Which of the following statements about the Boyle temperature $T_{B}$ of a van der Waals gas are correct?

AAt $T_{B}$ the second virial coefficient $B = b - a / RT$ is zero
BAt $T = T_{B}$ the initial slope of $Z$ versus $P$ is zero
CAt temperatures above $T_{B}$ the gas shows $Z$ greater than 1 at all pressures
DAt temperatures below $T_{B}$ the gas shows $Z$ greater than 1 at all pressures

Answer: A, B, C

Expanding the van der Waals equation gives

Z \approx 1 + \frac{1}{RT} (b - \frac{a}{RT}) P

, so the initial slope and the second virial coefficient both vanish when

RT = a / b

, i.e. at

T_{B}

(A and B true). Above

T_{B}

the repulsive term dominates at every pressure and

Z

stays above 1 (C true). Below

T_{B}

attractions win at low pressure and

Z

first dips below 1 (D false).