Due to the presence everywhere of the microwave radiation background, the minimum temperature possible 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 possess excited states of low excitation energy may, in fact, be in those excited states. Subsequent de-excitation leads to the emission of radiation that could be detected. We consider a (hypothetical) molecule with just one excited state. (a) We have to find the energy of the excited state so that 23% of the molecules are in the excited state. (b) We have to find the wavelength of the photon emitted in a transition to the ground state.

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(a)

We consider an ensemble of molecules of a given type with low excitation energy at temperature We have to find the energy of the excited state so that 23% of the molecules are in the excited state. Let be the number of molecules in the ground state of the molecule and let be the number of molecules in the first excited state of the molecule. Using the Boltzmann law we have

(b)

The wavelength of the photon emitted in transition from the first excited state to the ground state of the molecule will be