It soon becomes clear to any student of physiology that there are many systems of units and forms of terminology. For example, respiratory physiologists measure pressures in centimetres of water and cardiovascular physiologists use millimetres of mercury. As the study of any single branch of physiology becomes increasingly sophisticated, more and more use is made of other disciplines in science. As a result the range of units has increased to such an extent that conversion between systems takes time and can easily cause confusion and mistakes.
We see also frequent misuse of terminology which can only confuse; for example, the partial pressure of oxygen in blood is often referred to as the 'oxygen tension' when in reality tension means a tensile force and is hardly the appropriate word to use.
In order to combat a situation which is deteriorating, considerable effort is being made to reorganize and unify the systems of nomenclature and units as employed in physiology. For any agreed procedure to be of value, it must be self-consistent and widely applicable. Therefore it has to be based upon a proper understanding of mathematical principles and the laws of physics.
The system of units which has been adopted throughout the world and is now in use in most branches of science is known as the Systeme International or SI (see §3.8). It is a coherent system of units based on the metric system of units of the kilogram mass, metre, and second, and it provides a suitable basis for the unification of systems currently employed in the various branches of physiology. Conversion from the older c.g.s. system is, moreover, quite simple.
The need to state units when specifying a physical quantity is recognized in everyday life as well as in the field of scientific research. Thus distances are commonly measured in kilometres, centimetres, angstroms, etc, and velocities are measured in miles per hour or centimetres per second. The number which indicates the magnitude or amount of a given quantity in a particular set of units is conveniently called the measure and is inversely proportional to the size of the units used. Thus 1 kilometre is measured as 103 metres or 105 centimetres.
The word dimension, however, is used rather differently in physical science from the way in which it is commonly employed. In everyday usage dimensions indicate the physical size of an object; for example, the size of a piece of paper is 10 x 8 in, or a box has the dimensions a x b x c cm. Implicit in these descriptions are the concepts of area and volume respectively, which are formed as the square and the cube of the unit of length.
This leads us to the specific scientific concept of the word 'dimension', in which the dimensions of area are those of (length)2, and those of volume are (length)3. It is important to realize that we are here concerned solely with the nature of the quantity and not with its measure in any particular set of units. Convention has established the notation [L2] and [L3] for these two quantities. Similarly the dimensions of velocity are [L]/[T], or [LT-1], and those of density [ML-3].
The dimensions of the quantities so far considered are all self-evident from the nature of the quantity or follow at once from its definition. Often, however, the dimensions of a quantity can be related to those of another quantity only by inference from a physical law.
An example of this is the derivation of the dimensions of force from those of mass, length, and time. The law involved is Newton's second law (§2.4). Thus:
force = mass x acceleration
or
F=Ma,
where a denotes acceleration whose dimensions are [LT-2]. Hence we have:
[F]=Ma
=[MLT-2]
or conversely we may define the dimensions of mass as
[M]=[FL-1T2].
The consequence of this is that the dimensions of all quantities in mechanics can be expressed in terms of [M], [L], and [T] or in terms of [F], [L], and [T]. These alternative methods of deriving dimensional expressions are widely known as the 'mass-based' and 'force-based' systems, and each has its corresponding system of units.
Thus it can be seen that the dimensions of all physical quantities not involving temperature can be described in terms of the fundamental dimensions of mass, length, and time. In science we also generally use mass, length, and time as the fundamental or primary units. These are treated as being independent of one another and the units are defined on the basis of arbitrary accepted standards.
For example, we base our measurement of length on the metre; the metre standard used to be a metal bar kept in Paris, and the distance between two marks on the bar (under specified conditions of temperature) was defined as the reference standard. Today the standard is the 'optical metre'; it is defined as 1,650,763.73 vacuum wavelengths of orange light from a krypton-86 discharge lamp! Although this may seem a bizarre and difficult way of defining a metre, it has two important advantages. First the new standard is virtually the same length as the old one, and secondly any competent laboratory can set up its own reference standard without the need to travel to Paris.
From the fundamental units of mass, length, and time, other units in mechanics can be derived. Thus in the c.g.s. system the unit of area is the square centimetre or cm2; in SI it is the square metre or m2. The units of density in SI become kilogramme per cubic metre (kg m-3).
The unit of force is defined in Newton's second law as that which produces unit acceleration when acting on unit mass. The unit of force is then given a special name depending upon the system of units in which it is derived: in the c.g.s. system it is the dyne (1 dyne = 1 g cm s-2); in SI it is called the newton (1 newton = 1 kg m s-2), which is 105 dynes. Hence the units of stress or pressure (see §4.1), which is force per unit area, become, in SI, N m-2 or kg m-1 s-2.
The use of mass, length, and time as the fundamental units is not the only possible choice; different independent units would be equally permissible and indeed occasionally provide convenient solutions to particular problems. Thus engineering systems often use force, length and time as the fundamental units; the unit of mass then becomes a derived unit. However, the choice of mass, length, and time is practically universal in pure science and is both convenient and well founded.
In the force-based system of units, the units of force were originally defined as the weights of unit mass. Now the weight of a body results from the action of the force of gravity upon it; gravitational acceleration has been introduced and this varies slightly from place to place on the earth—and is considerably less on the moon. Thus the force-based system suffers from a severe disadvantage because of the variability of gravitational attraction with location.
The mass of a body is simply an indication of the amount of matter within it. We may measure this by measuring its resistance to a change of motion with the help of Newton's second law:
force = mass x acceleration.
Thus if we subject a body to a known acceleration we may quantify its mass by measuring the force required to maintain that acceleration. Alternatively we may compare the mass of two bodies by comparing the forces required to maintain the same acceleration in both of them.
The force-based units have become known as gravitational units in contrast with the absolute units of mass, length, and time. The implication of gravity in the definition and the casual use of the words 'weight' and 'mass' have led to inconsistencies in calculations on many occasions. One of the important aspects of SI is that it is based on mass, not force, and problems associated with the use of the weight of a material are overcome.
In mechanics it will be seen that we are continuously concerned with the energy content of a system; the dimensions of energy are [ML2T-2], and in SI we measure energy in the units of joules. In physiology, as in other branches of science, we are often concerned with the measurement of heat, which in fact is a form of energy and so has the above dimensions and is measured in the same units. Temperature, which is related to heat, is also considered as a fundamental quantity and is given the dimensions [q], the unit used being the degree Kelvin.
When we consider the mass of a body we are unable to make any statements about its molecular content. However, when we are interested in studying processes of chemical change we want to know something about the number of molecules we are dealing with. We want to know the amount of substance, not its mass. The amount is expressed as the number of moles of the material and in turn a mole is defined as the molecular weight, usually expressed in grams.
As we have already noted, physically meaningful equations express relationships between quantities of the same physical character, and are therefore dimensionally homogeneous. This is known as the principle of dimensional homogeneity, and it has been tacitly assumed above in deducing derived dimensions and units. It is by the application of this principle that we can guarantee the consistency of derived units.
In the past, the concepts of volume and flow-rate have been used in a loose manner which is unacceptable in precise scientific descriptions. Therefore it is worthwhile to consider these quantities in further detail.
Volume indicates a region of space whose dimensions are [L3], so that in SI units it is measured in cubic metres (m3). That volume of space may contain a certain amount of some substance or it may be completely empty. It is a mistake to consider that 'volume' implies anything about the amount of material within it. Of course, if an independent statement is made about the density of the material in that space, then we can specify its mass.
Flow-rate implies the rate of transport of a given amount of material from one region of space to another. We are thus concerned with the movement of a given mass of material in a given time—not the movement of a volume. The dimensions of flow-rate are mass per unit time [M/T] and its units in the SI kgs-1; the dimensions are not [L3/T] as is so often assumed (see Chapter 4).
SI is a rationalized selection of units from the metric system, so the individual units are not new. There are seven fundamental units (Table 3.1) and several derived units, some of which have special names (Table 3.2). Derived units are merely for our convenience and they can all be expressed in terms of the fundamental units.
Table 3.1 The fundamental SI units |
||
|---|---|---|
| Quantity | Name of unit | Symbol |
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Thermodynamic temperature | degree Kelvin | K |
| Electric current | ampere | A |
| Luminous intensity | candela | cd |
| Substance | mole | mol |
Table 3.2 Common derived SI units |
||
|---|---|---|
| Physical quantity | SI unit | Symbol |
| Force, tension | newton | N=kg m s-2 |
| Work , energy, quantity of heat | joule | J=N m |
| Power | watt | W=J s-1 |
| Frequency | hertz | Hz=s-1 |
| Area | metre2 | m2 |
| Volume | metre2 | m3 |
| Density (mass density) | kilogram/metre3 | kg m-3 |
| Velocity | metre/second | m s-1 |
| Angular velocity | radian/second | rad s-1 |
| Acceleration | metre/second2 | m s-2 |
| Pressure, stress | newton/metre2
(also called pascal) |
N m-2 |
| Surface tension | newton/metre | N m-1 |
| Dynamic viscosity | newton second/metre2 | N s m-2 |
| Kinematic viscosity | metre2/second | m2 s-1 |
| Diffusion coefficient | metre2/second | m2 s-1 |
Although SI is simply a development of the existing metre-kilogram-second system it is superior because it is coherent. This means that the product of unit quantities yields a unit resultant quantity. For example:
l N x l m = 1 J
and
I kg x I m / l s = l N.
No numerical factors are involved, and this makes calculation much more straightforward and eliminates the tedious problem of applying conversion factors. Table 3.3 lists a number of conversion factors for common units to the appropriate SI values.
Table 3.3 Conversion factors
|
||
| Quantity | Common unit | SI |
| Volume | 1 ft3 | 0.02832 m3 |
| Mass | 1 lb |
0.4536 kg |
| Force | 1 dyne | 10~5 N |
| Work | l erg | 10-7 N m |
| Pressure | 1 cm H2O 1 mmHg |
98.1 N m-2 133.3 N m-2 |
| Viscosity | 1 poise | 0.1 N s m-2 |
The system also possesses a clearly defined organization for describing multiples of the basic and derived units. The way in which it works can be seen from Table 3.4. Great care should be taken in the use of these prefixes.
Table 3.4 |
|||
| Factor by which unit is multiplied |
Prefix | Symbol | Example |
| 1012 | tera | T | |
| 109 | giga | G | |
| 106 | mega | M | megawatt (MW) |
| 103 | kilo | k | kilometre (km) |
| 102 | hecto | h | |
| 101 | deca | da | decagram (dag) |
| 10-1 | deci | d | decimetre (dm) |
| 10-2 | centi | c | centimetre (cm) |
| 10-3 | milli | m | milligram (mg) |
| 10-6 | micro | m | microsecond (ms) |
| 10-9 | nano | n | nanometre (nm) |
| 10-12 | pico | p | picogram (pg) |
The prefix should always be written immediately adjacent to the unit to be qualified, e.g. meganewton (MN), kilojoule (kJ), microsecond (ms). Only one prefix can be applied to a given unit at any one time, e.g. one thousand kilograms is 1 megagram (Mg) and not 1 kilo-kilogram.
The symbol m stands for the fundamental unit 'metre' and also for the prefix 'milli' so to avoid confusion it has to be used very carefully in certain circumstances. For example, mN stands for millinewton while m N denotes the metre-newton or unit of work. However, the subtle use of spacing between the letters can lead to confusion so it is better to write the metre as the second unit, i.e. newton-metre (N m).
Another important point to note is that when a multiple of a fundamental unit is raised to a power, the power applies to the whole multiple and not the fundamental unit alone, e.g. 1 km2 means 1 (km)2 = 106 m2 and not 1 k(m)2 = 103 m2.