We have constructed MOCASSIN photoionization plus dust radiative transfer models for the Crab Nebula core-collapse supernova (CCSN) remnant, using either smooth or clumped mass distributions, in order to determine the chemical composition and masses of the nebular gas and dust. We computed models for several different geometries suggested for the nebular matter distribution but found that the observed gas and dust spectra are relatively insensitive to these geometries, being determined mainly by the spectrum of the pulsar wind nebula which ionizes and heats the nebula. Smooth distribution models are ruled out since they require 16-49 Msun of gas to fit the integrated optical nebular line fluxes, whereas our clumped models re quire 7.0 Msun of gas. A global gas-phase C/O ratio of 1.65 by number is derived, along with a He/H number ratio of 1.85, neither of which can be matched by current CCSN yield predictions. A carbonaceous dust composition is favoured by the observed gas-phase C/O ratio: amorphous carbon clumped model fits to the Crabs Herschel and Spitzer infrared spectral energy distribution imply the presence of 0.18-0.27 Msun of dust, corresponding to a gas to dust mass ratio of 26-39. Mixed dust chemistry models can also be accommodated, comprising 0.11-0.13 Msun of amorphous carbon and 0.39-0.47 Msun of silicates. Power-law grain size distributions with mass distributions that are weighted towards the largest grain radii are derived, favouring their longer-term survival when they eventually interact with the interstellar medium. The total mass of gas plus dust in the Crab Nebula is 7.2 +/- 0.5 Msun, consistent with a progenitor star mass of 9 Msun.
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