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Guide for the Use of the International System of Units (SI)
21
the relative expanded uncertainty of the resistance R is U
r
= 3 ppm
Because the names of numbers 10
9
and larger are not uniform worldwide, it is best that they be
avoided entirely (in many countries, 1 billion = 1 × 10
12
, not 1 × 10
9
as in the United States); the preferred
way of expressing large numbers is to use powers of 10. This ambiguity in the names of numbers is one of
the reasons why the use of ppm, ppb, ppt, and the like is deprecated. Another, and a more important one, is
that it is inappropriate to use abbreviations that are language dependent together with internationally
recognized signs and symbols, such as MPa, ln, 10
13
, and %, to express the values of quantities and in
equations or other mathematical expressions (see also Sec. 7.6).
Note: This Guide recognizes that in certain cases the use of ppm, ppb, and the like may be required by a
law or a regulation. Under these circumstances, Secs. 2.1 and 2.1.1 apply.
7.10.4 Roman numerals
It is unacceptable to use Roman numerals to express the values of quantities. In particular, one
should not use C, M, and MM as substitutes for 10
2
, 10
3
, and 10
6
, respectively.
7.11 Quantity equations and numerical-value equations
A quantity equation expresses a relation among quantities. An example is l = υt, where l is the
distance a particle in uniform motion with velocity υ travels in the time t.
Because a quantity equation such as l = υt is independent of the units used to express the values of
the quantities that compose the equation, and because l, υ, and t represent quantities and not numerical
values of quantities, it is incorrect to associate the equation with a statement such as “where l is in meters, υ
is in meters per second, and t is in seconds.”
On the other hand, a numerical value equation expresses a relation among numerical values of
quantities and therefore does depend on the units used to express the values of the quantities. For example,
{l}
m
= 3.6
−1
{υ}
km/h
{t}
s
expresses the relation among the numerical values of l, υ, and t only when the
values of l, υ, and t are expressed in the units meter, kilometer per hour, and second, respectively. (Here
{A}
X
is the numerical value of quantity A when its value is expressed in the unit X—see Sec. 7.1, note 2.)
An alternative way of writing the above numerical value equation, and one that is preferred because
of its simplicity and generality, is l/m = 3.6
−1
[υ/(km/h)](t / s). NIST authors should consider using this
preferred form instead of the more traditional form “l = 3.6
−1
υt, where l is in meters, υ is in kilometers per
hour, and t is in seconds.” In fact, this form is still ambiguous because no clear distinction is made between
a quantity and its numerical value. The correct statement is, for example, “l* = 3.6
−1
υ* t *, where l* is the
numerical value of the distance l traveled by a particle in uniform motion when l is expressed in meters, υ*
is the numerical value of the velocity υ of the particle when υ is expressed in kilometers per hour, and t* is
the numerical value of the time of travel t of the particle when t is expressed in seconds.” Clearly, as is
done here, it is important to use different symbols for quantities and their numerical values to avoid
confusion.
It is the strong recommendation of this Guide that because of their universality, quantity equations
should be used in preference to numerical-value equations. Further, if a numerical-value equation is used, it
should be written in the preferred form given in the above paragraph, and if at all feasible the quantity
equation from which it was obtained should be given.
Notes:
1. Two other examples of numerical-value equations written in the preferred form are as follows,
where E
g
is the gap energy of a compound semiconductor and k is the conductivity of an
electrolytic solution:
134
Guide for the Use of the International System of Units (SI)
E
g
/eV = 1.425 − 1.337x + 0.270x
2
, 0 ≤ x ≤ 0.15,
where x is an appropriately defined amount-of-substance fraction (see Sec. 8.6.2).
k /(S / cm) = 0.065 135 + 1.7140 × 10
−3
(t / ºC) + 6.4141 × 10
−6
(t / ºC)
2
− 4.5028 × 10
−8
(t / ºC)
3
,
0 ºC ≤ t ≤ 50 ºC, where t is Celsius temperature.
2. Writing numerical-value equations for quantities expressed in inch-pound units in the preferred
form will simplify their conversion to numerical-value equations for the quantities expressed in SI
units.
7.12 Proper names of quotient quantities
Derived quantities formed from other quantities by division are written using the words “divided by” or per
rather than the words “per unit” in order to avoid the appearance of associating a particular unit with the
derived quantity.
Example: pressure is force divided by area
but not: pressure is force per unit area
or pressure is force per area
7.13 Distinction between an object and its attribute
To avoid confusion, when discussing quantities or reporting their values, one should distinguish
between a phenomenon, body, or substance, and an attribute ascribed to it. For example, one should
recognize the difference between a body and its mass, a surface and its area, a capacitor and its capacitance,
and a coil and its inductance. This means that although it is acceptable to say “an object of mass 1 kg was
attached to a string to form a pendulum,” it is not acceptable to say “a mass of 1 kg was attached to a string
to form a pendulum.”
7.14 Dimension of a quantity
Any SI derived quantity Q can be expressed in terms of the SI base quantities length (l) , mass (m),
time (t), electric current (l ) , thermodynamic temperature (T ) , amount of substance (n), and luminous
intensity (I
v
) by an equation of the form
Q = l
α
m
β
t
γ
I
δ
T
ε
n
ζ
I
v
η
a
k
,
∑
=
K
k 1
where the exponents α,
β
, γ, . . . are numbers and the factors a
k
are also numbers. The dimension of Q is
defined to be
dim Q = L
α
M
β
T
γ
I
δ
θ
ε
N
ζ
J
η
,
where L, M, T, I, θ, N, and J are the dimensions of the SI base quantities length, mass, time, electric
current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The
exponents α, β, γ, . . . are called “dimensional exponents.” The SI derived unit of Q is m
α
·kg
β
·
s
γ
·A
δ
·K
ε
·
mol
ζ
·
cd
η
, which is obtained by replacing the dimensions of the SI base quantities in the dimension of Q
with the symbols for the corresponding base units.
Example: Consider a nonrelativistic particle of mass m in uniform motion which travels a distance l in a
time t . Its velocity is υ = l / t and its kinetic energy is E
k
= mυ
2
/ 2 = l
.2
mt
.−2
/ 2. The
dimension of E
k
is dim E
k
= L
2
MT
−2
and the dimensional exponents are 2, 1, and −2. The SI
derived unit of E
k
is then m
2
·kg·s
−2
, which is given the special name “joule” and special
symbol J.
22
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74
Guide for the Use of the International System of Units (SI)
23
A derived quantity of dimension one, which is sometimes called a “dimensionless quantity,” is one
for which all of the dimensional exponents are zero: dim Q = 1. It therefore follows that the derived unit for
such a quantity is also the number one, symbol 1, which is sometimes called a “dimensionless derived
unit.”
Example: The mass fraction w
B
of a substance B in a mixture is given by w
B
= m
B
/ m, where m
B
is the
mass of B and m is the mass of the mixture (see Sec. 8.6.10). The dimension of w
B
is dim
w
B
= M
1
M
−1
= 1; all of the dimensional exponents of w
B
are zero, and its derived unit is
kg
1
·kg
−1
= 1 also.
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 (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
110
Guide for the Use of the International System of Units (SI)
24
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 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
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 A
r
(X) = m(X) / [m(
12
C) / 12]
and M
r
(B) = m(B) / [m(
12
C) / 12], where m(X) is the mass of X, m(B) is the mass of B, and m(
12
C)
is the mass of an atom of the nuclide
12
C. It should also be recognized that m(
12
C) / 12 = u, the
unified atomic mass unit, which is approximately equal to 1.66 × 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(H
2
) = 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
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 (ºC) 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 }ºC = {
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 ºC = ΔT = 29.7546 K.
80
Guide for the Use of the International System of Units (SI)
25
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(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.)
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
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; 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.
175
Guide for the Use of the International System of Units (SI)
Table 12. Summary description of nine quantities that are quotients involving amount of substance,
volume, or mass
(a)
Quantity in numerator
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
amount-of-substance
fraction
n
n
x
B
B
=
SI unit:
mol/mol = 1
molar volume
n
V
V
=
m
SI unit:
m3/mol
molar mass
n
m
M
=
SI unit:
kg/mol
Volume
Symbol: V
SI unit: m
3
amount-of-substance
concentration
V
n
c
B
B
=
SI unit:
mol/mol3
volume fraction
*
m,A
A
*
m,B
B
B
x V
x V
∑
=
ϕ
SI unit:
m3/m3 = 1
mass density
V
m
=
ρ
SI unit:
kg/m3
Quantity in denominator
Mass
Symbol: m
SI unit: kg
molality
A
B
B
m
n
b =
SI unit:
mol/kg
specific volume
m
V
=
υ
SI unit:
m3/kg
mass fraction
m
m
w
B
B
=
SI unit:
kg/kg = 1
(a)
Adapted from Canadian Metric Practice Guide (see Ref. [6], note 2; the book cited in Ref. [6], note 3, 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, . . . ,
∑
+ ≡
+
+
A
A
C
B
A
...
n
n
n
n
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.
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.”
26
117
Guide for the Use of the International System of Units (SI)
27
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
.)
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/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,
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 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, . . . ,
94
Guide for the Use of the International System of Units (SI)
*
m,A
A
*
m,B
B
B
x V
x V
∑
=
ϕ
where x
A
, x
B
, x
C
, . . . are the amount-of-substance fractions of A, B, 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
,
,
,
*
m,C
*
mB,
*
m,A
V
V
V
∑
=1.
A
x
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 / υ.
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: υ.
SI unit: cubic meter per kilogram (m
3
/kg).
Definition: volume of a substance divided by its mass: υ = V / m.
Note: Specific volume is the reciprocal of mass density (see Sec. 8.6.7): υ = 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 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
28
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