantities A and B have different dimensions, e.g., linear expansion coefficient α
_{l}, dl/l = α_{l}dT—its unit is not equal to 1; sometimes, the term “modulus” is used instead of the term “coefficient”, e.g., modulus of elasticity E, σ = Eε. - The term “factor” should be used when the two quantities A and B have the same dimension, e.g., friction factor, μ, F = μF
_{n}; its unit is 1. - The adjective “specific” is added to the name of a quantity to indicate the quotient of that quantity by mass, e.g., specific heat capacity c, c = C/m; specific volume v, v = V/m
- Combination of quantities which occur in equations are often considered to constitute a new quantity called “parameter”, e.g., Grüneisen parameter γ, γ = α
_{V}/(kc_{V}ρ). - The quotient of two quantities of the same dimension is often called “ratio”, e.g., ratio of specific heat capacities γ, γ = c
_{p}/c_{V}; its unit is 1;- ○
- The term “index” is sometimes used instead of ratio, e.g., refractive index n, n = c
_{0}/c - ○
- The term “fraction” is used for ratios smaller than one, e.g., mass fraction of a substance X, w
_{X}, w_{X}= m_{X}/m; amount of substance fraction x_{X}, y_{X}, x_{X}= n_{X}/n; volume fraction φ_{X}, φ_{X}= x_{X}V_{m,X}/(Σ_{i}x_{X}V_{m,i}) (V_{m,X}being molar volume); their unit is 1;

- The term “percentage” shall not be used in a quantity name, because it is misleading; if a mass fraction is 0.78 (w = 78%), is the percentage than 78 or 78% = 0.78? Instead, the unambiguous term (volume, number, time, or amount of substance) “fraction” shall be used; the terms “mole fraction” and “share in %” are deprecated, too.
- The noun “density” is added to the name of a quantity to indicate the quotient of that quantity by the volume, e.g., mass density ρ, ρ = m/V; energy density e, e = E/V
- ○
- The noun “density” is also used to express a flux or current to indicate the quotient of such a quantity by the surface area, e.g., density of heat flow rate q, q = Φ/A
- ○
- The term “surface … density” is added to the name of a quantity to indicate the quotient of that quantity by the area, e.g., surface mass density ρ
_{A}, ρ_{A}= dm/dA - ○
- The term “linear … density” or the adjective “linear” is added to the name of a quantity to indicate the quotient of that quantity by the length, e.g., linear mass density or linear density ρ
_{l}, ρ_{l}= dm/dl - ○
- The term “linear” is also added to the name of a quantity, solely to distinguish between similar quantities, e.g., linear expansion coefficient α
_{l}, α_{l}= (1/l)(dl/dT), and cubic expansion coefficient α_{V}, α_{V}= (1/V)(dV/dT)

- The adjective “molar” is added to the name of a quantity to indicate the quotient of that quantity by the amount of substance, e.g., molar volume V
_{m}, V_{m}= V/n; molar mass M, M = m/n. - The term “concentration” is added to the name of a quantity to indicate the quotient of that quantity by the total volume, e.g., amount of substance (B) concentration c
_{B}, c_{B}= n_{B}/V; mass concentration ρ_{B}, ρ_{B}= m_{B}/V; number (particle or molecular) concentration C_{B}, C_{B}= N_{B}/V; in chemistry, the name “amount of substance concentration” can be abbreviated to “concentration” or replaced by specifying the substance, e.g., concentration of benzene, but the adjective “mass” should never be omitted from the name “mass concentration”. “Molarity” with a symbol M and the unit mol/L shall be avoided—use (amount of substance) concentration and the symbol c.

#### 2.2. Symbols for Quantities

_{p}(p: pressure), c

_{i}(i: running number), but μ

_{r}(r: relative), S

_{m}(m: molar). The e or exp as a base of natural logarithms is printed in roman. The same is true for the constant π (Ludolf number), and for mathematical symbols in expressions lg x, ln x, min(a, b), max f(x), lim f(x), sin x, sinh x, erf x, etc.

_{A}c

_{B}

^{2}. Division of one quantity by another is indicated in one of the following ways: $\frac{a}{b}$, a/b, a b

^{−1}, a · b

^{−1}; the solidus (/) can be substituted by a horizontal bar. Do not write ab

^{−1}, without a thin space between a and b

^{−1}, as ab

^{−1}could be misinterpreted as (ab)

^{−1}. Parentheses shall be used in more complex expressions, e.g., a/(b · c), (a · b)(c + d).

_{C}for cash flow, i for interest rate, I

_{FC}for fixed capital investment, N for number, R

_{IR}for internal return rate, S for sales, t

_{pb}for payback time, V

_{B}for book value, V

_{NP}for net present value (or worth), etc.

#### 2.3. Frequent Mistakes

_{Na}, or its amount fraction, x(Na) or x

_{Na}.

- Specification of quantities, e.g., chemical oxygen demand is a part of the name of a quantity, it is not a part of the unit (mass concentration of O
_{2}in mg/l, not mg O_{2}per litre). - “Amount (of substance)” is a chemical quantity expressed in moles; therefore, the word is not to be used for values in general (e.g., amount of heat)—be specific (use e.g., volume, mass, amount of substance, number, heat) or use magnitude, value, or contents as a general term.
- Velocity is a vector (rate of change of a position vector); speed is the magnitude of a velocity.

- Mass, m in kg, and weight F
_{g}in N; - Mass density or density, ρ or ρ
_{m}in kg/m^{3}, surface (mass) density, ρ_{A}in kg/m^{2}, linear (mass) density, ρ_{l}in kg/m, and density of heat flow rate, q or ϕ in W/m^{2}; - Mass flow rate, q
_{m}in kg/s, and mass flow,**j**_{m}in kg/(m^{2}s); - Work, W in J, energy, E in J, heat, Q in J, enthalpy H in J, and pressure, p in Pa;
- Power, P in W, heat flow rate, Φ in W, and enthalpy flow rate, I in W;
- Heat capacity, C in J/K, specific heat capacity, c in J/(kg K), specific heat capacity at constant pressure, c
_{p}in J/(kg K), and specific heat capacity at constant volume, c_{V}in J/(kg K); - Depreciation (decrease in value of a property over an estimated period of time, e.g., per year), and amortization (decrease in value when the period of time is definitely known).

- Find the right symbol by examining each Index of ISO 80000 parts 3–13 (hopefully, a new edition of ISO Standards Handbook [13] with a joint index will be published soon);
- Any attachment to a unit symbol as a means of giving information about the special nature of the quantity or context of measurement under consideration is not permitted, e.g., P
_{e}= 700 kW, not P = 700 kW_{e}; U_{max}= 500 V, not U = 500 V_{max}; - Do not use different symbols for the same quantity; when, for one quantity, different applications or different values are of interest, a distinction can be made by the use of subscripts, e.g., F
_{F}for an amount flow rate of feed to the distillation column, F_{B}for bottom product one, F_{D}for distillate flow rate; - Do not use two different symbols for the same quantity, e.g., A and S for area;
- Do not use the same symbol for two different quantities instead of two different symbols, e.g., γ for mass concentration (g/L), w for mass fraction, and ζ for mass ratio (e.g., in g/kg);
- Use N for number of entities (not n, which is used for the amount of substance in moles);
- Use q
_{V}for volume flow rate (in m^{3}/s), but q_{m}for the mass flow rate (in kg/s).

## 3. Units

#### 3.1. SI Base Units

#### 3.2. SI Derived Units

**v**= d

**r**/dt. The kilometre per hour is not a coherent derived unit in SI. Centimetre per second is also not a coherent derived unit in SI but it is a coherent derived unit in the CGS (centimetre, gram, second) system.

^{α}M

^{β}T

^{γ}I

^{δ}Θ

^{ε}N

^{ζ}J

^{η}, omitting any numerical factor, e.g., the dimension of force is denoted by dim F = LMT

^{−2}. In ISQ, dim ρ

_{B}= ML

^{−3}is the quantity dimension of mass concentration of component B, ρ

_{B}, and also the dimension of mass density, ρ. Quantities with the same dimension are named quantities of the same kind. When all exponents of the factors corresponding to a base quantity in its quantity dimension are zero, a quantity of dimension number (“dimensionless quantity”) is obtained.

^{2}, which is rather inconvenient. Therefore, special names and symbols have been approved for the 22 most common SI-derived units (Table 3)—the last four of them are admitted for reasons of safeguarding human health. Symbols for units consist of one or two letters from the Latin or Greek alphabet. These letters are lower case, except that the initial letter is a capital when the unit is derived from a proper name of a person, e.g., V for volt, A for ampere.

_{2}O, calories (cal) are deprecated.

- Those expressed in terms of base units, e.g., m
^{3}for volume, m/s for velocity or speed, kg/m^{3}for (mass) density or mass concentration, A/m for magnetic field strength, mol/m^{3}for the amount of substance concentration, cd/m^{2}for luminance, etc. - Those whose names and symbols include SI coherent derived units with special names and symbols, e.g., Pa s for dynamic viscosity, N m for the moment of force, rad/s for angular acceleration, W/m
^{2}for heat flux density or irradiance, J/(kg K) for specific heat capacity, J/m^{3}for energy density, J/mol for molar energy, J/(mol K) for molar entropy or molar heat capacity, Gy/s for absorbed dose rate, kat/m^{3}for catalytic activity concentration.

^{−3}, and microgram per kilogram, equal to 10

^{−9}.

^{−1}, m · s

^{−1}. Exponentiation has priority over multiplication and division. A solidus (/) shall not be followed by a multiplication sign or a division sign on the same line unless parentheses are inserted to avoid any ambiguity, e.g., J/ (mol K).

#### 3.3. SI Prefixes

^{3}= (10

^{−2}m)

^{3}= 10

^{−6}m

^{3}.

^{−9}m, not mμm. For historical reasons, the name of the base unit of mass, the kilogram, contains the SI prefix “kilo”. The names of its multiples are formed by adding the prefixes to the gram with the symbol g, e.g., milligram, symbol mg instead of microkilogram (μkg). The SI prefixes shall not be used to denote binary multiples, e.g., 1 kbit = 1 000 bit, but 1 Kibit = 1 024 bit (other prefixes for binary multiples are presented in ISO 80000-1, p. 8).

^{6}US dollars ($), 1 GSEK = 10

^{9}Swedish crowns.

_{F}= 12 Np = 12. Such special names and symbols may be used in expressions for derived units, e.g., angular velocity ω = 17 rad/s; attenuation coefficient α = 0.83 Np/m.

_{B}= 78%. Abbreviations such as ppm, ppb, and ppt are ambiguous and shall not be used; instead of them, alternative mass or volume or amount of substance fractions can be expressed in units such as μg/g = 10

^{−6}, ml/m

^{3}= 10

^{−9}, or pmol/mol = 10

^{−12}.

#### 3.4. Frequent Mistakes and Some Recommendations

- When using an Arabic symbol for a number, the symbol of a unit shall be used, not the spelt-out name, e.g., 20% (not 20 per cent), 15 min (not 15 min). Spelt out numbers and names (e.g., fifteen minutes) are not familiar in science and engineering.
- The unit symbol rpm is not an SI unit symbol for rotational frequency (its symbol is n)—use min
^{−1}; - The symbol for the unit degree Celsius is °C (Insert/Symbol/°), do not use the superscript 0, °C, or superscript o, °C;
- The symbol for the unit year is a, e.g., kt/a, not kt/yr or kt per year;
- A submultiple of 10
^{−6}metre is micrometre, symbol μm, not micron; - The SI multiple of the unit for mass can be kt (1000 kg), or Gg (10
^{9}g) but not million kg, Mkg; - The SI unit for volume fraction is mL/m
^{3}or μL/L (not ppm), μL/m^{3}or nL/L (not ppb); - The SI unit for amount fraction is μmol/mol (not ppm), and nmol/mol (not ppb);
- The SI unit of concentration is mol/L or its submultiple mmol/L (not milli-equivalent).

## 4. Quantity Values and Numbers

#### 4.1. Quantity Values

^{−7}m = 589.6 nm. If the point is used as the decimal sign, the cross and not the half-high dot shall be used as the multiplication sign between two numbers. If the comma is used as the decimal sign, as practised by the ISO standards [14], both the cross and the half-high dot may be used as the multiplication sign between numbers. In some cases multiplication signs may be omitted, e.g., 4c – 5d, 6ab, 7(a + b), 3 ln 2.

**–**

**r**, ≥72. Other mathematical signs and symbols are given in ISO 80000-2.

#### 4.2. Numbers

^{3}m; A = 80 mm × 25 mm, also in vector products and Cartesian products. The half-high dot shall be used to indicate a scalar product of vectors and is preferred for the multiplication of letter symbols. If the point is used as a decimal sign, the cross and not the half-high dot shall be used as a multiplication sign between numbers, e.g., 4 711.32 × 0.351 2. Division of one number by another is indicated in one of the following ways: $\text{}\frac{1}{2}$, 1/2, 1 2

^{−1}, 1 · 2

^{−1}.

^{3}to 401.000 × 10

^{3}. A standard uncertainty can be expressed in terms of the least significant digits in parentheses, e.g., l = 23.478 2(32) has the numerical value 23.478 2 and the standard uncertainty 32. The mathematical format 23.478 2 ± 0.003 2 shall be avoided as it does not contain all values between 23.481 4 and 23.475 0. Engineering tolerances 23.478 2 ± 0.003 2 contain the (upper and lower) limits and also all the values between them, symmetrically dispersed around the numerical value 23.478 2.

#### 4.3. Chemical Elements and Nuclides

^{14}N. The number of atoms of a nuclide in a molecule is shown in the right subscript position, e.g.,

^{14}N

_{2}; the number of atoms is equal to 1, if it is not indicated, e.g., H

_{2}O. The proton (atomic) number of a nuclide is shown in the left subscript position, e.g.,

_{64}Gd. The state of ionization and the state of electrical excitation are shown in the right superscript position, e.g., Na

^{+}, (PO

_{4})

^{3–}. The state of nuclear excitation is shown with the symbol * in the left superscript position, e.g.,

^{127}*Xe.

## 5. Some Other Frequent Mistakes

- A hyphen (-) shall not be used instead of minus (
**–**), or dash (–) in cases like Cl^{−}, m^{−1}; pH = 6–9, γ(Fe^{2+}) = 3–4 mg/L, July 8–12. - The dash (not the hyphen), used with number range has no spaces before and after it, e.g., 80–97%, 1.5–1.8 USD/m
^{3}, pp. 23–29. Spaces before and after each of them change their meanings from number range to minus. - A range of values can be expressed by units at both bounds, e.g., from 20 MW to 35 MW, or a dash must be used, 20–35 MW.

- Journal names should be abbreviated according to the List of Title Word Abbreviations, e.g., [15].
- The www-references have to contain the date of access—according to ISO 8601 standard the calendar date is represented by the form YYYY-MM-DD, e.g., 2021-02-08.
- Initials of author names shall be separated by a space, e.g., Akal Solmaz, S. K., etc.

- Differentiate I/we will and I/we shall (future time).
- Make sure the pages are numbered.
- Several of your sentences are not properly referenced. Please make sure you attribute or reference them.
- Please avoid reference overkill/run-on, i.e., do not use more than three references per sentence. If you need to use more, make sure you state the key relevant idea of each reference.
- Make sure your conclusions section underscores the scientific value added by your paper, and/or the applicability of your findings/results.
- Mark that there is no spacing between the announcing sentence before the bulleted lines and the first bulleted line.

## 6. Conclusions and Outlook

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Labelling of axes in a graph [1].

Term | Definition | Examples | |||
---|---|---|---|---|---|

Name | Symbol | Defining eq. | Unit | ||

coefficient | quotient of quantities with different dimension | linear expansion coefficient diffusion coefficient Hall coefficient | α_{l}D A _{H} | dl/l = α_{l} dTJ = –D ∇n E _{H} = A_{H}(B × J) | 1/K m ^{2}/sm ^{3}/C |

factor | quotient of quantities with the same dimension | friction factor quality factor thermal diffusion factor | μ Q α _{T} | F = μF_{n}X = QR α _{T} = k_{T}/(x_{A} x_{B}) | 1 1 1 |

specific | divided by mass | specific heat capacity specific volume | C v | c = C/m v = V/m | J/(kg K) m ^{3}/kg |

ratio, index | quotient of two quantities of the same dimension | heat capacity ratio mass ratio of B and A volume ratio of B and A refractive index | γ ζ ψ n | γ = c_{p}/c_{V} ζ = m _{B}/m_{A}ψ = V _{B}/V_{A})n = c _{0}/c | 1 1 1 1 |

fraction | ratio, smaller than one | mass fraction amount of substance fraction volume fraction | w_{X}x _{X}, y_{X}φ _{X} | w_{X} = m_{X}/Σmx _{X} = n_{X}/Σnφ _{X} = V_{X}/ΣV | 1 1 1 |

density | divided by volume | mass density | ρ | ρ = m/V | kg/m^{3} |

molar | divided by amount of substance | molar volume molar mass | V_{m}M | V_{m} = V/nM = m/n | m^{3}/molkg/mol |

concentration, mass concentration | divided by volume | concentration of B mass concentration of B | c_{B}ρ _{B} | c_{B} = n_{B}/Vρ _{B} = m_{B}/V | mol/m^{3}kg/m ^{3} |

**Table 2.**Seven International System of Quantities (ISQ) base quantities, their International System of Units (SI) base units, and quantity dimensions.

ISQ Base Quantity | SI Base Unit | Symbol for Dimension | ||
---|---|---|---|---|

Name | Symbol | Name | Symbol | |

length | l, L | metre | m | L |

mass | m | kilogram | kg | M |

time duration | t | second | s | T |

electric current | I, i | ampere | A | I |

thermodynamic temperature | T, Θ | kelvin | K | Θ |

amount of substance X | n(X) | mole | mol | N |

luminous intensity | I_{v} | candela | cd | J |

ISQ Derived Quantity | SI Derived Unit | |||
---|---|---|---|---|

Name | Symbol | Special Name | Special Symbol | In SI Base and Derived Units |

plane angle | α,β,γ | radian | rad | rad = m/m |

solid angle | Ω | steradian | sr | sr = m^{2}/m^{2} |

frequency | f,ν | hertz | Hz | Hz = s^{−1} |

force | F | newton | N | N = kg m/s^{2} |

pressure, stress | p | pascal | Pa | Pa = N/m^{2} |

energy | E | joule | J | J = N m |

power | P | watt | W | W = J/s |

electric charge | Q, q | coulomb | C | C = A s |

electric potential difference | V_{ab} | volt | V | V = W/A |

capacitance | C | farad | F | F = C/V |

electric resistance | R | ohm | Ω | Ω = V/A |

electric conductance | B | siemens | S | S = Ω^{−1} = A/V |

magnetic flux | Φ | weber | Wb | Wb = V s |

magnetic flux density | J_{m} | tesla | T | T = Wb/m^{2} |

inductance | L | henry | H | H = Wb/A |

Celsius temperature | t,δ | degree Celsius | °C | °C = K |

luminous flux | Φ_{v} | lumen | lm | lm = cd sr |

illuminance | E_{v} | lux | lx | lx = lm/m^{2} |

activity (of a radionuclide) | A | becquerel | Bq | Bq = s^{−1} |

absorbed dose | D | Gray | Gy | Gy = J/kg |

dose equivalent | H | sievert | Sv | Sv = J/kg |

catalytic activity | ξ | katal | kat | kat = mol/s |

Quantity | Unit | ||
---|---|---|---|

Name | Symbol | Definition | |

time | minute hour day | min h d | 1 min = 60 s 1 h = 60 min = 3600 s 1 d = 24 h = 86 400 s |

length | astronomical unit | au | 1 au = 149 597 870 700 m |

plane and phase angle | degree minute second | ° ′ ″ | 1° = (π/180) rad 1′ = (1/60)° = (π/10 800) rad 1″ = (1/60)′ = (π/648 000) rad |

area | hectare | ha | 1 ha = 1 hm^{2} = 10^{4} m^{2} |

volume | litre | l, L | 1 l = 1 dm^{3} = 10^{−3} m^{3} |

mass | tonne dalton | t Da | 1 t = 1 000 kg = 10^{−3} kg1 Da = 1.660 539 066(50) × 10 ^{−27} kg |

energy | electronnvolt | eV | 1 eV = 1.602 176 634 × 10^{−19} J |

level | neper bel | Np B | 1 Np = ln e = 1 1 B = (1/2) ln 10 Np ≈ 1.1151293 Np |

Factor | Prefix | Factor | Prefix | ||
---|---|---|---|---|---|

Name | Symbol | Name | Symbol | ||

10^{1} | deca | da | 10^{−1} | deci | d |

10^{2} | hecto | h | 10^{−2} | centi | c |

10^{3} | kilo | k | 10^{−3} | milli | m |

10^{6} | mega | M | 10^{−6} | micro | μ |

10^{9} | giga | G | 10^{−9} | nano | n |

10^{12} | tera | T | 10^{−12} | pico | p |

10^{15} | peta | P | 10^{−15} | femto | f |

10^{18} | exa | E | 10^{−18} | atto | a |

10^{21} | zetta | Z | 10^{−21} | zepto | z |

10^{24} | yotta | Y | 10^{−24} | yocto | y |

**Table 6.**Expressing values of quantities using recommended column headings [1].

p/kPa | v^{2}/(m/s)^{2} |
---|---|

48.73 | 94 766 |

72.87 | 94 771 |

135.42 | 94 784 |

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