# The Systeme International Metric

### The International System of Units

Quantitative measurement is the cornerstone of modern science, but it has not always been so. The application of quantitative measurements to chemistry, for example, does not predate AD 1500. Quantitative measurement was developed for other purposes, as technology, and was only then adopted for scientific use. The system of weights and measures were developed on an ad hoc basis in different parts of the world. The most fundamental quantities measured were mass or weight, length or distance, and time. Systems of units for measuring these were developed from the very beginning of recorded history. Measurement of temperature was added in the sixteenth century, and measurement of electric current in the eighteenth century. More recently the amount of substance and luminous intensity have been added in the International System of Units, or SI.

The International System of Units or Systeme Internationale (SI) is an improved metric system adopted by the Eleventh General Conference of Weights and Measures in 1960. It is the universal measuring system used in all areas of science throughout the world. The entire SI system of measurement is constructed from seven base units, each of which represents a single physical quantity as shown in the table below.

#### Base Units of the International System

`Quantity`
`Name of Unit`
`Unit Symbol`
length
metre
m
mass
kilogram
kg
time
second
s
temperature
kelvin
K
amount of substance
mole
mol
electric current
ampere
A
luminous intensity
candela
cd

Like earlier versions of the metric system, the SI units can be designated as decimal fractions or multiples by the use of appropriate prefixes. The acceptable SI prefixes are given in the table below.

#### Prefixes of the International System

Factor
Prefix
Symbol
10+24
yotta
Y
10+21
zetta
Z
10+18
exa
E
10+15
peta
P
10+12
tera
T
10+9
giga
G
10+6
mega
M
10+3
kilo
k
10+2
hecta
h
10+1
deca
da
10-1
centi
c
10-2
deci
d
10-3
milli
m
10-6
micro
μ
10-9
nano
n
10-12
pico
p
10-15
femto
f
10-18
atto
a
10-21
zepto
z
10-24
yocto
y

Any prefix can be applied to any base unit except the kilogram; the kilogram takes prefixes as if the base unit were the gram. As a consequence 10-6 kg is written as 1 milligram (mg) rather than 1 microkilogram (ukg). Luminous intensity is rarely used in chemistry and we will not consider it further in this course, but the remaining six base units are essential to chemical studies.

The great advantage of the SI over other systems of units is that when any physical quantity whatever is written out in the SI base units or in units derived only from the SI base units, any mathematical manipulations performed with them will follow as well. No conversion factors will ever be required. This means that if the symbols in any equation are replaced by real numbers with their SI base units and algebraic manipulations are performed upon the units in exactly the same way as they are performed upon the numbers to which those units refer, the result will come out with the correct numbers and units.

Example. The mass of a sample of pure rhombic sulfur was 150.637 g and the volume of water it displaced was 72.8 mL. The density of sulfur is then (150.637 g)/(72.8 mL) = 2.07 g/mL, or g/cm3, or kg/dm3, or kg/L. This is 2.07(0.001 kg/g)(106cm3/m3) = 2.07 x 103 kg/m3.

Using the mass as 0.150637 kg and the volume as 72.8 x 10-6 m3. The density of sulfur is then (0.150637 kg)/(72.8 x 10-6 m3) = 2.07 x 103 kg/m3. The quantity calculated gives the result in SI base units without conversion. Reporting the answer as 2070 kg/m3, while arithmetically correct and in SI base units, would give the answer to one more significant figure than is justifiable from the measured data.

#### Base Units of the SI

##### The SI unit of length is the metre, a fundamental unit of the SI. The metre was once defined in terms of the circumference of the earth as part of the older metric system. Since 1983 the metre is by definition the length of the path travelled by light in vacuum in 1/299792458 of a second. The micron (u) is an obsolete name for the micrometre (um). Conversion factors between other units of length and the metre are:

1 Angstrom = 10.0 nm (exactly)

1 inch = 25.4 mm (exactly); 1 foot = 0.3048 m (exactly); 1 yard = 0.9144 m (exactly); 1 mile = 1.609344 km (exactly)

1 astronomical unit (A.U.) = 149.51 ñ 0.05 Gm

##### Mass
The SI unit of mass is the kilogram, a fundamental unit of the SI. The kilogram was once defined as the mass of one cubic decimetre of water. Since 1901 it is by definition the mass of the international prototype of the kilogram, a platinum-iridium mass which is stored at Sevres in France. The metric tonne is a common name for the megagram (Mg). Conversion factors between other units of mass and the kilogram, or its subdivision the gram, are:

1 unified atomic mass unit (u) = 1.66... yg

1 pound (lb) = 453.59237 g (exactly); 1 ton (short, 2000 lb) = 907.18474 kg (exactly); 1 ounce = 1/16 lb = 28.348523... g

##### Time
The SI unit of time is the second, a fundamental unit of the SI. Originally defined in terms of the rotation of the earth, the second is now defined in terms of atomic transitions in Cesium-133 because these are subject to more precise measurement. Specifically, since 1967 the second is defined as the duration of 9 192 631 770 periods of the electromagnetic radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cs-133 atom. Conversion factors between other units of time and the second are:

1 minute = 60 s (exactly); 1 hour = 60 min = 3600 s (exactly); 1 day = 24 hr = 86.4 ks (exactly); 1 week = 7 days = 604.8 ks (exactly)

1 month (28 d) = 2.5056 Ms (exactly); 1 month (29 d) = 2.5920 Ms (exactly); 1 month (30 d) = 2.6784 Ms (exactly); 1 month (31 d) = 2.7648 Ms (exactly)

1 year (normal, 365 d) = 31.5360 Ms (exactly); 1 year (leap, 366 d) = 31.6224 Ms (exactly); 1 year (sidereal) = 31.55815... Ms

##### Temperature
The SI unit of temperature is the kelvin, a fundamental unit of the SI. Since 1967, the kelvin has been by definition the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. The triple point of water is the temperature at which ice, water, and water vapor can all exist in equilibrium and its value is +0.01o Celsius.

The kelvin (which is correctly written without a degree sign) is used for measuring both temperature and temperature interval; thus one can say, "The temperature is 300 K" or "This pan is 20 K hotter than that one." Temperatures in kelvin can only be positive and so they require no sign. The kelvin scale of temperature is also known as the absolute scale and the thermodynamic scale.

The degree Celsius, the unit of the common metric temperature scale, is not part of the SI but its use is not discouraged. A temperature interval in degrees Celsius is identical to a temperature interval in kelvin, although a temperature in degrees Celsius is not identical to a temperature in kelvin.

##### Amount of Substance
The SI unit of quantity or amount of substance is the mole, a fundamental unit of the SI. There are no other modern units in which amount of substance is measured, so no conversion factors are required. Often, however, units of mass or volume are used to give the amount of substance. Conversion of these to the mole requires the use of appropriate measured physical constants, the molar mass or the molar volume. Since 1971, by definition one mole of entities is the same number of entities as there are atoms of carbon-12 in exactly 0.012 kilogram of carbon-12, which is Avogadro's number of entities (approximately 6.023 x 1023 entities).
##### Electric Current
The SI unit of electric current is the ampere, another fundamental unit of the SI. Since 1948, the ampere is by definition that constant current which, if maintained in two straight parallel conductors of infinite length, of neglegible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force exactly equal to 2 x 10-7 newton per metre of length. There are no other modern units in which current is measured, so no conversion factors are required.

#### SI Derived Units and Conversions

All physical quantities which are not those of the base units of the SI, such as volume, are measured in units derived from the base units. Many units can be derived from the seven base units of the SI, but only a comparatively small number need be introduced in an elementary course in chemistry. These derived units are introduced now, together with conversion factors which can be used to convert measurements made in older systems to appropriate SI units. The common SI derived units used in chemistry and physics are given in the table below.
##### Selected Derived Units of the International System
Quantity
Unit name
Unit Symbol
Definition
area
square metre
m2
m2
volume
cubic metre
m3
m3
force
newton
N
kg•m/s2
pressure
pascal
Pa
kg•m/s2
energy
joule
J
kg•m2/s2
power
watt
W
kg•m2/s3
charge
coulomb
C
A•s
potential difference
volt
V
kg•m2/s3•A
resistance
ohm
G
kg•m2/s3•A2
conductance
siemens
S
A2•s3/kg•m2
capacitance
F
A2•s4/kg•m2

Example. The derived unit volume, since it is the cube of a length, can be measured in the cube of the base unit for length, the cubic metre (m3). Density, mass per unit volume, is then measured in kilograms per cubic metre (kg/m3); the kg/m3 is identical to the g/dm3. A density of 1.25 g/cm3 is a density of 1250 kg/m3.

Some of these derived units are used often enough that special names and symbols are used for them. They are listed in the table above with their definitions in terms of base units of the SI. This list is not all that are available. The volt is often described as the unit of electromotive force as well as the unit of potential difference.

##### Volume
The SI unit of volume is the cube of the SI unit of length, and so it is the cubic metre (m3). The cubic centimetre (cm3) and cubic decimetre (dm3) are convenient units of volume which are widely used in chemistry; 1000 cm3 = 1 dm3 and 1000 dm3 = 1 m3. The litre is an older, but common, name for the cubic decimetre. Both the symbol l and the symbol L have been used for the litre, but the capital L symbol is preferred and will be used. The millilitre (mL) is identical to the cubic centimetre (cm3).
##### Force
The SI unit of force is the kilogram-metre per second squared which is called the newton (1 N = 1 kg-m/s2). The newton is obtained as a result of Newton's first law of motion, force equals mass times acceleration; one newton is that force which when applied to a mass of one kilogram imparts to it an acceleration of one metre/second2. Conversion factors between the other units of force and the newton are: 1 dyne = 1.0 x 10-5 N (exactly); 1 kilogram-force = 9.80665 N (exactly).

Both the pound-force and the kilogram-force are units which depend upon the force of terrestrial gravitation; the value of the kilogram-force is defined in terms of the standard terrestrial force of gravity. The relation between the actual force of gravity and mass is given by Newton's law of gravitation, F = gmm'/l2, where m and m' are the masses of the two attracting objects, l is the linear distance separating them, and g is the Newtonian constant of gravitation. The value of g is found to be 6.67259(85) x 10-11 m3 kg-1 s-2.

##### Pressure
The SI unit of pressure is the kilogram per metre-second squared, which is called the pascal (1 Pa = 1 kg/m s2). Since pressure is force per unit area, one pascal (Pa) is one newton per square metre. An alternative way of thinking of the pascal is as one joule/m3 which is helpful in compression work. Conversion between other units of pressure and the pascal are:

1 mmHg (0oC) = 133.322... Pa. The millimeter of mercury, mm of Hg, is almost the same as the exactly defined torr; 1 torr = (101325/760) Pa (exactly)

1 bar = 100000 Pa (exactly)

1 standard atmosphere (atm) = 101325 Pa (exactly) = 760 Torr = 760 mmHg

##### Energy
The SI unit of energy is the kilogram-metre2 per second squared which is called the joule (1 J = 1 kg-m2/s2). In the SI, work and energy have the same units since energy is the ability to do work. Work may be considered in either of two equivalent ways: as the product of a force and the distance over which that force is exerted, so that one joule equals one newton-metre, or as the product of a potential difference and the charge separated by that potential difference, so that one joule equals one volt-coulomb. Conversion factors between other units of energy and the joule are:

1 erg = 1.0 x 10-7 J (exactly)

1 foot-pound = 1.355818... J

1 calorie (cal) = 4.184 J (exactly)

1 litre-atmosphere = 101.325 J (exactly)

1 British Thermal Unit (BTU) = 1055.06 J (exactly)

1 kilowatt-hour (kWh) = 3.6 MJ (exactly)

The electron-volt is used as an energy unit in nuclear physics; 1 electron-volt (eV) = 1.6021773... x 10-19 J. One eV/particle corresponds to an energy of 96485.309... J/mol. The calorie and BTU given here are the modern thermochemical values. Other archaic calories and BTUs did exist and were of slightly different magnitude.

##### Power
The SI unit of power is the kilogram-meter2 per second cubed, which is called the watt (1 W = 1 kg-m2/s3). Since power is the energy used per unit of time, it is derived as the energy/time quotient. A power of one watt is used when an energy of one joule is expended in one second, so one watt equals one joule/second or one volt-ampere. The only significant unit of power used prior to the SI in English-speaking countries was the mechanical horsepower, defined as equal to 550 foot-pounds per second and 1 mechanical horsepower (hp) = 745.700 W. Other horsepower units used were not exact equivalents. Some of these were: 1 horsepower (boiler) = 9.80950 kW; 1 horsepower (electric) = 746.0 W; 1 horsepower (water) = 746.043 W; 1 horsepower (metric) = 735.499 W. There are also several different (and flexibly defined) horsepowers used in automotive measurements.
##### Electrical Charge
The SI unit of electrical charge is the ampere-second, which is called the coulomb. The coulomb is the amount of charge passed when a current of one ampere flows for one second. There are no other modern units of electrical charge, although the Faraday (amount of electrical charge possessed by one mole of electrons) is sometimes considered to be such a unit. The Faraday is actually the ratio of electrical charge/amount of substance and has the modern value and units of 96485.309... C/mol of elementary charges.
##### Electrical Potential
The SI unit of electrical potential or electrical potential difference, sometimes also known as the unit of electromotive force, is the volt. The volt is the kilogram-metre2 per second-cubed ampere(1 V = 1 kg m2/s3A) which is the ratio of energy to electrical charge (1 V = 1 J/C). The volt may be viewed as the electrical force which is equivalent to a physical force of one newton moving a charge of one coulomb over a distance of one metre.

1 V = 1 N m/C = 1 N/(C/m))

There are no other modern units of potential difference. The SI volt is identical to the "absolute volt" used in some non-SI systems of measurement.