NIST Guide to SI Units - 8 Comments on Some Quantities and Their Units

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) may 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. [6: ISO 31-1] 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 [6: ISO 31-5]. The SI unit of this quantity is thus the reciprocal second (s^-1). However, as pointed out in Ref. [6: ISO 31-5], 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 [2, 3]. 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 [6: ISO 31-3]. 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 [6: ISO 31-8], which better reflect their definitions. Like 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 [6: 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: A_r (Si) = 28.0855 M_r (H₂) = 2.0159 A_r (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 A_r(X) = m(X)/[m(¹²C)/12] and M_r(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 × 10^-27 kg [see Table 7, footnote (b)].

2 It follows from the examples and note 1 that the respective average masses of Si, H₂, and ¹²C are m(Si) = A_r(Si) u, m(H₂) = M_r(H₂) u, and m(¹²C) = A_r(¹²C) u.

3 In publications dealing with mass spectrometry, one may encounter a statement 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. [6: 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, 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(H₂SO₄). (For a detailed discussion of the use of the SI in physical chemistry, see the book cited in Ref. [8], note 5.)

8.6.1 Amount of substance

Quantity symbol:	n (also ν). 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)H₂SO₄] = 3n(H₂SO₄).
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(F₂) = 2A_r(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:	x_B (also y_B). 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: x_B = n_B/n.
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, ..., n = n_A + n_B + n_C + $\equiv \sum_{\rm A} n_{\rm A}$
3	A related quantity is amount-of-substance ratio of B (commonly called "mole ratio of solute B"), symbol r_B. 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 one-solute 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.

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

		Amount of substance Symbol: n SI unit: mol	Volume Symbol: V SI unit: m³	Mass Symbol: m SI unit: kg
	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).

8.6.3 Molar volume

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

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) = 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; amount-of-substance concentration of B

Quantity symbol:	c_B. SI unit: mole per cubic meter (mol/m³).
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 "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, ρ_B = 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 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:

φ_B. SI unit: one (1) (volume fraction is a quantity of dimension one).

Definition:

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

$\varphi_{\rm B} = \frac{x_{\rm B} V_{\rm m,B}^* }{\Sigma x_{\rm A} V_{\rm m,A}^*}$

where x_A, x_B, x_C, ... 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 Σ x_A = 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 Gude 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/v.

8.6.8 Molality of solute B

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.)

8.6.9 Specific volume

Quantity symbol:	v. SI unit: cubic meter per kilogram (m³/kg).
Definition:	volume of a substance divided by its mass: v = V/m.
Note:	Specific volume is the reciprocal of mass density (see Sec. 8.6.7): v = 1/ρ.

8.6.10 Mass fraction of B

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: w_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 Refs. [6: ISO 31-2] and [7: IEC 27-3], which should be consulted for further details.

Two of the most common logarithmic quantities are level of a field quantity, symbol L_F ,and level of a power quantity, 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).

Level of a field quantity is defined by the relation L_F = 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 L_P = (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 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 "level of electric field strength" and it is defined by the relation L_E = ln(E/E₀).

The difference between two levels of a field quantity (called "field level difference") having the same reference amplitude F₀ is ΔL_F = L_F1 - L_F₂ = 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: ΔL_P = L_P₁ - L_P₂ = ln(P₁/P₀) - ln(P₂/P₀) = ln(P₁/P₂).

It is clear from their definitions that both L_F and L_P 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 L_F when L_F is expressed in the unit neper is {L_F}_Np = ln(F/F₀), 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/P₀); that is

L_F = ln(F/F₀) Np

L_P = (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 = log₁₀ x - 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 L_F when L_F is expressed in the unit bel is {L_F}_B = 2 lg(F/F₀) and that the numerical value of L_P when L_P is expressed in the unit bel is {L_F}_B = lg(P/P₀); that is

L_F = 2 lg(F/F₀) B = 20 lg(F/F₀) dB

L_P = lg(P/P₀) B = 10 lg(P/P₀) dB .

Since the value of L_F (or L_P) is independent of the unit used to express that value, one may equate L_F 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 L_F and L_P , one must always give the reference level. According to Ref. [7: IEC 27-3], 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 E₀, and 20 µPa for p₀. 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 E₀ = 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 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. [7: IEC 27-3] 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 stokes (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 [6: 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:	v = V/m
Examples:	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
Examples:	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
Examples:	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
Examples:	lineic electric current or linear electric current density:	A = I/b

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.

Examples:	A_r (Si) = 28.0855	M_r (H₂) = 2.0159	A_r (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 A_r(X) = m(X)/[m(¹²C)/12] and M_r(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 × 10^-27 kg [see Table 7, footnote (b)].
2	It follows from the examples and note 1 that the respective average masses of Si, H₂, and ¹²C are m(Si) = A_r(Si) u, m(H₂) = M_r(H₂) u, and m(¹²C) = A_r(¹²C) u.
3	In publications dealing with mass spectrometry, one may encounter a statement 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.

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. [7: IEC 27-3] 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.