# Is the compressibility factor smaller or greater than 1 at low temperature and high pressure?

The compressibility factor of a gas is defined as $Z = pV/(nRT)$. If attractive intermolecular forces dominate then $Z$ tends to be smaller than 1, and vice versa if repulsive forces dominate.

In the case where temperature is low, attractive forces increase due to less kinetic energy, but at the same time repulsive forces also increase due to high pressure. How can I determine which force will dominate?

The figure shows such a plot of Z vs. reduced pressure $P_R$ at different reduced temperatures. The reduced temperature is $T_R=T/T_c$ where T is the actual temperature and $T_c$ the temperature at the critical point. From the figure when $T_R \lt 2$ and $P_R \lt 6.5$ then $Z \lt 1$ and this indicates that a real gas (in so far as such is described by the Van-der-Waals eqn) has a lower pressure than an ideal gas and so this indicates that the molecules are more influenced by the attractive part of the potential than its repulsive part, which is due to their finite molecular volume. If the temperature is higher, above $T_R \approx 2$ (not shown), then Z is positive for all pressures, and similarly above $Z \approx 7$ for all temperatures. In these cases repulsion dominates over attraction.