Mathematics for Chemistry/Units and dimensions

Units, multipliers and prefixes

It is usually necessary in chemistry to be familiar with at least three systems of units, Le Système International d'Unités (SI), atomic units as used in theoretical calculations and the unit system used by the experimentalists. Thus if we are dealing with the ionization energy, the units involved will be the Joule (J), the Hartree (E_h, the atomic unit of energy), and the electron volt (eV).

These units all have their own advantages;

• The SI unit should be understood by all scientists regardless of their field.
• The atomic unit system is the natural unit for theory as most of the fundamental constants are unity and equations can be cast in dimensionless forms.
• The electron volt comes from the operation of the ionization apparatus where individual electrons are accelerated between plates which have a potential difference in Volts.

An advantage of the SI system is that the dimensionality of each term is made clear as the fundamental constants have structure. This is a complicated way of saying that if you know the dimensionality of all the things you are working with you know an awful lot about the mathematics and properties such as scaling with size of your system. Also, the same system of units can describe both the output of a large power station (gigaJoules), or the interaction of two inert gas atoms, (a few kJ per mole or a very small number of Joules per molecule when it has been divided by Avogadro's number).

In SI the symbols for units are lower case unless derived from a person's name, e.g. ampere is A and kelvin is K.

SI base units

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

Derived units used in chemistry

Quantity	Unit	Name	Symbol
Area	m2
Volume	m3
Velocity	m s-1
Acceleration	m s-2
Density	kg m-3
Entropy	J mol-1 K-1
Force	kg m s-2	newton	N
Energy	N m	joule	J
Pressure	N m-2	pascal	Pa
Frequency	s-1	hertz	Hz

Approved prefixes for SI units

Prefix	Factor	Symbol
atto	10-18	a
femto	10-15	f
pico	10-12	p
nano	10-9	n
micro	10-6	μ
milli	10-3	m
centi	10-2	c
deci	10-1	d
kilo	103	k
mega	106	M
giga	109	G
tera	1012	T
peta	1015	P
exa	1018	E

Note the use of capitals and lower case and the increment on the exponent being factors of 3. Notice also centi and deci are supposed to disappear with time leaving only the powers of 1000.

Conversion factors

The ${\hat {}}$, (sometimes call caret or hat), sign is another notation for to the power of. E means times 10 to the power of, and is used a great deal in computer program output.

Energy

An approximation of how much of a chemical bond each energy corresponds to is placed next to each one. This indicates that light of energy 4 eV can break chemical bonds and possibly be dangerous to life, whereas infrared radiation of a few cm^-1 is harmless.

• 1 eV = 96.48530891 kJ mol^-1 (Near infrared), approximately 0.26 chemical bonds
• 1 kcal mol^-1 = 4.184000000 kJ mol^-1 (Near infrared), approximately 0.01 chemical bonds
• 1 MHz = 0.399031E-06 kJ mol^-1 (Radio waves), approximately 0.00 chemical bonds
• 1 cm^-1 = 0.01196265819 kJ mol^-1 (Far infrared), approximately 0.00 chemical bonds

Wavelength, generally measure in nanometres and used in UV spectroscopy is defined as an inverse and so has a reciprocal relationship.

Length

There is the metre, the Angstrom (10^-10 m), the micron (10^-6 m), the astronomical unit (AU)and many old units such as feet, inches and light years.

Angles

The radian to degree conversion is 57.2957795130824, (i.e. a little bit less than 60, remember your equilateral triangle and radian sector).

Dipole moment

1 Debye = 3.335640035 Cm x 10^-30 (coulomb metre)

Magnetic Susceptibility

1 cm³ mol^-1 = 16.60540984 JT^-2 x 10³⁰ (joule tesla²)

Old units

Occasionally, knowledge of older units may be required. Imperial units, or convert energies from BTUs in a thermodynamics project, etc.

In university laboratory classes you would probably be given material on the Quantity Calculus notation and methodology which is to be highly recommended for scientific work.

A good reference for units, quantity calculus and SI is: I. Mills, T. Cuitas, K. Homann, N. Kallay, K. Kuchitsu, Quantities, units and symbols in physical chemistry, 2nd edition, (Oxford:Blackwell Scientific Publications,1993).

Unit labels

The labelling of tables and axes of graphs should be done so that the numbers are dimensionless, e.g. temperature is $T/K$,

$\ln(k/k_{0})$ and energy mol / kJ etc.

This can look a little strange at first. Examine good text books like Atkins Physical Chemistry which follow SI carefully to see this in action.

The hardest thing with conversion factors is to get them the right way round. A common error is to divide when you should be multiplying, also another common error is to fail to raise a conversion factor to a power.

${\text{Conversion factor}}={\frac {\text{Units required}}{\text{Units given}}}$

1 eV = 96.48530891 kJ mol^-1

1 cm^-1 = 0.01196265819 kJ mol^-1

To convert eV to cm^-1, first convert to kJ per mole by multiplying by 96.48530891 / 1. Then convert to cm^-1 by multiplying by 1 / 0.01196265819 giving 8065.540901. If we tried to go directly to the conversion factor in 1 step it is easy to get it upside down. However, common sense tell us that there are a lot of cm^-1s in an eV so it should be obviously wrong.

1 inch = 2.54 centimetres. If there is a surface of nickel electrode of 2 * 1.5 square inches it must be 2 * 1.5 * 2.54² square centimetres.

To convert to square metres, the SI unit we must divide this by 100 * 100 not just 100.

Dimensional analysis

The technique of adding unit labels to numbers is especially useful, in that analysis of the units in an equation can be used to double-check the answer.

An aside on scaling

One of the reasons powers of variables are so important is because they relate to the way quantities scale. Physicists in particular are interested in the way variables scale in the limit of very large values. Take cooking the turkey for Christmas dinner. The amount of turkey you can afford is linear, (power 1), in your income. The size of an individual serving is quadratic, (power 2), in the radius of the plates being used. The cooking time will be something like cubic in the diameter of the turkey as it can be presumed to be linear in the mass.

(In the limit of a very large turkey, say one the diameter of the earth being heated up by a nearby star, the internal conductivity of the turkey would limit the cooking time and the time taken would be exponential. No power can go faster / steeper than exponential in the limit. The series expansion of $e^{x}$ goes on forever even though $1/n!$ gets very small.)

Another example of this is why dinosaurs had fatter legs than modern lizards. If dinosaurs had legs in proportion to small lizards the mass to be supported rises as length to the power 3 but the strength of the legs only rises as the area of the cross section, power 2. Therefore the bigger the animal the more enormous the legs must become, which is why a rhino is a very chunky looking version of a pig.

There is a very good article on this in Cooper, Necia Grant; West, Geoffrey B., Particle Physics: A Los Alamos Primer, ISBN 0521347807