**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 (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)].

**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 *A _{r}*, and relative molecular mass, symbol

*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,

*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}*(

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 ^{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, {Δ*t* }°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 Δ*t* = 29.7546 °C = Δ*T* = 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, *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*(*x*B) = *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}] = 3*n*(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

**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*.

Quantity in numerator | ||||
---|---|---|---|---|

Amount of substance
| Volume
| Mass
| ||

Quantity in denominator | Amount of substance
| 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: m^{3}/mol | molar mass $$ M = \frac{m}{n} $$ SI unit: kg/mol |

Volume
| amount-of-substance concentration $$ c_{\rm B} = \frac{n_{\rm B}}{V} $$ SI unit: mol/m^{3} | volume fraction $$\varphi_{\rm B} = \frac{x_{\rm B} V_{\rm m,B}^* }{\Sigma x_{\rm A} V_{\rm m,A}^*}$$ SI unit: m^{3}/m^{3} = 1 | mass density $$ \rho = \frac{m}{V}$$ SI unit: kg/m^{3} | |

Mass
| molality $$ b_{\rm B} = \frac{n_{\rm B}}{m_{\rm A}}$$ SI unit: mol/kg | specific volume $$v = \frac{V}{m}$$ SI unit: m^{3}/kg | mass fraction $$ w_{\rm B} = \frac{m_{\rm B}}{m}$$ 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, . . . , *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 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 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.

**8.6.3 Molar volume**

*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.

**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

**8.6.5 Concentration of B; amount-of-substance 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 "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 = *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 amount-of-substance 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 amount-of-substance 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.)

**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, . . . ,

$$\varphi_{\rm B} = x_{\rm B} V_{\rm m,B}^* /\sum x_{\rm A} V_{\rm m,A}^*$$

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/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*.

**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:* *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: *w*B_{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 60027-3], 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*/*F*_{0}), where *F*/*F*_{0} is the ratio of two amplitudes of the same kind, *F*_{0} being a reference amplitude. Level-of-a-power-quantity 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 "level-of-electric-field-strength" and it is defined by the relation *L*_{E} = ln(*E*/*E*_{0}).

The difference between two levels-of-a-field-quantity (called "field-level difference") having the same reference amplitude *F*_{0} is Δ*L*_{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 levels-of-a-power-quantity (called "power-level difference") having the same reference power *P*_{0}: Δ*L*_{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 *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*/*F*_{0} = e, that is, when ln(*F*/*F*_{0}) = 1. Equivalently, 1 Np is the level-of-a-power-quantity when *P*/*P*_{0} = *e*2, 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 level-of-a-field-quantity when $$F/F_0 = \sqrt{10}$$ 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- of-a-power-quantity 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

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

When reporting values of *L*_{F} and *L*_{P}, one must always give the reference level. According to Ref. 5:IEC 60027-3, this may be done in one of two ways: L* _{x}* (re

*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 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 (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 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: *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}* =

lineic electric current or linear electric current density: A = I / *b*

Created January 28, 2016, Updated June 17, 2024