Black Body vs Non-Black Body: What Sets Them Apart in Kirchoff's Law?

[Kqwert]

Homework Statement

For a black body all absorbed radiation is emitted. Kirchoff´s law states that at the same temperature T1 the emissivity and absorptivity of a surface are equal, which holds for nonblack bodies as well. So, what really differs a black body and a non-black body in this case?

 

[haruspex]

what really differs a black body and a non-black body in this case?

That it can absorb and emit freely at all wavelengths.

 

[Kqwert]

That it can absorb and emit freely at all wavelengths.

Alright, but the attribute of emitting all absorbed radiation is not unique for black bodies, rather it holds for all bodies?

 

[haruspex]

Alright, but the attribute of emitting all absorbed radiation is not unique for black bodies, rather it holds for all bodies?

That it can emit at all the wavelengths at which it absorbs, yes, but that is not quite what you wrote.
One can imagine a cool body only able to emit and absorb at a short wavelength. Incoming short wavelength radiation will warm the body, but it might not be able to emit much until its temperature rises.