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
andto 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 as
then we can write
where
is a constant. In fact is a universal constant, called the universal gas constant: