## The Equation of State For an Ideal Gas

The three gas laws that separately describe the behaviours of an ideal gas can be combined into a single equation called 'the equation of state'.

These three laws, with=pressure (Pa or),=Volume () and=temperature (Kelvin or K),

The Pressure Law,a constant, subject to the volume being constant

Charles's Law,a constant, subject to the pressure being constant

Boyles's Law,a constant, subject to the temperature being constant

can be combined into a single equation by applying the pressure law then Charles's Law for example.

Suppose a gas is in the initial state

Applying Boyle's Law giveswith the temperaturekept constant.

Now keep the volume constant atand apply the Pressure Law to get

Now keep the pressure constant atand apply the Pressure Law to get

Multiply these three equations together to get

Cancellation ofandfrom both sides givesor

We can writewhereis a constant. In general the constant will depend on the amount of gas present. In fact it will only depend on the number of mols of gas, not on the mass of gas. Further, by considering any two ofandto be constant in turn and allowing the remaining variable to vary, we see thatmust be directly proportional to the number of mols. If we label the number of mols asthen we can writewhereis a constant. In fact is a universal constant, called the universal gas constant: