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We know that mass can be  converted to energy and vice versa. So does the mass of molecules or atoms change when they absorb a radiation and go to the higher energy state?  

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Yes. For example, when a hydrogen atom absorbs a photon and transitions from the $1s$ ground state to the $2p$ excited state, with 10.2 electron-volts more energy, its mass increases by $1.8\times 10^{-35}$ kilograms, or about one part in 100 million. This is just $\Delta m=\Delta E/c^2$.

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