8 Comments on Some Quantities and Their Units
8.1 Time and rotational frequency
The SI unit of time (actually time interval) is the second (s) and should be used in all technical
calculations. When time relates to calendar cycles, the minute (min), hour (h), and day (d) might be
necessary. For example, the kilometer per hour (km/h) is the usual unit for expressing vehicular speeds.
Although there is no universally accepted symbol for the year, Ref. [4: ISO 800003] suggests the
symbol a.
The rotational frequency n of a rotating body is defined to be the number of revolutions it makes in a time interval divided by that time interval [4: ISO 800003]. The SI unit of this quantity is thus the reciprocal second (s^{1}). However, as pointed out in Ref. [4: ISO 800003], the designations “revolutions per second” (r/s) and “revolutions per minute” (r/min) are widely used as units for rotational frequency in specifications on rotating machinery.
The SI unit of volume is the cubic meter (m^{3}) and may be used to express the volume of any substance, whether solid, liquid, or gas. The liter (L) is a special name for the cubic decimeter (dm^{3}), but the CGPM recommends that the liter not be used to give the results of high accuracy measurements of volumes [1, 2]. Also, it is not common practice to use the liter to express the volumes of solids nor to use multiples of the liter such as the kiloliter (kL) [see Sec. 6.2.8, and also Table 6, footnote (b)].
In science and technology, the weight of a body in a particular reference frame is defined as the
force that gives the body an acceleration equal to the local acceleration of free fall in that reference frame
[4: ISO 800004]. Thus the SI unit of the quantity weight defined in this way is the newton (N). When the
reference frame is a celestial object, Earth for example, the weight of a body is commonly called the local
force of gravity on the body.
Example: The local force of gravity on a copper sphere of mass 10 kg located on the surface of the
Earth, which is its weight at that location, is approximately 98 N.
Note: The local force of gravity on a body, that is, its weight, consists of the resultant of all the
gravitational forces acting on the body and the local centrifugal force due to the rotation of the
celestial object. The effect of atmospheric buoyancy is usually excluded, and thus the weight of a
body is generally the local force of gravity on the body in vacuum.
In commercial and everyday use, and especially in common parlance, weight is usually used as a
synonym for mass. Thus the SI unit of the quantity weight used in this sense is the kilogram (kg) and the
verb “to weigh” means “to determine the mass of” or “to have a mass of.”
Examples: the child’s weight is 23 kg the briefcase weighs 6 kg Net wt. 227 g
Inasmuch as NIST is a scientific and technical organization, the word “weight” used in the everyday
sense (that is, to mean mass) should appear only occasionally in NIST publications; the word “mass”
should be used instead. In any case, in order to avoid confusion, whenever the word “weight” is used, it
should be made clear which meaning is intended.
8.4 Relative atomic mass and relative molecular mass
The terms atomic weight and molecular weight are obsolete and thus should be avoided. They have
been replaced by the equivalent but preferred terms relative atomic mass, symbol A_{r}, and relative molecular
mass, symbol M_{r}, respectively [4: ISO 318], which better reflect their definitions. Similar to atomic weight
and molecular weight, relative atomic mass and relative molecular mass are quantities of dimension one
and are expressed simply as numbers. The definitions of these quantities are as follows [4: ISO 318]:
Relative atomic mass (formerly atomic weight): ratio of the average mass per atom of an element to 1/12
of the mass of the atom of the nuclide ^{12}C.
Relative molecular mass (formerly molecular weight): ratio of the average mass per molecule or specified
entity of a substance to 1/12 of the mass of an atom of the nuclide ^{12}C.
Examples: A_{r}(Si) = 28.0855 M_{r}(H_{2}) = 2.0159 A_{r}(^{12}C) = 12 exactly
Notes:
1. It follows from these definitions that if X denotes a specified atom or nuclide and B a specified molecule or entity (or more generally, a specified substance), then Ar(X) = m(X) / [m(12C) / 12] and Mr(B) = m(B) / [m(12C) / 12], where m(X) is the mass of X, m(B) is the mass of B, and m(12C) is the mass of an atom of the nuclide 12C. It should also be recognized that m(12C) / 12 = u, the unified atomic mass unit, which is approximately equal to 1.66 3 10^{27} kg [see Table 7, footnote (d)].
2. It follows from the examples and note 1 that the respective average masses of Si, H_{2}, and ^{12}C are m(Si) = A_{r}(Si) u, m (H2) = M_{r}(H_{2}) u, and m(^{12}C) = A_{r}(^{12}C) u.
3. In publications dealing with mass spectrometry, one often encounters statements such as “the masstocharge ratio is 15.” What is usually meant in this case is that the ratio of the nucleon number (that is, mass number—see Sec. 10.4.2) of the ion to its number of charges is 15. Thus masstocharge ratio is a quantity of dimension one, even though it is commonly denoted by the symbol m / z. For example, the masstocharge ratio of the ion ^{12}C_{7}^{1}H_{7}^{+ +} is 91/2 = 45.5.
8.5 Temperature interval and temperature difference
As discussed in Sec. 4.2.1.1, Celsius temperature (t) is defined in terms of thermodynamic
temperature (T) by the equation t = T  T_{0}, where T_{0} = 273.15 K by definition. This implies that the
numerical value of a given temperature interval or temperature difference whose value is expressed in the
unit degree Celsius (°C) is equal to the numerical value of the same interval or difference when its value is
expressed in the unit kelvin (K); or in the notation of Sec. 7.1, note 2, {Δt }°C = {ΔT}_{K}.
Thus temperature
intervals or temperature differences may be expressed in either the degree Celsius or the kelvin using the
same numerical value.
Example: The difference in temperature between the freezing point of gallium and the triple point of
water is Δt = 29.7546 °C = ΔT = 29.7546 K.
8.6 Amount of substance, concentration, molality, and the like
The following section discusses amount of substance, and the subsequent nine sections, which are based on Ref. [4: ISO 318] and which are succinctly summarized in Table 12, discuss quantities that are quotients involving amount of substance, volume, or mass. In the table and its associated sections, symbols for substances are shown as subscripts, for example, x_{B}, n_{B}, b_{B}. However, it is generally preferable to place symbols for substances and their states in parentheses immediately after the quantity symbol, for example n(H_{2}SO_{4}). (For a detailed discussion of the use of the SI in physical chemistry, see the book cited in Ref.[6], note 3.)
Quantity symbol: n (also v). SI unit: mole (mol).
Definition: See Sec. A.7.
Notes:
1. Amount of substance is one of the seven base quantities upon which the SI is founded (see
Sec. 4.1 and Table 1).
2. In general, n(xB) = n(B) / x, where x is a number.
Thus, for example, if the amount of substance of
H_{2}SO_{4} is 5 mol, the amount of substance of (1/3)H_{2}SO_{4} is 15 mol:
n[(1/3) H_{2} SO _{4}] = 3n(H_{2}SO_{4}).
Example: The relative atomic mass of a fluorine atom is A_{r}(F) = 18.9984. The relative molecular mass
of a fluorine molecule may therefore be taken as M_{r}(F_{2}) = 2A_{r}(F) = 37.9968. The molar mass
of F_{2} is then M(F_{2}) = 37.9968 × 10^{3} kg/mol = 37.9968 g/mol (see Sec. 8.6.4). The amount of
substance of, for example, 100 g of F_{2} is then n(F_{2}) = 100 g / (37.9968 g/mol) = 2.63 mol.
8.6.2 Mole fraction of B; amountofsubstance fraction of B
Quantity symbol: xB (also yB). SI unit: one (1) (amountofsubstance fraction is a quantity of
dimension one).
Definition: ratio of the amount of substance of B to the amount of substance of the mixture: xB = nB/n.
Amount of substance Symbol: n SI unit: mol 
Volume Symbol: V SI unit: m^{3} 
Mass Symbol: m SI unit: kg 


Amount of substance Symbol: n SI unit: mol 
amountofsubstance fraction SI unit: mol/mol = 1 
molar volume SI unit: m^{3}/mol 
molar mass SI unit: kg/mol 

Volume Symbol: V SI unit: m^{3} 
amountofsubstance concentration SI unit: mol/m^{3} 
volume fraction SI unit: 
mass density SI unit: kg/m^{3} 

Mass Symbol: m SI unit: kg 
molality SI unit: mol/kg 
specific volume SI unit: m^{3}/kg 
mass fraction 

Adapted from Canadian Metric Practice Guide (see Ref. [8], note 3; the book cited in Ref. [8], note 5, may also be consulted). 
1. This quantity is commonly called “mole fraction of B” but this Guide prefers the name “amount of substance fraction of B,” because it does not contain the name of the unit mole (compare kilogram fraction to mass fraction).
2. For a mixture composed of substances A, B, C, . . . , n_{A} + n_{B} + n_{C} + ...
3. A related quantity is amountofsubstance ratio of B (commonly called “mole ratio of solute B”), symbol rB. It is the ratio of the amount of substance of B to the amount of substance of the solvent substance: r_{B} = n_{B}/n_{S}. For a single solute C in a solvent substance (a onesolute solution), r_{C} = x_{C}/(1  x_{C}). This follows from the relations n = n_{C} + n_{S}, x_{C} = n_{C} / n, and r_{C} = n_{C} / n_{S}, where the solvent substance S can itself be a mixture.
Quantity symbol: V_{m}. SI unit: cubic meter per mole (m^{3}/mol).
Definition: volume of a substance divided by its amount of substance: V_{m} = V/n.
Notes:
1. The word “molar” means “divided by amount of substance.”
2. For a mixture, this term is often called “mean molar volume.”
3. The amagat should not be used to express molar volumes or reciprocal molar volumes. (One amagat is the molar volume V_{m} of a real gas at p = 101 325 Pa and T = 273.15 K and is approximately equal to 22.4 × 10^{3} m^{3}/mol. The name “amagat” is also given to 1/V_{m} of a real gas at p = 101 325 Pa and T = 273.15 K and in this case is approximately equal to 44.6 mol/m^{3}.) solvent substance S can itself be a mixture.
Quantity symbol: M. SI unit: kilogram per mole (kg/mol).
Definition: mass of a substance divided by its amount of substance: M = m/n.
Notes:
1. For a mixture, this term is often called “mean molar mass.”
2. The molar mass of a substance B of definite chemical composition is given by M(B) = M_{r}(B) × 10^{3} kg/mol = M_{r}(B) kg/kmol = M_{r} g/mol, where M_{r}(B) is the relative molecular mass of B (see Sec. 8.4). The molar mass of an atom or nuclide X is M(X) = A_{r}(X) × 10^{3} kg/mol = A_{r}(X) kg/kmol = A_{r}(X) g/mol, where A_{r}(X) is the relative atomic mass of X (see Sec. 8.4).
8.6.5 Concentration of B; amountofsubstance concentration of B
Quantity symbol: c_{B}. SI unit: mole per cubic meter (mol/m^{3}).
Definition: amount of substance of B divided by the volume of the mixture: c_{B} = n_{B}/V.
Notes:
1. This Guide prefers the name “amountofsubstance concentration of B” for this quantity because it is unambiguous. However, in practice, it is often shortened to amount concentration of B, or even simply to concentration of B. Unfortunately, this last form can cause confusion because there are several different “concentrations,” for example, mass concentration of B, ρB = m_{B}/V; and molecular concentration of B, C_{B} = N_{B}/V, where N_{B} is the number of molecules of B.
2. The term normality and the symbol N should no longer be used because they are obsolete. One should avoid writing, for example, “a 0.5 N solution of H_{2}SO_{4}” and write instead “a solution having an amountofsubstance concentration of c [(1/2)H_{2}SO_{4}]) = 0.5 mol/dm^{3}” (or 0.5 kmol/m^{3} or 0.5 mol/L since 1 mol/dm^{3} = 1 kmol/m^{3} = 1 mol/L).
3. The term molarity and the symbol M should no longer be used because they, too, are obsolete. One should use instead amountofsubstance concentration of B and such units as mol/dm^{3}, kmol/m^{3}, or mol/L. (A solution of, for example, 0.1 mol/dm^{3} was often called a 0.1 molar solution, denoted 0.1 M solution. The molarity of the solution was said to be 0.1 M.)
Quantity symbol: φB. SI unit: one (1) (volume fraction is a quantity of dimension one).
Definition: for a mixture of substances A, B, C, . . . ,
where x_{A}, x_{B}, x_{C}, . . . are the amountofsubstance fractions of A, B, C, . . ., V^{*}_{m,A} , V^{*} _{m,B} , V^{*} _{m,C} , . . . are the
molar volumes of the pure substances A, B, C, . . . at the same temperature and pressure, and where the
summation is over all the substances A, B, C, . . . so that Σx_{A} = 1.
Quantity symbol: ρ. SI unit: kilogram per cubic meter (kg/m^{3}).
Definition: mass of a substance divided by its volume: ρ = m / V.
Notes:
1. This Guide prefers the name “mass density” for this quantity because there are several different
“densities,” for example, number density of particles, n = N / V; and charge density, ρ = Q / V.
2. Mass density is the reciprocal of specific volume (see Sec. 8.6.9): ρ = 1 / ν.
Quantity symbol: b_{B} (also m_{B}). SI unit: mole per kilogram (mol/kg).
Definition: amount of substance of solute B in a solution divided by the mass of the solvent: b_{B} = n_{B} / m_{A}.
Note: The term molal and the symbol m should no longer be used because they are obsolete. One should
use instead the term molality of solute B and the unit mol/kg or an appropriate decimal multiple or
submultiple of this unit. (A solution having, for example, a molality of 1 mol/kg was often called a
1 molal solution, written 1 m solution.)
Quantity symbol: ν. SI unit: cubic meter per kilogram (m^{3}/kg).
Definition: volume of a substance divided by its mass: ν = V / m.
Note: Specific volume is the reciprocal of mass density (see Sec. 8.6.7): ν = 1 / ρ.
Quantity symbol: w_{B}. SI unit: one (1) (mass fraction is a quantity of dimension one).
Definition: mass of substance B divided by the mass of the mixture: wB_{B} = m_{B} / m.
8.7 Logarithmic quantities and units: level, neper, bel
This section briefly introduces logarithmic quantities and units. It is based on Ref. [5: IEC 600273],
which should be consulted for further details. Two of the most common logarithmic quantities are levelof
afieldquantity, symbol L_{F}, and levelofapowerquantity, symbol L_{P}; and two of the most common
logarithmic units are the units in which the values of these quantities are expressed: the neper, symbol Np,
or the bel, symbol B, and decimal multiples and submultiples of the neper and bel formed by attaching SI
prefixes to them, such as the millineper, symbol mNp (1 mNp = 0.001 Np), and the decibel, symbol dB
(1 dB = 0.1 B).
Levelofafieldquantity is defined by the relation L_{F} = ln(F/F_{0}), where F/F_{0} is the ratio of two
amplitudes of the same kind, F_{0} being a reference amplitude. Levelofapowerquantity is defined by the
relation L_{P} = (1/2) ln(P/P_{0}), where P/P_{0} is the ratio of two powers, P_{0} being a reference power. (Note that if
P/P_{0} = (F/F_{0})^{2}, then L_{P} = L_{F}.) Similar names, symbols, and definitions apply to levels based on other
quantities which are linear or quadratic functions of the amplitudes, respectively. In practice, the name of
the field quantity forms the name of L_{F} and the symbol F is replaced by the symbol of the field quantity.
For example, if the field quantity in question is electric field strength, symbol E, the name of the quantity is
“levelofelectricfieldstrength” and it is defined by the relation L_{E} = ln(E/E_{0}).
The difference between two levelsofafieldquantity (called “fieldlevel difference”) having the
same reference amplitude F_{0} is ΔL_{F} = L_{F1}  L_{F2}
= ln(F_{1}/F_{0})  ln(F_{2}/F_{0}) = ln(F_{1}/F_{2}),
and is independent of F_{0}. This is also the case for the difference between two levelsofapowerquantity (called “powerlevel
difference”) having the same reference power P_{0}: ΔL_{P1} = L_{P2}
= ln(P_{1}/P_{0})  ln(P_{2}/P_{0}) = ln(P_{1}/P_{2}).
It is clear from their definitions that both LF and LP are quantities of dimension one and thus have as
their units the unit one, symbol 1. However, in this case, which recalls the case of plane angle and the
radian (and solid angle and the steradian), it is convenient to give the unit one the special name “neper” or
“bel” and to define these socalled dimensionless units as follows:
One neper (1 Np) is the levelofafieldquantity when F/F_{0} = e, that is, when ln(F/F_{0}) = 1.
Equivalently, 1 Np is the levelofapowerquantity when P/P_{0} = e2, that is, when (1/2) ln(P/P_{0}) = 1. These
definitions imply that the numerical value of L_{F} when L_{F} is expressed in the unit neper is {L_{F}}_{Np} = ln(F/F_{0}),
and that the numerical value of L_{P} when L_{P} is expressed in the unit neper is {L_{P}}_{Np} = (1/2) ln(P/P_{0}); that is
L_{F} = ln(F/F_{0}) Np
L_{P} = (1/2) ln(P/P_{0}) Np.
One bel (1 B) is the levelofafieldquantity when that is, when 2 lg(F/F_{0})
= 1 (note that lg x = log_{10}x – see Sec. 10.1.2).
Equivalently, 1 B is the level ofapowerquantity when P/P_{0} = 10, that is,
when lg(P/P_{0}) = 1. These definitions imply that the numerical value of LF when LF is expressed in the unit
bel is {L_{F}}_{B} = 2 lg(F/F_{0}) and that the numerical value of LP when LP is expressed in the
unit bel is {L_{P}}_{B} = lg(P/P_{0}); that is
L_{F} = 2 lg(F/F_{0}) B = 20 lg(F/F_{0}) dB
L_{P} = lg(P/P_{0}) B = 10 lg(P/P_{0}) dB.
Since the value of L_{F} (or L_{P}) is independent of the unit used to express that value, one may equate LF
in the above expressions to obtain ln(F/F_{0}) Np = 2 lg(F/F_{0}) B, which implies
When reporting values of L_{F} and L_{P}, one must always give the reference level. According to Ref. 5:IEC 600273,
this may be done in one of two ways: L_{x} (re x_{ref}) or L _{x / x}ref where x is the quantity symbol for
the quantity whose level is being reported, for example, electric field strength E or sound pressure p, and
x_{ref} is the value of the reference quantity, for example, 1 μV/m for E_{0}, and 20 μPa for p_{0}. Thus
L_{E} (re 1 μV/m) =  0.58 Np or L_{E/(1 μV/m)} =  0.58 Np
means that the level of a certain electric field strength is 0.58 Np below the reference electric field strength
E_{0} = 1 μV/m. Similarly
L_{p} (re 20 μPa) = 25 dB or L_{p/(20 μPa)} = 25 dB
means that the level of a certain sound pressure is 25 dB above the reference pressure p_{0} = 20 μPa.
Notes:
1. When such data are presented in a table or in a figure, the following condensed notation may be used instead:  0.58 Np (1 μV/m); 25 dB (20 μPa).
2. When the same reference level applies repeatedly in a given context, it may be omitted if its value is clearly stated initially and if its planned omission is pointed out.
3. The rules of Ref. [5: IEC 600273] preclude, for example, the use of the symbol dBm to indicate a reference level of power of 1 mW. This restriction is based on the rule of Sec. 7.4, which does not permit attachments to unit symbols.
The proper SI units for expressing values of viscosity η (also called dynamic viscosity) and values of
kinematic viscosity ν are, respectively, the pascal second (Pa·s) and the meter squared per second (m^{2}/s)
(and their decimal multiples and submultiples as appropriate). The CGS units commonly used to express
values of these quantities, the poise (P) and the stoke (St), respectively [and their decimal submultiples the
centipoise (cP) and the centistoke (cSt)], are not to be used; see Sec. 5.3.1 and Table 10, which gives the
relations 1 P = 0.1 Pa·s and 1 St = 10^{4} m^{2}/s.
8.9 Massic, volumic, areic, lineic
Reference [4: ISO 310] has introduced the new adjectives “massic,” “volumic,” “areic,” and
“lineic” into the English language based on their French counterparts: “massique,” “volumique,”
“surfacique,” and “linéique.” They are convenient and NIST authors may wish to use them. They are
equivalent, respectively, to “specific,” “density,” “surface . . . density,” and “linear . . . density,” as
explained below.
(a) The adjective massic, or the adjective specific, is used to modify the name of a quantity to indicate the
quotient of that quantity and its associated mass.
Examples: massic volume or specific volume: ν = V / m
massic entropy or specific entropy: s = S / m
(b) The adjective volumic is used to modify the name of a quantity, or the term density is added to it, to
indicate the quotient of that quantity and its associated volume.
Examples: volumic mass or (mass) density: ρ = m / V
volumic number or number density: n = N / V
Note: Parentheses around a word means that the word is often omitted.
(c) The adjective areic is used to modify the name of a quantity, or the terms surface . . . density are added
to it, to indicate the quotient of that quantity (a scalar) and its associated surface area.
Examples: areic mass or surface (mass) density: ρ_{A} = m / A
areic charge or surface charge density: σ = Q / A
(d) The adjective lineic is used to modify the name of a quantity, or the terms linear . . . density are added
to it, to indicate the quotient of that quantity and its associated length.
Examples: lineic mass or linear (mass) density: ρ_{l} = m / l
lineic electric current or linear electric current density: A = I / b