We analyze the far-infrared dust emission seen by the IRAS satellite in the Orion region as a function of the local radiation field intensity and compare the dust temperature and opacity with (CO)-C-12 and (CO)-C-13 emission. We interpret the infrared radiation in the framework of a single component large grain model and a multicomponent grain model consisting of subpopulations of grains with size-dependent temperatures. We find a strong dependence of the 100-mu-m optical depth derived using the large grain model on the average line-of-sight dust temperature and radiation field. In the hot environment surrounding high-luminosity sources and H II regions, all dust along the line-of-sight radiates at 100-mu-m, and the dust-to-gas ratio, based on the 100-mu-m opacity and I((CO)-C-13), appears to be in agreement with the standard value, about 1% by mass. However, in cold regions of the molecular cloud, which comprise over 90% of the projected area on the sky, only a few percent of the total dust is detectable by IRAS. The dust-to-gas ratio, based on the 100-mu-m intensity and the grain temperature derived from the 60-100-mu-m intensity ratio using a large grain model, is an order of magnitude or more below that found in the hot regions. The large grain model leads to an overestimate of the temperature of most of the dust in a cold cloud, which results in a severe underestimate of the total dust column density. We find a relationship between the inferred dust-to-gas ratio and the radiation field intensity responsible for heating the dust which can be used to estimate the gas column density from the dust opacity derived from the 60 and 100-mu-m IRAS fluxes. A multicomponent grain model is required to interpret 60 and 100-mu-m data when small warm grains make a major contribution to the 60-mu-m intensity. We attempted to use the 12-mu-m intensity as a measure of the contribution of small stochastically heated warm grains to the longer wavelength infrared flux. With this model the spatial structure of the resulting column density distribution is nonphysical; the column density increases toward the cloud exterior. This behavior may be due to errors in the zero point offset in the infrared intensity distributions, changes in small grain abundance, and variations in the radiation field intensity. The computed dust optical depth is extremely sensitive to small changes in grain temperature. Submillimeter wavelength observations directly sensitive to the large grain emission are required to study the dust column density in most molecular clouds. In cold regions, there is an inverse correlation between the peak (CO)-C-12 temperature and the dust temperature. We find that the dust opacity correlates better with peak (CO)-C-12 temperature than with I((CO)-C-13). These results may be explained by small stochastically heated grains dominating even the 100-mu-m flux in the irradiated surface layers of the cloud.