Any of the real gas equations of state can be written as a power series expansion. For the van der Waals
equation, we have
PV RT
= 1 +
(
b -
a RT
)
1 V
+
b^{2}
1 V^{2}
+ ...
At the Boyle temperature T_{Boyle} intermolecular attractive
and repulsive effects are balanced so that the terms linear in P (or 1/V) in the expansion to go to zero.
The Boyle temperature for the van der Waals gas is then
T_{Boyle} =
a bR
Inspection of the power series expansion for the Dieterici gas shows that
T_{Boyle} =
(
a bR
)^{2/3}
For most gases, experiment shows that the ratio of the Boyle temperature to the critical
temperature is consistently about 2.75 (except for He).