Units and Conversions

Further information on this topic can be found in Chapter 3 of:

Maths from Scratch for Biologists

Numerical ability is an essential skill for everyone studying the biological sciences but many students are frightened by the 'perceived' difficulty of mathematics, and are nervous about applying mathematical skills in their chosen field of study. Maths from Scratch for Biologists is a highly instructive, informal text that explains step by step how and why you need to tackle maths within the biological sciences. (Amazon.co.UK)

The Systeme International d'Unites (SI) (1960) is based upon seven principal units:

Derived S.I. Units:

farad [F]
The SI unit of the capacitance of an electrical system, i.e. its capacity to store electricity. This is a large unit as defined and is often used as a microfarad [µF]. (Michael Faraday, 1791-1867)

hertz [Hz]
The SI unit of the frequency of a periodic phenomenon. One hertz indicates that 1 cycle of the phenomenon occurs every second. Higher frequencies such as the kilohertz [kHz] and megahertz [MHz] are commonly used. (Heinrich Rudolph Hertz, 1857-94)

joule [J]
The SI unit of work or energy. One joule is the amount of work done when an applied force of 1 newton moves through a distance of 1 metre in the direction of the force. (James Prescott Joule, 1818-89)

Newton [N]
The SI unit of force. One Newton is the force required to give a mass of 1 kilogram an acceleration of 1 metre per second per second. (Isaac Newton, 1642-1727)

ohm [W]
The SI unit of resistance of an electrical conductor. Its symbol is the Greek letter known as 'omega' (W). (Georg Simon Ohm, 1789-1854)

pascal [Pa]
The SI unit of pressure. One Pascal is the pressure generated by a force of 1 Newton acting on an area of 1 square metre. A small unit, often used as the kilopascal [kPa]. (Blaise Pascal, 1623-62)

siemen [S]
The SI unit of conductance. One siemen is the conductance at which a potential of 1 volt causes a current of 1 ampere to flow. Often expressed as microsiemen [µS] for aqueous solutions. (Named after ???)

volt [V]
The SI unit of electric potential. One volt is the difference of potential between two points of an electrical conductor when a current of 1 ampere flowing between those points dissipates a power of 1 watt. (Count Alessandro Giuseppe Anastasio Volta, 1745-1827)

watt [W]
The unit used to measure power or the rate of doing work. One watt is a power of 1 joule per second. (James Watt, 1736-1819)

S.I. Prefixes:

• The farad is a rather large unit for most uses, so 1*10^-6 F = 1 microfarad (1 µF) is commonly used.

• 1*10³ Hz = 1 kilohertz [1 kHz] and 1*10⁶ Hz = 1 megahertz [1 MHz].

• The watt is a small unit, generally used in terms of 1000 watts at a time = 1 kilowatt [1 kW]. 1*10⁶ watts = 1 megawatt [MW] and 1*10⁹ watts = 1 gigawatt [GW].

Usage:

• Many units are eponymous, i.e. named after people who were prominent in early work done within the field in which the unit is used.

• A unit which is named after a person is written in lower case (newton, volt, pascal etc.) when named in full, but should start with a capital letter (N, V, Pa, etc.) when abbreviated.

• The unit or abbreviation should be separated from the number to which it refers by a space, e.g. "99 µF", not "99µF".

• Unit abbreviations (such as J, N, g, Pa, etc.) are NEVER followed by a full-stop unless at the end of a sentence.

• Units written in abbreviated form are NEVER pluralised.

• To make numbers easier to read they may be divided into groups of 3 separated by spaces (or half-spaces) but NOT commas.

• Any unit may take only ONE prefix. For example "millimillimetre" should be written as "micrometre" (µm).

Why do we need S.I. units?

If x = 1 m, volume of cube = 1 * 1 * 1 = 1 m³

Since 1 m = 100 cm, volume of cube = 100 * 100 * 100 = 1*10⁶ cm³

and 1 m = 1*10⁶ µm, volume of cube = 1*10⁶ * 1*10⁶ * 1*10⁶ = 1*10¹⁸ µm³

Therefore, 1 cm³ = 1*10¹² µm³, etc.

Centigrade and Celsius:

The Centigrade (from the Latin centum, "a hundred", plus gradus, "degree") scale of temperature (1801) is defined as "a thermometric scale on which the interval between the freezing point of water and the boiling point of water is divided into 100 degrees with 0° representing the freezing point and 100° the boiling point".

This has now been replaced by the Celsius scale (named after Anders Celsius, 1701-1744), "an international thermometric scale on which the interval between the triple point * of water and the boiling point of water is divided into 99.99 degrees with 0.01° representing the triple point and 100° the boiling point".

* The triple point of water is the condition of temperature and pressure under which the gaseous, liquid, and solid phases of a substance can exist in equilibrium.

However, the S.I. unit of temperature is the Kelvin:

Example:

25°C = 273 + 25 = 298 K

Work and energy

Energy is the ability of a system to perform work (the transfer of energy from one system to another).
Power is the rate at which work is done, i.e. the amount of work per unit time.
Force is an action which maintains or alters the position of a body, or distorts it. Because force has both magnitude and direction, it is known as a vector quantity.
The mass of an object is a measure of the object's resistance to changes in either the speed or direction. Weight and mass are not the same thing! The weight of an object is the force it exerts under a given gravitational force. Thus the weight of an astronaut is different on the moon than on the Earth, but their mass is constant. Weight should therefore be measured in newtons and mass in kilograms.

There are five main forms of energy:

• Mechanical energy: associated with motion.
• Heat: caused by the motion of the particles that make up matter.
• Electromagnetic energy: the energy given off by moving electric charges.
• Chemical energy: the energy released or absorbed when bonds between atoms are formed or broken.
• Nuclear energy: the energy stored in the nuclei of atoms.

The joule is the SI unit of work or energy, equal to the work done by a force of one newton acting through a distance of one metre.

J = N * m

"Work" measures the energy transfer that occurs when an object is moved over a distance by an external force at least part of which is applied in the direction of the movement. If the force is constant, the energy expended can be calculated by multiplying the length of the path by the force acting along the path.

Example:
If a pint of beer weighs 675 grams, how much work is done in raising the glass 0.75 metres from the bar to your mouth? The force exerted by the glass due to gravity is given by the equation:

force = mass * acceleration,  so in this case:
force = mass * acceleration due to gravity
= 0.675 kg * 9.8 m s-2 = 6.615 N
Work = force * distance (J = N * m), so:
6.615 N * 0.75 m = 4.96 J

Kinetic energy is the energy of motion. An object which is in motion has kinetic energy.

Kinetic energy = 0.5 * mass * velocity²

Example:

What is the kinetic energy of an 85 kg man running at 4.25 m^-2 ?
0.5(85 * 4.252) = 768 J
What is his kinetic energy if he sprints at 8.9 m^-2 ?
0.5(85 * 8.92) = 3366 J

Note that kinetic energy is directly proportional to the mass of the moving object, but proportional to the square of its speed, e.g. as speed doubles, kinetic energy increases four-fold, but tripling the speed increases the energy nine-fold! That's why high speed car crashes are so dangerous!

Potential energy is the energy stored in an object as the result of its position. The main sources of potential energy are elastic potential (as in bent branches and bows and arrows) and gravitational potential (as in falling down stairs).

Potential energy = mass * gravity * height (defined with respect to an arbitrary zero height)

There is a direct relationship between stored gravitational potential and the height which an object falls - as you will know if an apple falls on your head from five metres compared with one metre!

Electricity:

The potential difference (voltage) across an ideal conductor is proportional to the current through it.

The constant of proportionality is called the "resistance", R.

Ohm's Law is given by:

V = I * R

I = V/R

R = V/I

where V is the potential difference between two points which include a resistance R and I is the current flowing through the resistance.

Electrical power (in watts, W) = V * I

In biology, conductance is often quoted, S = 1/R; i.e. the ability of a solution to allow an electric current to flow through it; the reciprocal of resistance. Conductance (or conductivity) is usually expressed as microsiemens per cm (µS cm^-1).

Example:

S = 1/6666 = 1.5*10^-4 = 150 µS cm^-1

© MicrobiologyBytes 2007.