High resolution, approximately 50", measurements of the 170-mu-m and H-alpha luminosity have been compared at 36 locations within the spiral galaxy M51 in order to elucidate the origin of the far-infrared luminosity. After correcting the H-alpha luminosity for extinction and extrapolating the 170-mu-m luminosity to a 40-1000-mu-m far-infrared luminosity we find that the far-infrared luminosity is comparable to that expected from the O and B stars which are required to ionize the hydrogen gas. The results show that the far-infrared luminosity is in quantitative agreement with that expected from O and B stars at all locations within M51. Furthermore, IRAS HiRes 60 and 100-mu-m images of M51 indicate that the temperature of the dust responsible for the far-infrared luminosity is similar to that expected for dust in the vicinity of H II regions over the entire star forming disk of M51. The mass of dust required to radiate the 60 and 100-mu-m emission, measured for M51 by IRAS, is about a factor of 10 lower than the dust content of M51 estimated from the atomic and molecular gas masses and a Galactic gas/dust mass ratio. The most plausible explanation for the discrepancy is that the majority (approximately 90%) of the dust within M51 is radiating at wavelengths longward of 100-mu-m, and must therefore be cooler than the approximately 33 K dust that dominates the emission measured by IRAS. The 170-mu-m measurement of M51, obtained by Smith [ApJ, 261, 463 (1982)], provides independent support for large quantities of cold dust as the 170-mu-m flux is significantly higher than would be expected if all the dust were radiating at the same temperature as that measured by IRAS. The high gas/warm dust mass ratio is used to constrain the mass of dust that radiates at wavelengths longward of 100-mu-m, and the 60, 100, and 170-mu-m photometry is used to constrain the temperature of the dust. The constraints require the bulk, approximately 90%, of the dust mass within M51 to be cold with a temperature less-than-or-equal-to 16 K. Molecular gas dominates the interstellar medium within M51 and the temperature derived for the dust is consistent with that expected for dust in molecular clouds. The far-infrared luminosity of M51 is dominated by thermal emission from warm, approximately 33 K, dust even though most (approximately 90%) of the dust mass within M51 is much colder with a temperature of approximately 16 K. The ability for a small mass of warm dust to dominate the luminosity of significantly greater quantities of colder dust arises as a natural consequence of the strong temperature dependence of thermal emission.
