$(10)$ BONUS $($Must get all correct$)$ Wave$-$particle duality of light extends to matter particles by the de Broglie hypothesis. If an electron in free space is in thermal equilibrium with Cosmic Microwave Background $($CMB$)$ photons at $2\mathrm{.}73$K$,$ and assuming electron motion can be modeled as an $$ideal gas$$ particle, $($a$)(5)$ What is the electron$$s de Broglie wavelength? $($me $=9\mathrm{.}11$ x $10-31$ kg$)$ and $(3/2)$ kBT $=(1/2)$ mevavg $2.($b$)(5)$ What is the corresponding de Broglie frequency? $($with v $=$ f$)$