  Consider a Grothendieck category $\mathcal{G}$ along with a choice of
generator $G$, or equivalently a generating set $\{G_i\}$. We introduce the
derived category $\mathcal{D}(G)$, which kills all $G$-acyclic complexes, by
putting a suitable model structure on the category of chain complexes. It
follows that the category $\mathcal{D}(G)$ is always a well-generated
triangulated category. It is compactly generated whenever the generating set
$\{G_i\}$ has each $G_i$ finitely presented, and in this case we show that two
recollement situations hold. The first is when passing from the homotopy
category $K(\mathcal{G})$ to $\mathcal{D}(G)$. The second is a $G$-derived
analog to the recollement of Krause. We illustrate with several examples
ranging from pure and clean derived categories to quasi-coherent sheaves on the
projective line $P^1(k)$.
