## 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 gives with the temperature kept constant.

Now keep the volume constant at and apply the Pressure Law to get Now keep the pressure constant at and apply the Pressure Law to get Multiply these three equations together to get Cancellation of and from both sides gives or We can write where is 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 of and to be constant in turn and allowing the remaining variable to vary, we see that must be directly proportional to the number of mols. If we label the number of mols as then we can write where is a constant. In fact is a universal constant, called the universal gas constant: 