Question

when does the entropy of a substance become zero

Answer 1

when the reaction is under equilibrium!

Answer 2

Well to be exact the entropy of pure crystalline solid will be zero at absolute temperature and the reason behind this is

As in solids, the molecules only vibrate in their lattice and this vibration of solid molecules constitutes to the randomness of molecules and thus entropy. however this vibration decreases when temperature decreases. So we may assume safely that when the temperature becomes absolute, there will be no vibration of molecules and thus the randomness of molecules diminishes or entropy becomes zero. Now this is only applicable for pure crystalline solid only such as diamond