Due to the presence everywhere of the cosmic background radiation, the minimum possible temperature of a gas in interstellar or intergalactic space is not 0 K but 2.7 K. This implies that a significant fraction of the molecules in space that can be in a low level excited state may, in fact, be so. Subsequent de-excitation would lead to the emission of radiation that could be detected. Consider a (hypothetical) molecule with just one possible excited state.
(a) What would the excitation energy have to be for 25% of the molecules to be in the excited state? (See Eq. 40-29.)
(b) What would be the wavelength of the photon emitted in a transition back to the ground state?