1 PHYSICAL QUANTITIES AND UNITS

1.1 PHYSICAL QUANTITIES AND QUANTITY CALCULUS

The value of a physical quantity Q can be expressed as the product of a numerical value {Q} and a unit [Q] $Q = \{Q\}[Q] \quad (1)$ Neither the name of the physical quantity, nor the symbol used to denote it, implies a particular choice of unit. Physical quantities, numerical values, and units may all be manipulated by the ordinary rules of algebra. Thus we may write, for example, for the wavelength of one of the yellow sodium lines $\lambda = 5.896\ \times 10^{−7}\ \rm{m} = 589.6\ \rm{nm} \quad (2)$ where m is the symbol for the unit of length called the metre (see Section 2.2, p. 11), nm is the symbol for the nanometre, and the units metre and nanometre are related by $1\ \rm{nm} = 10^{−9}\ \rm{m}\ \rm{or}\ \rm{nm} = 10^{−9}\ \rm{m} \quad (3)$ The equivalence of the two expressions for $\lambda$ in Equation (2) follows at once when we treat the units by the rules of algebra and recognize the identity of 1 nm and $10^{−9}\ \rm{m}$ in Equation (3). The wavelength may equally well be expressed in the form $\lambda/\rm{m} = 5.896\ \times 10^{−7}\ \rm{or}\ \lambda/\rm{nm} = 589.6 \quad (4)$ It can be useful to work with variables that are defined by dividing the quantity by a particular unit. For instance, in tabulating the numerical values of physical quantities or labeling the axes of graphs, it is particularly convenient to use the quotient of a physical quantity and a unit in such a form that the values to be tabulated are numerical values, as in Equations (4).

Algebraically equivalent forms may be used in place of $10^3\ \rm{K/T}$ , such as $\rm{kK}/T$ or $10^3\ (T/\rm{K})^{−1}$. Equations between numerical values depend on the choice of units, whereas equations between quantities have the advantage of being independent of this choice. The method described here for handling physical quantities and their units is known as quantity calculus (see ISO [5.a] and [11–14]). It is recommended for use throughout science and technology. The use of quantity calculus does not imply any particular choice of units (see Section 3.1, p. 15 for quantity calculus).

1.2 BASE QUANTITIES AND DERIVED QUANTITIES

By convention physical quantities are organized in a dimensional system built upon seven base quantities, each of which is regarded as having its own dimension. These base quantities in the International System of Quantities (ISQ) on which the International System of units (SI) is based, and the principal symbols used to denote them and their dimensions are as follows:

• • Insert Table**

All other quantities are called derived quantities and are regarded as having dimensions derived algebraically from the seven base quantities by multiplication and division.

Example: dimension of energy is equal to dimension of $M\ L^2\ T^{−2}$. This can be written with the symbol dim for dimension (see footnote 1, below) $\dim(E) = \dim(m·l^2·t^{−2}) = M\ L^2\ T^{−2}$

The quantity amount of substance is of special importance to chemists. Amount of substance is proportional to the number of specified elementary entities of the substance considered. The proportionality factor is the same for all substances; its reciprocal is the Avogadro constant (see Section 4.10, p. 44). The SI unit of amount of substance is the mole. "Amount of substance" is also called “chemical amount”, and may be abbreviated to the single word "amount", particularly in such phrases as “amount concentration” (see footnote 2), and "amount of $\ce{N2}$". In the same sense one might for instance say “amount concentration of $\ce{N2}$.” A possible name for international usage has been suggested: “enplethy” [10] (from Greek, similar to enthalpy and entropy).

In the ISQ, electric current is chosen as base quantity and ampere is the SI base unit. In atomic and molecular physics, the so-called atomic units are useful (see Section 2.4, p. 12).

Ordinal quantities and nominal properties are outside the scope of the Green Book [9].

1.3 SYMBOLS FOR PHYSICAL QUANTITIES AND UNITS [5.a]

A clear distinction should be drawn between the names and symbols for physical quantities, and the names and symbols for units. Names and symbols for many quantities are given in Chapter 4, p. 21; the symbols given there are recommendations. If other symbols are used they should be clearly defined. Names and symbols for units are given in Chapter 2, p. 11; the symbols for units listed there are quoted from the Bureau International des Poids et Mesures (BIPM) and are mandatory.

1.3.1 General rules for symbols for quantities

The symbol for a physical quantity should be a capital or lower case single letter (see footnote 1) of the Latin or Greek alphabet (see Section 1.6, p. 5). The letter should be printed in italic (sloping) type. When necessary the symbol may be modified by subscripts and superscripts of specified meaning. Subscripts and superscripts that are themselves symbols for physical quantities or for numbers should be printed in italic type; other subscripts and superscripts should be printed in roman (upright) type.

Examples		$C_{\rm{p}}$	for heat capacity at constant pressure
		$p_{\rm{i}}$	or partial pressure of the i th substance
	but	$C_{\rm{p}}$	for heat capacity of substance $\ce{B}$
		$\mu_{\rm{B}}^{\alpha}$	for chemical potential of substance $\ce{B}$ in phase $\alpha$
		$\Delta_{\rm{r}} H^{⦵}$	for standard reaction enthalpy

The meaning of symbols for physical quantities may be further qualified by the use of one or more subscripts, or by information contained in parentheses.

Examples	$\Delta_{\rm{f}}S^{⦵} (\ce{HgCl2},\ \rm{cr},\ 25 ^{\circ}C) = −154.3\ \rm{J\ K}^{−1}\ \rm{mol}^{−1}$
	$\mu_i = (\partial G/{\partial}n_i)_{T,p,\ldots,n_j,\ldots;\ j\neq i}$ or $μ_i = (\partial G/{\partial}n_i)_{T,p,n_{j\neq i}}$

Vectors and matrices may be printed in bold-face italic type, e.g. $\boldsymbol{A}$, $\boldsymbol{a}$. Tensors may be printed in bold-face italic sans serif type, e.g. S, T. Vectors may alternatively be characterized by an arrow, $\overrightarrow{A}$, $\overrightarrow{a}$ and second-rank tensors by a double arrow, $\overset{\rightrightarrows}{S}$, $\overset{\rightrightarrows}{T}$.

1.3.2 General rules for symbols for units

Symbols for units should be printed in roman (upright) type. They should remain unaltered in the plural, and should not be followed by a full stop except at the end of a sentence.

Example

${\rm{r}} = 10\ {\rm{cm}}$, not cm. or cms.

When a quantity symbol is used as a symbol for a unit, it is written following the general rules for symbols for quantities (see Section 1.3.1, p. 3).

Example

${\rm{r}} = 1a_0$, where $a_0$ is the bohr, the atomic unit of length.

Symbols for units shall be printed in lower case letters, unless they are derived from a personal name when they shall begin with a capital letter. An exception is the symbol for the litre which may be either $\rm{L}$ or $\rm{l}$, i.e. either capital or lower case (see footnote 2).

Examples

$\rm{m}$ (metre), $\rm{s}$ (second), but $\rm{J}$ (joule), $\rm{Hz}$ (hertz)

Decimal multiples and submultiples of units may be indicated by the use of prefixes as defined in Section 2.5, p. 14.

Examples

$\rm{nm}$ (nanometre), $\rm{MHz} (megahertz)$, $\rm{kV}$ (kilovolt)

1.4 USE OF THE WORDS "EXTENSIVE", "INTENSIVE", "SPECIFIC", AND "MOLAR"

A quantity that is additive for independent, noninteracting subsystems is called extensive; examples are mass $m$, volume $V$, Gibbs energy $G$. A quantity that is independent of the extent of the system is called intensive; examples are temperature $T$, pressure $p$, chemical potential (partial molar Gibbs energy) $\mu$.

The adjective specific before the name of an extensive quantity is used to mean divided by mass. When the symbol for the extensive quantity is a capital letter, the symbol used for the specific quantity is often the corresponding lower case letter.

Examples	volume, $V$, and specific volume, $v = V/m = l/\rho$  (where $\rho$ is mass density);
	heat capacity at constant pressure, $C_{\rm{p}}$, and
	specific heat capacity at constant pressure, $c_{\rm{p}} = C_{\rm{p}}/m$

ISO [5.a] and the Chemistry and Human Health Division of IUPAC recommend systematic naming of physical quantities derived by division with mass, volume, area, and length by using the attributes massic or specific, volumic, areic, and lineic, respectively. Thus, for example, the specific volume could be called massic volume and the surface charge density would be areic charge. In addition, the Chemistry and Human Health Division of IUPAC recommends the use of the attribute entitic for quantities derived by division with the number of entities [15].

The adjective molar¹ before the name of an extensive quantity generally means divided by amount of substance. The subscript m on the symbol for the extensive quantity denotes the corresponding molar quantity.

Examples	volume, $V$ molar volume, $V_{\rm{m}} = V/n$ (Section 4.10, p. 44)
	enthalpy, $H$ molar enthalpy, $H_{\rm{m}} = H/n$

It is sometimes convenient to divide all extensive quantities by amount of substance, so that all quantities become intensive; the subscript m may then be omitted if this convention is stated and there is no risk of ambiguity. (See also the symbols recommended for partial molar quantities in Section 4.11, p. 49, and in Section 4.11.1 (iii), p. 51.)

Examples	molar absorption coefficient, $\varepsilon = a/c$ (see Section 4.7, p. 38)
	molar conductivity, $\Lambda = \kappa/c$ (see Section 4.13, p. 60)

Sometimes the adjective molar has a different meaning, namely divided by amount-of-substance concentration. The adjective molar is also used instead of the unit $\rm{mol\ dm}^{−3}$ for an amount concentration (see Section 4.10, note 11, p. 45) which is very different from the meaning explained above and should be avoided.

Despite this very common use, it should be noted that the word “molar” violates the rule [3, 5] that neither the name of a quantity, nor the symbol used to denote it, should imply a particular name of a unit ("mole" in this case). The same concern applies to related quantity names, such as "molar mass", "molar volume", "molar gas constant", etc., as well as their recommended symbols involving the letter "M" or "m" as main character or subscript. Other quantity names also violating this rule include “Celsius temperature”, "molality" and "mole fraction".