Physical quantities and units
Physical Quantities and Units
Department of Chemistry
University of Reading, RG6 6AD
1. Reference material
Establishing a system of quantities and units for use in science, technology, and
commerce is a subject of vital importance to the world community. There are many
choices and conventions involved in achieving such a system, and these choices have
grown up over the years with the history of science and technology. Today these
decisions are made by several different international committees. These committees
and their publications are an important source of information.
The Bureau International des Poids et Mesures, the BIPM (in English: the
International Bureau of Weights and Measures) is established at the Pavillon de
Breteuil at Sèvres, near Paris, and this is the international home of metrology. It is
also the home of the annual meeting of the Comité International des Poids et Mesures,
the CIPM, and the meeting once every four years of the Conférence Générale des
Poids et Mesures, the CGPM. These bodies are responsible for establishing the
Système International des Unités, the SI (in English: the International System of
Units). The most important publication from the BIPM is the SI Brochure , which
is prepared by the Comité Consultatif des Unités, one of the consultative committees
of the CIPM. The SI Brochure provides the official definition of the SI. It is revised
every few years; the current 7th edition appeared in 1998. Thus the BIPM is
responsible for the world’s units of measurement.
The system of quantities (or physical quantities), and the equations relating these
quantities, is also an important part of the SI. This is sometimes referred to as the
International System of Quantities, or the ISQ. It has developed and grown as science
has grown, and it is still growing and changing. The most commonly used quantities,
with their names and symbols, are reviewed by the Technical Committee 12 of the
International Organization for Standardization, ISO/TC 12, who publish a review of
quantities and their symbols every few years . They also collaborate with the
International Electrotechnical Commission, Technical Committee 25, IEC/TC 25, and
they are together at present preparing a revised agreed list of quantities and symbols
used in science and technology which will displace reference  when it is published.
The International Unions in the various specialized fields also publish valuable
guides to quantities and units. In particular the International Union of Pure and
Applied Chemistry, IUPAC, publishes the so-called Green Book, ‘Quantities, Units
and Symbols in Physical Chemistry’  which is revised every few years, and the
International Union of Pure and Applied Physics (IUPAP) has a similar but less
extensive publication .
The Joint Committee on Guides in Metrology is another international committee
with publications in this area. Their International Vocabulary of Metrology  is an
important guide to the specialized language of metrology.
Finally there are of course very many individual papers in this field. I shall
mention only two. The first is a paper by my old teacher Edward Guggenheim ,
published in 1942, which is a review of how a system of quantities and units has to be
established, with reference to the system we all use today. Although it is now dated,
this paper has always been an inspiration to me, and is the model for many of the ideas
that I shall express in these lectures. The second is a recent paper by Jan de Boer 
published in a special issue of Metrologia in 1995 devoted to the subject of physical
units. There is interest in many papers in this issue, but de Boer’s paper – which is the
first in the issue – is an important current review of the subject.
A leading physicist of the 19th century is quoted as having said that one can
multiply together only numbers, and that the idea of multiplying a length by a mass is
nonsense. I take the opposite view, that we are entitled to multiply together – or to
divide one by the other – any two quantities, provided that our definition of
multiplication and division is self-consistent and obeys the associative and distributive
laws. (In general the commutative law is also obeyed in multiplication of quantities,
although operator quantities in quantum mechanics are an exception.) Thus if a reader
asks me what is the product of a kilogram and a metre, I would say the answer is a
kilogram metre. And if the reader suggests that this answer is somehow
unsatisfactory, I would point out that when a quarter is multiplied by three the answer
is three quarters, and when ? is multiplied by ?3 the answer is ?3?, and no simpler
answer is possible.
Similarly we may say that the product of three metres and four metres is twelve
metres squared, and the ratio of eight metres and two seconds is four metres per
second. In these lectures I shall consistently make use of the concept of multiplying or
dividing physical quantities by one another.
Having said that, I should add that multiplying quantities like a kilogram and a
metre to get a kilogram metre is not quite the same as multiplying 3 by 4 to get 12.
We should recognize that we are extending the meaning of ‘multiply’ to cover the
multiplication of quantities, and even to cover the multiplication of operators in
quantum mechanics. But this extension of the meaning of multiplication leads to no
inconsistencies, and proves to be a valuable addition to the language of science.
In science and technology we frequently write equations of the form
m = 150 g , T = 273 K ,
In each case the symbol on the left of the equation represents a quantity (mass,
temperature, or electric current in the examples above), and the symbols on the right
of the equation give the value of the quantity expressed as the product of a number
(the numerical value) and a unit (gram, kelvin or ampere in these examples). There
are conventions regarding the symbols used to denote quantities and units which we
review below, but one that we note here is that the symbols for quantities are generally
single letters which are always printed in an italic (sloping) font, whereas the symbols
for units are always in a roman (upright) font, as in these examples.
I = 2.5 A (2.1)
To give some further examples, suppose that v denotes the speed of a ship. Then
we may write
v = 4.00 knots
= 2.06 m/s
= 7.40 km/h
where knot, m/s, and km/h are units of speed. A knot is a nautical mile per hour,
and a nautical mile is 1852 m. The symbols m, km, s, and h are the symbols for a
metre, a kilometre, a second and an hour. Note that although the numerical values
and the units differ in the three equations (2.2), the product which expresses the value
of v is always the same.
It would however be wrong to write v = 4.00, or v = 2.06, or v = 7.40, or any
other number. v is a quantity, and the value of a quantity is in general the product of a
number and a unit. (The only exception is for a dimensionless quantity, whose value
is indeed given by a pure number, the unit being simply the number one, 1.)
We thus adopt a system in which we use symbols to denote quantities, and the
value of a quantity is always expressed as the product of a number and a unit.
Moreover we allow ourselves to multiply and divide quantities at will – and
correspondingly multiply and divide both the numbers and the units. This system is
known as quantity calculus, but it might be better known as the algebra of quantities,
because it is more to do with algebraic manipulation than calculus.
Quantity calculus is not universally used in science and technology. There are still
eminent scientists who use the alternative system in which symbols are used to
represent the numerical values of quantities, expressed in units which are specified in
the accompanying text. However I hope to convince you of the advantage of always
using quantity calculus, and its use is certainly becoming more widespread.
Examples of quantities are length, volume, mass, time, velocity, energy, power,
electric charge, and electric current. Further examples that are more specifically
concerned with chemistry are concentration, pressure, temperature, molar mass,
amount of substance (or chemical amount), mole fraction, surface tension, and electric
dipole moment. There are quantities associated with each specialized field of science,
and many examples span all fields. I shall use the word quantity to mean both a
quantity in the general sense (e.g. length, mass, time) and a quantity in the particular
sense (e.g. the length of my pencil, the mass of my car, the time interval between
sunrise and sunset at Stonehenge on the 21st of June 2000). This is common practice,
and the context is usually sufficient to distinguish these different meanings. I shall not
attempt to define the word quantity, other than by giving examples.
There is usually a recommended symbol (or sometimes two alternative symbols)
for each quantity, as illustrated in Table 1. Quantity symbols are generally single
letters of the Latin or Greek alphabet, but they may be further specified by subscripts
or superscripts, or information in brackets. They are always type-set in an italic
(sloping) font. Subscript indices and other qualifying information are set in either an
italic or a roman font depending on whether they represent either other quantities or
mi for the mass of the ith particle
Ti and Tf for the initial and final temperatures
M(NaCl) for the molar mass of NaCl
?fHo(CH4) for the standard enthalpy of formation of
length l, x
electric charge Q, or q
electric current I
Quantities are sometimes called physical quantities, but there seems little need for
the extra adjective, since they are concerned with all fields of science – chemistry and
biology just as much as physics.
Examples of units are the metre, centimetre, kilometre, and the ångström, all of
which are units of length; the gram, kilogram, milligram, and the unified atomic mass
unit, all of which are units of mass; the pascal, bar, millibar, and torr, all of which are
units of pressure. For each unit there is an internationally agreed symbol, as for the
examples in Table 2.
The use of the recommended symbols for units is mandatory. They are always
written in a roman (upright) font. They may consist of more than one letter, which is
never followed by a stop (as might be used for an abbreviation), and is never
pluralised by adding an s. However when writing products of units it is important to
always leave one space between the units for clarity (for example, m s denotes the
product of a metre and a second, but ms denotes a millisecond). A half-high dot
between the units may also be used to indicate multiplication. Another rule about the
symbols for units is that they begin with a capital letter when the unit is named for a
person (J for joule, K for kelvin), but with a lower case letter otherwise (s for second,
m for metre).
amount of substance n
p, or P
?, or ?
p, or ?
Units are actually just particular examples of quantities, which have been chosen
for easy reference for the purpose of comparing the values of quantities of the same
kind. They are standard quantities, which should ideally be readily available, easily
realized in any laboratory, and internationally agreed and recognized.
atomic mass unit
5. The equations of physics, and definitional constants
The equations of physics play an important role in the way we define quantities and
units. The area A of a rectangle is related to its length l and its height h according to
A = hl
The area of a triangle of base length l and height h is
A = ½ hl
and the area of an ellipse of length (major axis) l and height (minor axis) h is
A = (?/4) hl
In principle it would be possible to define area in a different way, so that the
numerical factor became 1 instead of ½ in the second equation, or 1 instead of (?/4) in
the third equation. But once the definition of area is settled, the numerical factor in
each formula is uniquely determined, and whatever definition is chosen the numerical
factor can be unity in at most one of these three formulae. It is useful to refer to such
numerical factors as definitional constants.
symbol for unit
cm = 10–2 m
km = 103 m
Å = 10–10 m
kg = 103 g
mg = 10–3 g
u = (1/6.022 137 × 1023) g
bar = 105 Pa
mbar = 100 Pa
Torr = 133.3 Pa