The ground state properties of a one-dimensional system with particle-hole symmetry, consisting of a gate controlled dot coupled to an interacting reservoir, are explored using the numerical DMRG method. It has previously been shown that the system's thermodynamic properties as a function of the gate voltage in the Luttinger liquid phase are qualitatively similar to the behaviour of a non-interacting wire with an effective (renormalized) dot-lead coupling. Here we examine the thermodynamic properties of the wire in the charge density wave phase, and show that these properties behave quite differently. The number of electrons in the system remains constant as a function of the gate voltage, while the total energy becomes linear. Moreover, by tuning the gate voltage on the dot in the charge density wave phase it is possible to drive the wire through a first order quantum phase transition in which the population of each site in the wire is inverted.