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Written by Tim Sheppard MBBS BSc. Last updated 11/11/10

A small velvety animal, a spicy chocolate sauce, a highly-pigmented patch of skin, a spy, a structure to keep a beach in place... what do these things have in common? They are all given the name **mole**, but unfortunately none of them are particularly relevant for this page, as another meaning of the word *mole* is *the amount of substance which contains the same number of particles as there are in 12g of carbon-12.*

Although the above 'official' definition is difficult to apply, it can be adjusted slightly to sound more relevant. Since carbon has an atomic weight of 12, then 12g is its atomic weight in grams. So a mole is ** the amount of substance which has a mass equivalent to its molecular (or atomic) weight in grams** - i.e if something has a molecular weight of 32, a mole of that substance weighs 32g. Most importantly, this number of particles is the same every time.

An Italian physicist named Amedeo Avogadro came up with a hypothesis concerning molecules and volumes, which became known as 'Avogadro's Law'. He also considered that particles could be made up of molecules which could be made up of atoms (although these terms were not then in place). When scientists worked out how many atoms there were in a mole, they decided to commemorate Avogadro's intellectual achievements - and his contributions to science - by naming the number after him. Since then, the number has been termed Avogadro's number - despite the fact he never knew it himself! - and is immensely useful to scientists worldwide.

So what is it? Avogadro's number is given as

Therefore

So why is that so helpful? Well, as mentioned previously, it's all to do with reactions and calculations. Let's take the following reaction:

3H

One way of reading that is to say

Note that commonly 'moles' is abbreviated to

I suppose one way of looking at it would be to say that saying 'moles' is just like multiplying everything by 6.022... x 10

Although it sounds like a concept - and it is a concept - molarity is also a unit of measurement. You have '1 molarity' or '3 molarity' or '0.03452 molarity'. 1 molarity means 1 mole of a particular substance dissolved in a litre (or 1000cm^{3}) of water, and it is given the abbreviation **M**.

Once again it is of no interest to you until you have to work something out. Then it is very helpful indeed, because **it is a measure of concentration**. In fact, it's not just 'a measure of concentration', it is **the** measure which everyone uses. pH is -log_{10}[H^{+}] and the concentration of H^{+} ions is given in molarity (or moles per litre). When calculating the equilibrium constant **K _{c}**, the concentrations required are in molarity (or moles per litre).

So, put simply,

As mentioned previously, a physicist named Avogadro came up with the hypothesis that if different gases occupied the same volume under the same temperature and pressure, they had the same number of molecules in them. In other words, the volume (V) of a gas is proportional to the number of moles (n).

A slightly different observation was made concerning the effect of temperature. A scientist named Jacques Charles noticed that as temperature increased, so did the volume. Just as Avogadro suggested that volume was proportional to the number of moles, Charles came up with the theory that the volume (V) of a gas is proportional to the temperature (T).

Finally, a man named Robert Boyle realised that as you increased the pressure, the volume decreased. This is quite easy to see generally - if you squeeze a plastic bottle full of air, you will compress the bottle, effectively reducing the volume that the gas occupies. In other words, the volume (V) of a gas is inversely proportional (i.e. as one increases, the other decreases) to pressure (p).

These three proportionalities have one common feature - volume. It was decided, then, that you could bung them all together in something we like to call the **ideal gas equation**. If you put them all together, you see that volume is proportional to the product of **n** and **T**, divided by **p**. Multiply both sides by **p** and the right hand side cancels out, giving **p****V** is proportional to **n****T**, or **p****V** is equal to (**n****T** ? something).

Experiments have revealed this 'something' to have a value, which is approximately 8.31, and it is known as **R**, the gas constant. That is, assuming you have the units right. Temperature could be in degrees centigrade (celsius) or in Kelvin, and either of these would give completely different results. When using the ideal gas equation, the most difficult and important thing to remember is **units**! Temperature is measured, for the purposes of this equation, in Kelvin. Number of moles is quite simply a number - the number of moles. Pressure is measured in *Pascals* (Pa); not kilopascals (kPa) as it will often be given in. And volume is measured in metres cubed (m^{3}).

So, you know how it was derived, but what exactly is the point? Well, the ideal gas equation is a way of finding out the volume of a gas, if you know the temperature and pressure that it's in, and how many moles there are. In fact, if you know the volume, pressure and temperature, you could work out how many moles there are without even having to weigh the gas (which is a whole different challenge!); basically it is **a very useful equation for making calculations about gases**.

- Sutton R, Rockett B, Swindells P. (2000) Chemistry for the Life Sciences. New York: Taylor & Francis Ltd