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 106 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/cm^{3}, or kg/dm^{3},
or kg/L. This is 2.07(0.001 kg/g)(106cm^{3}/m^{3}) = 2.07
x 103 kg/m^{3}.
Using the mass as 0.150637 kg and the volume as 72.8 x 10^{6}
m^{3}. The density of sulfur is then (0.150637 kg)/(72.8 x 10^{6}
m^{3}) = 2.07 x 10^{3} kg/m^{3}. The quantity calculated
gives the result in SI base units without conversion. Reporting the answer
as 2070 kg/m^{3}, 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
Length
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 platinumiridium 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 Cesium133 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 Cs133 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 +0.01^{o} 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 carbon12 in
exactly 0.012 kilogram of carbon12, which is Avogadro's number of entities
(approximately 6.023 x 10^{23} 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 crosssection, 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

m^{2}

m^{2}

volume

cubic metre

m^{3}

m^{3}

force

newton

N

kg•m/s^{2}

pressure

pascal

Pa

kg•m/s^{2}

energy

joule

J

kg•m^{2}/s^{2}

power

watt

W

kg•m^{2}/s^{3}

charge

coulomb

C

A•s

potential difference

volt

V

kg•m^{2}/s^{3}•A

resistance

ohm

G

kg•m^{2}/s^{3}•A^{2}

conductance

siemens

S

A^{2}•s^{3}/kg•m^{2}

capacitance

farad

F

A^{2}•s^{4}/kg•m^{2}


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 (m^{3}). Density, mass per unit volume, is then measured
in kilograms per cubic metre (kg/m^{3}); the kg/m^{3} is identical
to the g/dm^{3}. A density of 1.25 g/cm^{3} is a density
of 1250 kg/m^{3}.
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.
Area
The SI unit of area is the square of the SI unit of length,
and so it is the square metre (m^{2}). Conversion factors between
other metric units and the square metre are: 1 cm^{2} = 10^{4}
m^{2} (exactly); 1 are = 100 m^{2} (exactly); 1 hectare =
10000 m^{2} (exactly). Conversion factors between English units and
the square metre are: 1 square foot = 0.09290304 m^{2} (exactly);
1 square yard = 0.83612736 m^{2 }(exactly).
Volume
The SI unit of volume is the cube of the SI unit of length, and so
it is the cubic metre (m^{3}). The cubic centimetre (cm^{3})
and cubic decimetre (dm^{3}) are convenient units of volume which
are widely used in chemistry; 1000 cm^{3} = 1 dm^{3} and
1000 dm^{3} = 1 m^{3}. 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 (cm^{3}).
Force
The SI unit of force is the kilogrammetre per second squared which
is called the newton (1 N = 1 kgm/s^{2}). 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/second^{2}. Conversion factors
between the other units of force and the newton are: 1 dyne = 1.0 x 10^{5}
N (exactly); 1 kilogramforce = 9.80665 N (exactly).
Both the poundforce and the kilogramforce are units which depend
upon the force of terrestrial gravitation; the value of the kilogramforce
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'/l^{2}, 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} m^{3} kg^{1} s^{2}.
Pressure
The SI unit of pressure is the kilogram per metresecond squared,
which is called the pascal (1 Pa = 1 kg/m s^{2}). 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/m^{3} which
is helpful in compression work. Conversion between other units of pressure
and the pascal are:
1 mmHg (0^{o}C) = 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 kilogrammetre^{2} per second
squared which is called the joule (1 J = 1 kgm^{2}/s^{2}).
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 newtonmetre, or as the product of a potential
difference and the charge separated by that potential difference, so that
one joule equals one voltcoulomb. Conversion factors between other units
of energy and the joule are:
1 erg = 1.0 x 10^{7} J (exactly)
1 footpound = 1.355818... J
1 calorie (cal) = 4.184 J (exactly)
1 litreatmosphere = 101.325 J (exactly)
1 British Thermal Unit (BTU) = 1055.06 J (exactly)
1 kilowatthour (kWh) = 3.6 MJ (exactly)
The electronvolt is used as an energy unit in nuclear physics;
1 electronvolt (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 kilogrammeter^{2} per second
cubed, which is called the watt (1 W = 1 kgm^{2}/s^{3}).
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 voltampere.
The only significant unit of power used prior to the SI in Englishspeaking
countries was the mechanical horsepower, defined as equal to 550 footpounds
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 amperesecond, 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 kilogrammetre^{2} per secondcubed ampere(1 V = 1
kg m^{2}/s^{3}A) 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 nonSI systems of
measurement.
