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Physical Quantities and Units

Ian Mills

Department of Chemistry

University of Reading, RG6 6AD

England

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 [1], 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 [2]. 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 [2] 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’ [3] which is revised every few years, and the

International Union of Pure and Applied Physics (IUPAP) has a similar but less

extensive publication [4].

The Joint Committee on Guides in Metrology is another international committee

with publications in this area. Their International Vocabulary of Metrology [5] is an

important guide to the specialized language of metrology.

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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 [6],

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 [7]

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.

2. Preamble

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)

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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.

(2.2a)

(2.2b)

(2.2c)

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.

3. Quantities

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

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italic or a roman font depending on whether they represent either other quantities or

mnemonic labels.

Examples:

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

methane

Table 1

quantity symbol

------------- --------

length l, x

volume V

mass m

time t

velocity

v

energy E

power P

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.

4. Units

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).

quantity

-------------

concentration

pressure

temperature

molar mass

amount of substance n

mole fraction

surface tension

electric dipole

moment

symbol

--------

c

p, or P

T

M

x

?, or ?

p, or ?

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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.

Table 2

quantity unit

---------- -------------

length metre

centimetre

kilometre

ångström

mass gram

kilogram

milligram

atomic mass unit

pressure pascal

bar

millibar

torr

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

the equation

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

-----------------------------------------

m

cm = 10–2 m

km = 103 m

Å = 10–10 m

g

kg = 103 g

mg = 10–3 g

u = (1/6.022 137 × 1023) g

Pa

bar = 105 Pa

mbar = 100 Pa

Torr = 133.3 Pa

(5.1)

(5.2)

(5.3)