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Does a black body (imagine like a square of solid black-body-matter) thermally conduct well at room temperature?

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If you are talking of "black body" as in "black body radiation" better read the link:

A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.

"idealized" is the crucial point here. All matter can be approximated to a "black body", i.e an emitter of black body radiation by the multiplication with a constant specific for the material, the emissivity.

Thermal conductivity is a different concept than the attributes of a black body. It is measured experimentally and depends on the molecular structure.

A black body will have a specific modified black body spectrum of radiation but can have any type of conductivity.For example dull wrought iron and black concrete have the same emissivity , .94, but very different thermal conductivity.

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