Boyle temperature

The Boyle temperature is formally defined as the temperature for which the second virial coefficient, formula_1 becomes zero.
It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out
formula_2
This is the virial equation of state and describes a real gas. 
Since higher order virial coefficients are generally much smaller than the second coefficient, the gas tends to behave as an ideal gas over a wider range of pressures when the temperature reaches the Boyle temperature (or when formula_3 or p are minimized). 
In any case, when the pressures are low, the second virial coefficient will be the only relevant one because the remaining concern terms of higher order on the pressure. Also at Boyle temperature the dip in a PV diagram tends to a straight line over a period of pressure. We then have
where formula_5 is the compressibility factor.
For negligible formula_6, using the van der Waals equation, one can prove formula_7.