The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain Ω⊂ℝd, d = 1, 2 or d = 3 is the minimizer of the total energy [Escr ]ελ; [Escr ]ελ involves the squares of the scaled Planck's constant ε and the scaled minimal Debye length λ. In applications one frequently has λ2[Lt ]1. In these cases the zero-space-charge approximation is rigorously justified. As λ → 0, the particle densities converge to the minimizer of a limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal energies one gets an differential-algebraic system for the limiting (λ = 0) particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit ε → 0 exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small ε, a reasonable approximation of the quantum model if and only if the quantum gap vanishes. The simultaneous limit ε = λ → 0 is analyzed.