  We study $P-V$ critical behavior of 4-dimensional AdS black hole in the
Einstein-Maxwell gravity with conformal anomaly by treating the cosmological
constant as a variable related to the thermodynamic pressure. It shows that
there is no phase transition if taking $k=0$ or -1. When the charge $q_1$ of
conformal field and the coefficient $\alpha$ satisfy a certain relation, the
Van der Waal like phase transition for the spherical black hole can occur in
case of the temperature is lower than the small critical temperature or higher
than the large one. We also evaluate the critical exponents of the phase
transitions and find that the thermodynamic exponents associated with this
4-dimensional AdS black hole coincide with those of the van der Waals fluid.
