Let $${\mathcal {G}}$$ G be any Grothendieck category along with a choice of generator $$G$$ G , or equivalently a generating set $$\{G_i\}$$ { G i } . We introduce the derived category $${\mathcal {D}}(G)$$ D ( G ) , which kills all $$G$$ G -acyclic complexes, by putting a suitable model structure on the category $$\text {Ch}({\mathcal {G}})$$ Ch ( G ) of chain complexes. It follows that the category $${\mathcal {D}}(G)$$ D ( G ) is always a well-generated triangulated category. It is compactly generated whenever the generating set $$\{G_i\}$$ { G i } has each $$G_i$$ G i finitely presented, and in this case, we show that two recollement situations hold. The first is when passing from the homotopy category $$K({\mathcal {G}})$$ K ( G ) to $${\mathcal {D}}(G)$$ D ( G ) . The second is a $$G$$ G -derived analog of a recollement due to Krause. We describe several examples ranging from pure and clean derived categories to quasi-coherent sheaves on the projective line.