Guide for the Use of the International System of Units (SI)

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 80000-3] 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 80000-3]. The SI unit of this quantity is thus the reciprocal second (s-1). However, as pointed out in Ref. [4: ISO 80000-3], 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.

8.2 Volume

The SI unit of volume is the cubic meter (m3) 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 (dm3), 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)].

8.3 Weight

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 80000-4]. 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 Ar, and relative molecular mass, symbol Mr, respectively [4: ISO 31-8], 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 31-8]:

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 12C.

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 12C.

Examples: Ar(Si) = 28.0855 Mr(H2) = 2.0159 Ar(12C) = 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, H2, and 12C are m(Si) = Ar(Si) u, m (H2) = Mr(H2) u, and m(12C) = Ar(12C) u.

    3. In publications dealing with mass spectrometry, one often encounters statements such as “the mass-to-charge 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 mass-to-charge ratio is a quantity of dimension one, even though it is commonly denoted by the symbol m / z. For example, the mass-to-charge ratio of the ion 12C71H7+ + 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 - T0, where T0 = 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 31-8] 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, xB, nB, bB. However, it is generally preferable to place symbols for substances and their states in parentheses immediately after the quantity symbol, for example n(H2SO4). (For a detailed discussion of the use of the SI in physical chemistry, see the book cited in Ref.[6], note 3.)

8.6.1 Amount of substance

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 H2SO4 is 5 mol, the amount of substance of (1/3)H2SO4 is 15 mol: n[(1/3) H2 SO 4] = 3n(H2SO4).

Example: The relative atomic mass of a fluorine atom is Ar(F) = 18.9984. The relative molecular mass of a fluorine molecule may therefore be taken as Mr(F2) = 2Ar(F) = 37.9968. The molar mass of F2 is then M(F2) = 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 F2 is then n(F2) = 100 g / (37.9968 g/mol) = 2.63 mol.

8.6.2 Mole fraction of B; amount-of-substance fraction of B

Quantity symbol: xB (also yB).       SI unit: one (1) (amount-of-substance 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.

Table 12. Summary description of nine quantities that are quotients involving amount of substance, volume, or mass

  Quantity in Images/numerator
Amount of substance

Symbol:   n

SI unit:   mol

Volume

Symbol:   V

SI unit:   m3

Mass

Symbol:   m

SI unit:   kg

Quantity in Denominator Amount of substance

Symbol:   n

SI unit:   mol

amount-of-substance
fraction
$ x_{\rm B} = \frac{n_{\rm B}}{n} $
SI unit: mol/mol = 1
molar volume
$V_{\rm m} = \frac{V}{n} $
SI unit: m3/mol
molar mass
$ M = \frac{m}{n} $
SI unit: kg/mol
Volume

Symbol:   V

SI unit:   m3

amount-of-substance
concentration
$ c_{\rm B} = \frac{n_{\rm B}}{V} $
SI unit: mol/m3
volume fraction
$\varphi_{\rm B} = \frac{x_{\rm B} V_{\rm m,B}^* }{\Sigma x_{\rm A} V_{\rm m,A}^*}$
SI unit:   m3/m3 = 1
mass density
$ \rho = \frac{m}{V}$
SI unit:   kg/m3
Mass

Symbol:   m

SI unit:   kg

molality
$ b_{\rm B} = \frac{n_{\rm B}}{m_{\rm A}}$
SI unit: mol/kg
specific volume
$v = \frac{V}{m}$
SI unit:   m3/kg
mass fraction
$v = \frac{V}{m}$
SI unit:   kg/kg = 1

Adapted from Canadian Metric Practice Guide (see Ref. [8], note 3; the book cited in Ref. [8], note 5, may also be consulted).



Notes:

    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, . . . , nA + nB + nC + ... $$\equiv \sum_{\rm A} n_{\rm A}$$

    3. A related quantity is amount-of-substance 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: rB = nB/nS. For a single solute C in a solvent substance (a one-solute solution), rC = xC/(1 - xC). This follows from the relations n = nC + nS, xC = nC / n, and rC = nC / nS, where the solvent substance S can itself be a mixture.

8.6.3 Molar volume

Quantity symbol: Vm.       SI unit: cubic meter per mole (m3/mol).

Definition: volume of a substance divided by its amount of substance: Vm = 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 Vm of a real gas at p = 101 325 Pa and T = 273.15 K and is approximately equal to 22.4 × 10-3 m3/mol. The name “amagat” is also given to 1/Vm 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/m3.) solvent substance S can itself be a mixture.

8.6.4 Molar mass

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) = Mr(B) × 10-3 kg/mol = Mr(B) kg/kmol = Mr g/mol, where Mr(B) is the relative molecular mass of B (see Sec. 8.4). The molar mass of an atom or nuclide X is M(X) = Ar(X) × 10-3 kg/mol = Ar(X) kg/kmol = Ar(X) g/mol, where Ar(X) is the relative atomic mass of X (see Sec. 8.4).

8.6.5 Concentration of B; amount-of-substance concentration of B

Quantity symbol: cB.       SI unit: mole per cubic meter (mol/m3).

Definition: amount of substance of B divided by the volume of the mixture: cB = nB/V.

Notes:

    1. This Guide prefers the name “amount-of-substance 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 = mB/V; and molecular concentration of B, CB = NB/V, where NB 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 H2SO4” and write instead “a solution having an amount-of-substance concentration of c [(1/2)H2SO4]) = 0.5 mol/dm3” (or 0.5 kmol/m3 or 0.5 mol/L since 1 mol/dm3 = 1 kmol/m3 = 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 amount-of-substance concentration of B and such units as mol/dm3, kmol/m3, or mol/L. (A solution of, for example, 0.1 mol/dm3 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.)

8.6.6 Volume fraction of B

Quantity symbol: φB.       SI unit: one (1) (volume fraction is a quantity of dimension one).

Definition: for a mixture of substances A, B, C, . . . ,

print -r '$\varphi_{\rm B} = x_{\rm B} V_{\rm m,B}^* /\sum x_{\rm A} V_{\rm m,A}^*$' | t2g 130

where xA, xB, xC, . . . are the amount-of-substance 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 ΣxA = 1.

8.6.7 Mass density; density

Quantity symbol: ρ.       SI unit: kilogram per cubic meter (kg/m3).

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.

8.6.8 Molality of solute B

Quantity symbol: bB (also mB).       SI unit: mole per kilogram (mol/kg).

Definition: amount of substance of solute B in a solution divided by the mass of the solvent: bB = nB / mA.

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.)

8.6.9 Specific volume

Quantity symbol: ν.      SI unit: cubic meter per kilogram (m3/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.

8.6.10 Mass fraction of B

Quantity symbol: wB.      SI unit: one (1) (mass fraction is a quantity of dimension one).

Definition: mass of substance B divided by the mass of the mixture: wBB = mB / 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 60027-3], which should be consulted for further details. Two of the most common logarithmic quantities are level-of a-field-quantity, symbol LF, and level-of-a-power-quantity, symbol LP; 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).

Level-of-a-field-quantity is defined by the relation LF = ln(F/F0), where F/F0 is the ratio of two amplitudes of the same kind, F0 being a reference amplitude. Level-of-a-power-quantity is defined by the relation LP = (1/2) ln(P/P0), where P/P0 is the ratio of two powers, P0 being a reference power. (Note that if P/P0 = (F/F0)2, then LP = LF.) 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 LF 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 “level-of-electric-field-strength” and it is defined by the relation LE = ln(E/E0).

The difference between two levels-of-a-field-quantity (called “field-level difference”) having the same reference amplitude F0 is ΔLF = LF1 - LF2 = ln(F1/F0) - ln(F2/F0) = ln(F1/F2), and is independent of F0. This is also the case for the difference between two levels-of-a-power-quantity (called “power-level difference”) having the same reference power P0: ΔLP1 = LP2 = ln(P1/P0) - ln(P2/P0) = ln(P1/P2).

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 so-called dimensionless units as follows:

One neper (1 Np) is the level-of-a-field-quantity when F/F0 = e, that is, when ln(F/F0) = 1. Equivalently, 1 Np is the level-of-a-power-quantity when P/P0 = e2, that is, when (1/2) ln(P/P0) = 1. These definitions imply that the numerical value of LF when LF is expressed in the unit neper is {LF}Np = ln(F/F0), and that the numerical value of LP when LP is expressed in the unit neper is {LP}Np = (1/2) ln(P/P0); that is

LF = ln(F/F0) Np
LP = (1/2) ln(P/P0) Np.

One bel (1 B) is the level-of-a-field-quantity when $F/F_0 = \sqrt{10}$ that is, when 2 lg(F/F0) = 1 (note that lg x = log10x – see Sec. 10.1.2). Equivalently, 1 B is the level- of-a-power-quantity when P/P0 = 10, that is, when lg(P/P0) = 1. These definitions imply that the numerical value of LF when LF is expressed in the unit bel is {LF}B = 2 lg(F/F0) and that the numerical value of LP when LP is expressed in the unit bel is {LP}B = lg(P/P0); that is

LF = 2 lg(F/F0) B = 20 lg(F/F0) dB LP = lg(P/P0) B = 10 lg(P/P0) dB.

Since the value of LF (or LP) is independent of the unit used to express that value, one may equate LF in the above expressions to obtain ln(F/F0) Np = 2 lg(F/F0) B, which implies

\begin{eqnarray*} 1~{\rm B}&=&\frac{\ln 10}{2} ~ {\rm Np~exactly} \\ & \approx&1.151 \, 293 ~ {\rm Np} \\ 1~{\rm dB} &\approx& 0.115 \, 129 \, 3 ~ {\rm Np} ~ . \end{eqnarray*}

When reporting values of LF and LP, one must always give the reference level. According to Ref. 5:IEC 60027-3, this may be done in one of two ways: Lx (re xref) or L x / xref 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 xref is the value of the reference quantity, for example, 1 μV/m for E0, and 20 μPa for p0. Thus

LE (re 1 μV/m) = - 0.58 Np or LE/(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 E0 = 1 μV/m. Similarly

Lp (re 20 μPa) = 25 dB or Lp/(20 μPa) = 25 dB

means that the level of a certain sound pressure is 25 dB above the reference pressure p0 = 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 60027-3] 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.

8.8 Viscosity

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 (m2/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 m2/s.

8.9 Massic, volumic, areic, lineic

Reference [4: ISO 31-0] 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: ρA = 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: ρl = m / l
                    lineic electric current or linear electric current density: A = I / b


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