We study the relation between the chemical composition and the type of dust present in a group of 20 Galactic planetary nebulae (PNe) that have high quality optical and infrared spectra. The optical spectra are used, together with the best available ionization correction factors, to calculate the abundances of Ar, C, Cl, He, N, Ne, and O relative to H. The infrared spectra are used to classify the PNe in two groups depending on whether the observed dust features are representative of oxygen-rich or carbon-rich environments. The sample contains one object from the halo, eight from the bulge, and eleven from the local disc. We compare their chemical abundances with nucleosynthesis model predictions and with the ones obtained in seven Galactic H II regions of the solar neighbourhood. We find evidence of O enrichment (by $sim$ 0.3 dex) in all but one of the PNe with carbon-rich dust (CRD). Our analysis shows that Ar, and especially Cl, are the best metallicity indicators of the progenitors of PNe. There is a tight correlation between the abundances of Ar and Cl in all the objects, in agreement with a lockstep evolution of both elements. The range of metallicities implied by the Cl abundances covers one order of magnitude and we find significant differences in the initial masses and metallicities of the PNe with CRD and oxygen-rich dust (ORD). The PNe with CRD tend to have intermediate masses and low metallicities, whereas most of the PNe with ORD show higher enrichments in N and He, suggesting that they had high-mass progenitors.
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