Classical novae commonly show evidence of rapid dust formation within months of the outburst. However, it is unclear how molecules and grains are able to condense within the ejecta, given the potentially harsh environment created by ionizing radiation from the white dwarf. Motivated by the evidence for powerful radiative shocks within nova outflows, we propose that dust formation occurs within the cool, dense shell behind these shocks. We incorporate a simple molecular chemistry network and classical nucleation theory with a model for the thermodynamic evolution of the post-shock gas, in order to demonstrate the formation of both carbon and forsterite ($rm Mg_2SiO_4$) grains. The high densities due to radiative shock compression ($n sim 10^{14}$ cm$^{-3}$) result in CO saturation and rapid dust nucleation. Grains grow efficiently to large sizes $gtrsim 0.1mu$m, in agreement with IR observations of dust-producing novae, and with total dust masses sufficient to explain massive extinction events such as V705 Cas. As in dense stellar winds, dust formation is CO-regulated, with carbon-rich flows producing carbon-rich grains and oxygen-rich flows primarily forming silicates. CO is destroyed by non-thermal particles accelerated at the shock, allowing additional grain formation at late times, but the efficiency of this process appears to be low. Given observations showing that individual novae produce both carbonaceous and silicate grains, we concur with previous works attributing this bimodality to chemical heterogeneity of the ejecta. Nova outflows are diverse and inhomogeneous, and the observed variety of dust formation events can be reconciled by different abundances, the range of shock properties, and the observer viewing angle. The latter may govern the magnitude of extinction, with the deepest extinction events occurring for observers within the binary equatorial plane.
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