[confusedstud]

When learning about compressibility factor, we were taught that at moderate pressures Z<1 because they are attraction dominant, so in equation form Preal/Pideal<1 which means Preal<Pideal. However, high pressure Z>1 because they are repulsion dominant so the gases repels each other rather than attracts each other. So in equation form it was expressed as Preal/Pideal=Z>1 So this suggested that Preal>Pideal

However, the derivation of the van der Waals equation seems to suggest that Pideal>Preal so isn't there a contradiction in the thoeries?

[mjc123]

No, the vdW equation accommodates both:
P = RT/(V-b) - a/V² ≈ RT/V + RTb/V² - a/V²
so depending on the relative magnitudes of a, b and T, P may be less or greater than RT/V.