Boyle temperature is the temperature at which a non-ideal gas behaves like an ideal gas over an appreciable range of pressure. At Boyle temperature, setting the compressibility factor to 1, and one can obtain
Where a and b are van der Waals parameters.
This is the virial equation of state and describes a real gas. The Boyle temperature is formally defined as the temperature for which the second virial coefficient, becomes zero. It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out. 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 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. We then have
where is the compressibility factor